The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity

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The stationary three-dimensional Navier-Stokes Equations with a non-zero constant velocity at infinity

Chérif Amrouche, Huy Hoang Nguyen

To cite this version:

Chérif Amrouche, Huy Hoang Nguyen. The stationary three-dimensional Navier-Stokes Equations

with a non-zero constant velocity at infinity. 2008. �hal-00206208�

with a non-zero onstant veloity at innity

Chérif AMROUCHE

⋆

and Huy Hoang NGUYEN

†

Laboratoire deMathématiques Appliquées

CNRSUMR5142

Université dePauetdesPaysde l'Adour

IPRA-Avenuedel'Université 64013Pau,Frane

⋆

herif.amrouheuniv-pau.fr

†

huy-hoang.nguyenetud.univ-pau.fr

Abstrat- Thispaperisdevoted to somemathematialquestions related

tothe3-dimensionalstationaryNavier-Stokes. Ourapproahisbasedonaom-

binationofpropertiesofOseen problemsin

R ³

.

Keywords: Navier-Stokesequations;Oseenequations;weightedSobolevspaes;

uidmehanis.

AMSlass: 35Q30,76D03,76D05,76D07

1 Introdution

Let

Ω

^{′ be a bounded open region of}

R ³

, not neessarily onneted, with a Lipshitz-ontinuous boundary and let

Ω

^{be the omplement of}

Ω

^{′. We sup-}

posethat

Ω

^{′ hasanitenumberofonnetedomponentsandeahonneted}

omponenthasaonnetedboundary,sothat

Ω

^{isonneted. Theproblemon-}

siststhen in ndingaveloityeld u

= (u 1 , u 2 , u 3 )

^{andthe pressure}

π

^satisfy

theNavier-Stokessystem:

(N S )

 

 



 

 

−ν∆

^u

+

^u

.∇

^u

+ ∇π =

^f

in Ω,

div

^u

= 0 in Ω,

u

= 0 on ∂Ω,

u

→

^u∞

at infinity,

where

ν > 0

^{, f and u}∞

∈ R ³

are respetively the visosity of the uid, the external foreeld ating onthe uidand agiven onstantvetor. Thethird

equationofthesystemstatesthat theuidadheresatthesurfaeofthebody,

whih is the ommon no-slip ondition. For the last equation, we have two

dierentasesonerningthebehaviorofuatinnity. Ifu∞

= 0

^,theowisat

restatinnityandintheremainingase,ifu∞

6= 0

^{,theowispastatinnity.}

In this paper, we areinterestedin onsidering the ase

Ω = R ³

and u∞

6= 0

^.

Ourpurposeistostudysomeregularitypropertiesoftheweaksolutionstothe

problem

(N S )

^.

aboutweaksolutions,weightedSobolevspaesandsomeresultsofOseensystem

in weghted Sobolev spaes. In Setion 2, a result about existene of weak

solutions for the problem

(N S)

^{will be presented. In next setions, we shall}

obtain some regularity properties of the weak solution u and the assoiated

pressure

π

^{. We shall also onsider the identity energy in the last setion. In}

thispaper,weuseboldtypeharaterstodenotevetordistributionsorspaes

ofvetordistributions with3omponentsand

C > 0

^{usuallydenotes ageneri}

onstantthevalueofwhih mayhangefrom linetoline.

Nowwereallthemainnotationsandresults,onerningtheweightedSobolev

spaes,whihweshalluselateron.

We dene

D(Ω)

^{to be thelinear spaeof innite} dierentiablefuntions with ompat support on

Ω

^{. Now, let}

D

^′

(Ω)

^{denote thedual spae of}

D(Ω)

^{, often}

alled the spae of distributions on

Ω

^{. We denote by}

h., .i

^{the duality pairing}

between

D(Ω)

^{′ and}

D(Ω)

^{. Remarkthatwhenf isaloallyintegrablefuntion,}

thenf anbeidentiedwithadistributionby

h

^f

, ϕi = Z

Ω

f

(

^x

) .ϕ(

^x

) d

^x

.

