boling guo institute of applied physics and computational

Differential and Integral Equations Volume 22, Numbers 3–4 (2009), 251–274

STOCHASTIC LANDAU-LIFSHITZ EQUATION

Boling GuoInstitute of Applied Physics and Computational Mathematics

P. O. Box 8009, Beijing, China, 100088

Xueke PuThe Graduate School of China Academy of Engineering Physics

P. O. Box 2101, Beijing, China, 100088

(Submitted by: Giuseppe Da Prato)

Abstract. In this paper, we establish the existence of global weaksolutions to the stochastic Landau-Lifshitz equation in Rd for any d > 0.When restricted to the real line R1, we show that the Cauchy problem ofthis equation admits a unique global smooth solution with the differencemethod.

1. Introduction

As is well known, the Landau-Lifshitz-Gilbert equation [6, 11] plays animportant role in understanding magnetic phenomena emerging from mag-netic materials. Many mathematicians have contributed their efforts to thestudy of this equation and there has been a broad study from both the math-ematical and physical point of view regarding existence and uniqueness ofstrong or weak solutions and regularities as well as its link with harmonicmaps, see [1, 2, 7, 12, 15, 18] to list only a few; the list can be expanded asone wishes.

On the other hand, when thermal activation is introduced in the Landau-Lifshitz-Gilbert equation by a stochastic thermal field Hth, which is addedto the effective field Heff , we are led to the following stochastic Landau-Lifshitz-Gilbert equation [4]:

dM

dt= −γM ∧ (Heff +Hth)− αγM ∧ (M ∧ (Heff +Hth)). (1.1)

When α = 0, we call this equation the stochastic Landau-Lifshitz equation.Physical evidence shows that the stochastic integral should be understood

Accepted for publication: August 2008.AMS Subject Classifications: 60H15, 35Qxx.

251

252 Boling Guo and Xueke Pu

in the sense of Stratonovich instead of Ito [13] to yield correct thermal-equilibrium properties, which we will illustrate mathematically rigorouslyunder the assumption that the magnetization field M is of unit length. Forweak damping α 1 we can drop the fluctuating field from the relaxationterm of the above equation to arrive at

dM

dt= −γM ∧ (Heff +Hth)− αγM ∧ (M ∧Heff ), (1.2)

which will be called the stochastic Landau-Lifshitz-Gilbert equation in thefollowing. In fact, the two models are completely equivalent in the averagesense when the condition of thermodynamic consistency is applied [4], thus inthe following we consider the formally simpler one (1.2). One more thing thatshould be illustrated here is that since a large number of microscopic degreesof freedom contribute to this mechanism, the thermal field is assumed to bea Gaussian random process with zero mean value. This observation showsthat the average effects of the thermal field taken over different realizationsvanish. In this paper, we restrict ourselves to the case when the effectivefield Heff = ∆M .

Let us put forward the stochastic Landau-Lifshitz equation mathemati-cally. Let (Ω,F , (Ft)t,P) be a complete filtration probability space, andWt = (W (1)

t ,W(2)t ,W

(3)t ) be a three dimensional Ft-adapted Brownian mo-

tion on (Ω,F ,P) with the following statistical properties:

EW (i)t = 0, E

[W

(i)t W (j)

s

]= δijt ∧ s, for i, j = 1, 2, 3. (1.3)

For fundamentals of stochastic differential equations, we refer the reader to[3, 9, 14, 13] for more details.

When α = 0, setting all the coefficients to be one, we can rewrite theabove stochastic Landau-Lifshitz equation (1.1) as

du = u ∧ (∆udt+ dWt), (1.4)

where d means that the stochastic integral is understood in the sense ofStratonovich, and u = (u1, u2, u3) is the magnetization vector from Rd → S2.This is a system of Langevin equations with multiplicative noise, because themultiplicative factor u ∧ is a function of the unknown u [8]. The reasonwhy we use Stratonovich integral is illustrated in Section 2. However, it isconvenient for us to use the equivalent Ito equation. By the relationshipbetween Ito integral and Stratonovich integral [5]

X dY = XdY + 12dXdY, (1.5)

Stochastic Landau-Lifshitz equation 253

for two Ito processes X,Y , the equivalent Ito equation may be written as

du = u ∧∆udt− udt+ u ∧ dW. (SLL)

We call this equation (SLL) the stochastic Landau-Lifshitz equation, whichcan be completed with a given initial data u(t = 0, x) = u0.

In what follows this paper is concerned with the mathematical study ofthe equation (SLL). We will focus on two aspects of its solutions, namely,weak solution and smooth solution. For this purpose, we define the weaksolution as follows.

Definition 1.1 (Weak Solution). We call an Ft-adapted stochastic processu(t) with |u(t)| = 1 for almost every x ∈ Rd and E‖∇u(t)‖2L2 < ∞ for allt ∈ [0, T ] a weak solution of the stochastic Landau-Lifshitz equation (SLL),provided that

〈u(t), v(t)〉 =〈u0, v(0)〉 −∫ t

0〈u ∧∇u,∇v〉ds

−∫ t

0〈u(s), v(s)〉ds+

∫ t

0〈v(s), u(s) ∧ dW (s)〉 a.s., (1.6)

for any v(s) ∈ L∞(0, T ;H1(Rd)).

This definition requires only that ∇u(t) ∈ L2, and |u(t)| = 1 for almostevery x ∈ Rd. In this case, we seek a solution of this equation in the Sobolevspace H1(Rd), and we can regard the solution (u(t))t≥0 as a stochastic pro-cess in infinite-dimensional space with continuous parameters t ≥ 0.

By smooth solutions [16] we mean that the process (u(t))t≥0 is in anySobolev space Hk for k = 1, 2, · · · . For the existence part, the method ismainly based on the vanishing viscosity method and the finite differencemethod. We consider the viscous equation

du = −εu ∧ u ∧∆udt+ u ∧∆udt− udt+ u ∧ dW (SLLε-1)

with damping parameter ε > 0. Equivalently, we rewrite this equation as

du = ε∆udt+ ε|∇u|2udt+ u ∧∆udt− udt+ u ∧ dW, (SLLε-2)

whose equivalence when u(t) is regular is stated in the Appendix. One cansee immediately that these two equations are two variants of the stochasticLandau-Lifshitz-Gilbert equation (1.2).

