a fourier series approach to solving the navier-stokes ...1 a fourier series approach to solving the...
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-
1
A Fourier Series approach to solving the Navier-Stokes equations withspatially periodic data
William McLeod RiveraCollege Park, MD 20740
January 15, 2011
Email address: [email protected]
Abstract: The symbol Û
denotes the velocity or momentum (the mass multiplied by thevelocity). Transform the Navier Stokes momentum and density equations into infinite systems ofordinary differential and linear equations for the classical Fourier coefficients. Prove theoremson existence, uniqueness and smoothness of solutions. Interpret the results using the Fourier
series representation PPUU ˆ̂,ˆ̂
.Key Words: Fluid Mechanics, incompressible Navier-Stokes equations, Fourier series
1. Introduction
The main result in this paper can be stated as follows. If the data are jointly smooth, spatiallyperiodic, and, furthermore, the body force and its higher order time derivatives satisfy thegeneralized sector conditions of theorem 2-2, a unique physical solution ),( PU
exists which is
separately smooth in ),( xt on 3,0 Rxt . This solution is the extension of the unique regular(smooth) short time solution determined by the data. The solution ),( PU
is also bounded for all
forward time.
In 1934 Leray (in [9]) formulated the regularity problem and related it to the smoothnessproblem. In the year 2000, Fefferman formulated the problem (in [5]). In that same year, Bardoswrote a monograph on the problem ([3]) which summarized the then literature. The authorinterprets the remarks in [3] Bardos to indicate that the problem of regularity/smoothness can besolved as formulated in (A) of [5] Fefferman.
What is new?
As far as this author knows the formulas for the Navier Stokes ordinary differential equations inthis paper are new. However for the problem defined on the aperiodic space domain Cannone-in the formula immediately prior to (27) on page 15 of Harmonic Analysis Tools for Solving theIncompressible Navier-Stokes Equations. (2003)- uses the Fourier transform to rewrite thevariation of constants formula solution of the Navier-Stokes evolution equations.
The vector form of the Navier-Stokes equations
-
2
The vector form of the Navier-Stokes equations for an incompressible fluid on the unit cube is
.0,]1,0[),(),,()),,1,(),,0,(
0,]1,0[),(),,(),1,,(),0,,(
0,]1,0[),(),,()1,,,()0,,,(
]1,0[),,,(),,,0(
]1,0[,0,0
]1,0[,0,0,
23
22
21
30
3
3
tzyzyhzytUzytU
tzxzxhzxtUzxtU
tyxyxhyxtUyxtU
xzyxUzyxU
xtU
xtFPUUUU t
(1-1)
One seeks a unique solution of (1-1) on the unit cube which is jointly smooth in time and spaceand bounded for all forward time given that the data 3,2,1,,,0 ihFU
i
is smooth. Such solutionsextend to periodic solutions of period 1 in each space variable holding the others fixed.
In (1-1) U
is the velocity vector field, P is the pressure gradient to be determined, U
is the
divergence of U
, and U
is the matrix tensor ),,(, WVUUWVU
UDx
.
The first equation of (1-1) determines the momentum. The body force F
is smooth on3]1,0[),0[ . The initial function 0U
is smooth on 3]1,0[ . The boundary functions 3,2,1, ih i
are smooth on ]1,0[]1,0[ .
The momentum equation can be interpreted as Newton’s second law of motion for fluidscombined with a dynamic version of Archimedes’ law of hydrostatics. If 0),(
xtU , the
equation reduces to Archimedes’ law ),(),( xtFxtP
.
The second equation of (1-1) is the equation of continuity. It is the Navier-Stokes equation forthe density for an incompressible fluid.
The following equations for the classical Fourier series coefficients are equivalent to (1-1). Theconditions on the data functions are equivalent to the conditions that their Fourier coefficientsbelong to discrete Schwartz frequency spaces
-
3
.0,,0),(ˆ),(ˆ),0(ˆ
0,,0
),,(ˆ),(ˆ2),(ˆ),(ˆ2),(ˆ||4),(ˆ
3
30
3
22
tNrrtUr
NrrUrU
Nrt
rtFrtPrrtUrrtUrtUrdt
rtUd t
(1-2a)
The boundary conditions for the Fourier coefficents are
.,0
0)(,0),,(ˆ)0,,,(ˆ0)(,0),,(ˆ),0,,(ˆ0)(,0),,(ˆ),,0,(ˆ
32121321
3131231
3232132
NrtrrtrrhrrtU
rrtrrhrrtU
rrtrrhrrtU
(1-2b)
In (1-2), 3,2,1,ˆ,ˆ,ˆ,ˆ ihFPU i
denote the classical Fourier sine series coefficients of
3,2,1,,,, ihFPU i
and 33,WN denote the set of natural (respectively whole) number triples.
The law governing the average mechanical energy of an incompressible fluid
Theorem 2-4 establishes the existence of a unique solution defined (bounded) for all forwardtime smooth in t uniformly in 3]1,0[x and smooth in x uniformly in .0t The followingformulas provide a smooth generalization of Leray’s mechanical energy law ([12] section 17formula 3.4) for the Navier-Stokes momentum equation and its time derivatives of order k
,...3,2,1
,0,0,||||21
||||21||
21
0 ]1,0[
)()(
0 ]1,0[
2)(
]1,0[
2)(
0 ]1,0[0 ]1,0[
2
]1,0[
20
]1,0[
2
333
3333
k
tdsydFUdsydUydU
dsydFUdsydUydUydU
tkk
tkk
tt
(1-3)
Equations (1-3) state that the difference of the space average of the kinetic energy (at time0t ) minus that at time 0t is equal to the viscosity times the potential energy minus the
average work done by the body force acting on the incompressible fluid where),,(,|| 2 wvuUwwvvuuU t
. The energy formulas (1-3) are equivalent to
the formulas
-
4
,...3,2,1,0,0,),(ˆ),(ˆ|ˆ|||4
|),(ˆ|
,),(ˆ),(ˆ|ˆ|||4
|),0(ˆ||),(ˆ|
0 0
)()(
0 0
2)(22
2
0
)(
0 00 0
222
2
0
2
0
ktdsrsFrsUdsUr
rtU
dsrsFrsUdsUr
rUrtU
t
r
kkt
r
k
r
k
t
r
t
r
rr
(1-4)
established in theorem 2-2.