Given aBanah spae

B

^{, with dual spae}

B

^{′ and a losed subspae}

X

^of

B

^,

wedenote by

B

^′

⊥ X

^{(or moresimply}

X

^{⊥, ifthere is noambiguityasto the}

dualityprodut)thesubspaeof

B

^{′ orthogonalto}

X

^{, i.e.}

B

^′

⊥ X = X

^⊥

= {f ∈ B

^′

|∀ v ∈ X, < f, v >= 0} = (B/X)

^′

.

The spae

X

^{⊥ is also alled thepolar spaeof}

X

ⁱⁿ

B

^{′. In1933, Jean Leray}

[13℄whointroduedtheoneptoftheweaksolution:

Denition 1.1. Aweaksolutionto the problem(

N S

^{)is aeld u}

∈ H ¹

loc (Ω)

vanishingon

∂Ω

^,with

∇

^u

∈ L ² (Ω)

^{, divu}

= 0

ⁱⁿ

Ω

^and

lim

|^x|→∞

Z

S

|

^u

(

^x

−

^u∞

)| =

|

^u

(

^x

) −

^u∞

| = 0

^where

S

^{istheunitsphereof}

R ³

suhthatforall

ϕ ∈ V (Ω) = {

^v

∈ D (Ω), div

^v

= 0}

^:

ν Z

Ω

∇

^u

. ∇ϕ d

^x

+ Z

Ω

u

. ∇

^u

. ϕ d

^x

= h

^f

, ϕi .

Atypialpointin

R ³

isdenotedbyx

= (x 1 , x 2 , x 3 )

^{anditsnormisgivenby}

|

^x

| = (x ² ₁ + x ² ₂ + x ² ₃ )

¹²

.

^{Wedene theweightfuntion}

ρ(

^x

) = (1+ |

^x

| ² )

^{12. For}

eah

p ∈ R

and

1 < p < ∞

^{,theonjugateexponent}

p

^{′ isgivenbytherelation}

1

p + 1

p

^′

= 1

^{. With}

α ∈ R

and

m ∈ N

, weset

k = k(m, p, α) =

 



 

−1, if 3

p + α 6∈ {1, ..., m}, m − 3

p − α, if 3

p + α ∈ {1, ..., m},

andweintroduethedenition oftheweightedSobolevspaes.

Denition1.2. Let

Ω

^{beeither anexteriordomainor}

Ω = R ³

. Then,

W _α ^m,p (Ω) = {u ∈ D

^′

(Ω); ∀ λ ∈ N ³ ,

0 ≤ |λ| ≤ k, ρ ^α−m+|λ| (ln(1 + ρ))

⁻¹

∂ ^λ u ∈ L ^p (Ω),

k + 1 ≤ |λ| ≤ m, ρ ^α−m+|λ| ∂ ^λ u ∈ L ^p (Ω)}.

k u k _W

_α^m,p

_(Ω) = ( X

0≤|λ|≤k

||ρ ^α−m+|λ| (ln(1 + ρ))

⁻¹

∂ ^λ u|| ^p _L

p

(Ω) +

+ X

k+1≤|λ|≤m

||ρ ^α−m+|λ| ∂ ^λ u|| ^p _L

p

(Ω) ) ^1/p .

Wealsodene thesemi-norm:

|u| _W

_α^m,p

(Ω) =



 X

|λ|=m

||ρ ^α ∂ ^λ u|| ^p _L

p

(Ω)





1/p

.

We note that the logarithmi weight only appears for the ase

3/p + α ∈ {1, ..., m}

^{andalltheloalpropertiesof}

W _α ^m,p (Ω)

^{oinidewiththoseofthelas-}

sialSobolevspae

W ^m,p (Ω)

^{. Weset}

W

^◦

^{m, p} α (Ω) = D(Ω) ^W

m, p α

(Ω)

andwedenote

thedualspaeof

◦

W ^{m, p} α (Ω)

^by

W

_−α^−m,p′

(Ω)

^{,whihisthespaeof}distributions.

When

Ω = R ³

, we have

W _α ^m,p ( R ³ ) = W

^◦

^{m, p} α ( R ³ )

^{. If}

3/p + α 6∈ {1, ..., m}

^{, we}

havethealgebraiandtopologialimbeddings

W _α ^m,p (Ω) ֒ → W _α−1 ^m−1,p (Ω) ֒ → ... ֒ → W _α−m ^0,p (Ω).

