On the steady Oseen problem in the whole space

35 (2005), 371–401

On the steady Oseen problem in the whole space

T. Z. Boulmezaoud* and U. Razafison^†

(Received May 11, 2004) (Revised December 3, 2004)

Abstract. We deal with Oseen’s equations in the whole space. A class of existence, uniqueness and regularity results for both the scalar and the vectorial equations are given. Isotropic weighted Sobolev spaces are used for describing the growth or the decay of functions at infinity.

1. Introduction

The Oseen equations are a linearized version of the Navier-Stokes equations describing a viscous and incompressible fluid in which a small body is moving. The purpose of this paper is to study the Oseen problem in the whole space Rⁿ, nb2:

nDuþk qu

qx₁þ‘p¼ f in Rⁿ; divu¼h in Rⁿ:

ð1:1Þ

Here the unknowns are the velocity u of the fluid and the pressure p. The data are the viscositynof the fluid, the external force f, the functionh and the positive real k. System (1.1) was proposed by Oseen (see [19]) in order to remove some physical paradoxes of the Stokes system, which corresponds to the case k¼0. One of the first works devoted to these equations is due to Finn [10]. Specifically, Finn treated Oseen’s equations in three dimensional exterior domains when ð1þ jxjÞf is square integrable and h¼0. He proved the existence of a solution u such that ð1þ jxjÞ¹u is square integrable.

Farwig [9] proved, among other results, the existence of a solution ðu;pÞ of (1.1) when f AL^pðRⁿÞⁿ and hAW^1;pðRⁿÞ. In that case the solution ðu;pÞ satisfies uAL_loc^p ðRⁿÞⁿ, qiqjuk AL^pðRⁿÞ, qipAL^pðRⁿÞ, i¼1;2;. . .;n. In [11]

* Laboratoire de Mathe´matiques, Universite´ de Versailles Saint-Quentin-en-Yvelines 45 avenue des Etats-Unis—Baˆtiment Fermat, 78035 Versailles, France.

†Laboratoire de Mathe´matiques Applique´es, Universite´ de Pau et des pays de l’Adour, 64000, Pau, France.

2000 Mathematics Subject Classification. 76D03, 35Qxx, 35N05, 35A05, 35A08.

Key words and phrases. Oseen’s equations, weighted Sobolev spaces, unbounded domains, fluid mechanics.

Galdi stated that if f AW^m;pðRⁿÞⁿ, hAW^mþ1;pðRⁿÞ, mb0, then the problem (1.1) has a solution in W_loc^mþ2;pðRⁿÞ W_loc^mþ1;pðRⁿÞ. In [7] Farwig investigates the system (1.1), set in a three dimensional exterior domain, using anisotropic weighted L² spaces. The use of anisotropic weights seems to be a natural approach because of the anisotropy introduced by the term q₁u. However, such an approach contains some serious technical complications. The reader can refer to [16], [20], [8], [7], [3] for existence results in anisotropic weighted spaces. To our knowledge, most of the existing results in the literature concern the case f AL^pðRⁿÞⁿ or are around that case. Several questions concerning the existence, the uniqueness and regularity of the solution remain not treated, especially when the data f and h are slowly decreasing or have a polynomial behavior at infinity. Among the results we present in this paper, we shall prove that if f ¼ ðfi;. . .;fnÞ and h satisfies the conditions

ð1þ jxj²Þ^ðjmjmÞ=2D^mfk AL^pðRⁿÞ for jmjam;

ð1þ jxj²Þ^ðjmjm1Þ=2D^mhAL^pðRⁿÞ for jmjamþ1;

for some integer mb0, then Problem (1.1) admits a solutionðu;pÞ, unique up to a class of polynomials, and satisfying

ð1þ jxj²Þ^ðjmjm2Þ=2D^muk AL^pðRⁿÞ for jmjamþ2;

ð1þ jxj²Þ^ðjmjm1Þ=2D^mpAL^pðRⁿÞ for jmjamþ1:

In all the paper, we deal with following problem obtained from (1.1) by means of a simple scaling argument

Duþ2 qu qx1

þ‘p¼ f in Rⁿ; divu¼h in Rⁿ:

ð1:2Þ

We are also interested in the scalar equation Duþ2 qu

qx₁¼ f in Rⁿ; ð1:3Þ which is intimately linked to the system (1.2). The relation between this scalar equation and the general vectorial system (1.2) as well as the relation between their fundamental solutions are discussed in Section 3 hereafter. Observe for the moment that Oseen’s system (1.2) can be formally decomposed into two problems: a Laplace equation for the pressure

Dp¼divf þDh2 qh qx1

; ð1:4Þ

and a scalar equation on each component of the velocity Duiþ2qu_i

qx1

¼ ff~_i; ð1:5Þ

where ff~_i¼ fiqp

qx_i. Consequently, one must choose a functional framework which allows to solve both the equations (1.4) and (1.5) for several behaviors at infinity. The use of weighted Sobolev spaces turned out to be convenient for treating problems in unbounded domains, and consequently seems to be the natural framework for treating Problem (1.4) (see for instance [2]). The main di‰culty here lies in the choice of the weights since the convective term

qu qx1

in Equation (1.5) induces an anisotropic behavior of the velocity, while the pressure keeps an isotropic behavior as in the Stokes problem. Another di‰culty is due to divergence condition divu¼h which complicates seriously the problem. In Section 3 we expose how the system (1.2) can be treated by solving only the scalar equation (1.3) in such a way that the divergence condition is automatically fulfilled. For all these reasons, we shall treat in a first time and independently the scalar equation (1.3). We prove that there exists at least two kinds of solutions; tempered solutions, which are tempered distributions, and quasi-tempered solutions which are not necessarily tempered distributions. Only tempered solutions seem to be useful for solving the vectorial system (1.2).

In a forthcoming paper, we will use our present results in order to solve Oseen’s equations in an exterior domain.

In the sequel, we set

T¼ Dþ2 q qx1

and

T¼ D2 q qx1

The remaining of this paper is organized as follows

– Section 2 is devoted to a brief presentation of some basic definitions and properties of weighted Sobolev spaces, used as a functional framework for solving both the scalar and the vectorial Oseen equations.

– In Section 3, the relation between the scalar equation (1.5) and the vectorial system (1.2) is discussed. Some properties of their fundamental solutions are shown.

– Section 4 deals with the scalar equation (1.5). Existence of solutions and the well posedness of the problem are treated in several functional frameworks.

– Section 5 is devoted to the study of the vectorial system (1.2). After giving a characterization to the solutions of homogeneous probelem, i.e. with f ¼0 and h¼0, we prove a complete class of existence, uniqueness and regularity results for the nonhomogeneous problem.

2. Notation and functional framework 2.1 Notation

In the sequel, nb2 is an integer and p is a real in the interval 1;þy½. The dual number of p denoted p⁰ is defined by the relation 1=pþ 1=p⁰¼1. We use bold characters for vector functions or distributions. For x¼ ðx1;. . .;x_nÞARⁿ we write

jxj ¼ ðx₁²þ þx_n²Þ¹⁼²:

Given a real a, we denote by ½a its integer part. For any kAZ, P_k stands for the space of polynomials of degree lower than or equal to k and P_k^D the subspace of harmonic polynomials of P_k. If k is a negative integer, we set by convention P_k ¼ f0g. We recall that DðRⁿÞ is the well-known space of C^yðRⁿÞ functions with a compact support and D⁰ðRⁿÞ its dual space, namely the space of distributions. We denote by SðRⁿÞ the Schwartz space of functions fAC^yðRⁿÞ with rapid decrease at infinity, by S⁰ðRⁿÞ its dual, i.e.

the space of tempered distributions, and by S₁⁰ðRⁿÞ the space of all the distributions uAD⁰ðRⁿÞ such that e^x1uAS⁰ðRⁿÞ.

The Fourier transform of any complex valued Lebesgue integrable function u:R!C is defined by

u^

uðxÞ ¼ 1 ð2pÞⁿ⁼²

ð

Rⁿ

e^ihx;xiuðxÞdx;

where xARⁿ. If uAS⁰ðRⁿÞ then its Fourier transform uu^AS⁰ðRⁿÞ is defined by h^uu;fi¼hu;ff^i for any function fASðRⁿÞ. The Fourier transform is an invertible mapping from SðRⁿÞ into SðRⁿÞ and from S⁰ðRⁿÞ into S⁰ðRⁿÞ.