By Parseval’s theorem, the quantities on the left side of (1-3) and (1-4) are both equal to theaverage kinetic energy.
The problem of finite time blow up
In theorem 2-3 the author shows that finite time blow up of solutions of (1-1) is impossible givensmooth data, the equation of continuity, and the following conditions
,...2,1,0,0,0,)(,||0 ]1,0[
)()(
0 ]1,0[
2
33
kttMdsxdFUdsxdUt
kkk
t
(1-5)
In the frequency domain, inequalities (1-5) become
,...2,1,0,0,0
),(ˆˆ),(ˆ,|),(ˆ|||40 0
)()(
0 0
2)(22
kt
tMdsFrsUdsrsUr kt
r
kkt
r
k
. (1-6)
as proved in lemma 2-3.
Under the conditions (1-5) on F
the solutions of (1-1) are absolutely bounded. Absolutestability/boundedness extends the concept of Lyapunov stability/boundedness fromhomogeneous nonlinear systems to nonlinear systems with a forcing function.
2. Existence of a unique smooth short time solution and its forward time extension
The Navier-Stokes ordinary differential equations for the momentum coefficients in the discretefrequency domain comprise an infinite system of ordinary differential equations for the time
dependent Fourier coefficients ),(ˆ rtU
of the velocity ),( xtU
. It is possible to use the classicalFourier sine series coefficients because the inhomogeneous Dirichlet boundary conditions matchon opposite faces of the unit cube.
The notation
-
5
}0),(ˆ{}]),,0([{ˆ 33 tNSNrTCU
indicates the space of Fourier coefficients of momentum vectors which are smooth on ],0[ Tt and discrete Schwartz in the vector of frequency parameters. A Fourier coefficient is discreteSchwartz in a parameter if it decays more rapidly than any fixed power of the norm of the
frequency Wpr p ,|| grows. Thus WptrtUr pr ,0,0),(ˆ|||lim ||
. Equivalently the
discrete Schwartz coefficients can be defined with upper bounds replacing limits (see definition2-2 below).
Lemma 2-1. The Navier-Stokes ordinary differential equations for the Fourier coefficients of thesolution of the spatially periodic problem (1-1) are
.),(ˆ),0(ˆ0,,0),(ˆ
0,,0
),,(ˆ),(ˆ2),(ˆ),(ˆ2),(ˆ||4),(ˆ
30
3
3
22
NrrUrU
tNrrtUr
Nrt
rtFrtPrrtUrrtUrtUrdt
rtUd t
(2-1a)
where
.0)(,0),,(ˆ),,0,(ˆ0)(,0),,(ˆ),0,,(ˆ0)(,0),,(ˆ)0,,,(ˆ
3232132
3131231
2121321
rrtrrhrrtU
rrtrrhrrtU
rrtrrhrrtU
(2-1b)
In line 1 of (2-1a) denotes the discrete convolution
.),(ˆ]))(,(ˆ[),(ˆ*),(ˆ0
q
tt qtUqrqrtUrtUrrtU (2-2)
The discrete Fourier transform (the Fourier coefficient) of the velocity is
.0,
,2sin2sin2sin),()},({),(ˆ
3
321]1,0[ 3
tNr
zdxdydzryrxrxtUxtUrtU
(2-3)
and similarly for ),(ˆ),,(ˆ rtPrtF
.
PROOF
-
6
Note that all triple integrals over 3]1,0[ in the definition of the discrete Fourier transform defined
by (2-3) are well defined for all forward time since 0,UF
are smooth in Dx and smooth andbounded in t with all time derivatives continuous and bounded on ),0[ t . The boundary
functions 3,2,1, ih i
are smooth on 2]1,0[ .
The terms FPrdtUd ˆ,ˆ,ˆ
are calculated from the discrete Fourier transform to the terms of (1-1) and
the differentiation property of discrete Fourier transforms applied to the first partial derivativesin the case of the pressure gradient term.
The Fourier series coefficient of the diffusion term can be calculated using integration by partstwice. Since the three calculations are identical, it suffices to calculate the transform of thesecond partial derivative of U
with respect to the first component x,
.,0),,(ˆ42sin2sin4
2sin2sin2cos2
2sin2sin|2sin
2sin2sin2sin
321
232
]1,0[
21
2
32]1,0[
11
321
01
1
0
1
0
321]1,0[
3
3
3
NrtrtUrzdxdydzryrUr
zdxdydzryrxrUr
zdydzryrxrU
zdxdydzryrxrU
x
xy z
x
xx
(2-4)
Thus the diffusion term is
.,0,0),(ˆ)(
)0,,,()1,,,(),0,,(),1,,(),,0,(),,1,(
])0,,,()1,,,(
[]),0,,(),1,,(
[
]),,0,(),,1,(
[),(ˆ)(}{
323
22
21
212131313232
3
21
3
213
2
31
2
312
1
32
1
321
23
22
21
NrtrtUrrr
xxtUxxtUxxtUxxtUxxtUxxtU
xxxtU
xxxtU
rx
xxtUx
xxtUr
xxxtU
xxxtU
rrtUrrrU
(2-5)
It follows that
0,,ˆ||4}{ 322 tNrUrU . (2-6)
The transform of the Euler term is
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7
.,0,),(ˆ]))(,(ˆ[2}{ 33
NrtqtUqrqrtUUUNq
t
(2-7)
The transform of the pressure gradient term is
.,0,ˆ22sin2sin2sin 3321]1,0[ 3
NrtPrxzdryrxrP (2-8)
The transform of the equation of continuity is
.,0
,0][22sin2sin2sin)(
3
321321]1,0[ 3
Nrt
WrVrUrxzdryrxrWVU zyx
(2-9)
The boundary conditions (lines 3-6 of formula (2-1)) are derived by solving the homogeneousLaplace’s equation constrained by the face matching boundary conditions using transform(Fourier series) methods.
END PROOF
Remark 2-1. Equations (2-1) specify an infinite system of ordinary differential equations witha discrete vector parameter whose solutions are the time dependent Fourier coefficients in theFourier expansions of ),(),(),,(),,( 0 xtPxUxtFxtU
. In fact formulas (2-4) through (2-9) hold on
3Wr . But, since the boundary conditions are Dirichlet and the Fourier sine series is used,there is no constant term corresponding to 0
r while cases with one or two indices ir equal to
0 occur on the boundary of the frequency domain.