Forall

λ ∈ N ⁿ

with

|λ| ≥ 0

^,themapping

u ∈ W _α,β ^m,p (Ω) → ∂ ^λ u ∈ W _α,β ^m−|λ|,p (Ω)

is ontinuous. Moreover,if

3

p + α 6∈ {1, ..., m}

^{, then forany}

γ

ⁱⁿ

R

suh that

3

p + α − γ 6∈ {1, ..., m}

^themapping

u → ρ ^γ u

^{is an} isomorphismof

W _α ^m,p (Ω)

onto

W _α−γ ^m,p (Ω)

^{. Note that ifweonly suppose}

3

p + α 6∈ {1, ..., m}

^,themapping

isontinuous.

We denote by

[q]

^{the integer part of}

q

^{. For any}

k ∈ N

,

P _k

(respetively,

P ^∆

k

^{)standsforthespae of}polynomials(respetively,harmonipolynomials) ofdegree

≤ k

^{. If}

k

^{isstritlynegativeinteger,wesetbyonvention}

P _k = {0}.

Let

k

^{beaninteger,then}

P _k

isinludedin

W _α ^m,p (Ω)

^with

k =

 



 

m − 3 p + α

, if 3

p + α 6∈ Z

⁻

, m − 3

p − α − 1, otherwise.

Weintroduethespae

W f ₀ ^1,p (Ω) =

u ∈ W ₀ ^1,p (Ω), ∂u

∂x 1

∈ W ₀

^−1,p

(Ω)

whihisaBanahspaeequippedwiththefollowingnorm

||u|| _W

_f^1,p

0

(Ω) = ||u|| _W

^0,p

−1

(Ω) + X 3 i=1

|| ∂u

∂x i

|| L

^p

(Ω) + || ∂u

∂x 1

|| _W

^−1,p

0

(Ω) ,

^if

p 6= 3,

||u||

_f

_W

^1,3

0

(Ω) = ||(

^ln

(1 + ρ))

⁻¹

u|| _W

^0,3

−1

(Ω) + X 3 i=1

|| ∂u

∂x i

|| L

³

(Ω) + || ∂u

∂x 1

|| _W

^−1,3

0

(Ω) ,

and

W f ₀

^−1,p′

( R ³ )

^{isitsdualspae. Thepreviousnormisequivalenttothenatural}

one and it allows to prove the density of

D(Ω)

ⁱⁿ

W f ₀ ^1,p (Ω)

^{. This result is}

announedin[7℄. Weintroduealsothespae

V

(Ω) = n

v

∈ W

^◦

^1,2 0 (Ω),

^divv

= 0

ⁱⁿ

Ω o .

Inorder to understandbetter thethe onditionu

→

^u∞ ^{at innityof the}

Navier-Stokessystem,weintrodueafollowinglemma(f[8℄):

Lemma1.3. Assume

1 < p < 3

^and

u ∈ D

^′

( R ³ )

^suhthat

∇u ∈ L ^p ( R ³ )

^{. Then}

thereexistsauniqueonstant

u

∞

∈ R

suhthat

u − u

∞

∈ W ₀ ^1,p ( R ³ )

^,where

u

∞

isdenedby

u

∞

= lim

|x|→∞

1 ω

Z

S

u(σ(|x|)) dσ

where S is the unit sphere of

R ³

and

ω

^{is the area of}

S

^{. Moreover, we have}

u − u

∞

∈ L

^3−p3p

( R ³ )

^{with the estimate}

||u − u

∞

||

L

3p

3−p

(

R3

) ≤ C||∇u||

Lp

(

R3

) ,

^(1.1)

lim

|x|→∞

Z

S

|u(σ|x|) − u

∞

|dσ = lim

|x|→∞

Z

S

|u(σ|x|) − u

∞

| ^p dσ = 0

^(1.2)

and

Z

S

|u(rσ) − u

∞

| ^p dσ ≤ Cr ^p−3 Z

{x∈R³

,|x|>r}

|∇u| ^p dx.

^(1.3)

ReallalsothefollowingSobolevembeddings

W ₀ ^1,p ( R ³ ) ֒ → L ^p∗ ( R ³ )

^where

p∗ = 3p

3 − p

^and

1 < p < 3, W ₀ ^1,3 ( R ³ ) ֒ → V M O( R ³ )

^where

V M O( R ³ ) = D( R ³ )

^{||.||BM O}

.