Our main results are the existence of global weak solutions on Rd andthe existence of a unique global smooth solution on the real line R1 for thestochastic Landau-Lifshitz equation for any initial data (without a smallnessrestriction), see Theorem 3.1 and Theorem 5.1 for a clearer statement. As a


corollary of Section 4 and Section 5, we get the existence of global in timesmooth solutions for the Cauchy problem of the stochastic Landau-Lifshitz-Gilbert equation (1.2) on R1 for α > 0 fixed. See the corollary for details.

This paper is organized as follows: in Section 2, we show mathematicallywhy the Ito integral is not appropriate for our model and thus we shouldchoose the Stratonovich integral to keep our model physically meaningfuland to yield correct thermal equilibrium properties. In Section 3, we studythe existence of weak solutions to our equation (SLL) by difference methodin Rd for any dimension d = 1, 2, · · · . In Section 4, we study the local intime existence of smooth solutions for the approximating equation (SLLε-2)by a finite difference method and finally, in Section 5, based on the localtheory we get the existence of smooth solutions of the Cauchy problem ofthe equation (SLL) on R1. As a corollary stated at the end of this paper,we also get the existence of global smooth solutions of the Cauchy problemof the stochastic Landau-Lifshitz-Gilbert equation on R1.Convention: the letters C, c will denote some positive constants, which areunimportant and may change from line to line.Notation: we use ‖·‖m and 〈u, v〉m = 〈(−∆)m/2u, (−∆)m/2v〉0 respectivelyto denote the norm and the inner product of the homogeneous Hilbert spaceHm. When m = 0 we use ‖ · ‖ and 〈·, ·〉 for simplicity.

2. Why the Ito integral is not appropriate

In this section, we show why the Ito integral is not appropriate for ourmodel. By physical intuition, we require that the magnetization field u(t)lies on the unit sphere S2 for all time when it initially lies on the sphere.Starting from this observation, we convince the reader that the Ito integralis not appropriate for our model and we should choose the Stratonovichintegral for the study of the model.

Assume that the Ito integral is appropriate for our model. We can rewritethe equation as

du = u ∧∆udt+B(u)dW,where B(u) is antisymmetric:

B(u) =

0 −u3 u2

u3 0 −u1

−u2 u1 0

.

By the Ito formula, we have

|u|2(t) = |u0|2 + 2∫ t

0< u, u ∧∆u > ds


+ 2∫ t

0< u, u ∧ dW > +

∫ t

0Tr(BB∗)ds =

∫ t

0Tr(BB∗)ds,

where |u|2 = |u1|2 + |u2|2 + |u3|2 denotes the vector norm in R3. One cancompute directly that Tr(B(u)B∗(u)) = 2|u|2. Inserting this into the Itoformula, we have

|u|2(t) = |u0|2 + 2∫ t

0|u|2ds,

which implies that |u|2(t) = |u0|2e2t = e2t. This shows that, in the Ito sense,the solution is not on the sphere S2 although the magnetization field initiallylies on the unit sphere.

On the contrary, when the integral is understood in the Stratonovich sense,correspondingly for (SLL), one can show by the same procedure as abovethat the solution u(t) lies on the unit sphere S2 when the magnetizationfield initially lies on the unit sphere. The above analysis shows from themathematical point of view that the Stratonovich integral is appropriate forthe study of the stochastic Landau-Lifshitz equation.

3. Global weak solution for (SLL)

In this section, we establish the existence of global weak solutions tothe stochastic Landau-Lifshitz equation (SLL). More precisely, we have thefollowing theorem.

Theorem 3.1. Let u0 be such that u0 ∈ S2 for almost every x ∈ Rd, u0 ∈H1(Rd) and is measurable with respect to F. There exists a global weaksolution u(t) of the stochastic Landau-Lifshitz equation, which is an adaptedprocess satisfying the stochastic Landau-Lifshitz equation in the weak sensein Definition 1.1. The solution remains on the unit sphere S2 as time evolves,and keeps E‖∇u‖2 bounded for all time.

We utilize the difference method to derive the existence of weak solutionsof this equation. Let uh(xi) denote the magnetization vector located atthe grid point xi = λih in Rd, for λi = (λ(1)

i , · · · , λ(d)i ) ∈ Zd. Denoting

hj = (0, · · · , 0, h︸︷︷︸jth

, 0, · · · , 0), we use the forward difference quotient

D+j u

h(xi) =uh(t, xi + hj)− uh(t, xi)

h(3.1)


to approximate the space derivatives ∂xju. Similarly one can define thebackward difference quotient D−j u

h(xi). The Laplacian operator ∆ is ap-proximated by

∆uh(xi) =d∑

j=1

D+j D−j u

h(xi) =d∑

j=1

D−j D+j u

h(xi).