Definition 2-1. The family of linear operators defined by the transformed equation of continuityis
.0,),,(ˆ)],(ˆ[ 3 tNrrtFrrtFLr
(2-10)
Remark 2-2. For any 3Nr , ),0[),0[: CCLr
. The linear operator rL is a map fromvectors of smooth (infinitely continuous differentiable) functions on ),0[ t to smooth scalarfunctions on ),0[ .
Proposition 2-1. Any derivative of finite order of the velocity ,...2,1,0,ˆ
kdt
Udk
k
is in )( rL (the
null space of rL ).
PROOF
-
8
By (1-1) this relation holds for the discrete Fourier transform of the equation of continuity (0k ). Take derivatives of order ,...3,2,1k with respect to t of both sides of
0,0,0),(ˆ
rtrrtU (2-11)
to complete the proof.
END PROOF
Since differentiation of the momentum function with respect to the space variables correspondsto multiplication of its transform by frequency variables, the following Banach space is the oneneeded to establish that solutions of the Navier-Stokes equations are smooth in 3),0[),( Rxt .
Definition 2-2. The discrete Schwartz space (uniform in ],0[ Tt ) for Fourier coefficients Û
defined on 3],0[ NT is
.0
},,),(|ˆ|sup||ˆ:||]),0([ˆ{ˆ 33)3(3)2(
2)1(
1)3(
3)2(
2)1(
1 3
TtWsNrsMUrrrUrrrTCUS sssNr
sss
(2-12)
The following lemma establishes that P̂ is an auxiliary function for (2-1)- i.e. a function whichcan be removed in an equivalent form of the equation.
Lemma 2-2. Equations (2-1) can be placed into the following equivalent form
).0),(ˆˆ,),,0)[ˆ(ˆ:ˆ{ˆ),(ˆ),0(ˆ
,0,0),(ˆ
,0],ˆˆ)ˆ(2[100010001
||ˆ||4ˆ
33
30
3
32
22
tNSUNrLCUUU
NrrUrU
NrtrrtU
NrtFUrUrrrUrU t
t
t
(2-13a)
The boundary conditions for the velocity coefficients are
.0)(,0),,(ˆ),,0,(ˆ0)(,0),,(ˆ),0,,(ˆ0)(,0),,(ˆ)0,,,(ˆ
3232132
3131231
2121321
rrtrrhrrtU
rrtrrhrrtU
rrtrrhrrtU
(2-13b)
The pressure coefficients satisfy the following equations
-
9
.},,0(ˆ)(ˆ*))(ˆ(2{||2
1),0(ˆ
,0)},,(ˆˆ*)ˆ(2{||2
1),(ˆ
3002
32
NrrFrrUrrUrr
rP
NrtrtFrUrUrr
rtP
t
t
(2-14a)
The pressure coefficients satisfy the following boundary conditions
.0),0,0(),()},,,0,(ˆ),,0(
)],,0,(ˆ][),,0)(0,,,(ˆ[),,0(2{||2
1),,0,(ˆ
,0),0,0(),()},,0,,(ˆ),0,(
)],0,,(ˆ][),0,)(,0,,(ˆ[),0,(2{||2
1),0,,(ˆ
,0),0,0(),()},0,,,(ˆ)0,,(
)]0,,,(ˆ][)0,,)(0,,,(ˆ[)0,,(2{||2
1)0,,,(ˆ
323232
3322322132232
313131
3311313131231
212121
2211212121221
2
2
2
trrrrtFrr
qrqrtUqqqqtUrrr
rrtP
trrrrtFrr
qrqrtUqqqqtUrrr
rrtP
trrrrtFrr
qrqrtUqqqqtUrrr
rrtP
Nq
t
Nq
t
Nq
t
(2-14b)
PROOF
The equations in the first line of (2-13a) form an infinite system of vector ordinary differentialequations-one for each Fourier coefficient as a function of time. The equations in the second lineof (2-13a) define an infinite set of homogeneous linear equations.
Apply the linear operator rL to each side of equation (2-1) by forming the dot product of eachterm with the vector r to obtain
.0,),,(ˆ),(ˆ2),(ˆ]))(,(ˆ2[
),(ˆ||),(ˆ
3
22
3
tNrrtFrrtPrrqtUqrqrtUr
rtUrrdt
rtUdr
Nq
t
(2-15)
By proposition 2-1, 3),(),(ˆ),,(ˆ WrLrtUrtU rt
hence (2-15) reduces to
),(ˆ),(ˆ2),(ˆ]))(,(ˆ[203
rtFrrtPrrqtUqrqrtUrNq
t
. (2-16)
Solve (2-16) for ),(ˆ rtP to obtain the first line of (2-14a). Insert the first line of the pressureformula into (2-1) to obtain the first line of (2-13a). The second line of (2-14a) is obtained from
the transform of the initial function )(ˆ 0 rU . The coefficients 3,2,1,ˆ ih i
on the boundary of the
-
10
frequency domain can be inserted for Û
to further reduce the formulas (2-14b). The pressurecoefficients are independent of time on the boundary.
END PROOF
Remark 2-3. From higher order derivatives of (2-1) and the projection defined by the kth orderequation of continuity, one can calculate formulas for ),(ˆ )( rtP k similar to (2-14).
The equation that results from calculating any finite order time derivative of equations (2-13a) is
.,,0,0),(ˆ,,0),0(ˆ
,,0
],ˆ)ˆ)ˆ(2[(100010001
||ˆ||4ˆ
3)(
3)(
3
)()(2
)(22)1(
NkNrtrrtU
NkNrrU
NkNrt
FUrUrrrUrU
k
k
kktt
kk
(2-17)
The following theorem establishes the existence of smooth short time solutions of equation (2-13).
Theorem 2-1. Suppose 3,2,1,0),(ˆˆ),(ˆˆ),(ˆ),0([ˆ,ˆ 2303 itNShNSUNSCUF i
,then
there exists 0T such that ),(ˆ)),0([ˆ 3NSTCU
satisfies (2-11) and
).(ˆ)),0([ˆ 3NSTCP
PROOF
The goal is to establish
a. For any fixed 3Nr , and any non negative integer k, )(ˆ kU
is continuous for sufficientlyshort time.
