Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 757824 9 pageshttpdxdoiorg1011552013757824

Research ArticleExistence and Houmllder Regularity of the FractionalLandau-Lifshitz Equation without Gilbert Damping Term

Lijun Wang1 Jingna Li1 and Li Xia2

1 Department of Mathematics Jinan University Guangzhou 510632 China2Department of Mathematics Guangdong University of Finance amp Economics Guangzhou 510320 China

Correspondence should be addressed to Jingna Li jingna8005hotmailcom

Received 15 September 2013 Accepted 6 November 2013

Academic Editor Natig M Atakishiyev

Copyright copy 2013 Lijun Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Existence and Holder regularity of weak solutions to the fractional Landau-Lifshitz equation without Gilbert damping term isproved through viscosity approximation Since the nonlinear term is nonlocal and of full order of the equation a commutator isconstructed to get the convergence of the approximating solutions

1 Introduction

We study the fractional Landau-Lifshitz equation

120597119898

120597119905= minus1205821119898 times Λ

2120572119898 + 120582

2119898 times (119898 times Λ

2120572119898) (1)

where 119898(119909 119905) is a three-dimensional vector representing themagnetization and 120572 120582

1 1205822

ge 0 are real numbers Λ =

(minusΔ)12 is the square root of the Laplacian and the so-called

Zygmund operator and times denotes the cross product of R3-valued vectors The first term 119898 times Λ

2120572119898 is the gyromagnetic

term and the second term 119898 times (119898 times Λ2120572

119898) is called theGilbert damping term The fractional diffusion operator Λ

2120572

is nonlocal except120572 = 0 1 2 3 whichmeans thatΛ2120572

119906(119909)

depends not only on 119906(119910) for 119910 near 119909 but also on 119906(119910) for all119910

Equation (1) plays a fundamental role in the understand-ing of nonequilibrium magnetism which is an interestingproblem from both scientific and technological points ofview Besides their traditional applications in the magneticrecording industry these films are also currently beingexplored as alternatives to semiconductors asmagneticmem-ory devices (MRAMs) which has given greater incentiveto study this subject Since defects impurities and thermalnoise play important roles in the dynamics of the mag-netization field in nanometer thick films they also make

an ideal playground for studying some of the nanoscalephysics branches [1ndash4]

Fractional differential equations which appear in severalbranches of physics such as viscoelasticity electrochemistrycontrol porous media and electromagnetic now attract theinterests of many mathematicians see for example [5 6]A good case in point is the quasi-geostrophic equationwith fractional dissipation which has been extensively stud-ied in the last decade see [7ndash9] The fractional Landau-Lifshitz equation shares some similar difficulties with quasi-geostrophic equation however the equation studied here ismuchmore complicated in several waysThe derivative in thenonlinear convective term is local in the quasi-geostrophicequation and the fluid velocity is divergence free but here for1205822

= 0 (1) is degenerate and even worse the derivative inthe nonlinear term is nonlocal and of the same order as theequation which brings new difficulties in the convergence ofthe approximate solutions Hence subtle techniques must beused to overcome the difficulties

Let us recall some previous results of the equation When120572 = 1 (1) becomes the standard Landau-Lifshitz equationintroduced first by Landau and Lifshitz in [10] which waswidely studied in [11ndash16] For general 120572 isin (0 1] the interestedreader can refer to [17] for mathematical theory When 120582

2=

0 (1) corresponds to Schrodinger flow which representsthe conservation of angular momentum [18ndash21] Numericaltreatments can be found in [22 23]

2 Abstract and Applied Analysis

In this paper we will study local existence of weaksolutions in the spatial domain (0 2120587) with 120582

2= 0 and 120572 isin

[12 1] The main difficulty as in many partial differentialequations is the convergence of the nonlinear terms In oursituation we even face the problem of nonlocal differentialoperators degeneracy and nonlocal nonlinear term Forthese reasons the structure of (1) must be explored in detail

Without loss of generality we assume that 1205821

= 1Actually (1) can be written as

120597119898

120597119905= 119861 (119898) Λ

2120572119898 (2)

in which

119898 = (

1198981

1198982

1198983

) 119861 (119898) = (

0 1198983

minus1198982

minus1198983

0 1198981

1198982

minus1198981

0

) (3)

with initial condition

119898 (119909 0) = 1198980

(119909) (4)

and the periodic boundary condition

119898 (119909 119905) = 119898 (119909 + 2120587 119905) (5)

It is straightforward to check the following conclusions

(1) The matrix 119861(119898) is ldquozero definiterdquo namely

120585120591119861 (119898) 120585 = 0 forall120585 119898 isin R

3 (6)

(2) The matrix 119861(119898) is singular that is

det119861 (119898) = 0 forall119898 isin R3 (7)

Hence (2) is quite different from usual quasilinear parabolicequations for the above reasons

To approximate (2) we consider the following mollifiedequation

120597119898

120597119905= minus120576Λ

2120572119898 minus 119898 times Λ

2120572119898 (8)

which can be written as

120597119898

120597119905+ (120576119864 minus 119861 (119898)) Λ

2120572119898 = 0 (9)

The rest of this paper is divided into three parts firstwe consider the corresponding linear equation and get theregularity as a preparation to deal with (9) second positive-definition and uniform ellipticity of matrix 120576119864 minus 119861(119898) andchoice of norm space 119871

infin ensure that Leray-Schauder fixed-point theorem can be applied to prove the existence of weaksolution to (9) and the necessary a priori estimates in orderto guarantee convergence are obtained finally existence andHolder regularity of weak solution to (2) is proved by takingthe limit of the solution to (9) in which a commutator isconstructed to get the convergence

2 Cauchy Problem for the CorrespondingLinear Equation

Our starting point is the linear equation

120597119898

120597119905+ 119860 (119909 119905) Λ

2120572119898 = 119891 (119909 119905) in T

119899times (0 119879) (10)

119898 (119909 0) = 1198980

(119909) on T119899 (11)

where T119899 = R119899119885119899 is the flat torus and 119898(119909 119905) and 119898

0(119909)

are N-dimensional vector-valued functions We have thefollowing theorem about existence of solution to (10)-(11)

Theorem 1 Suppose that119873times119873matrix119860(119909 119905) defined on T119899times(0 119879) is measurable bounded and uniformly elliptic namelythere exists a constant 119870 such that

119860 (119909 119905) 120578 sdot 120578 ge 1198701003816100381610038161003816120578

10038161003816100381610038162

(12)

for all 119873-dimensional vectors 119891(119909 119905) isin 1198712(0 119879 119871

2(T119899))

and 1198980(119909) isin 119867

120572(T119899)Then there exists a unique vector-valued

solution to (10)-(11) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(T119899)) ⋂ 119871

2(0 119879 119867

2120572(T119899))

120597119898

120597119905isin 1198712

(0 119879 1198712

(T119899))

(13)

Proof of Theorem 1 Weapply theGalerkinmethod let 120593119895 be

an orthogonal basis of 1198712(T119899) consisting of all the eigenfunc-

tions for the operator

Λ2120572

120593119895

= 120582119895120593119895

120593119895

(0) = 120593119895

(2120587)

(14)

We are looking for approximate solutions 119898119899(119909 119905) to (10)-(11)

under the form

119898119899

(119909 119905) =

119899

sum

119895=1

119892119895

(119905) 120593119895

(119909) (15)

where 119892119895are vector-valued functions such that for 1 ⩽ 119894 ⩽ 119899

there holds

intΩ

[120597119898119899

120597119905+ 119860 (119909 119905) Λ

2120572119898119899

minus 119891 (119909 119905)] 120593119894119889119909 = 0 (16)

intΩ

[119898119899

(119909 0) minus 1198980

(119909)] 120593119894119889119909 = 0 (17)

These relations produce an ordinary differential system thatcan be writeen as

120597119892

120597119905= 119865 (119892) 119892 (0) = 119892

0 (18)

where 119892 = (1198921 1198922 119892

119899) and 119892

0is the projection of

1198980on (120593

1 1205932 120593119899) The existence of a local solution to

system (18) is a classical matter We now proceed to estimate

Abstract and Applied Analysis 3

the approximate solution 119898119899 Multiplying equality (16) by 119892

119894

and summing for 1 le 119894 le 119899 we have

1

2

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

= int

119879

0

intT119899

(119891 sdot 119898119899

minus 119860 (119909 119905) Λ2120572

119898119899

sdot 119898119899) 119889119909 119889119905

+1

2

10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

(19)

Multiplying equality (16) by 120582119894119892119894and summing for 1 le 119894 le 119899

we have

1

2

1003817100381710038171003817Λ120572

119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+ int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot Λ2120572

119898119899

119889119909 119889119905

= int

119879

0

intT119899

119891 sdot Λ2120572

119898119899

119889119909 119889119905 +1

2

1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

(20)

Adding (19) to (20) we get

1

2

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+1

2

1003817100381710038171003817Λ120572

119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+ int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot Λ2120572

119898119899119889119909 119889119905

=1

2(10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

+1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

)

+ int

119879

0

intT119899

(119891 sdot 119898119899

+ 119891 sdot Λ2120572

119898119899) 119889119909 119889119905

minus int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot 119898119899

119889119909 119889119905

le1

2(10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

+1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

) + 1198621003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

+ 119862 int

119879

0

intT119899

100381610038161003816100381611989811989910038161003816100381610038162

119889119909 119889119905 +119870

2

10038171003817100381710038171003817Λ2120572

119898119899

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(21)

Since int119879

0intT119899

119860(119909 119905)Λ2120572

119898119899

sdot Λ2120572

119898119899119889119909 119889119905 ge

119870Λ2120572

1198981198992

1198712(T119899times(0119879)) by Gronwallrsquos inequality we have

sup0le119905le119879

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119898119899

10038171003817100381710038172

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

)

10038171003817100381710038171003817Λ2120572

119898119899

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

)

(22)

Taking the inner product of 120597119898119899120597119905 and (10) and integrating

over T119899 times (0 119879) we have

int

119879

0

intT119899

10038161003816100381610038161003816100381610038161003816

120597119898119899

120597119905

10038161003816100381610038161003816100381610038161003816

2

119889119909 119889119905

= int

119879

0

intT119899

(119891 minus 119860 (119909 119905) Λ2120572

119898119899) sdot

120597119898119899

120597119905119889119909 119889119905

le 119862 int

119879

0

intT119899

100381610038161003816100381611989110038161003816100381610038162

119889119909 119889119905 + 119862 int

119879

0

intT119899

10038161003816100381610038161003816Λ2120572

119898119899

10038161003816100381610038161003816

2

119889119909 119889119905

+1

2int

119879

0

intT119899

10038161003816100381610038161003816100381610038161003816

120597119898119899

120597119905

10038161003816100381610038161003816100381610038161003816

2

119889119909 119889119905

(23)

Hence10038171003817100381710038171003817100381710038171003817

120597119898119899

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

) (24)

Actually if the matrix 119860(119909 119905) is retrained to a small classof good function matrix one can get higher regularity ofsolution to (10)-(11)

Since the right-hand member of equality (22) and (24) isuniformly bounded thus the solution g can be extended to alltime andwe can extract from119898

119899a subsequence (still denoted

by 119898119899) such that

119898119899

119898119904119908 in 1198712

(0 119879 1198672120572

(T119899))weaklylowast

120597119898119899

120597119905

120597119898

120597119905 in 119871

2(0 119879 119871

2(T119899))weakly

(25)

Hence we know that [24]

119898119899

997888rarr 119898 strongly in 1198712

(0 119879 1198712

(T119899))

ae in T119899

times [0 119879]

(26)

Passing to the limit (119899 rarr infin) we find a weak solution to(10)-(11) From (16) and taking the limit 119899 rarr infin we deducethat for all 120593 in vectors (120593

1 1205932 120593

119899) there holds

intT119899

120597119898119899

120597119905120593 + int

T119899119860 (119909 119905) Λ

2120572119898119899120593 minus int

T119899119891 (119909 119905) 120593 119889119909 = 0

(27)

By a density argument we also obtain formula (27) for all 120593

in 119867120572

(119876119879

) 119876119879

= T119899 times [0 119879]

Theorem 2 Suppose that(i) 119873 times 119873 matrix 119860(119909 119905) defined on T119899 times (0 119879) is

measurable bounded and uniformly elliptic namely thereexists a constant 119870 such that

119860 (119909 119905) 120578 sdot 120578 ge 1198701003816100381610038161003816120578

10038161003816100381610038162

(28)

for all N-dimensional vectors 120578 Δ119860(119909 119905) is bounded and|nabla119860(119909 119905)|

infinle 119870radic5120575 where 120575 is the Sobolev embedding

constant satisfying1003817100381710038171003817nabla119891

10038171003817100381710038171198712(T119899times(0119879)) le 1205751003817100381710038171003817Δ119891

10038171003817100381710038171198712(T119899times(0119879)) (29)

4 Abstract and Applied Analysis

(ii) 119891(119909 119905) isin 1198712(0 119879 119867

2(T119899)) 119898

0(119909) isin 119867

120572+2(T119899) Then

there exists a unique vector-valued solution to (10)-(11) suchthat

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(T119899)) ⋂ 119867

1(T119899

times (0 119879))

sup0le119905le119879

119898 (sdot 119905)2

119867120572+2(T119899) +

10038171003817100381710038171003817100381710038171003817

120597119898

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

+ 1198982

1198712(01198791198672120572+2(T119899))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(30)

in which constant 119862 is dependent 119860(119909 119905) independent of119898(119909 119905)