Denote‖uh‖p

Lph

=∑i∈Zd

|uh(xi)|phd,

when p =∞, ‖uh‖L∞h = supi∈Zd |uh(xi)|. Higher-order Sobolev norms can bedefined by higher-order difference quotients similarly. Rewriting the equa-tion (SLL) in the discrete form for the unknowns uh(t, xi)i, we obtain

duh(t, xi) = uh(t, xi) ∧ ∆uh(t, xi)dt− uh(t, xi)dt+B(uh(t, xi))dWt. (3.2)

This is a standard stochastic differential system (with respect to t) of thefollowing form

dXt = σ(t,Xt)dt+ b(t,Xt)dWt, (3.3)

whose coefficients

uh 7→ uh ∧ ∆uh − uh(t, xi), uh 7→ B(uh(t, xi))

are locally Lipschitz. Then by standard SDE theory, we know that thereexists a local solution for uh(t, xi)i∈Zd .• Zeroth estimates. Using the Ito formula, we deduce that

d|uh(xj)|2 =2(uh(t, xi), uh(t, xi) ∧ ∆uh(t, xi))dt

− 2(uh(t, xi), uh(t, xi))dt+ 2(uh(t, xi), uh(t, xi) ∧ dW )

+ Tr[B(uh(t, xi))B∗(uh(t, xi))

]dt = 0,

where B(u) is defined in Section 2 and

Tr [B(u)B∗(u)] = 2|u|2. (3.4)

This shows that uh(t, xi) remains of unit length for almost surely ω ∈ Ω.• First estimates. Applying D+

j to the equation (3.2), we have

dD+j u

h(t, xi) =D+j

[uh(t, xi) ∧ ∆uh(t, xi)

]dt

−D+j u

h(t, xi)dt+D+j u

h(t, xi) ∧ dW (t).


Using the Ito formula and summation we have∑j,xi

d|D+j u

h(t, xi)|2 =2∑j,xi

< D+j u

h(t, xi), D+j

[uh(t, xi) ∧ ∆uh(t, xi)

]> dt

− 2∑j,xi

< D+j u

h(t, xi), D+j u

h(t, xi) > dt

+ 2∑j,xi

< D+j u

h(t, xi), D+j u

h(t, xi) ∧ dW >

+∑j,xi

Tr[B(D+

j uh(t, xi))B∗(D+

j uh(t, xi))

]dt

= 0,

thanks to the well-known integration by parts formula in the discrete form∑xi

vh(xi) ·D+j u

h(xi) = −∑xi

D−j vh(xi) · uh(xi). (3.5)

This implies that‖uh‖2

H1h

= ‖uh0‖2H1

h, (3.6)

for almost surely ω ∈ Ω.In the above, we constructed a solution uh defined on the mesh Zd

h :=xi : xi = λih, λi ∈ Zd in Rd which satisfies the stochastic equation (3.2).To extend the solution to the whole space, we use an interpolation processintroduced in [10]. Denote by Ci the cell

Ci = xi;1, xi;1 + h1 × · · · × xi;d, xi;d + hd (3.7)

of the ithlattice xi. For any x ∈ Ci, let qh, ph, r(m)h be the interpolation

operators introduced below.• qhuh(x) = uh(xi).•

phuh(x) =uh(xi) +

d∑j=1

D+j u

h(xi)hj

h· (x− xi) + · · ·

+d∑

j=1

D+1 · · ·D

+j−1D

+j+1 · · ·D

+d u

h(xi)d∏

l=1,l 6=j

hl

h· (x− xi)

+D+1 · · ·D

+d u

h(xi)d∏

l=1

hl

h· (x− xi).


•

r(m)h uh(x) =uh(xi) +

d∑j=1,j 6=m

D+j u

h(xi)hj

h· (x− xi)

+D+1 · · ·D

+m−1D

+m+1 · · ·D

+d u

h(xi)d∏

l=1,l 6=m

hl

h· (x− xi).

For these operators, we have ∂∂xm

phuh = r

(m)h (D+

muh). More importantly,

if one of the interpolates phuh, qhu

h or r(m)h converges strongly (respectively

weakly) in L2 as h→ 0, then the other two also converge to the same limitin L2 strongly (respectively weakly), see [10].

We can see from the above estimates that

phuh(x)→ u(x) in L2(Ω;L∞(0, T ; H1)) weak ∗

phuh(x)→ u(x) in L2(Ω;L2(0, T ;L2

loc)) strongly,

where the second one is due to the Sobolev compactness embedding theorem.To show that the sequence uh(t, xi)h converges to a weak solution u(t)

of the stochastic equation, we consider the interpolation sequence phuhh.

Indeed, for this sequence,

〈phuh(x), phv

h(x)〉 (3.8)

= 〈phuh0(x), phv

h0 (x)〉+

∫ t

0〈phD

+i (uh ∧D−i u

h), phvh(x)〉ds

−∫ t

0〈phu

h(x), phvh(x)〉ds+

∫ t

0〈phv

h(x), phuh(x)dW 〉

holds for almost surely ω ∈ Ω, and for any sequence vh such that phvh

converges to v in L∞(0, T ;H1), thanks to (3.2) and the definition of theinterpolation operators.

We now consider the convergence of the second term and the fourth termon the right; the other terms are similar.Convergence of the second term: We need only to show that phD

+i (uh∧

D−i uh) converges to ∂xi(u∧ ∂xi) in L∞(0, T ;H−1

loc ) weak *. Since uh ∧D−i uh

is in a bounded subset of L∞(0, T ;L2h), ph(uh ∧D−i uh), qh(uh ∧D−i uh) and

r(m)h (uh∧D−i uh) converge to the same limit in L∞(0, T ;L2). Rewrite qh(uh∧D−i u

h) = qhuh∧qhD−i uh. By the previous strong convergence result, qhuh →

u for almost every x ∈ Rd up to a subsequence by the Riesz theorem and


qhD−i u

h → ∂xiu weak *, one then has

ph(uh ∧D+i u

h)→ u ∧ ∂u

∂xiin L∞(0, T ;L2) weak ∗;

∂

∂xiph(uh ∧D+

i uh)→ ∂

∂xi(u ∧ ∂u

∂xi) in L∞(0, T ;L2) weak ∗ .

This implies that r(i)h D+

i (uh ∧D−i uh) and thus phD+i (uh ∧D−i uh) converges

to the same limit ∂∂xi

(u ∧ ∂u∂xi

) in L∞(0, T ;H−1loc ) weak *.

Convergence of the fourth term: Denote

Qh −Q :=∫ t

0〈phv

h, phuh ∧ dWs〉 −

∫ t

0〈v, u ∧ dWs〉

=∫ t

0〈v, (phu

h − u) ∧ dWs〉+∫ t

0〈phv

h − v, phuhdWs〉 =:Mh +Nh.