)(ˆsup],,0[,,||),(|)(ˆ)(ˆ:|0 )1(021121)(
2)( tUMTttttrkMtUtUT kTt
kk
(2-18)
b. The continuity of )(ˆ kU
is uniform in Wk
)(ˆsupsup~],,0[,,||)(|)(ˆ)(ˆ:|0
)1(0
21121)(
2)(
tUM
TttttrMtUtUTk
TtWk
kk
(2-19)
c. The continuity of )(ˆ kU
is uniform in 3Nr
-
11
)(ˆsupsupsup
~~],,0[,,|||)(ˆ)(ˆ:|0)1(
0
21121)(
2)(
3 tU
MTttttMtUtUTk
TtWkNr
kk
(2-20)
PROOF
For derivatives of the momentum coefficients of any finite order ,...3,2,1,0k the variation ofconstants formula yields
,...3,2,1,0,,0
,,0,.]ˆ)ˆ)ˆ(2[(100010001
||
ˆ||)4(),(ˆ
3
3
0
)()(2
)(||4
0||422)(
22
22
kNrt
NrtdsFUrUrrre
UerrtU
tkkt
tstr
trkkkk
(2-21)
To prove continuity of the kth derivative of the momentum coefficient in time, calculate
,..3,2,1,0,,0
]ˆ)ˆ)ˆ(2[(100010001
||
]ˆ)ˆ)ˆ(2[(100010001
||
ˆ||)4(ˆ||)4(),(ˆ),(ˆ
312
)1(
0
)()(2
))1((||4
)2(
0
)()(2
))2((||4
0)1(||422
0)2(||422
1)(
2)(
22
22
2222
kNrtt
dsFUrUrrre
dsFUrUrrre
UerUerrtUrtU
tkkt
tstr
tkkt
tstr
trkkktrkkkkk
(2-22)
Simplify the difference of )(ˆ kU
at two distinct times
-
12
,...3,2,1,0,,0
]ˆ)ˆ)ˆ(2[(100010001
||][
]ˆ)ˆ)ˆ(2[(100010001
||
ˆ][||)4(),(ˆ),(ˆ
312
)1(
0
)()(2
))1((||4))2((||4
)2(
)1(
)()(2
))2((||4
0)1(||4)2(||422
1)(
2)(
2222
22
2222
kNrtt
dsFUrUrrree
dsFUrUrrre
UeerrtUrtU
tkkt
tstrstr
t
t
kktt
str
trtrkkkkk
(2-23)
To investigate behavior near time 0, evaluate (2-22) at ttt 21 ,0
,...3,2,1,0,,0
]ˆ)ˆ)ˆ(2[(100010001
||
ˆ]1[||)4(),(ˆ
3
0
)()(2
)(||4
0||422)(
22
22
kNrt
dsFUrUrrre
UerrtU
tkkt
tstr
trkkkk
(2-24)
The upper bound on the momentum coefficient is
,...3,2,1,0,,0
|]ˆ||)ˆ)ˆ(2([||100010001
|||
|ˆ|]1[|||4||),(ˆ|
3
0
)()(2
)(||4
0||422)(
22
22
kNrt
dsFUrUrrre
UerrtU
tkkt
tstr
trkkkk
(2-25)
The upper bound can be simplified as follows.
,...3,2,1,0,,0
|]ˆ||)ˆ)ˆ(2([||100010001
||1|
|ˆ|||4|||4||),(ˆ|
3
0
)()(
233231
322
221
31212
1
2)(||4
02222)(
22
kNrt
dsFUrUrrrrrrrrrrrrrrr
re
tUrrrtU
tkktstr
kkkk
(2-26)
Use the trace norm to bound the matrix operator inside the integral above
-
13
.2||
||||3|
||||
00
0||
||0
00||
||
||100010001
|||
12
22
2
223
2
222
2
221
233231
322
221
31212
1
2
rrr
rrr
rrr
rrr
rrrrrrrrrrrrrrr
r
(2-27)
By the hypothesis on the data
LrLrUt )(|)(ˆ|sup 00
(2-28a)
},4max{||,)(|),(ˆ||||),(ˆ||||4|sup 22)(5)(22
0 rMrMrsFrrsFrkkkkkk
s (2-28b)
Since the variation of constants operator is defined on coefficients which are smooth time/
discrete Schwartz frequency, trrsU
),(ˆ is discrete Schwartz in the frequency. The convolution
of discrete Schwartz functions is discrete Schwartz in the frequency 3Nr which is alsosmooth in the time variable,
.)(|)],(ˆ)),(ˆ[(2(|sup 11)(
0 MrMrsUrrsUkt
s (2-29)
It follows that
,...3,2,1,,0
}{2)||4
1(
}{2
|),(ˆ|
3
2122
||4
021
)(||4
)(
22
22
kNrt
MMr
eLt
dsMMe
LtrtU
tr
tstr
k
(2-30)
Not only are the coefficients |),(ˆ| )( rtU k
bounded for sufficiently short time but for all finiteforward time.
The formulas (2-13b) and (2-14b) for PU ˆ,̂
on the integer lattice “boundaries” of the frequency
domain and their discrete Schwartz property in 3N for all forward time follow automaticallyfrom the given conditions on the boundary data.
END PROOF
-
14
Proposition 2-2. The inner product of ),(ˆ )( rtU k
with the transformed Euler (convolution) term
.0,0,...,3,2,1,0,0)],(ˆˆ)[,(ˆ )()(
rtkrtUrUrtU ktk
PROOF
By the Liebnitz rule, the kth order Euler term can be written
.))(ˆ()))((ˆ()(ˆ
))(ˆ())(ˆ()](ˆ)(ˆ[ˆ
)()(
0
)(
)()(
0
)()(
qr
lkq
ltqr
k
l
kr
lkr
ltr
k
l
krr
trk
kk
r
tUqrtUlk
tU
tUrtUlk
UtUrtUdtdU
(2-31)
The equality in the second line follows by the definition of convolution and the fact that thecoefficients for the classical Fourier sign series are one sided 0
r .