Proof Let the operator Λ2 act on (10) we have

minus Δ119898119905

minus Δ119860 (119909 119905) Λ2120572

119898 + 119860 (119909 119905) Λ2120572+2

119898

minus nabla119860 (119909 119905) sdot nablaΛ2120572

119898 = minusΔ119891 (119909 119905)

(31)

Taking the inner product of Λ2120572+2

119898 and (31) and integratingover T119899 times (0 119879) we have

1

2intT119899

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

= minus int

119879

0

intT119899

Δ119891 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

Δ119860 (119909 119905) Λ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

nabla119860 (119909 119905) nablaΛ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+1

2intT119899

10038161003816100381610038161003816Λ120572+2

1198980

10038161003816100381610038161003816

2

119889119909

le1

2

10038171003817100381710038171003817Λ120572+2

1198980

10038171003817100381710038171003817

2

1198712(T119899)

+1

119870

1003817100381710038171003817Δ11989110038171003817100381710038172

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817Δ119860 (119909 119905) Λ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817nabla119860 (119909 119905) nablaΛ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

(32)

Note that

|nabla119860 (119909 119905)|infin lt119870

radic5120575

10038171003817100381710038171003817nablaΛ2120572

119898100381710038171003817100381710038171198712(T119899times(0119879))

le 12057510038171003817100381710038171003817Λ2120572+2

119898100381710038171003817100381710038171198712(T119899times(0119879))

and

Δ119860 (119909 119905) is bounded

(33)

Reporting (33) in (32) and taking into account

int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

ge 11987010038171003817100381710038171003817Λ2120572+2119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(34)

we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

119898 (sdot 119905)10038171003817100381710038171003817

2

1198712(T119899)

+10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(35)

3 Cauchy Problem for the Mollified Equation

To get existence of weak solution to (2) we consider the fol-lowing approximate equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 119898 times Λ

2120572119898 (36)

which is called mollified equation In this section and nextsection we assume that the spatial variable 119909 isin (0 2120587) ByLeray-Schauder fixed-point theorem we have the followingtheorem

Theorem3 Suppose that1198980(119909) isin 119867

120572(0 2120587) then there exists

a unique weak solution to (36)with initial-boundary condition(4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587)) (37)

where 119876119879

= (0 2120587) times (0 119879)

Proof First themapping119879120582

119871infin

(119876119879

) rarr 119871infin

(119876119879

) is definedas follows For each 119906 isin 119871

infin(119876119879

) 119898 = 119879120582(119906) is a solution to

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119906 times Λ

2120572119898 (38)

with initial condition (4) in which 0 le 120582 le 1 By Theorem 1we know that 119898 = 119879

120582(119906) is the unique solution to (38)

with initial-boundary condition (4) and (5) moreover 119898 isin

119871infin

(0 119879 1198671(0 2120587))

Obviously for all 120582 the mapping 119879120582is continuous and

for any bounded closed set of 119871infin

(119876119879

) 119879120582is uniformly

continuous with respect to 0 le 120582 le 1To apply Leray-Schauder fixed-point theorem wemake a

priori estimate on all fixed points of 119879120582

Abstract and Applied Analysis 5

Taking the inner product of 119898(119909 119905) and equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119898 times Λ

2120572119898 (39)

we have

119898 sdot120597119898

120597119905= minus120576119898 sdot Λ

2120572119898 minus 120582 (119898 times Λ

2120572119898) sdot 119898 (40)

Integrating (40) over 119876120591(0 le 120591 le 119879) we get

119898 (sdot 120591)2

1198712(02120587)

+ 21205761003817100381710038171003817Λ120572

11989810038171003817100381710038172

1198712(119876120591)

le10038171003817100381710038171198980

10038171003817100381710038172

1198712(02120587)

(41)

in which 0 le 120582 le 1 0 le 120591 le 119879 Hence

sup0le119905le119879

119898 (sdot 119905)1198712(02120587)

le 1198621 (42)

in which 1198621is a constant independent of 120576 120582

Taking the inner product of Λ2120572

119898(119909 119905) and (39) weobtain

Λ2120572

119898 sdot120597119898

120597119905= minus120576Λ

2120572119898 sdot Λ2120572

119898 minus 120582Λ2120572

119898 sdot (119898 times Λ2120572

119898)

(43)

Integrating (43) over (0 2120587) with respect to variable 119909 leadsto

1

2

119889

119889119905

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038172

1198712(02120587)

+ 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572

11989810038161003816100381610038161003816

2

119889119909 le 0 (44)

Obviously

sup0le119905le119879

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038171198712(02120587) le 119862

2 (45)

in which 1198622is a constant independent of 120582 120576 From (42) for

each 120576 gt 0 we have

1198981198712(01198791198672120572(02120587)) le 119862 (46)

In view of (42) (45) and (46) Sobolev embedding theoremgives the desired result

For small initial data 1198980(119909) we can get higher regularity

of the solution to (36)

Theorem 4 Suppose that 1198980(119909) isin 119867

120572+2(0 2120587) 120572 isin (12 1]

and 1198980(119909)2

119867120572+2 le 1119872119879 where 119872 = 119900(1120576) is a certain

constant then there exists a unique weak solution to (36) withinitial-boundary condition (4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(0 2120587)) (47)

Proof Let the operator Λ2 act on (39) we get

minusΔ119898119905

= minus 120576Λ2120572+2

119898 + 120582Δ119898 times Λ2120572

119898

minus 120582119898 times Λ2120572+2

119898 + 120582nabla119898 times nablaΛ2120572

119898

(48)

Taking the inner product of Λ2120572+2

119898 and (48) and integratingover (0 2120587) we have

1

2

119889

119889119905int

2120587

0

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572+2

11989810038161003816100381610038161003816

2

119889119909

= 120582 int

2120587

0

(Δ119898 times Λ2120572

119898) sdot Λ2120572+2

119898 119889119909

+ 120582 int

2120587

0

(nabla119898 times nablaΛ2120572

119898) sdot Λ2120572+2

119898 119889119909

le1

120576

10038171003817100381710038171003817Λ2120572

11989810038171003817100381710038171003817

2

119871infin

Δ1198982

1198712

+1

120576nabla119898

2

119871infin

10038171003817100381710038171003817nablaΛ2120572

11989810038171003817100381710038171003817

2

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

le119872

2

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

4

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

(49)

Note that119872 = 119900(1120576) is a constant hence by Lemma 5 whichwill be proved later we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

2

1198712(02120587)

le 1198623 (50)

in which 1198623is independent of 120576 120582 From (49) for each 120576 gt 0

we have

1198981198712(01198791198672120572+2(02120587)) le 119862 (51)

Lemma 5 Let 119891(119905) be nonnegative continuous functions for0 le 119905 le 119879 Suppose that 119891(0) lt 1119896119879 and

119891 (119905) le 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) 0 le 119905 le 119879 (52)

where 119896 is a constant Then

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0)(53)

holds for 0 le 119905 le 119879

Proof Define

V (119905) = 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) (54)

Then the function V(119905) is nondecreasing V(0) = 119891(0) and

119889V (119905)

119889119905= 1198961198912

(119905) le 119896V2 (119905) (55)

since 119891(119905) le V(119905) le V(119879) According to (55) the function119911(119905) = minus1V(119905) satisfies

119889119911 (119905)

119889119905=

V1015840 (119905)

V2 (119905)=

1198961198912

(119905)

V2 (119905)le 119896 (56)

6 Abstract and Applied Analysis

Integrating (56) from 0 to 119905 yields

119911 (119905) le 119896119905 + 119911 (0) (57)

orminus

1

V (119905)le 119896119905 minus

1

V (0) (58)

that isV (119905) le

V (0)

1 minus 119896119905V (0) (59)

Hence

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0) (60)

4 Convergence Process

Before we prove existence of weak solution to the fractionalLandau-Lifshitz model without Gilbert term (2) we firstrecall two Lemmas in [25 26] respectively

Lemma 6 Suppose that 119904 gt 0 and 119901 isin (1 +infin) If 119891 119892 isin Sthe Schwartz class then

1003817100381710038171003817Λ119904

(119891119892) minus 119891Λ119904119892

1003817100381710038171003817119871119901

le 119862 (1003817100381710038171003817nabla119891

100381710038171003817100381711987111990111003817100381710038171003817119892

1003817100381710038171003817119904minus11199012 +1003817100381710038171003817119891

100381710038171003817100381711990411990131003817100381710038171003817119892

10038171003817100381710038171198711199014 )

1003817100381710038171003817Λ119904

(119891119892)1003817100381710038171003817119871119901 le 119862 (

100381710038171003817100381711989110038171003817100381710038171198711199011

100381710038171003817100381711989210038171003817100381710038171199041199012 +

100381710038171003817100381711989110038171003817100381710038171199041199013

100381710038171003817100381711989210038171003817100381710038171198711199014 )

(61)

with 1199012 1199013

isin (1 +infin) such that

1

119901=

1

1199011

+1

1199012

=1

1199013

+1

1199014

(62)

where 119896119901 is the homogeneous Sobolev space and the

119896119901-norm of 119891 is defined by

10038171003817100381710038171198911003817100381710038171003817119896119901 =

10038171003817100381710038171003817Fminus1

(1003816100381610038161003816120585

1003816100381610038161003816119904

119891 (120585))10038171003817100381710038171003817119871119901

(63)

The following Lemma is due to Gagliardo and Nirenbergsee [11]

Lemma 7 Let Ω be R119899 or a bounded Lipschitz domain in R119899

with 120597Ω and let 119906 be any function in 119882119898119903

(Ω) ⋂ 119871119902(Ω) 1 le

119903 119902 le +infin For any integer 119895 0 le 119895 lt 119898 and for any number119886 in the interval 119895119898 le 119886 le 1 set

1

119901=

119895

119899+ 119886 (

1

119903minus

119898

119899) + (1 minus 119886)

1

119902 (64)

If 119898 minus 119895 minus (119899119903) is not a nonnegative integer then10038171003817100381710038171003817nabla119895119906

10038171003817100381710038171003817119871119901(Ω)le 119862119906

119886

119882119898119903(Ω)

1199061minus119886

119871119902(Ω)

(65)

If 119898 minus 119895 minus (119899119903) is a nonnegative integer then (65) holds for119886 = 119895119898 The constant 119862 depends only on 119903 119902 119898 119895 119886 and theshape of Ω

From (45) we conclude the following

Lemma 8 Solutions to (36) satisfy

sup0le119905le119879

1003817100381710038171003817119898120576

(sdot 119905)1003817100381710038171003817119867120572(02120587) le 119862 (66)

in which 119862 is independent of 120576

For the uniform bound of 120597119898120576120597119905 we have the following

lemma

Lemma 9 120597119898120576120597119905 in (36) satisfies

sup0le119905le119879

10038171003817100381710038171003817100381710038171003817

120597119898120576

120597119905(sdot 119905)

10038171003817100381710038171003817100381710038171003817119867minus119896(02120587)le 119862 (67)

in which 12 lt 119896 lt 1 and 119862 is independent of 120576

Proof For all 120595(119909) isin (119867120572

(0 2120587))3 we have

int

+infin

minusinfin

120595 sdot120597119898120576

120597119905119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(119898120576

times Λ2120572

119898120576) sdot 120595 (119909) 119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(120595 (119909) times 119898120576) sdot Λ2120572

119898120576119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

Λ120572

(120595 (119909) times 119898120576) sdot Λ120572

119898120576119889119909

le1003817100381710038171003817120595

1003817100381710038171003817119871infin1003817100381710038171003817Λ120572

11989812057610038171003817100381710038172

1198712 +

1003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712

1003817100381710038171003817Λ120572

12059510038171003817100381710038171198714

100381710038171003817100381711989812057610038171003817100381710038171198714

le 1198621003817100381710038171003817120595

1003817100381710038171003817119867119896(02120587)

(68)

in which formula ( 119886 times ) sdot 119888 = ( 119888 times 119886) sdot Sobolev embeddingtheorem and Lemma 6 are used

The following lemma is about the uniform bound of theHolder norm of 119898

120576

Lemma 10 Solutions to (36) satisfy

1003817100381710038171003817119898120576

(119909 119905)1003817100381710038171003817119862(2120572minus1)(120572+1)(2120572minus1)(2(120572+1))(119876

119879)

le 119862 (69)

where 119862 is independent of 120576

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

In this paper we will study local existence of weaksolutions in the spatial domain (0 2120587) with 120582

2= 0 and 120572 isin

[12 1] The main difficulty as in many partial differentialequations is the convergence of the nonlinear terms In oursituation we even face the problem of nonlocal differentialoperators degeneracy and nonlocal nonlinear term Forthese reasons the structure of (1) must be explored in detail

Without loss of generality we assume that 1205821

= 1Actually (1) can be written as

120597119898

120597119905= 119861 (119898) Λ

2120572119898 (2)

in which

119898 = (

1198981

1198982

1198983

) 119861 (119898) = (

0 1198983

minus1198982

minus1198983

0 1198981

1198982

minus1198981

0

) (3)

with initial condition

119898 (119909 0) = 1198980

(119909) (4)

and the periodic boundary condition

119898 (119909 119905) = 119898 (119909 + 2120587 119905) (5)

