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: WILLIAM_RIVERA35@comcast.net

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

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.

3. Bardos, C., A Basic Example of Nonlinear Equations, The Navier-Stokes Equations,2000, http://www.its.Caltech/~siam/tutorials/Bardos_The_NSE.pdf

4. Cafferelli, L. Kohn, R. Nirenberg, L. Partial Regularity of Suitable Weak Solutions of theNavier-Stokes Equations, Comm. Pure. Appl. Math 35 (1982).

5. Fefferman C., Existence and Smoothness of the Navier-Stokes Equation, 2000, http://www.claymath.org/millenium/NavierStokes_equations/

6. Goldberg, R., Methods of Real Analysis, 2nd edition, John Wiley &Sons, New York,1976.

7. Hale J. , Ordinary Differential Equations, Krieger, Malabar, FLA , 1980.8. Hirsch M. and Smale S., Differential Equations, Dynamical Systems, and Linear Algebra,

Academic Press, N.Y, 1974.9. Leray J., Essai sur les mouvement d’un liquide visqueux Emplissant L’Espace,

J.Math.Pures Appl., se’rie 9, 13, 331-418,1934.10. Powers, D.L., Boundary Value Problems, Second Edition, 1970.11. Rivera, W.M., A New Invariant Manifold with an application to smooth conjugacy at a

node, Dynamic Systems and Applications 2 (1993) 491-503.12. Stein E. and Weiss G., Introduction to Fourier Analysis, Princeton University Press,

Princeton, NJ, 1971.13. Sternberg S., Local contractions and a theorem of 'Poincare , Amer. J. Math. 79, 1957,

809-824.14. Strauss, W. A., Partial Differential Equations: An Introduction, John Wiley&Sons, New

York, 1992.15. Vanderbauwhede, A. Centre Manifolds, Normal Forms and Elementary Bifurcations,

Dynamics Reported, Volume II, Kirchgraber and Walther,1989.