Here,

BM O

^{isthespaeofloallyintegrablefuntionsin}

R ³

andsuhthat,on allubes

Q

^,

|| f || BMO = sup

Q

1

|Q|

Z

Q

|f (x) − f (Q)|dx < ∞.

Note alsothatif

∇u ∈ L ^p

with

p > 3

^and

u ∈ L ^r ( R ³ )

^forsome

r ≥ 1

^{,then we}

have

u ∈ L

^∞

( R ³ )

^.

If

Ω

^{isanexteriordomain, wehaveaorollaryasfollows:}

Corollary 1.4. Let

Ω ⊂ R ³

be an exterior domain. Assume

1 < p < 3

^and

u ∈ D

^′

(Ω)

^suhthat

∇u ∈ L ^p (Ω)

^{. Then thereexistsaunique onstant}

u

∞

∈ R

suhthat

u − u

∞

∈ W ₀ ^1,p (Ω)

^{andwehave the properties} (1.1)-(1.3).

Proof. Let

u ∈ D

^′

(Ω)

^{suh that}

∇u ∈ L ^p (Ω)

^{. Then, therestritionof}

u

^to

Ω R

with asuientlylarge

R

^satisfy

u ∈ D

^′

(Ω R )

^and

∇u ∈ L ^p (Ω R ).

^Therefore,

wehave

u ∈ W ^1,p (Ω R )

^and

u| ∂B

R

∈ W ^1−1/p,p (∂Ω R )

^(see Proposition2.10[4℄).

Thenthereexists

u 0 ∈ W ^1,p (Ω R )

^suhthat

u 0 = u

^on

Γ

^and

u 0 = 0

^on

∂B R .

^We

extend

u 0

^{byzerooutside}

B R

^anddenote

u

∼

0

^{theextendedfuntionthatbelongs}

to thelassialSobolevspae

W ^1,p (Ω)

^{and hasompat support in}

Ω R

^{. Note}

that

v = u− u

^∼

0

^{, then}

∇v ∈ L ^p (Ω)

^and

v = 0

^on

Γ

^{. Weset that ∼}

v = v

ⁱⁿ

Ω

^and

∼

v = 0

^outside

Ω.

^{Thenweandeduethat}

∇

^∼

v∈ L ^p ( R ³ )

^{. Thereforethereexists}

auniqueonstant

u

∞^suhthat

∼

v −v

∞

∈ W ₀ ^1,p ( R ³ )

^,or

u− u

^∼

0 −v

∞

∈ W ₀ ^1,p ( R ³ )

^.

Then

u − v

∞

∈ W ₀ ^1,p (Ω)

^.

NowweshallintroduethefollowinglemmabyombiningaresultofBabenko

(1973,Proposition3)withTheoremII.5.1[11℄. Theproofofthislemmaanbe

foundin [11℄.

Lemma 1.5. Let

Ω ⊂ R ³

bea Lipshitzexterior domainor

Ω = R ³

. Assume that

u ∈ W ₀ ^1,2 (Ω) and ∂u

∂x 1

∈ L ^q (Ω) where 1 < q < 2.

Then

u ∈ L ^3q (Ω)

^{andthe following inequalityholds:}

||u|| L

^3q

(Ω) ≤ C(|| ∂u

∂x 1

|| L

^q

(Ω) + ||∇u|| L

²

(Ω) ).

Thenextlemmagivesananotherversionofthisresult.

Lemma1.6. Let

1 < p < 3

^{. Assumethat}

u ∈ f W ₀ ^1,p ( R ³ )

^{. Then}

u ∈ L

^44p−p

( R ³ ) ∩ L

^33p−p

( R ³ )

^{andfollowing inequality holds:}

||u||

L

4p

4−p

(

R³

) + ||u||

L

3p

3−p

(

R³

) ≤ C||u||

_f

_W

1,p

0

(

R3

) .

^(1.4)

Proof. We already showed that if

u ∈ W ₀ ^1,p ( R ³ )

^with

1 < p < 3

^{, then}

u ∈ L

^33p−p

( R ³ )

^satisfying

||u||

L

3p

3−p

(

R³

) ≤ C||∇u|| _L

^p

₍

R³

) .