We use the BDG inequality to deduce that

E sup0≤s≤t

|Qh −Q|2 ≤ E sup0≤s≤t

|Mh|2 + E sup0≤s≤t

|Nh|2

≤ sup0≤s≤t

E∫ t

0

∣∣∣ ∫Rd

v ∧ (phuh − u)dx

∣∣∣2ds+ sup

0≤s≤tE∫ t

0

∣∣∣ ∫Rd

(phvh − v) ∧ phu

hdx∣∣∣2ds→ 0,

where the first one is due to the weak convergence of phuh − u to 0 in

L2(0, t; H1) for almost surely ω ∈ Ω, and v is bounded in L2(0, t;L2) andthe second one is due to the fact that phu

h is bounded in L2(0, t; H1) foralmost surely ω ∈ Ω and phvh − v is convergent strongly in L2(0, t;H1).This shows that E sup0≤s≤t |Qh − Q|2 → 0, which implies that there existsa subsequence of phu

h(x)h such that∫ t

0〈phv

h, phuh ∧ dWs〉 →

∫ t

0〈v, u ∧ dWs〉, (3.9)

up to a subsequence for almost surely ω ∈ Ω by the Risez theorem; i.e., everysequence an → a in norm has a subsequence an′ such that an′ → a almosteverywhere.

Then passing to the limit of (3.8) as h→ 0, we have that

〈u, v〉 = 〈u0, v0〉+∫ t

0〈u ∧∆u, v〉ds−

∫ t

0〈u, v〉ds+

∫ t

0〈v, u ∧ dW 〉 (3.10)


for any v ∈ L∞(0, T ;H1). This shows that u constructed as the limit ofthe approximating sequence uh is a weak solution of the stochastic Landau-Lifshitz equation and concludes the proof of the theorem.

4. Local theory for ε > 0

In this section, we establish the local in time existence of solutions tothe periodic approximating stochastic equation on the finite interval Λ =(−D,D)

du = ε∆udt+ ε|∇u|2udt+ u ∧∆udt− udt+ u ∧ dW, (SLLε-2)

for fixed ε > 0 with initial data u0. To this end, we consider the followingdifference equation:

duh(t, xi) = εD+D−uh(t, xi)dt+ ε|D+uh(t, xi)|2uh(t, xi)dt

+ uh(t, xi) ∧D+D−uh(t, xi)dt− uh(t, xi)dt+ uh(t, xi) ∧ dW, (4.1)

where i ∈ J := 0,±1, · · · ,±J, h = D/J is the step size and D+, D−

are one-dimensional difference quotient operators defined in (3.1). The pe-riodic condition implies that uh(t, x−J) = uh(t, xJ) for any t ≥ 0. This isa system of stochastic ordinary differential equations with respect to t, andthe unknowns are uh(t, xi)i∈J . The coefficients of this system are locallyLipschitz thus, by the standard SODE theory, we know that there exists alocal solution uh(t, xi)i∈J on [0, Th] of the stochastic system (4.1). In thefollowing, we will give some a priori estimates independent of the step sizeh > 0, then letting h → 0, one can get a local in time solution of the ap-proximating stochastic Landau-Lifshitz equation (SLLε-2) on some publicinterval [0, T0].

Before starting our exposition, we introduce some notation. Defining thefirst-order difference quotient as in (3.1), one can then define the discretefunction δuh as

δuh := D+uh(xi) : i ∈ J ,the Lp

h-norm of which can be defined by

‖δuh‖pLp

h=∑j∈J|D+uh(xi)|ph.

Applying the difference operator D+ to the first-order difference quotient,one can define the second-order difference quotient of the discrete functionuh(xi) : i ∈ J as

D+D−uh(xi) =uh(xi+1)− 2uh(xi) + uh(xi−1)

h2, for all −J+1 ≤ i ≤ J−1.


Since we are given the periodic boundary condition, the definition can beeasily extended to all i ∈ J . One needs only to use uh(x−J+1) to replaceuh(xJ+1) and uh(xJ−1) to replace uh(x−J−1) in the above formula, so thatthe second-order difference quotient is well defined for all i ∈ J . By induc-tion, one can define the kth-order difference quotient for all i ∈ J and allk ∈ N and similarly the discrete function δkuh and their Lp

h-norms. For theinterpolation inequality in the discrete form, we have the following:

Lemma 4.1 (Theorem 3, PP.6, [17]). For any discrete function uh =uj |j = 0,±1, · · · ,±J on the finite interval Λ, there holds

‖δkuh‖Lph≤ c‖uh‖1−

k+12−

1p

n

L2h

(‖δnuh‖L2

h+‖uh‖L2

h

Dn

) k+12−

1p

n, (4.2)

where 2 ≤ p ≤ ∞, 0 ≤ k ≤ n for any n ∈ N and c is a constant independentof uh.

Lemma 4.2. Let uh(t, xi) be a local solution of the equation (4.1) with F0

measurable initial data u0 ∈ S2 and ∇u0(x) ∈ L2. Then there exist a T0 > 0and a constant C > 0 such that

sup0≤t≤T0

E‖uh(t)‖2L2h

+ sup0≤t≤T0

E‖δuh(t)‖2L2h≤ C, (4.3)

where T0 and C are independent of h.

Proof. Applying the Ito formula for F (x) = |x|2, we have

d|uh(t, xi)|2

= 2εuh(t, xi) ·D+D−uh(t, xi)dt+ 2εuh(t, xi) · |D+uh(t, xi)|2uh(t, xi)dt

+ 2uh(t, xi) · uh(t, xi) ∧D+D−uh(t, xi)dt− 2uh(t, xi) · uh(t, xi)dt

+ 2uh(t, xi) · uh(t, xi) ∧ dW + Tr(B(uh(t, xi))B∗(uh(t, xi)))dt.