The next equality follows by matrix vector multiplication,
.))(ˆ())(ˆ())(ˆ()()](ˆ)(ˆ[ˆ )()()(0
)(
qr
lkq
lqr
lkq
k
lr
trk
kk
r tUtUtUqrlk
tUrtUdtdU
(2-32)
The next inequality follows by the Schwartz inequality for vectors in 3R applied to each termand both dot products,
.|)()ˆ(||)()ˆ(||)()ˆ(|||
)](ˆ)(ˆ[ˆ
)(2)(2)(2
0
)(
qr
lkq
lqr
lkq
k
l
rt
rk
kk
r
tUtUtUqrlk
tUrtUdtdU
(2-33)
The next equality follows by the definition of the inner product of a vector with itself 2|| xxx .
.,0|)()()ˆ(||)()ˆ(||)()ˆ(|
)()ˆ(||)()ˆ(||)()ˆ(|||
2)(2)(2)(
0
)(2)(2)(2
0
WkqrtUtUtUlk
tUtUtUqrlk
qr
lqr
lkq
lkq
k
l
qr
lkq
lqr
lkq
k
l
(2-34)
The final inequality follows by applying the higher order equation of continuity to each term
0,0,,...,2,1,0,0)(ˆ )(
rtklrtU lr . (2-35)
Similarly
-
15
.,0|)()()ˆ(||)()ˆ(||)()ˆ(|
)](ˆ)(ˆ[ˆ
2)(2)(2)(
0
)(
NkqrtUtUtUlk
tUrtUdtdU
qr
lqr
lkq
lkq
k
l
rt
rk
kk
r
(2-36)
Hence
NkrttUrtUdtdU r
trk
kk
r ,0,0,0)](ˆ)(ˆ[ˆ )(
. (2-37)
END PROOF
The notation 0
r indicates that all discrete frequency indices are non negative integers and notall indices are 0.
The domain of definition of solutions of (2-13) can be extended by showing that they and all oftheir finite time derivatives are bounded and continuous for all forward time. A frequencydomain formula for the total mechanical energy of the average velocity and any time derivativeof it also appears. This is the frequency domain analog of the extension of Leray’s energy lawaugmented by the body force.
In theorem 2-2 I assume that the velocity vector fields of interest are those whose time integral ofthe potential energy of order k is bounded below (this suffices because the potential energy is adecreasing function of time). I also assume that the time integral of the work done by the bodyforce on the incompressible fluid converges.
Theorem 2-2. If
,...2,1,0,0,0
),(ˆˆ),(ˆ,|),(ˆ|||40 0
)()(
0 0
2)(22
kt
tMdsFrsUdsrsUr kt
r
kkt
r
k
(2-38)
then
a. the following formulas for the total mechanical energy are well defined
-
16
.0,...,3,2,1,),(ˆ),(ˆ
|),(ˆ
|||4|),(ˆ
|
,),(ˆ),(ˆ
|),(ˆ|||4|),0(ˆ|)|,(ˆ|
0 0
0 0
2222
0
0 0
0 0
2222
0
2
0
tkdss
rsFs
rsU
dst
rtUrt
rtU
dsrtFrtU
dsrtUrrUrtU
t
rk
tk
k
tk
t
rk
k
rk
k
t
r
t
rrr
(2-39)
b. The solution of (2-1)/ (2-13a) and every finite time derivative 0,...,2,1,0),,(ˆ )( tkrtU k
of
it is unique, continuous, and bounded in t for all forward time, square summable in 3Nr forall forward time and jointly bounded in 3),0[),( Nrt .
PROOF
a. First note that, by continuity, the hypothesis implies ,...2,1,0,0)(|| 2)(0 kxdxUD
k hence
,...2,1,0,0|)(ˆ|0
2)(0
krUr
k
. (2-40)
By Parseval’s theorem
,...2,1,0,|)(ˆ|)(|| 20
)(0
2)(0
krUxdxUr
k
D
k
(2-41)
Construct formulas for the average energy )(kU
in terms of ,...2,1,0),,(ˆ )( krtU k
Then apply
proposition 2-2 to eliminate the ),(ˆ )( rtU k
dotted time derivative of the Euler terms to simplify
them. Form the dot product of each equation (2-13a) with ),(ˆ )( rtU k
, sum over }0{3
Wr andintegrate from 0 to t to obtain
-
17
,...3,2,1,),(ˆ),(ˆ
)},(ˆ),(ˆ))((),(ˆ{2
|ˆ|||4|),(ˆ|
,),(ˆ),(ˆ
|),(ˆ|||4|),0(ˆ|)|,(ˆ|
0 0
)()(
0 0
)(
0
)(
0 0
2)(222
0
)(
0 0
0 0
2222
0
2
0
kdsrsFrsU
dsqsUrsUqrqrsUt
dsUrrtU
dsrtFrtU
dsrtUrrUrtU
t
r
kk
t
qr
k
q
kk
k
t
r
k
r
k
t
r
t
rrr
(2-42)
By proposition 2-2 the Û
dotted transformed Euler term summed over 0
r satisfies
.0),(ˆ)),(ˆ(),(ˆ20 0
qr q
qsUqqrsUrsU (2-43)
By proposition 2-2 the )(ˆ kU
dotted transformed Euler term of the kth order Navier-Stokesordinary differential equations, likewise summed over 3Wr satisfies
.0]),(ˆ)),(ˆ[()],(ˆ[20
)(
0
)(
qr
k
q
k qsUqrqrsUrsU (2-44)
It follows by (2-38) that the space average of the kinetic energy is bounded for all forward time.Thus all terms appearing in (2-39) are well defined for all forward time.
b. If a series converges absolutely (for all 0t ) then each term is finite. It converges uniformlywith respect to t treated as a parameter in its full range.
}0{),,0[ˆ}0{),,0[,|),(ˆ|sup
}0{),,0[,|),(ˆ|sup
),0[,|),(ˆ|21
3
30
320
0
2
WrLU
WrtrtU
WrtrtU
trtU
t
t
r
(2-45)
-
18
By the fundamental (extension) theory of ordinary differential equations it follows that for each3Nr , ),(ˆ )( rtU k
is unique and continuous in t for all forward time uniformly with respect to
3Nr .
END PROOF
The following lemma establishes a link between the inequalities (2-18) in the hypothesis oftheorem 2-2 and inequality (5) and its higher order analogs.