Weknowthat

D( R ³ )

^isdensein

W ₀ ^1,p ( R ³ )

^{,thenthereexistsasequene}

(ϕ k ) k∈

N

∈ D( R ³ )

^{whih onverges towards}

1

ⁱⁿ

W ₀ ^1,p

^′

( R ³ ).

^By hypothesis, we dedue

∆u ∈ W ₀

^−1,p

( R ³ )

^{. Then, wehave}

h∆u, 1i _W

−1,p

0

(

R3

)×W

₀^1,p′

(

R3

) = lim

k→+∞ h∆u, ϕ k i _W

−1,p

0

(

R3

)×W

₀^1,p′

(

R3

)

= − lim

k→+∞ h∇u, ∇ϕ k i _L

p

(

R3

)×L

^p′

(

R3

) = 0.

Analogously, sine

D( R ³ )

^{is dense in}

W f ₀ ^1,p ( R ³ )

^{(see [7℄), then we an dedue}

that

∂u

∂x 1

, 1

W

₀^−1,p

(

R³

)×W

₀^1,p′

(

R³

)

= 0.

Weset

−∆u + ∂u

∂x 1

= f.

^(1.5)

Thenbyhypothesisand[5℄,wehave

f ∈ W ₀

^−1,p

( R ³ )

^{satisfyingthe}ompatibility onditionasfollows

hf, 1i _W

^−1,p

0

(

R3

)×W

₀^1,p

(

R3

) = 0.

−∆w + ∂w

∂x 1

= f in R ³

(1.6)

has a unique solution

w ∈ L

^33p−p

( R ³ ) ∩ L

^44p−p

( R ³ )

^{suh that}

∇w ∈ L ^p ( R ³ )

^,

∂w

∂x 1

∈ W ₀

^−1,p

( R ³ )

^{alsosatisfying}

||w||

L

3p

3−p

(

R3

) + ||w||

L

4p

4−p

(

R3

) + ||∇w||

L^p

(

R³

) + ∂w

∂x 1

W

₀^−1,p

(

R3

)

≤ C|| f || _W

^−1,p

0

(

R3

) .

^(1.7)

Weset

z = u−w

^{. Subtrating(1.5)to(1.6),weget}

−∆z + ∂z

∂x 1

= 0

ⁱⁿ

R ³

. Sine

z ∈ L ^3p/(3−p) ( R ³ )

^{,then,from Lemma4.1[8℄,wededuethat}

z

^{isapolynomial}

andthen

z = 0

^{. From(1.7),wehave(1.4). Theproofisomplete.}

AnalogouslyasinLemma1.6, itiseasyto deduethefollowing.

Lemma 1.7. Let

1 < p < 2

^{. Assumethat}

u ∈ W ₀ ^2,p ( R ³ )

^and

∂u

∂x 1

∈ L ^p ( R ³ )

^.

Then we have

u ∈ L

^2−p2p

( R ³ ) ∩ L

^3−2p3p

( R ³ )

^if

1 < p < 3/2

^and

u ∈ L ^s ( R ³ )

^for

all

s ≥ 2p

2 − p

^if

3/2 ≤ p < 2

^.

Denition 1.8. Let

1 < p < ∞

^{. Let}

γ, δ ∈ R

besuh that

γ ∈ [3, 4]

^,

γ > p

^,

δ ∈ [ ³ ₂ , 2], δ > p

^{. Wedenetworeals}

r = r(p, γ)

^and

s = s(p, δ)

^asfollow

1 r = 1

p − 1

γ and 1

s = 1 p − 1

δ .

Remark 1.9. FromDenition1.8, weandeduethat

i) If

1 < p < 3

^,then

4p

4 − p ≤ r ≤ 3p 3 − p

^,

ii) If

3 ≤ p < 4

^,then

4p

4 − p ≤ r < ∞,

iii)If

1 < p < 3/2

^,then

2p

2 − p ≤ s ≤ 3p 3 − 2p

^,

iv)If

3/2 ≤ p < 2

^,then

2p

2 − p ≤ s < ∞

^.

Finally, we introdue thepropertiesonerningthe Oseenequations whih

will be useful in the next parts. We onsider the non homogeneous Oseen

problem : givenavetoreld fand afuntion

g

^{, welook forasolution}

(

^u

, π)

tothesystem

(OS)

 



−∆

^u

+ ∂

^u

∂x 1

+ ∇π =

^f

in R ³ , div

^u

= g in R ³ .