One can easily observe that the last four terms vanish. Multiplying the stepsize h, summing over all indices i and integrating over [0, t], we obtain∑

i

|uh(t, xi)|2h =∑

i

|uh0(t, xi)|2h+ 2ε

∫ t

0

∑i

uh(t, xi) ·D+D−uh(t, xi)dτ

+ 2ε∫ t

0

∑i

uh(t, xi) · |D+uh(t, xi)|2uh(t, xi)hdτ.

We thus have for almost surely ω ∈ Ω

‖uh‖2L2h

+ 2ε∫ t

0‖δuh‖2L2

hdτ ≤ ‖uh

0‖2L2h

+ 2Cε∫ t

0‖uh‖4L2

h+ ‖δuh‖4L2

hdτ, (4.4)


thanks to the interpolation inequality in Lemma 4.1 for p =∞.On the other hand, applying the difference operator D+ to the equation

(4.1), and then applying the same procedure as above, one has∑i

|D+D−uh(t, xi)|2h

≤∑

i

|D+D−uh0(t, xi)|2h− 2ε

∫ t

0

∑i

|D+D−uh(t, xi)|2hdτ

+2ε4

∫ t

0

∑i

|D+D−uh(t, xi)|2hdτ + 2ε∫ t

0max

i|uh(t, xi)|2

∑i

|D+uh|4hdτ.

Applying the interpolation inequality for p =∞ and p = 4, we have

‖δuh‖2L2h

+ 2ε∫ t

0‖δ2uh‖2L2

hdτ ≤ ‖δuh

0‖2L2h

+2ε4

∫ t

0‖δ2uh‖2L2

hdτ

+ 2ε∫ t

0‖uh‖

32

L2h

(‖δ2uh‖L2

h+ ‖uh‖L2

h

) 12 · ‖δuh‖3L2

h

(‖δ2uh‖L2

h+ ‖δuh‖L2

h

)dτ.

Then, applying the Holder inequality, one finally obtains for almost surelyω ∈ Ω

‖δuh‖2L2h

+ ε

∫ t

0‖δ2uh‖2L2

hdτ (4.5)

≤ ‖δuh0‖2L2

h+ Cε

∫ t

0(1 + ‖uh‖18

L2h

+ ‖δuh‖18L2

h)dτ.

This together with (4.4) implies that

‖uh‖2L2h

+ ‖δuh‖2L2h

+ ε

∫ t

0‖δ2uh‖2L2

hdτ (4.6)

≤ ‖δuh0‖2L2

h+ Cε

∫ t

0(1 + ‖uh‖18

L2h

+ ‖δuh‖18L2

h)dτ,

which shows that there exist constants C and T0 such that for almost surelyω ∈ Ω

‖uh‖2L2h

+ ‖δuh‖2L2h≤ C, for any t ∈ [0, T0],∫ T0

0‖δuh‖2L2

hdτ ≤ C.

Then taking expectations of the above inequality finishes the proof of thelemma.


Lemma 4.3. Let uh(t, xi) be a solution of the equation (4.1) with F0 mea-surable initial data u0 ∈ S2 and ∇u0(x) ∈ H1. Then there exist a T0 > 0and a constant C > 0 such that

sup0≤t≤T0

E‖δ2uh(t)‖2L2h

+∫ T0

0E‖δ3uh(t)‖2L2

hdt ≤ C,


Proof. The proof is similar to the one above. Applying the difference oper-ator D+D− to the equation (4.1), and then applying the Ito formula to thefunction F (x) = |x|2, one obtains

d|D+D−uh(t, xi)|2 = 2εD+D−uh(t, xi) ·D+D−D+D−uh(t, xi)dt

+ 2εD+D−uh(t, xi) ·D+D−(|D+uh(t, xi)|2uh(t, xi))dt

+ 2D+D−uh(t, xi) ·D+D−(uh(t, xi) ∧D+D−uh(t, xi))dt

− 2D+D−uh(t, xi) ·D+D−uh(t, xi)dt

+ 2D+D−uh(t, xi) ·D+D−uh(t, xi) ∧ dW

+ Tr(B(D+D−uh(t, xi))B∗(D+D−uh(t, xi)))dt.

Also, the sum of the last three terms vanishes. Summation over all indicesi gives that∑

i

|D+D−uh(t, xi)|2h

=∑

i

|D+D−uh0(t, xi)|2h− 2ε

∫ t

0

∑i

|D+D−D+uh(t, xi)|2hdτ

− 2ε∫ t

0

∑i

D−D+D−uh(t, xi) ·D−(|D+uh(t, xi)|2uh(t, xi))hdτ

+ 2∫ t

0

∑i

D−D+D−uh(t, xi) ·D−(uh(t, xi) ∧D+D−uh(t, xi))hdτ.

Observe that∫ t

0

∑i

D−D+D−uh(t, xi) ·D−(|D+uh(t, xi)|2uh(t, xi))hdτ

≤14

∫ t

0

∑i

|D−D+D−uh(t, xi)|2hdτ + c

∫ t

0max

i|D+uh(t, xi)|2


×∑

i

|D+D−uh(t, xi)|2h+∑

i

|D+uh(t, xi)|6hdτ

and

2∫ t

0

∑i

D−D+D−uh(t, xi) ·D−(uh(t, xi) ∧D+D−uh(t, xi))hdτ

≤2ε4

∫ t

0

∑i

|D−D+D−uh(t, xi)|2hdτ

+2ε

∫ t

0|D−uh(t, xi) ∧D+D−uh(t, xi)|2hdτ

≤2ε4

∫ t

0

∑i

|D−D+D−uh(t, xi)|2hdτ

+2ε

∫ t

0max

i|D−uh(t, xi)|2

∑i

|D+D−uh(t, xi)|2hdτ.

By the previous Lemma 4.2 and the interpolation inequality in Lemma 4.1,we have

supi,0≤t≤T0

|uh(t, xi)| ≤ C. (4.7)

Combining all the above estimates, one finally obtains

‖δ2uh‖2L2h

+ ε

∫ t

0‖δ3uh‖2L2

h≤ ‖δ2uh

0‖2L2h

+ c

∫ t

0(1 + ‖δ2uh‖2L2

h)dτ.