Lemma 2-3. If ,...2,1,0,)( kU k
satisfy the Navier-Stokes partial differential equations suchthat
0,...,2,1,0,]1,0[, 32)()()( tkLUFU kkk
(2-46)
such that the velocity, the body force and any finite order time derivatives of them satisfy
,...2,1,0,0,0,)(,||0 ]1,0[
)()(
0 ]1,0[
2)(
33
kttMdsxdFUdsxdUt
kkk
tk
(2-47)
for all forward time if and only if 0,...,2,1,0),(ˆ,ˆˆ 32)()()( tkNlUFU kkk
such that the
momentum ,...2,1,0,ˆ )( kU k
satisfy the Navier-Stokes ordinary differential equations. Moreoverthe finite order time derivatives of the momentum and the body force satisfy
,...2,1,0,0,0
),(ˆˆ),(ˆ,|),(ˆ|||40 0
)()(
0 0
2)(22
kt
tMdsFrsUdsrsUr kt
r
kkt
r
k
(2-48)
PROOF
By the strict inequality, the integral on the left of (2-47) over 3R must be finite. In order for theFourier transforms which appear in (2-48) to be well defined
0,...,2,1,0),]1,0([, 32)()()( tkLUUF kkk
if and only if
.0,...,2,1,0),(ˆ||,ˆˆ 32)(2)()( tkWlUFU kkk
Note that, since ,...2,1,0,)( kF k
is smooth in 3]1,0[x , )()( kk FU
is integrable by slderoH '
inequality if only )]1,0([ 31)( LU k
.
Now suppose
,...2,1,0,0,0,)(,||0 ]1,0[
)()(
0 ]1,0[
2)(
33
kttMdsxdFUdsxdUt
kkk
tk
(2-49)
-
19
By the definition of the discrete Fourier transform (i.e. the Fourier coefficient) the previousinequality is equivalent to
,...2,1,0,0,|2sin2sin2sin||),(|4
|2sin2sin2sin||),(),(
0 ]1,0[
2321
2)(2
0 ]1,0[
2321
)()(
3 3
3 3
ktdsxdzryrxrxsU
dsxdzryrxrxsFxsU
t
Nr
k
t
Nr
kk
(2-50)
Since 332321 ,,1|2sin2sin2sin|0 NrRxzryrxr (2-50) holds if and only if
,...2,1,0,0,0
),(ˆˆ),(ˆ,|),(ˆ|||40 0
)()(
0 0
2)(22
kt
tMdsFrsUdsrsUr kt
r
kkt
r
k
(2-51)
One can argue that this follows immediately by Parseval’s theorem.
Note that )(ˆ)( tMtM kk
END PROOF
Theorem 2-3. If 3,2,1,ˆ),,(ˆ ihrtF i
, and all time derivatives are discrete Schwartz in 3Nr for
all forward time and )(ˆ 0 rU is discrete Schwartz in 3Nr then any finite time derivative of
),(ˆ rtU
(thus ),(ˆ rtP ) ( the solution of the equation of lemma 2-2) is discrete Schwartz in 3Nr
.
PROOF
a. For any fixed kt, )(ˆ kU
is discrete Schwartz in 3Nr
WpUr kpNr ,|ˆ|||sup )(3
(2-52)
b. The upper bound for )(ˆ kU
is uniform in k
WpUr kpNrWk ,|ˆ|||supsup )(3
(2-53)
c. The upper bound for )(ˆ kU
is uniform for all 0t .
WpUr kpNrWkt ,|ˆ|||supsupsup )(0 3
(2-54)
-
20
By the variation of constants formula, it suffices to show that the convolution integral is discrete
Schwartz since 0||4 ˆ),(ˆ),(ˆ22
Uertr tr are discrete Schwartz in 3Nr by inspection given
the hypotheses on the initial and boundary conditions.
Multiply the time convolution by any monomial formed by the product of any finite powers ofthe frequency components
,3,2,1,,0,,0
|]ˆ)ˆ)ˆ((2[100010001
|||||sup
|||ˆ|sup
|)(ˆ|||sup
|ˆ|||sup
3
02
)(||4)3(3
)2(2
)1(1
)3(3
)2(2
)1(10
||4
)3(3
)2(2
)1(1
)3(3
)2(2
)1(1
22
3
22
3
3
3
kNrtWp
dsFUrUrrrerrr
rrrUe
rrrr
Urrr
tt
tstrppp
Nr
ppptrNr
pppNr
pppNr
(2-55)
Simplify the upper bound on the Schwartz weighted momentum coefficient of (2-55)
,3,2,1,,0,,0
|]ˆ)ˆ)ˆ(2[(100010001
|||||sup
|ˆ||||)4(|||sup
|),(ˆ|||sup
3
0
)()(2
)(||4)3(3
)2(2
)1(1
0||422)3(
3)2(
2)1(
1
)()3(3
)2(2
)1(1
22
3
22
3
3
kNrtWp
dsFUrUrrrerrr
Uerrrr
rtUrrr
tkkt
tstrppp
Nr
trkkkpppNr
kpppNr
(2-56)
The Fourier series coefficient of the solution of Laplace’s equation is discrete Schwartz
.|)(ˆ|||sup,
)(ˆ)(ˆ)(ˆ)(
1)3(
3)2(
2)1(
13
32
3 MrrrrWp
NSrNSrhppp
Nr
(2-57)
Since )(ˆˆ 30 NSU
-
21
2)3(
3)2(
2)1(
10)3(
3)2(
2)1(
10||4
3
|ˆ|sup|ˆ|sup
,0
3
22
3 MrrrUrrrUe
Wptppp
Nrppptr
Nr
(2-58)
Since
)(ˆˆ),(ˆˆ)ˆ()(ˆˆ 333 NSFNSUrUNSU t
and
.,2|100010001
||| 32 Nrr
rr t
Use the norm on the matrix operator immediately above to get the following upper bound
.|ˆ|||sup2
|)ˆ)ˆ((||supsup22
|]ˆ)ˆ)ˆ((2[100010001
|||||sup
|ˆ|||sup
0
)3(3
)2(2
)1(10
)(||4
0
)3(3
)2(2
)1(10
)(||4
02
)(||4)3(3
)2(2
)1(1
)()3(3
)2(2
)1(1
22
3
22
22
3
3
dsFrrre
dsUrUrrre
dsFUrUrrrerrr
Urrr
tppp
sstr
ttppp
sNrstr
tt
tstrppp
Nr
kpppNr
(2-59)
The convolution and body force terms in the integrand have upper bounds which are uniformover all time and any order k of the derivative.