Given a Banach space Bwith its dual spaceB⁰and a closed subspaceX of B, we denote by B⁰?X the subspace of B⁰ orthogonal to X, namely

B⁰?X ¼ ff AB⁰;EvAX;hf;vi¼0g ¼ ðB=XÞ⁰:

For any reala>0, the Bessel kernelg_a is defined as the function whose Fourier transform is

g^

g_aðxÞ ¼ ð2pÞⁿ⁼²ð1þ jxj²Þ^a=2:

The kernel g_a can be expressed in terms of Bessel functions by the formula g_aðxÞ ¼g_ajxj^ðanÞ=2K_ðnaÞ=2ðjxjÞ:

with g_a ¼ ð2pÞⁿ⁼²2^a=2þ1Gða=2Þ. Here G denotes the classical Gamma func- tion and K_l denotes the modified Bessel function of third kind. Since for any integer mb0, we know that

K_mþ1=2ðtÞ ¼ ffiffiffiffip 2t r

e^tX^m

k¼0

ðmþkÞ! k!ðmkÞ!

1 ð2tÞ^k;

we deduce an explicit expression of g_a when a<n and na is odd. In particular if a¼2 and n¼3 one has

g₂ðxÞ ¼ 1 4pjxje^jxj:

More generally, the following estimate holds (see [24] or [23]) ExARⁿ; jgaðxÞja ca

jxj^nað1þ jxjÞ^ðaþ1nÞ=2e^jxj: ð2:1Þ Hence,

gaAW_s^0;pðRⁿÞ for sAR; 1apaþy and ðnaÞp<n: ð2:2Þ In the sequel, the expression akb (resp. aFb) means that there exists a constant c not depending on the functions a and b such that aacb.

2.2 Weighted Sobolev spaces. Some basic results

In the sequel, hxi denotes the basic weight defined by

hxi¼ ð1þ jxj²Þ¹⁼²: ð2:3Þ For 1apaþy, L^pðRⁿÞ will refer to the space of (equivalence classes of ) all measurable functions that are p^th power integrable on Rⁿ. This space is equipped with the norm

kuk_LpðRⁿÞ¼ ð

Rⁿ

juj^pdx 1=p

:

Given two integers mb0 and kAZ, we consider the weighted spaces

W^m;pðRⁿÞ ¼ fuAD⁰ðRⁿÞ;EmANⁿ;jmjam;D^muAL^pðRⁿÞg V_k^m;pðRⁿÞ ¼ fuAD⁰ðRⁿÞ;EmANⁿ;jmjam; hxi^kD^muAL^pðRⁿÞg W_k^m;pðRⁿÞ ¼ fuAD⁰ðRⁿÞ;EmANⁿ;jmjam;hxi^kmþjmjD^muAL^pðRⁿÞg H_k^m;pðRⁿÞ ¼ fuAD⁰ðRⁿÞ;EmANⁿ;jmjam;e^x1hxi^kD^muAL^pðRⁿÞg:

These spaces are equipped with the norms kuk_Wm;pðRⁿÞ¼ X

jmjam

kD^muk_L^ppðRⁿÞ

0

@

1 A

1=p

:

kuk_V^m;p

k ðRⁿÞ¼ X

jmjam

khxi^kD^muk_L^ppðRⁿÞ

0

@

1 A

1=p

:

kuk_W^m;p

k ðRⁿÞ¼ X

jmjam

khxi^kmþjmjD^muk_L^ppðRⁿÞ

0

@

1 A

1=p

:

kuk_H^m;p

k ðRⁿÞ¼ X

jmjam

khxi^ke^x1D^muk_L^ppðRⁿÞ

0

@

1 A

1=p

:

The spaces W^m;pðRⁿÞ, W_k^m;pðRⁿÞ, V_k^m;pðRⁿÞ and H_k^m;pðRⁿÞ are Banach spaces.