Taking expectations, and using the Gronwall inequality, we know that thereexist some constants C and T0 independent of the step size h such that

sup0≤t≤T0

E‖δ2uh‖2L2h

+∫ T0

0E‖δ3uh‖2L2

hdτ ≤ C. (4.8)

One finishes the proof of the lemma.

Similarly, one can establish the higher-order estimates of the solutions andwe are led to the following.

Lemma 4.4. Let uh(t, xi) be a solution of the equation (4.1) with the F0-measurable initial data uh

0(xi) ∈ S2 and ∇u0(x) ∈ Hk for any k ≥ 2. Thereexist a T0 > 0 and a constant C > 0 such that

sup0≤t≤T0

E‖δk+1uh(t)‖2L2h≤ C,



Gathering together the above lemmas, we see that the a priori estimatesare independent of the step size h > 0, thus one can let h→ 0 to get a localin time solution u(t, x) of the periodic approximating stochastic Landau-Lifshitz equation (SLLε-2) on the finite interval Λ on some public interval[0, T0] for ε > 0 fixed. We summarize as follows.

Proposition 4.1. Let ε > 0 be fixed. The periodic approximating stochasticLandau-Lifshitz equation (SLLε-2) on the finite interval Λ admits a local intime solution u(t, x). This solution is constructed as the limit of solutions ofthe difference schemes (4.1) of the approximating stochastic system (SLLε-2)as h→ 0.

5. Global solution for (SLL)

In this section, we derive some global a priori estimates independent ofthe viscous parameter ε > 0 of the solutions constructed in Section 4 indimension d = 1. Then passing to the limit ε → 0, we get a global smoothsolution of the equation (SLL). We summarize our result in the followingtheorem.

Theorem 5.1. Let ∇u0 ∈ Hm for m ≥ 1 with u0 ∈ S2 and F0 measur-able. There exists a unique continuous Ft-adapted solution (u(t))t≥0 suchthat u(t) ∈ S2 for almost every x ∈ R1 and ∇u(t) ∈ L∞(0, T ;Hm) for al-most surely ω ∈ Ω for the Cauchy problem of the stochastic Landau-Lifshitzequation (SLL) on the real line R1; furthermore the solution is global in time.

Lemma 5.1. Let u(t) be a smooth solution of the approximating stochasticLandau-Lifshitz equation (SLLε-2) with F0-measurable smooth initial datau0; then the following estimates hold:

|u(t, x)| = 1 for any (t, x) ∈ [0, T ]× Λ (5.1)

sup0≤t≤T

E‖∇u‖2L2(Λ) ≤ C, (5.2)

where C may depend on the initial data, but is independent of ε,D.

Proof. By the Ito formula, we have that

‖∇u(t)‖2 = ‖∇u0‖2 + I11 + I12 + I13, (5.3)

where

I11 = 2∫ t

0〈∇u,∇(−εu ∧ u ∧∆u+ u ∧∆u− u)〉ds,

I12 = 2∫ t

0〈∇u,∇u ∧ dW 〉, I13 =

∫ t

0‖Tr(BB∗)‖21ds.


Let us compute one by one.

I11 = 2∫ t

0〈∆u, εu ∧ u ∧∆u〉ds− 2

∫ t

0‖∇u‖2ds

= −2ε∫ t

0‖u ∧∆u‖2ds− 2

∫ t

0‖∇u‖2ds;

I12 = 0; I13 = 2∫ t

0‖∇u‖2ds.

Then (5.3) becomes

‖∇u(t)‖2 = ‖∇u0‖2 − 2ε∫ t

0‖u ∧∆u‖2ds. (5.4)

This leads to‖∇u(t)‖2 ≤ ‖∇u0‖2, a.s. ω ∈ Ω, (5.5)

and thusE‖∇u(t)‖2 ≤ E‖∇u0‖2, (5.6)

which finishes the proof of the lemma.

Remark. Indeed we can expect more about the estimates. By (5.5), onehas sup0≤t≤T ‖∇u(t)‖2 ≤ ‖∇u0‖2, then taking expectations implies that

E sup0≤t≤T

‖∇u(t)‖2 ≤ ‖∇u0‖2. (5.7)

Similar remarks hold in the following lemmas.

Lemma 5.2. Let u(t) be a smooth solution of the approximating stochasticLandau-Lifshitz equation (SLLε-2) with F0-measurable initial data u0; thenfor any T > 0 the following estimates hold:

sup0≤t≤T

E‖uxx‖2L2(Λ) ≤ C, (5.8)


Proof. Since |u(t, x)| = 1 for any (t, x) ∈ [0, T ] × Λ, u, ux, u ∧ ux form anorthogonal basis in R3. Write

uxx = αu+ βux + γu ∧ ux. (5.9)

By simple computation, one has

α = −|ux|2, β =ux · uxx

|ux|2, γ =

(u ∧ ux) · uxx

|ux|2.


By the Ito formula, one has for (SLLε-2) that

‖∂2xu(t)‖2 = ‖∂2

xu0‖2 + I21 + I22 + I23, (5.10)

where

I21 = 2∫ t

0〈∂2

xu, ∂2x(ε∆u+ ε|∇u|2u+ u ∧∆u− u)〉ds,

I22 = 2∫ t

0〈∂2

xu,B(∂2xu)〉dW, I23 =

∫ t

0‖Tr(BB∗)‖22ds.

For the term I23, we have

I23 = 2∫ t

0‖uxx‖2ds.

For the term I21, we have

I21 =− 2ε∫ t

0‖uxxx‖2ds− 2

∫ t

0〈uxxx, ux ∧ uxx〉ds

− 2ε∫ t

0〈uxxx, |ux|2ux + 2(ux · uxx)u〉ds− 2

∫ t

0‖uxx‖2ds

= : R1 +R2 +R3 +R4.