It is straightforward to check the following conclusions

(1) The matrix 119861(119898) is ldquozero definiterdquo namely

120585120591119861 (119898) 120585 = 0 forall120585 119898 isin R

3 (6)

(2) The matrix 119861(119898) is singular that is

det119861 (119898) = 0 forall119898 isin R3 (7)

Hence (2) is quite different from usual quasilinear parabolicequations for the above reasons

To approximate (2) we consider the following mollifiedequation

120597119898

120597119905= minus120576Λ

2120572119898 minus 119898 times Λ

2120572119898 (8)

which can be written as

120597119898

120597119905+ (120576119864 minus 119861 (119898)) Λ

2120572119898 = 0 (9)

The rest of this paper is divided into three parts firstwe consider the corresponding linear equation and get theregularity as a preparation to deal with (9) second positive-definition and uniform ellipticity of matrix 120576119864 minus 119861(119898) andchoice of norm space 119871

infin ensure that Leray-Schauder fixed-point theorem can be applied to prove the existence of weaksolution to (9) and the necessary a priori estimates in orderto guarantee convergence are obtained finally existence andHolder regularity of weak solution to (2) is proved by takingthe limit of the solution to (9) in which a commutator isconstructed to get the convergence

2 Cauchy Problem for the CorrespondingLinear Equation

Our starting point is the linear equation

120597119898

120597119905+ 119860 (119909 119905) Λ

2120572119898 = 119891 (119909 119905) in T

119899times (0 119879) (10)

119898 (119909 0) = 1198980

(119909) on T119899 (11)

where T119899 = R119899119885119899 is the flat torus and 119898(119909 119905) and 119898

0(119909)

are N-dimensional vector-valued functions We have thefollowing theorem about existence of solution to (10)-(11)

Theorem 1 Suppose that119873times119873matrix119860(119909 119905) defined on T119899times(0 119879) is measurable bounded and uniformly elliptic namelythere exists a constant 119870 such that

119860 (119909 119905) 120578 sdot 120578 ge 1198701003816100381610038161003816120578

10038161003816100381610038162

(12)

for all 119873-dimensional vectors 119891(119909 119905) isin 1198712(0 119879 119871

2(T119899))

and 1198980(119909) isin 119867

120572(T119899)Then there exists a unique vector-valued

solution to (10)-(11) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(T119899)) ⋂ 119871

2(0 119879 119867

2120572(T119899))

120597119898

120597119905isin 1198712

(0 119879 1198712

(T119899))

(13)

Proof of Theorem 1 Weapply theGalerkinmethod let 120593119895 be

an orthogonal basis of 1198712(T119899) consisting of all the eigenfunc-

tions for the operator

Λ2120572

120593119895

= 120582119895120593119895

120593119895

(0) = 120593119895

(2120587)

(14)

We are looking for approximate solutions 119898119899(119909 119905) to (10)-(11)

under the form

119898119899

(119909 119905) =

119899

sum

119895=1

119892119895

(119905) 120593119895

(119909) (15)

where 119892119895are vector-valued functions such that for 1 ⩽ 119894 ⩽ 119899

there holds

intΩ

[120597119898119899

120597119905+ 119860 (119909 119905) Λ

2120572119898119899

minus 119891 (119909 119905)] 120593119894119889119909 = 0 (16)

intΩ

[119898119899

(119909 0) minus 1198980

(119909)] 120593119894119889119909 = 0 (17)

These relations produce an ordinary differential system thatcan be writeen as

120597119892

120597119905= 119865 (119892) 119892 (0) = 119892

0 (18)

where 119892 = (1198921 1198922 119892

119899) and 119892

0is the projection of

1198980on (120593

1 1205932 120593119899) The existence of a local solution to

system (18) is a classical matter We now proceed to estimate

Abstract and Applied Analysis 3

the approximate solution 119898119899 Multiplying equality (16) by 119892

119894

and summing for 1 le 119894 le 119899 we have

1

2

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

= int

119879

0

intT119899

(119891 sdot 119898119899

minus 119860 (119909 119905) Λ2120572

119898119899

sdot 119898119899) 119889119909 119889119905

+1

2

10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

(19)

Multiplying equality (16) by 120582119894119892119894and summing for 1 le 119894 le 119899

we have

1

2

1003817100381710038171003817Λ120572

119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+ int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot Λ2120572

119898119899

119889119909 119889119905

= int

119879

0

intT119899

119891 sdot Λ2120572

119898119899

119889119909 119889119905 +1

2

1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

(20)

Adding (19) to (20) we get

1

2

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+1

2

1003817100381710038171003817Λ120572

119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+ int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot Λ2120572

119898119899119889119909 119889119905

=1

2(10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

+1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

)

+ int

119879

0

intT119899

(119891 sdot 119898119899

+ 119891 sdot Λ2120572

119898119899) 119889119909 119889119905

minus int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot 119898119899

119889119909 119889119905

le1

2(10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

+1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

) + 1198621003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

+ 119862 int

119879

0

intT119899

100381610038161003816100381611989811989910038161003816100381610038162

119889119909 119889119905 +119870

2

10038171003817100381710038171003817Λ2120572

119898119899

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(21)

Since int119879

0intT119899

119860(119909 119905)Λ2120572

119898119899

sdot Λ2120572

119898119899119889119909 119889119905 ge

119870Λ2120572

1198981198992

1198712(T119899times(0119879)) by Gronwallrsquos inequality we have

sup0le119905le119879

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119898119899

10038171003817100381710038172

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

)

10038171003817100381710038171003817Λ2120572

119898119899

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

)

(22)

Taking the inner product of 120597119898119899120597119905 and (10) and integrating

over T119899 times (0 119879) we have

int

119879

0

intT119899

10038161003816100381610038161003816100381610038161003816

120597119898119899

120597119905

10038161003816100381610038161003816100381610038161003816

2

119889119909 119889119905

= int

119879

0

intT119899

(119891 minus 119860 (119909 119905) Λ2120572

119898119899) sdot

120597119898119899

120597119905119889119909 119889119905

le 119862 int

119879

0

intT119899

100381610038161003816100381611989110038161003816100381610038162

119889119909 119889119905 + 119862 int

119879

0

intT119899

10038161003816100381610038161003816Λ2120572

119898119899

10038161003816100381610038161003816

2

119889119909 119889119905

+1

2int

119879

0

intT119899

10038161003816100381610038161003816100381610038161003816

120597119898119899

120597119905

10038161003816100381610038161003816100381610038161003816

2

119889119909 119889119905

(23)

Hence10038171003817100381710038171003817100381710038171003817

120597119898119899

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

) (24)

Actually if the matrix 119860(119909 119905) is retrained to a small classof good function matrix one can get higher regularity ofsolution to (10)-(11)

Since the right-hand member of equality (22) and (24) isuniformly bounded thus the solution g can be extended to alltime andwe can extract from119898

119899a subsequence (still denoted

by 119898119899) such that

119898119899

119898119904119908 in 1198712

(0 119879 1198672120572

(T119899))weaklylowast

120597119898119899

120597119905

120597119898

120597119905 in 119871

2(0 119879 119871

2(T119899))weakly

(25)

Hence we know that [24]

119898119899

997888rarr 119898 strongly in 1198712

(0 119879 1198712

(T119899))

ae in T119899

times [0 119879]

(26)

Passing to the limit (119899 rarr infin) we find a weak solution to(10)-(11) From (16) and taking the limit 119899 rarr infin we deducethat for all 120593 in vectors (120593

1 1205932 120593

119899) there holds

intT119899

120597119898119899

120597119905120593 + int

T119899119860 (119909 119905) Λ

2120572119898119899120593 minus int

T119899119891 (119909 119905) 120593 119889119909 = 0

(27)

By a density argument we also obtain formula (27) for all 120593

in 119867120572

(119876119879

) 119876119879

= T119899 times [0 119879]

Theorem 2 Suppose that(i) 119873 times 119873 matrix 119860(119909 119905) defined on T119899 times (0 119879) is

measurable bounded and uniformly elliptic namely thereexists a constant 119870 such that

119860 (119909 119905) 120578 sdot 120578 ge 1198701003816100381610038161003816120578

10038161003816100381610038162

(28)

for all N-dimensional vectors 120578 Δ119860(119909 119905) is bounded and|nabla119860(119909 119905)|

infinle 119870radic5120575 where 120575 is the Sobolev embedding

constant satisfying1003817100381710038171003817nabla119891

10038171003817100381710038171198712(T119899times(0119879)) le 1205751003817100381710038171003817Δ119891

10038171003817100381710038171198712(T119899times(0119879)) (29)

4 Abstract and Applied Analysis

(ii) 119891(119909 119905) isin 1198712(0 119879 119867

2(T119899)) 119898

0(119909) isin 119867

120572+2(T119899) Then

there exists a unique vector-valued solution to (10)-(11) suchthat

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(T119899)) ⋂ 119867

1(T119899

times (0 119879))

sup0le119905le119879

119898 (sdot 119905)2

119867120572+2(T119899) +

10038171003817100381710038171003817100381710038171003817

120597119898

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

+ 1198982

1198712(01198791198672120572+2(T119899))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(30)

in which constant 119862 is dependent 119860(119909 119905) independent of119898(119909 119905)

Proof Let the operator Λ2 act on (10) we have

minus Δ119898119905

minus Δ119860 (119909 119905) Λ2120572

119898 + 119860 (119909 119905) Λ2120572+2

119898

minus nabla119860 (119909 119905) sdot nablaΛ2120572

119898 = minusΔ119891 (119909 119905)

(31)

Taking the inner product of Λ2120572+2

119898 and (31) and integratingover T119899 times (0 119879) we have

1

2intT119899

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

= minus int

119879

0

intT119899

Δ119891 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

Δ119860 (119909 119905) Λ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

nabla119860 (119909 119905) nablaΛ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+1

2intT119899

10038161003816100381610038161003816Λ120572+2

1198980

10038161003816100381610038161003816

2

119889119909

le1

2

10038171003817100381710038171003817Λ120572+2

1198980

10038171003817100381710038171003817

2

1198712(T119899)

+1

119870

1003817100381710038171003817Δ11989110038171003817100381710038172

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817Δ119860 (119909 119905) Λ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817nabla119860 (119909 119905) nablaΛ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

(32)

Note that

|nabla119860 (119909 119905)|infin lt119870

radic5120575

10038171003817100381710038171003817nablaΛ2120572

119898100381710038171003817100381710038171198712(T119899times(0119879))

le 12057510038171003817100381710038171003817Λ2120572+2

119898100381710038171003817100381710038171198712(T119899times(0119879))

and

Δ119860 (119909 119905) is bounded

(33)

Reporting (33) in (32) and taking into account

int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

ge 11987010038171003817100381710038171003817Λ2120572+2119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(34)

we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

119898 (sdot 119905)10038171003817100381710038171003817

2

1198712(T119899)

+10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(35)

3 Cauchy Problem for the Mollified Equation

To get existence of weak solution to (2) we consider the fol-lowing approximate equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 119898 times Λ

2120572119898 (36)

which is called mollified equation In this section and nextsection we assume that the spatial variable 119909 isin (0 2120587) ByLeray-Schauder fixed-point theorem we have the followingtheorem

Theorem3 Suppose that1198980(119909) isin 119867

120572(0 2120587) then there exists

a unique weak solution to (36)with initial-boundary condition(4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587)) (37)

where 119876119879

= (0 2120587) times (0 119879)

Proof First themapping119879120582

119871infin

(119876119879

) rarr 119871infin

(119876119879

) is definedas follows For each 119906 isin 119871

infin(119876119879

) 119898 = 119879120582(119906) is a solution to

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119906 times Λ

2120572119898 (38)

with initial condition (4) in which 0 le 120582 le 1 By Theorem 1we know that 119898 = 119879

120582(119906) is the unique solution to (38)

with initial-boundary condition (4) and (5) moreover 119898 isin

119871infin

(0 119879 1198671(0 2120587))

Obviously for all 120582 the mapping 119879120582is continuous and

for any bounded closed set of 119871infin

(119876119879

) 119879120582is uniformly

continuous with respect to 0 le 120582 le 1To apply Leray-Schauder fixed-point theorem wemake a

priori estimate on all fixed points of 119879120582

Abstract and Applied Analysis 5

Taking the inner product of 119898(119909 119905) and equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119898 times Λ

2120572119898 (39)

we have

119898 sdot120597119898

120597119905= minus120576119898 sdot Λ

2120572119898 minus 120582 (119898 times Λ

2120572119898) sdot 119898 (40)

Integrating (40) over 119876120591(0 le 120591 le 119879) we get

119898 (sdot 120591)2

1198712(02120587)

+ 21205761003817100381710038171003817Λ120572

11989810038171003817100381710038172

1198712(119876120591)

le10038171003817100381710038171198980

10038171003817100381710038172

1198712(02120587)

(41)

in which 0 le 120582 le 1 0 le 120591 le 119879 Hence

sup0le119905le119879

119898 (sdot 119905)1198712(02120587)

le 1198621 (42)

in which 1198621is a constant independent of 120576 120582

Taking the inner product of Λ2120572

119898(119909 119905) and (39) weobtain

Λ2120572

119898 sdot120597119898

120597119905= minus120576Λ

2120572119898 sdot Λ2120572

119898 minus 120582Λ2120572

119898 sdot (119898 times Λ2120572

119898)

(43)