3||,12
1][2||4
1][2
][2
|ˆ|||sup
2212221
210
)(||4
)()3(3
)2(2
)1(1
22
3
rNNr
NN
dsNNe
Urrrt
str
kpppNr
(2-60)
For the kth derivative with respect to time of the classical Fourier coefficient
-
22
.3||,12
1][2||4
1][2
][2
|ˆ|||sup2
|)ˆ)ˆ(2(||supsup22
]ˆ)ˆ)ˆ(2[(100010001
|||||sup
2212221
210
)(||4
0
)()3(3
)2(2
)1(10
)(||4
0
)()3(3
)2(2
)1(10
)(||4
0
)()(2
)(||4)3(3
)2(2
)1(1
22
22
3
22
22
3
rPPr
PP
dsPPe
dsFrrre
dsUrUrrre
dsFUrUrrrerrr
tstr
tkppp
sstr
tktppp
sNrstr
tkkt
tstrppp
Nr
(2-61)
For coefficients in discrete Schwartz spaces the integer weights are unrestricted. Hence any sumof squares can be exceeded by a product which is a single square. Thus the Schwartz norm hastwo equivalent formulations
|ˆ|||sup|ˆ|sup 0)3(
3)2(
2)1(
103 UrrrrUp
Wpppp
Wp
. (2-62)
The homogeneous term has a uniform upper bound
2)3(
3)2(
2)1(
10)3(
3)2(
2)1(
10||4
3
|ˆ|sup|ˆ|sup
,0
3
22
3 MrrrUrrrUe
Wptppp
Nrppptr
Nr
(2-63)
The pressure function
32 ,0)]},,(
ˆˆˆ2[{||2
1),(ˆ NrtrtFUrUrr
rtP t
(2-64)
is discrete Schwartz in r since trUU ˆ,ˆ are discrete Schwartz by hypothesis, the convolution of
discrete Schwartz coefficient is discrete Schwartz and the sum of discrete Schwartz functions
)],(ˆˆˆ[ rtFUrU t
is discrete Schwartz.
By the same reasoning any finite order time derivative of the pressure transform ),(ˆ rtP isdiscrete Schwartz.
END PROOF
The following theorem interprets the results obtained for the Fourier coefficients PU ˆ,̂
which
-
23
solve the Navier-Stokes ODE to the Fourier series representation of the functions PU ˆ̂,ˆ̂
whichsolve the Navier-Stokes PDE.
Theorem 2-4. Suppose 0,...,2,1,0,]1,0[, 32)()()( tkLUFU kkk
where the ,...2,1,0,)( kU k
satisfy the Navier-Stokes partial differential equations and its finite order time derivatives suchthat
,...2,1,0,0,0,||||0
2)(
0
)()( ktdsxdUdsxdFUt
D
kt
D
kk
(2-65)
where )(kF
is smooth in ),( xt and bounded in ),0[ t . Then every finite time derivative of the
solution )ˆ̂,ˆ̂
( PU
of the Navier-Stokes momentum equation is bounded, continuous and uniquely
determined in t –that is to say it is smooth and bounded in ),0[ t for each fixed 3]1,0[x It is
also smooth in 3]1,0[x for each fixed ),0[ t .
PROOF
The conclusion of the theorem follows directly by the properties of the Fourier series
representation Uˆ̂
of the velocity function and theorem 2-2. In particular,
33
33
]1,0[)),,0)([(ˆ̂
)),,0)([(ˆ
0),]1,0([ˆ̂
0),(ˆˆ
xLCUUNrLCU
tCUUtNSU
. (2-66)
By the projection formula (or by solving the Navier-Stokes momentum equation for P ) andthe first part of this theorem, the pressure gradient satisfies the same properties as eachcomponent of the momentum vector (marginal smoothness in xt , , uniqueness, and boundednessfor all forward time).
END PROOF
The variation of constants formulas for the incompressible Navier Stokes equation on the unitcube for all forward time are
-
24
dSn
xyGyhx
xdxUxxtKxt
dnmlxnzmylxe
nznzmymylxlxexxtK
tDxdsxdFPUUxxstK
xtxxtU
D
D
nml
tnml
nml
tnml
t
D
),()()(
)(),,(),(
sinsinsin)(sinsinsin
'sinsin'sinsin'sinsin),,(
0,0,,}){,,(
),()(),(
0
32]1,0[
1,,
)(
,,
)(
0
3
2222
2222
(2-67)
)(x is the Poisson form with vector Green’s function kernel solution of the solution of theLaplacian with opposite face matching boundary conditions and D consists of the faces of theunit cube. The variation of constants formula (2-67) generalizes that in [18] Strauss.
To obtain a second variation of constants formula equivalent to the one above, it is possible tointegrate over D to obtain a simplification of formula (2-67) in the domain of sequences ofvector Fourier coefficients.
nzmylxeUxt
nzmylxextK
tDxdssFrsPsUrUxstK
xtxxtU
tnml
nmlnml
nml
tnml
t
nmlnmlnml
sinsinsin)0(ˆ),(
sinsinsin),(
0,0,,)}(ˆ)(ˆ)(}ˆˆ){,(
),()(),(
)(
1,,,,
,,
)(
0,.,,.,
2222
2222
(2-68)
with
defined in series form as
-
25
.sinsin]sinh
)1(sinhsinh[ˆ
sinsin]sinh
)1(sinhsinh[ˆ
sinsin]sinh
)1(sinhsinh[ˆ
,22
22223,
,22
22222,
,22
22221
,
mylxml
mlzmlzh
nzlxnl
nlynlyh
nzmynm
nmxnmxh
nmml
nlnl
nmnm
(2-69)
In (2-69) a doubly indexed coefficient with an arrow overhead means
,...3,2,1,,),( 3,2,
1, jiAAAA
tjijijiij
while
nm ,denotes
m n 1and similarly for triple series.
Formulas (2-67) and (2-68)/(2-69) are equivalent except for the fact that, in (2-67), the initialvalue at 0t must be established by continuous extension 00 ),(lim UxtUt
since the
diffusion kernel contains a (removable) singularity at 0t .