Theorem1.10. [7℄LetrandsbethenumbersgiveninDenition 1.8. Assume

(

^f

, g) ∈ L ^p ( R ³ ) × W f ₀ ^1,p ( R ³ )

^.

(i) If

1 < p < 2

^{, thenProblem}

(OS)

^{has a uniquesolution}

(

^u

, π) ∈ L ^s ( R ³ ) ×

W ₀ ^1,p ( R ³ )

^suhthat

∇

^u

∈ L ^r ( R ³ )

^,

∇ ²

^u

∈ L ^p ( R ³ )

^and

∂

^u

∂x 1

∈ L ^p ( R ³ )

^{. More-}

over, the following estimate holds

||

^u

||

L^s

(

R³

) + ||∇

^u

||

L^r

(

R³

) + ||∇ ²

^u

||

L^p

(

R³

) + || ∂

^u

∂x 1

||

L^p

(

R³

) + ||π|| _W

^1,p

0

(

R³

)

≤ C(||

^f

||

L^p

(

R³

) + ||g|| _W

_f^1,p

0

(

R3

) ).

(ii) If

2 ≤ p < 3

^{, then Problem}

(OS)

^{has a solution}

(

^u

, π) ∈ W ^1,r

0 ( R ³ ) × W ₀ ^1,p ( R ³ )

^{,uniqueuptoanelementof}

N ₀

,suhthat

∇ ²

^u

∈ L ^p ( R ³ )

^and

∂

^u

∂x 1

∈ L ^p ( R ³ )

^{alsosatisfying}

K∈ inf

R³

||

^u

+ K ||

W^1,r

0

(

R3

) + ||∇ ²

^u

||

Lp

(

R3

) + || ∂

^u

∂x 1

||

Lp

(

R3

) + ||π|| _W

^1,p

0

(

R3

)

≤ C(||

^f

||

Lp

(

R3

) + ||g|| _W

_f^1,p

0

(

R³

) ).

(iii)If

p ≥ 3

^,thenProblem

(OS )

^hasasolution

(

^u

, π) ∈ W ^2,r

0 ( R ³ ) ×W ₀ ^1,p ( R ³ )

^,

uniqueuptoan elementof

N ₁

,suhthat

∂

^u

∂x 1

∈ L ^p ( R ³ )

^{. Moreover,wehave}

(

λ

,µ)∈ inf

N₁

(||

^u

+ λ||

W^2,p

0

(

R³

) + ||π + µ|| _W

^1,p

0

(

R³

) ) + || ∂

^u

∂x 1

||

L^p

(

R³

)

≤ C(||

^f

||

Lp

(

R3

) + ||g||

_f

_W

1,p 0

(

R3

) ).

Theorem1.11. [7℄ Let

r

^{bethe number given inDenition 1.8. Assumethat}

f

∈ W

^−1,p

0 ( R ³ )

^{andsatisesthe} ompatibility ondition

∀λ ∈ P _[1−3/p

_′

_] , h

^f

, λi

_W−1,p

0

(

R3

)×

W^1,p′

0

(

R3

) = 0.

Let

g ∈ L ^p ( R ³ )

^suhthat

∂g

∂x 1

∈ W ₀

^−2,p

( R ³ )

^,satisestheompatibilityondition

∀λ ∈ P _[2−3/p

_′

_] , ∂g

∂x 1

, λ

W

₀^−2,p

(

R3

)×W

₀^2,p′

(

R3

)

= 0.

(i) If

1 < p < 4

^{, thenthe Oseen system}

(OS )

^{has a unique solution}

(

^u

, π) ∈ L ^r ( R ³ )× L ^p ( R ³ )

^suhthat

∇

^u

∈ L ^p ( R ³ )

^and

∂

^u

∂x 1

∈ W

^−1,p

0 ( R ³ )

^{. Moreover,the}

following estimateholds

||

^u

||

Lr

(

R3

) + ||∇

^u

||

Lp

(

R3

) + || ∂

^u

∂x 1

||

W^−1,p

0

(

R3

) + ||π|| L

^p

(

R3

)

≤ C(||

^f

||

W^−1,p

0

(

R3

) + ||g|| L

^p

(

R³

) + || ∂g

∂x 1

|| _W

^−2,p

0

(

R3

) ).