Integrating (43) over (0 2120587) with respect to variable 119909 leadsto

1

2

119889

119889119905

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038172

1198712(02120587)

+ 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572

11989810038161003816100381610038161003816

2

119889119909 le 0 (44)

Obviously

sup0le119905le119879

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038171198712(02120587) le 119862

2 (45)

in which 1198622is a constant independent of 120582 120576 From (42) for

each 120576 gt 0 we have

1198981198712(01198791198672120572(02120587)) le 119862 (46)

In view of (42) (45) and (46) Sobolev embedding theoremgives the desired result

For small initial data 1198980(119909) we can get higher regularity

of the solution to (36)

Theorem 4 Suppose that 1198980(119909) isin 119867

120572+2(0 2120587) 120572 isin (12 1]

and 1198980(119909)2

119867120572+2 le 1119872119879 where 119872 = 119900(1120576) is a certain

constant then there exists a unique weak solution to (36) withinitial-boundary condition (4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(0 2120587)) (47)

Proof Let the operator Λ2 act on (39) we get

minusΔ119898119905

= minus 120576Λ2120572+2

119898 + 120582Δ119898 times Λ2120572

119898

minus 120582119898 times Λ2120572+2

119898 + 120582nabla119898 times nablaΛ2120572

119898

(48)

Taking the inner product of Λ2120572+2

119898 and (48) and integratingover (0 2120587) we have

1

2

119889

119889119905int

2120587

0

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572+2

11989810038161003816100381610038161003816

2

119889119909

= 120582 int

2120587

0

(Δ119898 times Λ2120572

119898) sdot Λ2120572+2

119898 119889119909

+ 120582 int

2120587

0

(nabla119898 times nablaΛ2120572

119898) sdot Λ2120572+2

119898 119889119909

le1

120576

10038171003817100381710038171003817Λ2120572

11989810038171003817100381710038171003817

2

119871infin

Δ1198982

1198712

+1

120576nabla119898

2

119871infin

10038171003817100381710038171003817nablaΛ2120572

11989810038171003817100381710038171003817

2

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

le119872

2

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

4

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

(49)

Note that119872 = 119900(1120576) is a constant hence by Lemma 5 whichwill be proved later we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

2

1198712(02120587)

le 1198623 (50)

in which 1198623is independent of 120576 120582 From (49) for each 120576 gt 0

we have

1198981198712(01198791198672120572+2(02120587)) le 119862 (51)

Lemma 5 Let 119891(119905) be nonnegative continuous functions for0 le 119905 le 119879 Suppose that 119891(0) lt 1119896119879 and

119891 (119905) le 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) 0 le 119905 le 119879 (52)

where 119896 is a constant Then

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0)(53)

holds for 0 le 119905 le 119879

Proof Define

V (119905) = 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) (54)

Then the function V(119905) is nondecreasing V(0) = 119891(0) and

119889V (119905)

119889119905= 1198961198912

(119905) le 119896V2 (119905) (55)

since 119891(119905) le V(119905) le V(119879) According to (55) the function119911(119905) = minus1V(119905) satisfies

119889119911 (119905)

119889119905=

V1015840 (119905)

V2 (119905)=

1198961198912

(119905)

V2 (119905)le 119896 (56)

6 Abstract and Applied Analysis

Integrating (56) from 0 to 119905 yields

119911 (119905) le 119896119905 + 119911 (0) (57)

orminus

1

V (119905)le 119896119905 minus

1

V (0) (58)

that isV (119905) le

V (0)

1 minus 119896119905V (0) (59)

Hence

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0) (60)

4 Convergence Process

Before we prove existence of weak solution to the fractionalLandau-Lifshitz model without Gilbert term (2) we firstrecall two Lemmas in [25 26] respectively

Lemma 6 Suppose that 119904 gt 0 and 119901 isin (1 +infin) If 119891 119892 isin Sthe Schwartz class then

1003817100381710038171003817Λ119904

(119891119892) minus 119891Λ119904119892

1003817100381710038171003817119871119901

le 119862 (1003817100381710038171003817nabla119891

100381710038171003817100381711987111990111003817100381710038171003817119892

1003817100381710038171003817119904minus11199012 +1003817100381710038171003817119891

100381710038171003817100381711990411990131003817100381710038171003817119892

10038171003817100381710038171198711199014 )

1003817100381710038171003817Λ119904

(119891119892)1003817100381710038171003817119871119901 le 119862 (

100381710038171003817100381711989110038171003817100381710038171198711199011

100381710038171003817100381711989210038171003817100381710038171199041199012 +

100381710038171003817100381711989110038171003817100381710038171199041199013

100381710038171003817100381711989210038171003817100381710038171198711199014 )

(61)

with 1199012 1199013

isin (1 +infin) such that

1

119901=

1

1199011

+1

1199012

=1

1199013

+1

1199014

(62)

where 119896119901 is the homogeneous Sobolev space and the

119896119901-norm of 119891 is defined by

10038171003817100381710038171198911003817100381710038171003817119896119901 =

10038171003817100381710038171003817Fminus1

(1003816100381610038161003816120585

1003816100381610038161003816119904

119891 (120585))10038171003817100381710038171003817119871119901

(63)

The following Lemma is due to Gagliardo and Nirenbergsee [11]

Lemma 7 Let Ω be R119899 or a bounded Lipschitz domain in R119899

with 120597Ω and let 119906 be any function in 119882119898119903

(Ω) ⋂ 119871119902(Ω) 1 le

119903 119902 le +infin For any integer 119895 0 le 119895 lt 119898 and for any number119886 in the interval 119895119898 le 119886 le 1 set

1

119901=

119895

119899+ 119886 (

1

119903minus

119898

119899) + (1 minus 119886)

1

119902 (64)

If 119898 minus 119895 minus (119899119903) is not a nonnegative integer then10038171003817100381710038171003817nabla119895119906

10038171003817100381710038171003817119871119901(Ω)le 119862119906

119886

119882119898119903(Ω)

1199061minus119886

119871119902(Ω)

(65)

If 119898 minus 119895 minus (119899119903) is a nonnegative integer then (65) holds for119886 = 119895119898 The constant 119862 depends only on 119903 119902 119898 119895 119886 and theshape of Ω

From (45) we conclude the following

Lemma 8 Solutions to (36) satisfy

sup0le119905le119879

1003817100381710038171003817119898120576

(sdot 119905)1003817100381710038171003817119867120572(02120587) le 119862 (66)

in which 119862 is independent of 120576

For the uniform bound of 120597119898120576120597119905 we have the following

lemma

Lemma 9 120597119898120576120597119905 in (36) satisfies

sup0le119905le119879

10038171003817100381710038171003817100381710038171003817

120597119898120576

120597119905(sdot 119905)

10038171003817100381710038171003817100381710038171003817119867minus119896(02120587)le 119862 (67)

in which 12 lt 119896 lt 1 and 119862 is independent of 120576

Proof For all 120595(119909) isin (119867120572

(0 2120587))3 we have

int

+infin

minusinfin

120595 sdot120597119898120576

120597119905119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(119898120576

times Λ2120572

119898120576) sdot 120595 (119909) 119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(120595 (119909) times 119898120576) sdot Λ2120572

119898120576119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

Λ120572

(120595 (119909) times 119898120576) sdot Λ120572

119898120576119889119909

le1003817100381710038171003817120595

1003817100381710038171003817119871infin1003817100381710038171003817Λ120572

11989812057610038171003817100381710038172

1198712 +

1003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712

1003817100381710038171003817Λ120572

12059510038171003817100381710038171198714

100381710038171003817100381711989812057610038171003817100381710038171198714

le 1198621003817100381710038171003817120595

1003817100381710038171003817119867119896(02120587)

(68)

in which formula ( 119886 times ) sdot 119888 = ( 119888 times 119886) sdot Sobolev embeddingtheorem and Lemma 6 are used

The following lemma is about the uniform bound of theHolder norm of 119898

120576

Lemma 10 Solutions to (36) satisfy

1003817100381710038171003817119898120576

(119909 119905)1003817100381710038171003817119862(2120572minus1)(120572+1)(2120572minus1)(2(120572+1))(119876

119879)

le 119862 (69)

where 119862 is independent of 120576

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

the approximate solution 119898119899 Multiplying equality (16) by 119892

119894

and summing for 1 le 119894 le 119899 we have

1

2

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

= int

119879

0

intT119899

(119891 sdot 119898119899

minus 119860 (119909 119905) Λ2120572

119898119899

sdot 119898119899) 119889119909 119889119905

+1

2

10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

(19)

Multiplying equality (16) by 120582119894119892119894and summing for 1 le 119894 le 119899

we have

1

2

1003817100381710038171003817Λ120572

119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+ int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot Λ2120572

119898119899

119889119909 119889119905

= int

119879

0

intT119899

119891 sdot Λ2120572

119898119899

119889119909 119889119905 +1

2

1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

(20)

Adding (19) to (20) we get

1

2

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+1

2

1003817100381710038171003817Λ120572

119898119899

(sdot 119905)10038171003817100381710038172

1198712(T119899)

+ int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot Λ2120572

119898119899119889119909 119889119905

=1

2(10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

+1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

)

+ int

119879

0

intT119899

(119891 sdot 119898119899

+ 119891 sdot Λ2120572

119898119899) 119889119909 119889119905

minus int

119879

0

intT119899

119860 (119909 119905) Λ2120572

119898119899

sdot 119898119899

119889119909 119889119905

le1

2(10038171003817100381710038171198980

10038171003817100381710038172

1198712(T119899)

+1003817100381710038171003817Λ120572

1198980

10038171003817100381710038172

1198712(T119899)

) + 1198621003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

+ 119862 int

119879

0

intT119899

100381610038161003816100381611989811989910038161003816100381610038162

119889119909 119889119905 +119870

2

10038171003817100381710038171003817Λ2120572

119898119899

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(21)

Since int119879

0intT119899

119860(119909 119905)Λ2120572

119898119899

sdot Λ2120572

119898119899119889119909 119889119905 ge

119870Λ2120572

1198981198992

1198712(T119899times(0119879)) by Gronwallrsquos inequality we have

sup0le119905le119879

1003817100381710038171003817119898119899

(sdot 119905)10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119898119899

10038171003817100381710038172

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

)

10038171003817100381710038171003817Λ2120572

119898119899

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

)

(22)

Taking the inner product of 120597119898119899120597119905 and (10) and integrating

over T119899 times (0 119879) we have

int

119879

0

intT119899

10038161003816100381610038161003816100381610038161003816

120597119898119899

120597119905

10038161003816100381610038161003816100381610038161003816

2

119889119909 119889119905

= int

119879

0

intT119899

(119891 minus 119860 (119909 119905) Λ2120572

119898119899) sdot

120597119898119899

120597119905119889119909 119889119905

le 119862 int

119879

0

intT119899

100381610038161003816100381611989110038161003816100381610038162

119889119909 119889119905 + 119862 int

119879

0

intT119899

10038161003816100381610038161003816Λ2120572

119898119899

10038161003816100381610038161003816

2

119889119909 119889119905

+1

2int

119879

0

intT119899

10038161003816100381610038161003816100381610038161003816

120597119898119899

120597119905

10038161003816100381610038161003816100381610038161003816

2

119889119909 119889119905

(23)

Hence10038171003817100381710038171003817100381710038171003817

120597119898119899

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(T119899times(0119879))

) (24)

Actually if the matrix 119860(119909 119905) is retrained to a small classof good function matrix one can get higher regularity ofsolution to (10)-(11)

Since the right-hand member of equality (22) and (24) isuniformly bounded thus the solution g can be extended to alltime andwe can extract from119898

119899a subsequence (still denoted

by 119898119899) such that

119898119899

119898119904119908 in 1198712

(0 119879 1198672120572

(T119899))weaklylowast

120597119898119899

120597119905

120597119898

120597119905 in 119871

2(0 119879 119871

2(T119899))weakly

(25)

Hence we know that [24]

119898119899

997888rarr 119898 strongly in 1198712

(0 119879 1198712

(T119899))

ae in T119899

times [0 119879]

(26)

Passing to the limit (119899 rarr infin) we find a weak solution to(10)-(11) From (16) and taking the limit 119899 rarr infin we deducethat for all 120593 in vectors (120593

1 1205932 120593

119899) there holds

intT119899

120597119898119899

120597119905120593 + int

T119899119860 (119909 119905) Λ

2120572119898119899120593 minus int

T119899119891 (119909 119905) 120593 119889119909 = 0

(27)

By a density argument we also obtain formula (27) for all 120593

in 119867120572

(119876119879

) 119876119879

= T119899 times [0 119879]

Theorem 2 Suppose that(i) 119873 times 119873 matrix 119860(119909 119905) defined on T119899 times (0 119879) is

measurable bounded and uniformly elliptic namely thereexists a constant 119870 such that

119860 (119909 119905) 120578 sdot 120578 ge 1198701003816100381610038161003816120578

10038161003816100381610038162

(28)

for all N-dimensional vectors 120578 Δ119860(119909 119905) is bounded and|nabla119860(119909 119905)|

infinle 119870radic5120575 where 120575 is the Sobolev embedding

constant satisfying1003817100381710038171003817nabla119891

10038171003817100381710038171198712(T119899times(0119879)) le 1205751003817100381710038171003817Δ119891

10038171003817100381710038171198712(T119899times(0119879)) (29)