For these terms, we have

R2 =− 2∫ t

0

∫ D

−D

[ux ∧

(− |ux|2u+

(u ∧ ux) · uxx

|ux|2u ∧ ux

)]· uxxxdxds

=− 2∫ t

0

∫ D

−D|ux|2(u ∧ ux) · uxxxdx+

∫ D

−D

(u ∧ ux) · uxx

|ux|2|ux|2u · uxxxdxds

=− 5∫ t

0

∫ D

−D|ux|2(u ∧ ux) · uxxxdxds,

where, in the last step, we have used the relationship

u · uxxx = −32(|ux|2)x;

R3 = 2ε∫ t

0

∫ D

−D|ux|2|uxx|2dxds+ 16ε

∫ t

0

∫ D

−D|ux · uxx|2dxds.

On the other hand, applying the operator ∂x to the stochastic Landau-Lifshitz equation (SLLε-2), we have

dux = ε∂xuxxdt+ ε∂x(|ux|2u)dt+ ∂x(u∧uxx)dt− ∂xudt+ ∂xu∧ dW. (5.11)


Let F (p) = |p|4 = (|p1|2 + |p2|2 + |p3|2)2 for a vector p ∈ R3. By the Itoformula we have

‖∂xu‖4L4 =∫ D

−DF (∂xu(t))dx = 4

∫ t

0〈|∂xu|2∂xu, dux〉ds+ 4

∫ t

0

∫ D

−D|∂xu|4dxds

= 4∫ t

0

∫ D

−D

(− ε|∂xu|2|∂xxu|2dx− 2ε|∂xu · ∂xxu|2 + ε|ux|6

− |ux|2(u ∧ ux) · uxxx

)dxds+ 4

∫ t

0

∫ D

−D|ux|2ux · ux ∧ dxdW,

which can be rewritten clearly as

‖∂xu‖4L4 = 4∫ t

0

∫ D

−D

(− ε|∂xu|2|∂xxu|2dx− 2ε|∂xu · ∂xxu|2

+ ε|ux|6 − |ux|2(u ∧ ux) · uxxx

)dxds. (5.12)

Putting together (5.10)-(5.12), we have

4‖uxx(t)‖2 + 8ε∫ t

0‖uxxx(s)‖2ds+ 20ε

∫ t

0

∫ D

−D|ux(s)|6dxds

= 5‖ux(t)‖4L4 − 5‖u0x‖4L4 + 4‖u0xx‖+ 28ε∫ t

0

∫ D

−D|ux|2|uxx|2dxds

+ 104ε∫ t

0

∫ D

−D|ux · uxx|2dxds. (5.13)

Denote A = 28∫ D−D |ux|2|uxx|2dx + 104

∫ D−D |ux · uxx|2dx. We can estimate

this term by

A ≤ 132∫ D

−D|ux|2|uxx|2dx ≤ δ

∫|ux|6 + c(δ)

∫|uxx|3dx

≤ δ∫ D

D

|ux|6dx+ c(δ)‖ux‖54 ‖uxxx‖

74

≤δ∫ D

D

|ux|6dx+ δ‖uxxx‖2 + c(δ)‖ux‖10.

Let δ = 4. From (5.13), we have

4‖uxx(t)‖2 + 4ε∫ t

0‖uxxx(s)‖2ds+ 16ε

∫ t

0

∫ D

−D|ux(s)|6dxds


=5‖ux(t)‖4L4 − 5‖u0x‖4L4 + 4‖u0xx‖2 + c(δ)∫ t

0‖ux‖10ds

≤2‖uxx(t)‖2 + c‖ux(t)‖6 + 4‖u0xx‖2 + c(δ)∫ t

0‖ux‖10ds, (5.14)

where in the last step we have used the interpolation inequality in dimensionone

‖ux‖4L4 ≤ c‖ux‖3‖uxx‖.Thanks to Lemma 5.1, one then derives from (5.14) that for almost surelyω ∈ Ω,

‖uxx‖2L2(Λ) ≤ C(T, ‖u0‖H2), (5.15)

which followssup

0≤t≤TE‖uxx‖2L2(Λ) ≤ C, (5.16)

where the constant C may depend on ‖u0‖H2 but is independent of ε andD.

Lemma 5.3. Let u(t) be a smooth solution of the approximating stochasticLandau-Lifshitz equation (SLLε-2); then the following estimates hold

sup0≤t≤T

E‖∂3xu‖2L2(Λ) ≤ C, (5.17)

where C depends on the initial data, but independent of ε,D.

Proof. The proof is similar to that in Lemma 5.2, thus we only sketch theproof. Applying the operator L = ∂3

x to the equation (SLLε-2), we have

d(∂3xu) = ε∂5

xu(t)dt+ ε∂3x(|ux|2u)dt+ ∂3

x(u ∧∆u)dt− ∂3xudt+ ∂3

xu ∧ dW.

Then, using the Ito formula for F (u) = |u|2, we can derive that

‖∂3xu‖2 = ‖∂3

xu0‖2 + I31 + I32 + I33, (5.18)

where

I31 =2ε∫ t

0〈∂3

xu(s), ∂5xu(s)〉ds+ 2ε

∫ t

0〈∂3

xu, ∂3x(|ux|2u)〉ds

+ 2∫ t

0〈∂3

xu, ∂3x(u ∧ uxx)〉ds− 2

∫ t

0〈∂3

xu, ∂3xu〉ds

= : R1 +R2 +R3 +R4,

I32 =2∫ t

0〈∂3

xu, ∂3xu ∧ dW 〉 = 0,


≤ c∫ t

0

∫|uxxx|2dxds+ c

∫ t

0‖uxx‖6ds,

where in the third equality we have used the fact that

u · uxxxx = −3|uxx|2 − 4ux · uxxx,

and in the last step we have used the interpolation inequality

‖uxx‖L4 ≤ C‖uxx‖34 ‖uxxx‖

14 .