(ii) If

p ≥ 4

^{, then the Oseen system}

(OS)

^{has a unique solution}

(

^u

, π) ∈ f

W ^1,p

0 ( R ³ ) × L ^p ( R ³ )

^{, uniqueup to an element of}

N ₀

. Moreover, the following estimate holds

K

inf

∈R³

||

^u

+ K ||

_Wf^1,p

0

(

R3

) + ||π|| _L

^p

₍

R3

)

≤ C(||

^f

||

_W−1,p

0

(

R³

) + ||g|| _L

^p

₍

R3

) + || ∂g

∂x 1

|| _W

−2,p 0

(

R³

) ).

spaes

WeshallonsidertheNavier-Stokesproblemin

R ³

:

(N S )

 



 

−ν ∆

^u

+

^u

.∇

^u

+ ∇π =

^f

in R ³ ,

div

^u

= 0 in R ³ ,

u

−→

^u∞

if |x| → ∞,

where u∞ ^{is a onstant vetor in}

R ³

. Without lossof generality, we anset u∞

= λ

^e

1

^{with e}

1 = (1, 0, 0)

^and

λ ≥ 0.

^{From now on, we onsider the ase}

of axed

λ > 0

^{. First, weprovetheexisteneof weaksolutionsand then, we}

shalltheregularityofthesesolutionsindimention3. Weonsiderthefollowing

lemma.

Lemma 2.1. If

f ∈ W ₀

^−1,2

( R ³ )

^{, then there exists F}

∈ L ² ( R ³ )

^{suh that}

f = div

^Fin

R ³

withthe estimate

||

^F

||

L²

(

R³

) ≤ C||f || _W

−1,2

0

(

R³

) .

^(2.1)

Additionallysupposethat

f ∈ W ₀

^−1,p

( R ³ )

^,andfurthermoreassumethat

hf, 1i = 0

^if

p ≤ 3

2

^,thenF

∈ L ^p ( R ³ )

^{andwehave theestimate}

||

^F

||

Lp

(

R3

) ≤ C

^′

||f || _W

^−1,p

0

(

R3

) .

^(2.2)

Proof. If

f ∈ W ₀

^−1,2

( R ³ )

^{, from Theorem 9.5 [5℄, there exists a unique}

z ∈ W ₀ ^1,2 ( R ³ )

^suhthat

∆z = f

ⁱⁿ

R ³

and

||z|| _W

^1,2

0

(

R3

) ≤ C||f || _W

^−1,2

0

(

R³

) .

We set that F

= ∇z

^{, but}

z ∈ W ₀ ^1,2 ( R ³ )

^{, from} Proposition 9.2 [5℄, we have F

∈ L ² ( R ³ )

^{and (2.1). Moreover,if}

f ∈ W ₀

^−1,p

( R ³ )

^{then there existsaunique}

h ∈ W ₀ ^1,p ( R ³ )/ P

[1−

³_p

]

^suhthat

f = ∆h

ⁱⁿ

R ³

and

||h|| _W

^1,p

0

(

R³

)/

P

[1−3

p]

≤ C

^′

||f || _W

−1,p 0

(

R³

) .

Then

∇(z − h)

^isharmoniin

L ² ( R ³ ) + L ^p ( R ³ )

^andonsequently,

∇z = ∇h

^and

F

∈ L ² ( R ³ ) ∩ L ^p ( R ³ )

^{withtheestimate(2.2).}

We now return to the question of the existene of weak solutions of the

Navier-StokesEquationsin

R ³

. Thenexttheorem iswell known,then wegive hereaskethoftheproof.

Theorem2.2. Givenaforef

∈ W

^−1,2

0 ( R ³ )

^,theproblem

(N S )

^hasaweakso-

lutionusatisfyingu

−

^u∞

∈ W ^1,2

0 ( R ³ )

^{andthereexistsafuntion}

π ∈ L ² _loc ( R ³ )

^,

uniqueup toaonstant,suhthat

(

^u

, π)

^{solves the problem}

(N S )

^{inthe sense}

of distributionsandwehavethe following estimation

||

^u

−

^u∞

||

W^1,2

0

(

R³

) ≤ C||

^f

||

W^−1,2

0

(

R³

) .

^(2.3)