4 Abstract and Applied Analysis

(ii) 119891(119909 119905) isin 1198712(0 119879 119867

2(T119899)) 119898

0(119909) isin 119867

120572+2(T119899) Then

there exists a unique vector-valued solution to (10)-(11) suchthat

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(T119899)) ⋂ 119867

1(T119899

times (0 119879))

sup0le119905le119879

119898 (sdot 119905)2

119867120572+2(T119899) +

10038171003817100381710038171003817100381710038171003817

120597119898

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

+ 1198982

1198712(01198791198672120572+2(T119899))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(30)

in which constant 119862 is dependent 119860(119909 119905) independent of119898(119909 119905)

Proof Let the operator Λ2 act on (10) we have

minus Δ119898119905

minus Δ119860 (119909 119905) Λ2120572

119898 + 119860 (119909 119905) Λ2120572+2

119898

minus nabla119860 (119909 119905) sdot nablaΛ2120572

119898 = minusΔ119891 (119909 119905)

(31)

Taking the inner product of Λ2120572+2

119898 and (31) and integratingover T119899 times (0 119879) we have

1

2intT119899

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

= minus int

119879

0

intT119899

Δ119891 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

Δ119860 (119909 119905) Λ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

nabla119860 (119909 119905) nablaΛ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+1

2intT119899

10038161003816100381610038161003816Λ120572+2

1198980

10038161003816100381610038161003816

2

119889119909

le1

2

10038171003817100381710038171003817Λ120572+2

1198980

10038171003817100381710038171003817

2

1198712(T119899)

+1

119870

1003817100381710038171003817Δ11989110038171003817100381710038172

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817Δ119860 (119909 119905) Λ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817nabla119860 (119909 119905) nablaΛ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

(32)

Note that

|nabla119860 (119909 119905)|infin lt119870

radic5120575

10038171003817100381710038171003817nablaΛ2120572

119898100381710038171003817100381710038171198712(T119899times(0119879))

le 12057510038171003817100381710038171003817Λ2120572+2

119898100381710038171003817100381710038171198712(T119899times(0119879))

and

Δ119860 (119909 119905) is bounded

(33)

Reporting (33) in (32) and taking into account

int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

ge 11987010038171003817100381710038171003817Λ2120572+2119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(34)

we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

119898 (sdot 119905)10038171003817100381710038171003817

2

1198712(T119899)

+10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(35)

3 Cauchy Problem for the Mollified Equation

To get existence of weak solution to (2) we consider the fol-lowing approximate equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 119898 times Λ

2120572119898 (36)

which is called mollified equation In this section and nextsection we assume that the spatial variable 119909 isin (0 2120587) ByLeray-Schauder fixed-point theorem we have the followingtheorem

Theorem3 Suppose that1198980(119909) isin 119867

120572(0 2120587) then there exists

a unique weak solution to (36)with initial-boundary condition(4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587)) (37)

where 119876119879

= (0 2120587) times (0 119879)

Proof First themapping119879120582

119871infin

(119876119879

) rarr 119871infin

(119876119879

) is definedas follows For each 119906 isin 119871

infin(119876119879

) 119898 = 119879120582(119906) is a solution to

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119906 times Λ

2120572119898 (38)

with initial condition (4) in which 0 le 120582 le 1 By Theorem 1we know that 119898 = 119879

120582(119906) is the unique solution to (38)

with initial-boundary condition (4) and (5) moreover 119898 isin

119871infin

(0 119879 1198671(0 2120587))

Obviously for all 120582 the mapping 119879120582is continuous and

for any bounded closed set of 119871infin

(119876119879

) 119879120582is uniformly

continuous with respect to 0 le 120582 le 1To apply Leray-Schauder fixed-point theorem wemake a

priori estimate on all fixed points of 119879120582

Abstract and Applied Analysis 5

Taking the inner product of 119898(119909 119905) and equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119898 times Λ

2120572119898 (39)

we have

119898 sdot120597119898

120597119905= minus120576119898 sdot Λ

2120572119898 minus 120582 (119898 times Λ

2120572119898) sdot 119898 (40)

Integrating (40) over 119876120591(0 le 120591 le 119879) we get

119898 (sdot 120591)2

1198712(02120587)

+ 21205761003817100381710038171003817Λ120572

11989810038171003817100381710038172

1198712(119876120591)

le10038171003817100381710038171198980

10038171003817100381710038172

1198712(02120587)

(41)

in which 0 le 120582 le 1 0 le 120591 le 119879 Hence

sup0le119905le119879

119898 (sdot 119905)1198712(02120587)

le 1198621 (42)

in which 1198621is a constant independent of 120576 120582

Taking the inner product of Λ2120572

119898(119909 119905) and (39) weobtain

Λ2120572

119898 sdot120597119898

120597119905= minus120576Λ

2120572119898 sdot Λ2120572

119898 minus 120582Λ2120572

119898 sdot (119898 times Λ2120572

119898)

(43)

Integrating (43) over (0 2120587) with respect to variable 119909 leadsto

1

2

119889

119889119905

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038172

1198712(02120587)

+ 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572

11989810038161003816100381610038161003816

2

119889119909 le 0 (44)

Obviously

sup0le119905le119879

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038171198712(02120587) le 119862

2 (45)

in which 1198622is a constant independent of 120582 120576 From (42) for

each 120576 gt 0 we have

1198981198712(01198791198672120572(02120587)) le 119862 (46)

In view of (42) (45) and (46) Sobolev embedding theoremgives the desired result

For small initial data 1198980(119909) we can get higher regularity

of the solution to (36)

Theorem 4 Suppose that 1198980(119909) isin 119867

120572+2(0 2120587) 120572 isin (12 1]

and 1198980(119909)2

119867120572+2 le 1119872119879 where 119872 = 119900(1120576) is a certain

constant then there exists a unique weak solution to (36) withinitial-boundary condition (4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(0 2120587)) (47)

Proof Let the operator Λ2 act on (39) we get

minusΔ119898119905

= minus 120576Λ2120572+2

119898 + 120582Δ119898 times Λ2120572

119898

minus 120582119898 times Λ2120572+2

119898 + 120582nabla119898 times nablaΛ2120572

119898

(48)

Taking the inner product of Λ2120572+2

119898 and (48) and integratingover (0 2120587) we have

1

2

119889

119889119905int

2120587

0

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572+2

11989810038161003816100381610038161003816

2

119889119909

= 120582 int

2120587

0

(Δ119898 times Λ2120572

119898) sdot Λ2120572+2

119898 119889119909

+ 120582 int

2120587

0

(nabla119898 times nablaΛ2120572

119898) sdot Λ2120572+2

119898 119889119909

le1

120576

10038171003817100381710038171003817Λ2120572

11989810038171003817100381710038171003817

2

119871infin

Δ1198982

1198712

+1

120576nabla119898

2

119871infin

10038171003817100381710038171003817nablaΛ2120572

11989810038171003817100381710038171003817

2

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

le119872

2

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

4

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

(49)

Note that119872 = 119900(1120576) is a constant hence by Lemma 5 whichwill be proved later we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

2

1198712(02120587)

le 1198623 (50)

in which 1198623is independent of 120576 120582 From (49) for each 120576 gt 0

we have

1198981198712(01198791198672120572+2(02120587)) le 119862 (51)

Lemma 5 Let 119891(119905) be nonnegative continuous functions for0 le 119905 le 119879 Suppose that 119891(0) lt 1119896119879 and

119891 (119905) le 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) 0 le 119905 le 119879 (52)

where 119896 is a constant Then

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0)(53)

holds for 0 le 119905 le 119879

Proof Define

V (119905) = 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) (54)

Then the function V(119905) is nondecreasing V(0) = 119891(0) and

119889V (119905)

119889119905= 1198961198912

(119905) le 119896V2 (119905) (55)

since 119891(119905) le V(119905) le V(119879) According to (55) the function119911(119905) = minus1V(119905) satisfies

119889119911 (119905)

119889119905=

V1015840 (119905)

V2 (119905)=

1198961198912

(119905)

V2 (119905)le 119896 (56)

6 Abstract and Applied Analysis

Integrating (56) from 0 to 119905 yields

119911 (119905) le 119896119905 + 119911 (0) (57)

orminus

1

V (119905)le 119896119905 minus

1

V (0) (58)

that isV (119905) le

V (0)

1 minus 119896119905V (0) (59)

Hence

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0) (60)

4 Convergence Process

Before we prove existence of weak solution to the fractionalLandau-Lifshitz model without Gilbert term (2) we firstrecall two Lemmas in [25 26] respectively

Lemma 6 Suppose that 119904 gt 0 and 119901 isin (1 +infin) If 119891 119892 isin Sthe Schwartz class then

1003817100381710038171003817Λ119904

(119891119892) minus 119891Λ119904119892

1003817100381710038171003817119871119901

le 119862 (1003817100381710038171003817nabla119891

100381710038171003817100381711987111990111003817100381710038171003817119892

1003817100381710038171003817119904minus11199012 +1003817100381710038171003817119891

100381710038171003817100381711990411990131003817100381710038171003817119892

10038171003817100381710038171198711199014 )

1003817100381710038171003817Λ119904

(119891119892)1003817100381710038171003817119871119901 le 119862 (

100381710038171003817100381711989110038171003817100381710038171198711199011

100381710038171003817100381711989210038171003817100381710038171199041199012 +

100381710038171003817100381711989110038171003817100381710038171199041199013

100381710038171003817100381711989210038171003817100381710038171198711199014 )

(61)

with 1199012 1199013

isin (1 +infin) such that

1

119901=

1

1199011

+1

1199012

=1

1199013

+1

1199014

(62)

where 119896119901 is the homogeneous Sobolev space and the

119896119901-norm of 119891 is defined by

10038171003817100381710038171198911003817100381710038171003817119896119901 =

10038171003817100381710038171003817Fminus1

(1003816100381610038161003816120585

1003816100381610038161003816119904

119891 (120585))10038171003817100381710038171003817119871119901

(63)

The following Lemma is due to Gagliardo and Nirenbergsee [11]

Lemma 7 Let Ω be R119899 or a bounded Lipschitz domain in R119899

with 120597Ω and let 119906 be any function in 119882119898119903

(Ω) ⋂ 119871119902(Ω) 1 le

119903 119902 le +infin For any integer 119895 0 le 119895 lt 119898 and for any number119886 in the interval 119895119898 le 119886 le 1 set

1

119901=

119895

119899+ 119886 (

1

119903minus

119898

119899) + (1 minus 119886)

1

119902 (64)

If 119898 minus 119895 minus (119899119903) is not a nonnegative integer then10038171003817100381710038171003817nabla119895119906

10038171003817100381710038171003817119871119901(Ω)le 119862119906

119886

119882119898119903(Ω)

1199061minus119886

119871119902(Ω)

(65)

If 119898 minus 119895 minus (119899119903) is a nonnegative integer then (65) holds for119886 = 119895119898 The constant 119862 depends only on 119903 119902 119898 119895 119886 and theshape of Ω

From (45) we conclude the following

Lemma 8 Solutions to (36) satisfy

sup0le119905le119879

1003817100381710038171003817119898120576

(sdot 119905)1003817100381710038171003817119867120572(02120587) le 119862 (66)

in which 119862 is independent of 120576

For the uniform bound of 120597119898120576120597119905 we have the following

lemma

Lemma 9 120597119898120576120597119905 in (36) satisfies

sup0le119905le119879

10038171003817100381710038171003817100381710038171003817

120597119898120576

120597119905(sdot 119905)

10038171003817100381710038171003817100381710038171003817119867minus119896(02120587)le 119862 (67)

in which 12 lt 119896 lt 1 and 119862 is independent of 120576

Proof For all 120595(119909) isin (119867120572

(0 2120587))3 we have

int

+infin

minusinfin

120595 sdot120597119898120576

120597119905119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(119898120576

times Λ2120572

119898120576) sdot 120595 (119909) 119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(120595 (119909) times 119898120576) sdot Λ2120572

119898120576119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

Λ120572

(120595 (119909) times 119898120576) sdot Λ120572

119898120576119889119909

le1003817100381710038171003817120595

1003817100381710038171003817119871infin1003817100381710038171003817Λ120572

11989812057610038171003817100381710038172

1198712 +

1003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712

1003817100381710038171003817Λ120572

12059510038171003817100381710038171198714

100381710038171003817100381711989812057610038171003817100381710038171198714

le 1198621003817100381710038171003817120595

1003817100381710038171003817119867119896(02120587)

(68)

in which formula ( 119886 times ) sdot 119888 = ( 119888 times 119886) sdot Sobolev embeddingtheorem and Lemma 6 are used

The following lemma is about the uniform bound of theHolder norm of 119898

120576

Lemma 10 Solutions to (36) satisfy

1003817100381710038171003817119898120576

(119909 119905)1003817100381710038171003817119862(2120572minus1)(120572+1)(2120572minus1)(2(120572+1))(119876

119879)

le 119862 (69)

where 119862 is independent of 120576

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

(ii) 119891(119909 119905) isin 1198712(0 119879 119867

2(T119899)) 119898

0(119909) isin 119867

120572+2(T119899) Then

there exists a unique vector-valued solution to (10)-(11) suchthat

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(T119899)) ⋂ 119867

1(T119899

times (0 119879))

sup0le119905le119879

119898 (sdot 119905)2

119867120572+2(T119899) +

10038171003817100381710038171003817100381710038171003817

120597119898

120597119905

10038171003817100381710038171003817100381710038171003817

2

1198712(T119899times(0119879))