Putting together (5.5) and the estimates for R2,R3 we are led to

‖uxxx‖2 + ε

∫ t

0‖uxxxx‖2 ≤ ‖u0xxx‖2 + c

∫ t

0‖uxxx‖2ds+ C, (5.19)

which shows that

‖uxxx‖2 ≤ C(T, ‖u0xxx‖2), a.s. ω ∈ Ω. (5.20)

Taking expectations then finishes the proof of the lemma.

Lemma 5.4. Let u(t) be a smooth solution of the approximating stochasticLandau-Lifshitz equation (SLLε-2); then the following estimates hold for anyk ≥ 2 :

sup0≤t≤T

E‖∂kxu‖2L2(Λ) ≤ C, (5.21)


Proof. Applying the operator L = (−∆)k/2 to the equation, and then usingthe Ito formula as above, one can get the estimates.

These estimates and the local (in time) theory obtained in the previousSection 4 then finish the proof of Theorem 5.1: the existence of global (intime) smooth solutions of the Cauchy problem of the stochastic Landau-Lifshitz equation (SLL) on R1. Indeed since the above estimates are inde-pendent of ε > 0 and D > 0, one can let ε → 0 and D → ∞ to get alocal in time smooth solution of the Cauchy problem (SLL) on the real lineR1, thanks to the local in time existence result of the approximating systemestablished in Proposition 4.1. The globality in time of the solution is astandard application of the continuity method and the global estimates es-tablished in the above lemmas. This finishes the proof of Theorem 5.1 oncewe show its uniqueness. However, uniqueness is implied by the regularity wehave proved in the above lemmas. We sketch it in the following.

Let u1 and u2 be two solutions with the same initial data. Setting w =u− v, then one has

dw = (u ∧∆w + w ∧∆v)dt− wdt+ w ∧ dW. (5.22)


By the Ito formula, and integration in the x-variable, we have

‖w(t)‖2 =∫ t

0〈w(τ), u(τ) ∧∆w(τ)〉dτ ≤ C

∫ t

0‖w(τ)‖2 + ‖∇w(τ)‖2dτ,

as well as

‖∇w(t)‖2L2 = −〈w(t),∆w(t)〉 = −2∫ t

0〈w(τ) ∧∆v(τ),∆w(τ)〉dτ

≤ C∫ t

0‖w(τ)‖2 + ‖∇w(τ)‖2dτ.

Putting together the two inequalities and using the Gronwall inequality theuniqueness of the solution of the (SLL) equation follows.

As a byproduct, we have the existence of smooth solutions for the Cauchyproblem of the stochastic Landau-Lifshitz-Gilbert equation (1.2) on R1 withdamping parameter α > 0 fixed. Noting that α plays the same role as theviscosity parameter ε plays in (SLLε-2), we regard them as the same.

Corollary 5.1. Let the Gilbert damping parameter α > 0 be fixed. For anyF0 measurable smooth initial data u0, the stochastic Landau-Lifshitz-Gilbertequation (1.2) admits a global in time smooth solution, which is a continuousFt-adapted process in the Sobolev space Hk for any k ∈ N.

Proof. Indeed, for fixed ε = α > 0, Proposition 4.1 gives the existenceof local smooth solutions to the periodic problem on the domain Λ. Sincethe estimates established in this section are independent of D and global,one can let D → ∞ to get a global in time smooth solution to the Cauchyproblem of the stochastic Landau-Lifshitz-Gilbert equation (1.2) on the realline R1.

One can also let D > 0 be fixed, and let ε → 0 to get a global in timesmooth solution for both the stochastic Landau-Lifshitz equation and thestochastic Landau-Lifshitz-Gilbert equation in Λ under periodic boundaryconditions. We omit the statement of the details here.

Appendix: the equivalence between the two approximatingequations (SLLε-1) and (SLLε-2)

To be complete, we show that the following two equations are equivalentwith the same initial data u0 ∈ S2:

du = u ∧ u ∧∆udt+ u ∧∆udt− udt+ u ∧ dW, (A.1)

du = ∆udt+ |∇u|2udt+ u ∧∆udt− udt+ u ∧ dW. (A.2)The equivalence for a general viscous parameter ε > 0 can be shown similarly.


• (A.1) =⇒ (A.2)

As in Section 2, we can show that, for any (t, x), |u(t, x)| = 1 holds, whichimplies that u·∆u = −|∇u|2. Then by the formula a∧(b∧c) = (a·c)b−(a·b)c,we have

u ∧ (u ∧∆u) = −|∇u|2u−∆u,

which shows that (A.1) implies (A.2) for regular enough solutions.

• (A.2) =⇒ (A.1)

We only need to show that the solution of (A.2) remains on the sphere S2

as time evolves. In fact, by the Ito formula,

d|u|2 =2u ·∆udt+ 2u · (|∇u|2u)dt+ 2u · (u ∧∆u)

− 2u · udt+ 2u · (u ∧ dW ) + Tr(BB∗)dt

=2u ·∆udt+ 2|∇u|2|u|2,

which can be rewritten as

d|u|2 = ∆|u|2dt− 2|∇u|2dt+ 2|∇u|2|u|2dt,

by the fact ∆|u|2 = 2|∇u|2 + 2u ·∆u.Denote U(t) = |u(t)|2 − 1. One has

dU(t) = ∆U(t)dt+ 2|∇u(t)|2U(t)dt

with initial data U(0) = 0. Applying the Ito formula to this equation, onegets

|U(t)|2 = |U(0)|2 + 2∫ T

0U(t) ·∆U(t)dt+ 4

∫ T

0|∇u|2|U(t)|2dt.

This implies that

|U(t)|2 + 2∫ T

0|∇U(t)|2dt = 4 sup

t,x|∇u|2

∫ T

0|U(t)|2dt,

which implies U(t) ≡ 0. This shows that |u(t)| ≡ 1 for all T > 0 and foralmost every ω ∈ Ω, thus the equivalence of the two approximating stochasticequations follows.

Acknowledgement. The authors thank the anonymous referees for theirhelpful suggestions and comments, as well as the journal editors for theirhard work.


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boling guo institute of applied physics and computational

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