The variation of constants formulas above are used to prove the main result of this paper. Sincethe space domain is bounded, smoothness at the boundary means smoothness on one side relativeto the interior of the unit cube.
Theorem 2-5. If the data is smooth and bounded for all forward time, then the solution ),( PU
of
the incompressible Navier-Stokes equations is jointly smooth on 3]1,0[),0[ and bounded forall forward time.
PROOF
It suffices to show that the arbitrary tensor derivatives of any non negative integer order m of theFourier series representation of the solution are jointly continuous on 3]1,0[),0[ and boundedfor all forward time.
The first (steady state) term (which solves the Laplacian and satisfies the boundary conditions) isindependent of time. Thus it suffices to show that the general tensor of order m is jointlycontinuous in 3Rx .
-
26
)12mod),1)(cosh)cosh
02mod),1)(sinh)sinh(
)}3,1{4mod,cos}2,0{4mod,sin
)(}3,1{4mod,cos}2,0{4mod,sin
()()(sinh
)(
),12mod),1)(cosh)cosh
02mod),1)(sinh)sinh(
)}3,1{4mod,cos}2,0{4mod,sin
)(}3,1{4mod,cos}2,0{4mod,sin
()()(sinh
)(
),12mod),1)(cosh)cosh
02mod),1)(sinh)sinh(
)}3,1{4mod,cos}2,0{4mod,sin
)(}3,1{4mod,cos}2,0{4mod,sin
()()(sinh
)(
32222
32222
, 3
3
1
1)2()1(
22
)3(223,
22222
22222
, 3
3
1
1)3()1(
22
)2(222,
12222
12222
, 2
2
2
2)3()2(
22
)1(221
,
kzmlzml
kzmlzml
kmykmy
klxklx
mlml
mlh
kynlynl
kynlynl
knzknz
klxklx
nlnl
nlh
kxnmxnm
kxnmxnm
knzknz
kmykmy
nmnm
nmh
ml
kkk
ml
nl
kkk
nl
nm
kkk
nm
(2-70)
Since ih ,
are discrete Schwartz in two integer indices each the series tensor component
converges and the (norm) of the tensor is finite on the boundaries 2]1,0[ of the unit cube.
By direct calculation (inspection), the tensor in (2-70) is smooth and bounded in 3Rx .
The generic term of the tensor of order m is
),,(),,,(
)}3,1{4mod,cos}2,0{4mod,sin
)(}3,1{4mod,cos}2,0{4mod,sin
)(}3,1{4mod,cos}2,0{4mod,sin
(
)()()()||)(0(ˆ),(
3
3
2
2
1
1
)3()1()1()(22
1,,,,)3()2()1(
)3()2()1(2222
nmlrzyxxknzknz
kmykmy
klxklx
mmlerUzyxt
xt kkktnmllnml
nmlkkkl
kkkl
(2-71)
Since )(0 rU
is discrete Schwartz in 3Nr the series converges. The homogeneous part of the
solution is bounded for all forward time since the series converges and 1sup )(02222
tnml
t e .
The tensor mklxtkl xt ||),,(,,
is jointly continuous (jointly smooth) in 3]1,0[),0[),( xt
because nzmylxe tnml sinsinsin)(2222 is jointly smooth in 3]1,0[),0[),( xt .
-
27
To summarize progress to this point, for the steady state and homogeneous part, the proof thatthe momentum is jointly smooth and bounded on the domain of interest follows automaticallyby inspection (recognition of known smoothness of exponential, hyperbolic, and trigonometricfunctions) and the given conditions on the data.
The generic component of the tensor of order m is
})}3,1{4mod,cos}2,0{4mod,sin
(sinsin)}(ˆ)(ˆ)(}ˆˆ{{[)(
}sin)}3,1{4mod,cos}2,0{4mod,sin
(sin)}(ˆ)(ˆ)(}ˆˆ{{[)(
}sinsin)}3,1{4mod,cos}2,0{4mod,sin
()}(ˆ)(ˆ)(}ˆˆ{{[)(
sinsin])}(ˆ)(ˆ)(}ˆˆ{[][
||
,, 0 3
3,.,,.,
)(||)3(
,, 0 3
3,.,,.,
)(||)2(
,, 0 1
1,.,,.,
)(||)1(
,, 0,.,,.,
||)(||
0
321
)3()2()1(
22
22
22
2222
nml
t
nmlnmlnmlstrk
nml
t
nmlnmlnmlstrk
nml
t
nmlnmlnmlstrk
nml
tpl
nmlnmlnmlsrptr
l
p
kkkl
knzknz
mylxdssFrsPsUrUen
nzkmykmy
lxdssFrsPsUrUem
nzmyklxklx
dssFrsPsUrUel
nzmydssFrsPsUrUeepl
xxxtU
mkl
(2-72)
Note that
111,1
,)],}(ˆˆˆˆ{])],}(ˆˆˆˆ{[
])||[(][
)1(||)(
0
||
||2)(||
22
22
mpll
tFPjUUjedssFPjUUje
ee
pltplt
s
tppt
(2-73)
The previous formula shows that strong mathematical induction (theorem 2-3) that all derivativesup to order 110 mpl are continuous and bounded can be applied to prove thatderivatives of order m are jointly continuous and bounded. By mathematical induction it followsthat general component of the tensor of the inhomogeneous term is jointly smooth and boundedon 3]1,0[),0[ .
-
28
The remaining portion of the solution ),( xtP is automatically smooth and bounded on3]1,0[),0[),( xt since, in 2-14, its transform is defined in terms of 3,2,1,,,, 0 iUhFU
i
which have all of those properties.
END PROOF
Acknowledgments: The author studied the problem formulation in [7]. He received helpfulcomments from an anonymous reviewer for the Proceedings of the Royal Society of Edinburgh,Steve Krantz and the Journal of Mathematical Analysis and Applications, Gene Wayne, andCharles Meneveau.
Selected Bibliography1. Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover, NY,
1962 (Dover 1989).2. Avrin, J., A One Point Attractor Theory for the Navier-Stokes Equation on Thin Domains
with No-Slip Boundary Conditions, Proceedings of the American Mathematical Society,Volume 127, Number 3, March1999, pp. 725-735.
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