+ 1198982

1198712(01198791198672120572+2(T119899))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(30)

in which constant 119862 is dependent 119860(119909 119905) independent of119898(119909 119905)

Proof Let the operator Λ2 act on (10) we have

minus Δ119898119905

minus Δ119860 (119909 119905) Λ2120572

119898 + 119860 (119909 119905) Λ2120572+2

119898

minus nabla119860 (119909 119905) sdot nablaΛ2120572

119898 = minusΔ119891 (119909 119905)

(31)

Taking the inner product of Λ2120572+2

119898 and (31) and integratingover T119899 times (0 119879) we have

1

2intT119899

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

= minus int

119879

0

intT119899

Δ119891 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

Δ119860 (119909 119905) Λ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+ int

119879

0

intT119899

nabla119860 (119909 119905) nablaΛ2120572

119898 sdot Λ2120572+2

119898 119889119909 119889119905

+1

2intT119899

10038161003816100381610038161003816Λ120572+2

1198980

10038161003816100381610038161003816

2

119889119909

le1

2

10038171003817100381710038171003817Λ120572+2

1198980

10038171003817100381710038171003817

2

1198712(T119899)

+1

119870

1003817100381710038171003817Δ11989110038171003817100381710038172

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817Δ119860 (119909 119905) Λ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

+1

119870

10038171003817100381710038171003817nabla119860 (119909 119905) nablaΛ

2120572119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

+119870

4

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

(32)

Note that

|nabla119860 (119909 119905)|infin lt119870

radic5120575

10038171003817100381710038171003817nablaΛ2120572

119898100381710038171003817100381710038171198712(T119899times(0119879))

le 12057510038171003817100381710038171003817Λ2120572+2

119898100381710038171003817100381710038171198712(T119899times(0119879))

and

Δ119860 (119909 119905) is bounded

(33)

Reporting (33) in (32) and taking into account

int

119879

0

intT119899

119860 (119909 119905) Λ2120572+2

119898 sdot Λ2120572+2

119898 119889119909 119889119905

ge 11987010038171003817100381710038171003817Λ2120572+2119898

10038171003817100381710038171003817

2

1198712(T119899times(0119879))

(34)

we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

119898 (sdot 119905)10038171003817100381710038171003817

2

1198712(T119899)

+10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712(T119899times(0119879))

le 119862 (10038171003817100381710038171198980

10038171003817100381710038172

119867120572+2(T119899)

+1003817100381710038171003817119891

10038171003817100381710038172

1198712(01198791198672(T119899))

)

(35)

3 Cauchy Problem for the Mollified Equation

To get existence of weak solution to (2) we consider the fol-lowing approximate equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 119898 times Λ

2120572119898 (36)

which is called mollified equation In this section and nextsection we assume that the spatial variable 119909 isin (0 2120587) ByLeray-Schauder fixed-point theorem we have the followingtheorem

Theorem3 Suppose that1198980(119909) isin 119867

120572(0 2120587) then there exists

a unique weak solution to (36)with initial-boundary condition(4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587)) (37)

where 119876119879

= (0 2120587) times (0 119879)

Proof First themapping119879120582

119871infin

(119876119879

) rarr 119871infin

(119876119879

) is definedas follows For each 119906 isin 119871

infin(119876119879

) 119898 = 119879120582(119906) is a solution to

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119906 times Λ

2120572119898 (38)

with initial condition (4) in which 0 le 120582 le 1 By Theorem 1we know that 119898 = 119879

120582(119906) is the unique solution to (38)

with initial-boundary condition (4) and (5) moreover 119898 isin

119871infin

(0 119879 1198671(0 2120587))

Obviously for all 120582 the mapping 119879120582is continuous and

for any bounded closed set of 119871infin

(119876119879

) 119879120582is uniformly

continuous with respect to 0 le 120582 le 1To apply Leray-Schauder fixed-point theorem wemake a

priori estimate on all fixed points of 119879120582

Abstract and Applied Analysis 5

Taking the inner product of 119898(119909 119905) and equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119898 times Λ

2120572119898 (39)

we have

119898 sdot120597119898

120597119905= minus120576119898 sdot Λ

2120572119898 minus 120582 (119898 times Λ

2120572119898) sdot 119898 (40)

Integrating (40) over 119876120591(0 le 120591 le 119879) we get

119898 (sdot 120591)2

1198712(02120587)

+ 21205761003817100381710038171003817Λ120572

11989810038171003817100381710038172

1198712(119876120591)

le10038171003817100381710038171198980

10038171003817100381710038172

1198712(02120587)

(41)

in which 0 le 120582 le 1 0 le 120591 le 119879 Hence

sup0le119905le119879

119898 (sdot 119905)1198712(02120587)

le 1198621 (42)

in which 1198621is a constant independent of 120576 120582

Taking the inner product of Λ2120572

119898(119909 119905) and (39) weobtain

Λ2120572

119898 sdot120597119898

120597119905= minus120576Λ

2120572119898 sdot Λ2120572

119898 minus 120582Λ2120572

119898 sdot (119898 times Λ2120572

119898)

(43)

Integrating (43) over (0 2120587) with respect to variable 119909 leadsto

1

2

119889

119889119905

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038172

1198712(02120587)

+ 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572

11989810038161003816100381610038161003816

2

119889119909 le 0 (44)

Obviously

sup0le119905le119879

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038171198712(02120587) le 119862

2 (45)

in which 1198622is a constant independent of 120582 120576 From (42) for

each 120576 gt 0 we have

1198981198712(01198791198672120572(02120587)) le 119862 (46)

In view of (42) (45) and (46) Sobolev embedding theoremgives the desired result

For small initial data 1198980(119909) we can get higher regularity

of the solution to (36)

Theorem 4 Suppose that 1198980(119909) isin 119867

120572+2(0 2120587) 120572 isin (12 1]

and 1198980(119909)2

119867120572+2 le 1119872119879 where 119872 = 119900(1120576) is a certain

constant then there exists a unique weak solution to (36) withinitial-boundary condition (4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(0 2120587)) (47)

Proof Let the operator Λ2 act on (39) we get

minusΔ119898119905

= minus 120576Λ2120572+2

119898 + 120582Δ119898 times Λ2120572

119898

minus 120582119898 times Λ2120572+2

119898 + 120582nabla119898 times nablaΛ2120572

119898

(48)

Taking the inner product of Λ2120572+2

119898 and (48) and integratingover (0 2120587) we have

1

2

119889

119889119905int

2120587

0

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572+2

11989810038161003816100381610038161003816

2

119889119909

= 120582 int

2120587

0

(Δ119898 times Λ2120572

119898) sdot Λ2120572+2

119898 119889119909

+ 120582 int

2120587

0

(nabla119898 times nablaΛ2120572

119898) sdot Λ2120572+2

119898 119889119909

le1

120576

10038171003817100381710038171003817Λ2120572

11989810038171003817100381710038171003817

2

119871infin

Δ1198982

1198712

+1

120576nabla119898

2

119871infin

10038171003817100381710038171003817nablaΛ2120572

11989810038171003817100381710038171003817

2

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

le119872

2

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

4

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

(49)

Note that119872 = 119900(1120576) is a constant hence by Lemma 5 whichwill be proved later we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

2

1198712(02120587)

le 1198623 (50)

in which 1198623is independent of 120576 120582 From (49) for each 120576 gt 0

we have

1198981198712(01198791198672120572+2(02120587)) le 119862 (51)

Lemma 5 Let 119891(119905) be nonnegative continuous functions for0 le 119905 le 119879 Suppose that 119891(0) lt 1119896119879 and

119891 (119905) le 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) 0 le 119905 le 119879 (52)

where 119896 is a constant Then

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0)(53)

holds for 0 le 119905 le 119879

Proof Define

V (119905) = 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) (54)

Then the function V(119905) is nondecreasing V(0) = 119891(0) and

119889V (119905)

119889119905= 1198961198912

(119905) le 119896V2 (119905) (55)

since 119891(119905) le V(119905) le V(119879) According to (55) the function119911(119905) = minus1V(119905) satisfies

119889119911 (119905)

119889119905=

V1015840 (119905)

V2 (119905)=

1198961198912

(119905)

V2 (119905)le 119896 (56)

6 Abstract and Applied Analysis

Integrating (56) from 0 to 119905 yields

119911 (119905) le 119896119905 + 119911 (0) (57)

orminus

1

V (119905)le 119896119905 minus

1

V (0) (58)

that isV (119905) le

V (0)

1 minus 119896119905V (0) (59)

Hence

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0) (60)

4 Convergence Process

Before we prove existence of weak solution to the fractionalLandau-Lifshitz model without Gilbert term (2) we firstrecall two Lemmas in [25 26] respectively

Lemma 6 Suppose that 119904 gt 0 and 119901 isin (1 +infin) If 119891 119892 isin Sthe Schwartz class then

1003817100381710038171003817Λ119904

(119891119892) minus 119891Λ119904119892

1003817100381710038171003817119871119901

le 119862 (1003817100381710038171003817nabla119891

100381710038171003817100381711987111990111003817100381710038171003817119892

1003817100381710038171003817119904minus11199012 +1003817100381710038171003817119891

100381710038171003817100381711990411990131003817100381710038171003817119892

10038171003817100381710038171198711199014 )

1003817100381710038171003817Λ119904

(119891119892)1003817100381710038171003817119871119901 le 119862 (

100381710038171003817100381711989110038171003817100381710038171198711199011

100381710038171003817100381711989210038171003817100381710038171199041199012 +

100381710038171003817100381711989110038171003817100381710038171199041199013

100381710038171003817100381711989210038171003817100381710038171198711199014 )

(61)

with 1199012 1199013

isin (1 +infin) such that

1

119901=

1

1199011

+1

1199012

=1

1199013

+1

1199014

(62)

where 119896119901 is the homogeneous Sobolev space and the

119896119901-norm of 119891 is defined by

10038171003817100381710038171198911003817100381710038171003817119896119901 =

10038171003817100381710038171003817Fminus1

(1003816100381610038161003816120585

1003816100381610038161003816119904

119891 (120585))10038171003817100381710038171003817119871119901

(63)

The following Lemma is due to Gagliardo and Nirenbergsee [11]

Lemma 7 Let Ω be R119899 or a bounded Lipschitz domain in R119899

with 120597Ω and let 119906 be any function in 119882119898119903

(Ω) ⋂ 119871119902(Ω) 1 le

119903 119902 le +infin For any integer 119895 0 le 119895 lt 119898 and for any number119886 in the interval 119895119898 le 119886 le 1 set

1

119901=

119895

119899+ 119886 (

1

119903minus

119898

119899) + (1 minus 119886)

1

119902 (64)

If 119898 minus 119895 minus (119899119903) is not a nonnegative integer then10038171003817100381710038171003817nabla119895119906

10038171003817100381710038171003817119871119901(Ω)le 119862119906

119886

119882119898119903(Ω)

1199061minus119886

119871119902(Ω)

(65)

If 119898 minus 119895 minus (119899119903) is a nonnegative integer then (65) holds for119886 = 119895119898 The constant 119862 depends only on 119903 119902 119898 119895 119886 and theshape of Ω

From (45) we conclude the following

Lemma 8 Solutions to (36) satisfy

sup0le119905le119879

1003817100381710038171003817119898120576

(sdot 119905)1003817100381710038171003817119867120572(02120587) le 119862 (66)

in which 119862 is independent of 120576

For the uniform bound of 120597119898120576120597119905 we have the following

lemma

Lemma 9 120597119898120576120597119905 in (36) satisfies

sup0le119905le119879

10038171003817100381710038171003817100381710038171003817

120597119898120576

120597119905(sdot 119905)

10038171003817100381710038171003817100381710038171003817119867minus119896(02120587)le 119862 (67)

in which 12 lt 119896 lt 1 and 119862 is independent of 120576

Proof For all 120595(119909) isin (119867120572

(0 2120587))3 we have

int

+infin

minusinfin

120595 sdot120597119898120576

120597119905119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(119898120576

times Λ2120572

119898120576) sdot 120595 (119909) 119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(120595 (119909) times 119898120576) sdot Λ2120572

119898120576119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

Λ120572

(120595 (119909) times 119898120576) sdot Λ120572

119898120576119889119909

le1003817100381710038171003817120595

1003817100381710038171003817119871infin1003817100381710038171003817Λ120572

11989812057610038171003817100381710038172

1198712 +

1003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712

1003817100381710038171003817Λ120572

12059510038171003817100381710038171198714

100381710038171003817100381711989812057610038171003817100381710038171198714

le 1198621003817100381710038171003817120595

1003817100381710038171003817119867119896(02120587)

(68)

in which formula ( 119886 times ) sdot 119888 = ( 119888 times 119886) sdot Sobolev embeddingtheorem and Lemma 6 are used

The following lemma is about the uniform bound of theHolder norm of 119898

120576

Lemma 10 Solutions to (36) satisfy

1003817100381710038171003817119898120576

(119909 119905)1003817100381710038171003817119862(2120572minus1)(120572+1)(2120572minus1)(2(120572+1))(119876

119879)

le 119862 (69)

where 119862 is independent of 120576

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Taking the inner product of 119898(119909 119905) and equation

120597119898

120597119905= minus120576Λ

2120572119898 minus 120582119898 times Λ

2120572119898 (39)

we have

119898 sdot120597119898

120597119905= minus120576119898 sdot Λ

2120572119898 minus 120582 (119898 times Λ

2120572119898) sdot 119898 (40)

Integrating (40) over 119876120591(0 le 120591 le 119879) we get

119898 (sdot 120591)2

1198712(02120587)

+ 21205761003817100381710038171003817Λ120572

11989810038171003817100381710038172

1198712(119876120591)

le10038171003817100381710038171198980

10038171003817100381710038172

1198712(02120587)

(41)

in which 0 le 120582 le 1 0 le 120591 le 119879 Hence

sup0le119905le119879

119898 (sdot 119905)1198712(02120587)

le 1198621 (42)

in which 1198621is a constant independent of 120576 120582

Taking the inner product of Λ2120572

119898(119909 119905) and (39) weobtain

Λ2120572

119898 sdot120597119898

120597119905= minus120576Λ

2120572119898 sdot Λ2120572

119898 minus 120582Λ2120572

119898 sdot (119898 times Λ2120572

119898)

(43)

Integrating (43) over (0 2120587) with respect to variable 119909 leadsto

1

2

119889

119889119905

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038172

1198712(02120587)

+ 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572

11989810038161003816100381610038161003816

2

119889119909 le 0 (44)

Obviously

sup0le119905le119879

1003817100381710038171003817Λ120572

119898 (sdot 119905)10038171003817100381710038171198712(02120587) le 119862

2 (45)

in which 1198622is a constant independent of 120582 120576 From (42) for

each 120576 gt 0 we have

1198981198712(01198791198672120572(02120587)) le 119862 (46)

In view of (42) (45) and (46) Sobolev embedding theoremgives the desired result

For small initial data 1198980(119909) we can get higher regularity

of the solution to (36)

Theorem 4 Suppose that 1198980(119909) isin 119867

120572+2(0 2120587) 120572 isin (12 1]

and 1198980(119909)2

119867120572+2 le 1119872119879 where 119872 = 119900(1120576) is a certain

constant then there exists a unique weak solution to (36) withinitial-boundary condition (4) and (5) such that

119898 (119909 119905) isin 119871infin

(0 119879 119867120572+2

(0 2120587)) (47)

Proof Let the operator Λ2 act on (39) we get

minusΔ119898119905

= minus 120576Λ2120572+2

119898 + 120582Δ119898 times Λ2120572

119898

minus 120582119898 times Λ2120572+2

119898 + 120582nabla119898 times nablaΛ2120572

119898

(48)

Taking the inner product of Λ2120572+2

119898 and (48) and integratingover (0 2120587) we have

1

2

119889

119889119905int

2120587

0

10038161003816100381610038161003816Λ120572+2

11989810038161003816100381610038161003816

2

119889119909 + 120576 int

2120587

0

10038161003816100381610038161003816Λ2120572+2

11989810038161003816100381610038161003816

2

119889119909

= 120582 int

2120587

0

(Δ119898 times Λ2120572

119898) sdot Λ2120572+2

119898 119889119909

+ 120582 int

2120587

0

(nabla119898 times nablaΛ2120572

119898) sdot Λ2120572+2

119898 119889119909

le1

120576

10038171003817100381710038171003817Λ2120572

11989810038171003817100381710038171003817

2

119871infin

Δ1198982

1198712

+1

120576nabla119898

2

119871infin

10038171003817100381710038171003817nablaΛ2120572

11989810038171003817100381710038171003817

2

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

le119872

2

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

4

1198712

+120576

2

10038171003817100381710038171003817Λ2120572+2

11989810038171003817100381710038171003817

2

1198712

(49)

Note that119872 = 119900(1120576) is a constant hence by Lemma 5 whichwill be proved later we have

sup0le119905le119879

10038171003817100381710038171003817Λ120572+2

11989810038171003817100381710038171003817

2

1198712(02120587)

le 1198623 (50)

in which 1198623is independent of 120576 120582 From (49) for each 120576 gt 0

we have

1198981198712(01198791198672120572+2(02120587)) le 119862 (51)

Lemma 5 Let 119891(119905) be nonnegative continuous functions for0 le 119905 le 119879 Suppose that 119891(0) lt 1119896119879 and

119891 (119905) le 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) 0 le 119905 le 119879 (52)

where 119896 is a constant Then

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0)(53)

holds for 0 le 119905 le 119879

Proof Define

V (119905) = 119896 int

119905

0

1198912

(120591) 119889120591 + 119891 (0) (54)

Then the function V(119905) is nondecreasing V(0) = 119891(0) and

119889V (119905)

119889119905= 1198961198912

(119905) le 119896V2 (119905) (55)

since 119891(119905) le V(119905) le V(119879) According to (55) the function119911(119905) = minus1V(119905) satisfies

119889119911 (119905)

119889119905=

V1015840 (119905)

V2 (119905)=

1198961198912

(119905)

V2 (119905)le 119896 (56)

6 Abstract and Applied Analysis

Integrating (56) from 0 to 119905 yields

119911 (119905) le 119896119905 + 119911 (0) (57)

orminus

1

V (119905)le 119896119905 minus

1

V (0) (58)

that isV (119905) le

V (0)

1 minus 119896119905V (0) (59)

Hence

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0) (60)

4 Convergence Process

Before we prove existence of weak solution to the fractionalLandau-Lifshitz model without Gilbert term (2) we firstrecall two Lemmas in [25 26] respectively

Lemma 6 Suppose that 119904 gt 0 and 119901 isin (1 +infin) If 119891 119892 isin Sthe Schwartz class then

1003817100381710038171003817Λ119904

(119891119892) minus 119891Λ119904119892

1003817100381710038171003817119871119901

le 119862 (1003817100381710038171003817nabla119891

100381710038171003817100381711987111990111003817100381710038171003817119892

1003817100381710038171003817119904minus11199012 +1003817100381710038171003817119891

100381710038171003817100381711990411990131003817100381710038171003817119892

10038171003817100381710038171198711199014 )

1003817100381710038171003817Λ119904

(119891119892)1003817100381710038171003817119871119901 le 119862 (

100381710038171003817100381711989110038171003817100381710038171198711199011

100381710038171003817100381711989210038171003817100381710038171199041199012 +

100381710038171003817100381711989110038171003817100381710038171199041199013

100381710038171003817100381711989210038171003817100381710038171198711199014 )

(61)

with 1199012 1199013

isin (1 +infin) such that

1

119901=

1

1199011

+1

1199012

=1

1199013

+1

1199014

(62)

where 119896119901 is the homogeneous Sobolev space and the

119896119901-norm of 119891 is defined by

10038171003817100381710038171198911003817100381710038171003817119896119901 =

10038171003817100381710038171003817Fminus1

(1003816100381610038161003816120585

1003816100381610038161003816119904

119891 (120585))10038171003817100381710038171003817119871119901

(63)

The following Lemma is due to Gagliardo and Nirenbergsee [11]

Lemma 7 Let Ω be R119899 or a bounded Lipschitz domain in R119899

with 120597Ω and let 119906 be any function in 119882119898119903

(Ω) ⋂ 119871119902(Ω) 1 le

119903 119902 le +infin For any integer 119895 0 le 119895 lt 119898 and for any number119886 in the interval 119895119898 le 119886 le 1 set

1

119901=

119895

119899+ 119886 (

1

119903minus

119898

119899) + (1 minus 119886)

1

119902 (64)

If 119898 minus 119895 minus (119899119903) is not a nonnegative integer then10038171003817100381710038171003817nabla119895119906

10038171003817100381710038171003817119871119901(Ω)le 119862119906

119886

119882119898119903(Ω)

1199061minus119886

119871119902(Ω)

(65)

If 119898 minus 119895 minus (119899119903) is a nonnegative integer then (65) holds for119886 = 119895119898 The constant 119862 depends only on 119903 119902 119898 119895 119886 and theshape of Ω

From (45) we conclude the following

Lemma 8 Solutions to (36) satisfy

sup0le119905le119879

1003817100381710038171003817119898120576

(sdot 119905)1003817100381710038171003817119867120572(02120587) le 119862 (66)

in which 119862 is independent of 120576

For the uniform bound of 120597119898120576120597119905 we have the following

lemma

Lemma 9 120597119898120576120597119905 in (36) satisfies

sup0le119905le119879

10038171003817100381710038171003817100381710038171003817

120597119898120576

120597119905(sdot 119905)

10038171003817100381710038171003817100381710038171003817119867minus119896(02120587)le 119862 (67)

in which 12 lt 119896 lt 1 and 119862 is independent of 120576

Proof For all 120595(119909) isin (119867120572

(0 2120587))3 we have

int

+infin

minusinfin

120595 sdot120597119898120576

120597119905119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(119898120576

times Λ2120572

119898120576) sdot 120595 (119909) 119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(120595 (119909) times 119898120576) sdot Λ2120572

119898120576119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

Λ120572

(120595 (119909) times 119898120576) sdot Λ120572

119898120576119889119909

le1003817100381710038171003817120595

1003817100381710038171003817119871infin1003817100381710038171003817Λ120572

11989812057610038171003817100381710038172

1198712 +

1003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712

1003817100381710038171003817Λ120572

12059510038171003817100381710038171198714

100381710038171003817100381711989812057610038171003817100381710038171198714

le 1198621003817100381710038171003817120595

1003817100381710038171003817119867119896(02120587)

(68)

in which formula ( 119886 times ) sdot 119888 = ( 119888 times 119886) sdot Sobolev embeddingtheorem and Lemma 6 are used

The following lemma is about the uniform bound of theHolder norm of 119898

120576

Lemma 10 Solutions to (36) satisfy

1003817100381710038171003817119898120576

(119909 119905)1003817100381710038171003817119862(2120572minus1)(120572+1)(2120572minus1)(2(120572+1))(119876

119879)

le 119862 (69)

where 119862 is independent of 120576

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Integrating (56) from 0 to 119905 yields

119911 (119905) le 119896119905 + 119911 (0) (57)

orminus

1

V (119905)le 119896119905 minus

1

V (0) (58)

that isV (119905) le

V (0)

1 minus 119896119905V (0) (59)

Hence

119891 (119905) le119891 (0)

1 minus 119896119879119891 (0) (60)

4 Convergence Process

Before we prove existence of weak solution to the fractionalLandau-Lifshitz model without Gilbert term (2) we firstrecall two Lemmas in [25 26] respectively

Lemma 6 Suppose that 119904 gt 0 and 119901 isin (1 +infin) If 119891 119892 isin Sthe Schwartz class then

1003817100381710038171003817Λ119904

(119891119892) minus 119891Λ119904119892

1003817100381710038171003817119871119901

le 119862 (1003817100381710038171003817nabla119891

100381710038171003817100381711987111990111003817100381710038171003817119892

1003817100381710038171003817119904minus11199012 +1003817100381710038171003817119891

100381710038171003817100381711990411990131003817100381710038171003817119892

10038171003817100381710038171198711199014 )

1003817100381710038171003817Λ119904

(119891119892)1003817100381710038171003817119871119901 le 119862 (

100381710038171003817100381711989110038171003817100381710038171198711199011

100381710038171003817100381711989210038171003817100381710038171199041199012 +

100381710038171003817100381711989110038171003817100381710038171199041199013

100381710038171003817100381711989210038171003817100381710038171198711199014 )

(61)

with 1199012 1199013

isin (1 +infin) such that

1

119901=

1

1199011

+1

1199012

=1

1199013

+1

1199014

(62)

where 119896119901 is the homogeneous Sobolev space and the

119896119901-norm of 119891 is defined by

10038171003817100381710038171198911003817100381710038171003817119896119901 =

10038171003817100381710038171003817Fminus1

(1003816100381610038161003816120585

1003816100381610038161003816119904

119891 (120585))10038171003817100381710038171003817119871119901

(63)

The following Lemma is due to Gagliardo and Nirenbergsee [11]

Lemma 7 Let Ω be R119899 or a bounded Lipschitz domain in R119899

with 120597Ω and let 119906 be any function in 119882119898119903

(Ω) ⋂ 119871119902(Ω) 1 le

119903 119902 le +infin For any integer 119895 0 le 119895 lt 119898 and for any number119886 in the interval 119895119898 le 119886 le 1 set

1

119901=

119895

119899+ 119886 (

1

119903minus

119898

119899) + (1 minus 119886)

1

119902 (64)

If 119898 minus 119895 minus (119899119903) is not a nonnegative integer then10038171003817100381710038171003817nabla119895119906

10038171003817100381710038171003817119871119901(Ω)le 119862119906

119886

119882119898119903(Ω)

1199061minus119886

119871119902(Ω)

(65)

If 119898 minus 119895 minus (119899119903) is a nonnegative integer then (65) holds for119886 = 119895119898 The constant 119862 depends only on 119903 119902 119898 119895 119886 and theshape of Ω

From (45) we conclude the following

Lemma 8 Solutions to (36) satisfy

sup0le119905le119879

1003817100381710038171003817119898120576

(sdot 119905)1003817100381710038171003817119867120572(02120587) le 119862 (66)

in which 119862 is independent of 120576

For the uniform bound of 120597119898120576120597119905 we have the following

lemma

Lemma 9 120597119898120576120597119905 in (36) satisfies

sup0le119905le119879

10038171003817100381710038171003817100381710038171003817

120597119898120576

120597119905(sdot 119905)

10038171003817100381710038171003817100381710038171003817119867minus119896(02120587)le 119862 (67)

in which 12 lt 119896 lt 1 and 119862 is independent of 120576

Proof For all 120595(119909) isin (119867120572

(0 2120587))3 we have

int

+infin

minusinfin

120595 sdot120597119898120576

120597119905119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(119898120576

times Λ2120572

119898120576) sdot 120595 (119909) 119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

(120595 (119909) times 119898120576) sdot Λ2120572

119898120576119889119909

= minus120576 int

+infin

minusinfin

Λ120572

120595 (119909) sdot Λ120572

119898120576119889119909

minus int

+infin

minusinfin

Λ120572

(120595 (119909) times 119898120576) sdot Λ120572

119898120576119889119909

le1003817100381710038171003817120595

1003817100381710038171003817119871infin1003817100381710038171003817Λ120572

11989812057610038171003817100381710038172

1198712 +

1003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712

1003817100381710038171003817Λ120572

12059510038171003817100381710038171198714

100381710038171003817100381711989812057610038171003817100381710038171198714

le 1198621003817100381710038171003817120595

1003817100381710038171003817119867119896(02120587)

(68)

in which formula ( 119886 times ) sdot 119888 = ( 119888 times 119886) sdot Sobolev embeddingtheorem and Lemma 6 are used

The following lemma is about the uniform bound of theHolder norm of 119898

120576

Lemma 10 Solutions to (36) satisfy

1003817100381710038171003817119898120576

(119909 119905)1003817100381710038171003817119862(2120572minus1)(120572+1)(2120572minus1)(2(120572+1))(119876

119879)

le 119862 (69)

where 119862 is independent of 120576

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Proof Denote 120596120576

= int119909

0119898120576(120585 119905)119889120585

forall120595 (119909) isin 119867119896

0(0 2120587)

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

1205951015840

(119909) 120596120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816

int

2120587

0

120595 (119909) 119898120576

119905(119909 119905) 119889119909

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817120595 (119909)

1003817100381710038171003817119867119896(02120587)

le 119862100381710038171003817100381710038171205951015840

(119909)100381710038171003817100381710038171198712(02120587)

(70)

Therefore

sup0le119905le119879

1003817100381710038171003817120596120576

119905(119909 119905)

10038171003817100381710038171198712 le 119862 (71)

Hence

120596120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572+1

(0 2120587))

⋂ 1198821infin

(0 119879 1198671minus120572

(0 2120587))

(72)

Therefore by Lemma 7

1003816100381610038161003816119898120576

(119909 1199052) minus 119898120576

(119909 1199051)1003816100381610038161003816

=1003816100381610038161003816120596120576

119909(119909 1199052) minus 120596120576

119909(119909 1199051)1003816100381610038161003816

le 1198621003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus 120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

le 11986210038161003816100381610038161199052 minus 119905

1

1003816100381610038161003816(2120572minus1)2(120572+1)

times sup0le119905le119879

1003817100381710038171003817119908120576

119905(sdot 119905)

1003817100381710038171003817(2120572minus1)2(120572+1)

1198712(02120587)

times1003817100381710038171003817120596120576

(sdot 1199052) minus120596120576

(sdot 1199051)100381710038171003817100381732(120572+1)

119867120572+1(02120587)

(73)

Weak solution to (2) with initial-boundary condition (4)and (5) is defined as follows

Definition 11 3-dimensional vector-valued function 119898(119909 119905)

is called a weak solution to (2) with initial-boundary condi-tion (4) and (5) if

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

int int119876119879

[120597119898

120597119905sdot 120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(74)

for all test functions 120593 isin (119862infin

(119876119879

))3

Theorem 12 Suppose that 1198980(119909) isin 119867

120572(0 2120587) then there

exists a weak solution to (2) with initial-boundary condition(4) and (5) satisfying

119898 (119909 119905) isin 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898

120597119905isin 119871infin

(0 119879 119867minus120572

(0 2120587))

(75)

Proof For a solution to the mollified equation (36) we have

int int119876119879

[120597119898120576

120597119905sdot 120593 + 120576Λ

120572120593 sdot Λ120572

119898120576

+ Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) 1198980

(119909) 119889119909 = 0 forall120593 isin (119862infin

(119876119879

))3

(76)

By Lemmas 8 and 9

119898120576

(119909 119905) is bounded in 119871infin

(0 119879 119867120572

(0 2120587))

⋂ 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

)

120597119898120576

120597119905(119909 119905) is bounded in 119871

infin(0 119879 119867

minus119896(0 2120587))

(77)

Hence 119898(119909 119905) isin 119862(2120572minus1)(120572+1)(2120572minus1)2(120572+1)

(119876119879

) We select asubsequence (still denoted as 119898

120576(119909 119905)) such that

119898120576

(119909 119905) is weaklowast convengent to 119898 (119909 119905)

in 119871infin

(0 119879 119867120572

(0 2120587))

119898120576

(119909 119905) is uniformly convergent to 119898 (119909 119905) in 119876119879

and

120597119898120576

120597119905(119909 119905) is weaklowast convengent to 120597119898

120597119905

in 119871infin

(0 119879 119867minus119896

(0 2120587))

(78)

By a theorem in [24] we have that

119898120576

(119909 119905) is strongly convengent to 119898 (119909 119905)

in 1198712

(0 119879 1198712

(0 2120587))

(79)

To prove that

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

997888rarr int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

(80)

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

we have to construct a commutator Actually

int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

Λ120572

(120593 times 119898120576) sdot Λ120572

119898120576119889119909 119889119905

minus int int119876119879

(120593 times Λ120572

119898120576) sdot Λ120572

119898120576119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898120576) minus 120593 times Λ

120572119898120576] sdot Λ120572

119898120576119889119909 119889119905

(81)

Similarly

int int119876119879

Λ120572

(120593 times 119898) sdot Λ120572

119898 119889119909 119889119905

= int int119876119879

[Λ120572

(120593 times 119898) minus 120593 times Λ120572

119898] sdot Λ120572

119898 119889119909 119889119905

(82)

Denote Λ120572

(120593 times 119906) minus 120593 times Λ120572

119906 byL120593

(119906) To get (80) it sufficesto prove that

int int119876119879

L120593

(119898120576) Λ120572

119898120576

minus L120593

(119898) Λ120572

119898 119889119909 119889119905

= int int119876119879

[L120593

(119898120576) minus L

120593(119898)] sdot Λ

120572119898120576

+ int int119876119879

L120593

(119898) [Λ120572

119898120576

minus Λ120572

119898] 119889119909 119889119905

= 1198681

+ 1198682

997888rarr 0 as 120576 997888rarr 0

(83)

Obviously 1198682

rarr 0 as 120576 rarr 0 We need to prove that 1198681

rarr 0as 120576 rarr 0 Actually by Lemma 6 we can get

10038171003817100381710038171003817L120593

(119898120576) minus L

120593(119898)

100381710038171003817100381710038171198712(02120587)

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 1198981003817100381710038171003817120572minus11199012 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198711199014 )

le 119862 (1003817100381710038171003817nabla120593

100381710038171003817100381711987111990111003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712 +

100381710038171003817100381712059310038171003817100381710038171205721199013

1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573)

le 1198621003817100381710038171003817120593

1003817100381710038171003817119867120574(02120587)1003817100381710038171003817119898120576

minus 1198981003817100381710038171003817119867120573(02120587)

(84)

where 1199012 1199013

isin (1 +infin) are such that

1

1199011

+1

1199012

=1

2

1

1199013

+1

1199014

=1

2

0 lt 120573 lt 120572 lt 11199012

1205721199012

minus 1= 2

1199014

=2

1 minus 2120573 2 (120574 minus 1) gt 1

(85)

Therefore 120574 gt 120572 + (12) minus (11199013) automatically

By Holderrsquos inequality we can show that

100381610038161003816100381610038161003816100381610038161003816

int int119876119879

Λ120572

119898120576L120593

(119898120576

minus 119898) 119889119909 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621003817100381710038171003817Λ120572

11989812057610038171003817100381710038171198712(119876

119879)

100381710038171003817100381712059310038171003817100381710038171198712(0119879119867120574(02120587))

times1003817100381710038171003817119898120576

minus 11989810038171003817100381710038171198712(0119879119867120573(02120587)) 997888rarr 0 as 120576 997888rarr 0

(86)

As 120576 rarr 0 in (76) we have

int int119876119879

[120597119898

120597119905120593 + Λ

120572(120593 times 119898) sdot Λ

120572119898] 119889119909 119889119905

+ int

2120587

0

120593 (119909 0) sdot 1198980

(119909) 119889119909 = 0

(87)

for all test functions 120593 isin (119862infin

(119876119879

))3

Acknowledgment

The paper is supported by the National Natural ScienceFoundation of China (no 11201181 no 11201311)

References

[1] A Hubert and R Schafer Magnetic Domains The Analysis ofMagnetic Microstructures Springer Berlin Germany 1998

[2] J M Daughton ldquoMagnetoresistive memory technologyrdquo ThinSolid Films vol 216 no 1 pp 162ndash168 1992

[3] B Heinrich and J A C Bland Ultrathin Magnetic Structures ISpringer Berlin Germany 1994

[4] G A Prinz ldquoMagnetoelectronicsrdquo Science vol 282 no 5394pp 1660ndash1663 1998

[5] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations JohnWiley New Youk NYUSA 1993

[6] I Podlubny Fractional Differential Equations Mathematics inScience and Engeineering 198 Academic Press San Diego CalifUSA 1999

[7] P Constantin D Cordoba and J Wu ldquoOn the critical dissipa-tive quasi-geostrophic equationrdquo Indiana University Mathemat-ics Journal vol 50 pp 97ndash106 2001

[8] P Constantin A J Majda and E Tabak ldquoFormation ofstrong fronts in the 2-D quasigeostrophic thermal active scalarrdquoNonlinearity vol 7 no 6 pp 1495ndash1533 1994

[9] P Constantin and J Wu ldquoBehavior of solutions of 2D quasi-geostrophic equationsrdquo SIAM Journal on Mathematical Analy-sis vol 30 no 5 pp 937ndash948 1999

[10] L Landau and E Lifshitz ldquoOn the theory of the dispersionof magnetic permeability in ferromagnetic bodiesrdquo UkrainianJournal of Physics vol 8 pp 153ndash169 1935

[11] B L Guo and M C Hong ldquoThe Landau-Lifshitz equation ofthe ferromagnetic spin chain and harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 1 no 3 pp311ndash334 1993

[12] M C Yunmei J D Shijin and L G Boling ldquoPartial regularityfor two dimensional Landau-Lifshitz equationsrdquo Acta Mathe-matica Sinica vol 14 no 3 pp 423ndash432 1998

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

[13] F Alouges and A Soyeur ldquoOn global weak solutions forLandau-Lifshitz equations existence and nonuniquenessrdquoNon-linear Analysis vol 18 no 11 pp 1071ndash1084 1992

[14] A Visintin ldquoOn Landau-Lifshitzrsquo equations for ferromag-netismrdquo Japan Journal of Applied Mathematics vol 2 no 1 pp69ndash84 1985

[15] J L Joly GMetivier and J Rauch ldquoGlobal solutions toMaxwellequations in a ferromagnetic mediumrdquoAnnales Henri Poincarevol 1 no 2 pp 307ndash340 2000

[16] G Carbou and P Fabrie ldquoRegular solutions for Landau-Lifshitzequation in a bounded domainrdquoDifferential Integral Equationsvol 14 pp 213ndash229 2001

[17] B Guo and X Pu ldquoThe fractional Landau-Lifshitz-Gilbertequation and the heat flow of harmonic mapsrdquo Calculus ofVariations and Partial Differential Equations vol 42 no 1-2 pp1ndash19 2011

[18] P L Sulem C Sulem and C Bardos ldquoOn the continuous limitfor a system of classical spinsrdquo Communications in Mathemati-cal Physics vol 107 no 3 pp 431ndash454 1986

[19] N Chang J Shatah and K Uhlenbeck ldquoSchrodinger mapsrdquoCommunications on Pure and Applied Mathematics vol 53 no5 pp 590ndash602 2000

[20] J Shatah and C Zeng ldquoSchrodinger maps and anti-ferromagnetic chainsrdquo Communications in MathematicalPhysics vol 262 no 2 pp 299ndash315 2006

[21] W Ding H Tang and C Zeng ldquoSelf-similar solutions ofSchrodinger flowsrdquo Calculus of Variations and Partial Differen-tial Equations vol 34 no 2 pp 267ndash277 2009

[22] N E Weinan and X P Wang ldquoNumerical methods for theLandau-Lifshitz equationrdquo SIAM Journal on Numerical Analy-sis vol 38 no 5 pp 1647ndash1665 2001

[23] C J Garcıa-Cervera and X Wang ldquoSpin-polarized currents inferromagnetic multilayersrdquo Journal of Computational Physicsvol 224 no 2 pp 699ndash711 2007

[24] J L Lions Quelques Methodes de Resolution des Problems AuxLimites Non Lineaire Chinese Edition Sun Yat-sen UniversityPress GuangZhou China 1996

[25] R Coifman and Y MeyerNolinear Harmonic Analysis OpertorTheory and PDE Princeton University Press Princeton NJUSA 1986

[26] A Friedman Partial Differential Equations Holt Rinehart andWinston New York NY USA 1969

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of