The spaceDðRⁿÞis dense inW^m;pðRⁿÞ, inV_k^m;pðRⁿÞand inW_k^m;pðRⁿÞ(see, e.g., Hanouzet [15]). We denote by W^m;p0ðRⁿÞ, V_k^m;p0ðRⁿÞ and W_k^m;p0ðRⁿÞ the dual spaces of W^m;pðRⁿÞ, W_k^m;pðRⁿÞ and V_k^m;pðRⁿÞ respectively. They are spaces of tempered distributions. It is quite clear that the local properties of the spaces W_k^m;pðRⁿÞ and V_k^m;pðRⁿÞ coincide with those of the Sobolev space W_k^m;pðRⁿÞ. We also have the obvious identities

V₀^m;pðRⁿÞ ¼W^m;pðRⁿÞ; V_k^0;pðRⁿÞ ¼W_k^0;pðRⁿÞ:

The spaces W_k^m;pðRⁿÞwill play a particular role here. For a detailed study of these spaces, one can refer to [2], [15] and [17]. In this paper we need the semi-norm

juj_W^m;p

k ðRⁿÞ¼ X

jmj¼m

khxi^kD^muk_LpðRⁿÞ

0

@

1 A

1=p

:

and the Green’s formula

EuAW_k^1;pðRⁿÞ; EvAW_kþ1^1;p0 ðRⁿÞ;

ð

Rⁿ

qu

qx_iv dx¼ ð

Rⁿ

uqv

qx_i dx; ð2:4Þ

where 1aian, 1<p<þy and kAZ. We also have the inclusion P_lHW_k^m;pðRⁿÞ; if l<mn=pk: ð2:5Þ In what follows, the space W_k^m;pðRⁿÞ will be considered often in the case

n

pþkBf1;. . .;mg: ð2:6Þ

Indeed, this condition is su‰cient to get some Hardy’s type inequalities.

Namely, if (2.6) is fulfilled, then (see [2]) EuAW_k^m;pðRⁿÞ; inf

lAP_j0

kuþlk_W^m;p

k ðRⁿÞkjuj_W^m;p

k ðRⁿÞ;

where j⁰¼minðm1;jÞ and j¼ ½kþn=pm 1 is the highest degree of polynomials contained in W_k^m;pðRⁿÞ. If (2.6) does not hold, namely if n

pþkAf1;. . .;mg, then similar inequalities can be obtained by adding loga- rithmic factor to the weights in the definition of W_k^m;pðRⁿÞ (see [2]). In that case, all the forthcoming results remains valid provided some minor corrections are given.

We have the following algebraic and topological inclusions ðmb0Þ V_k^m;pðRⁿÞHW_k^m;pðRⁿÞHW_k1^m1;pðRⁿÞH HW_km^0;p ðRⁿÞ: ð2:7Þ For any mANⁿ, the mapping

uAW_k^m;pðRⁿÞ !D^muAW_k^mjmj;pðRⁿÞ ð2:8Þ is continuous (see [15]).

The spaces W_k^m;pðRⁿÞ have proved to be adequate for treating several elliptic problems in unbounded regions of spaces and for several kinds of behavior at infinity (see [2], [21], [4, 5], [13, 14]). Let us recall some basic but fundamental results concerning the Laplace operator in the whole space (see [2]):

Theorem 2.1. Let mAZ and lAN¼Nnf0g.

1. If n=pBf1;. . .;lþ1g, then the operator

D:W_ml^mþ1;pðRⁿÞ=P_½lþ1n=p^D 7!W_ml^m1;pðRⁿÞ;

is an isomorphism.

2. If n=p⁰Bf1;. . .;lþ1g, then the operator

D:W_mþl^mþ1;pðRⁿÞ 7!W_mþl^m1;pðRⁿÞ ?P_½lþ1n=p^D 0

is an isomorphism.

3. If n=p⁰01 and n=p01, then the operator

D:W_m^mþ1;pðRⁿÞ ?P_½1n=p7!W_m^m1;pðRⁿÞ ?P_½1n=p⁰ is an isomorphism.

3. Relation between scalar and vectorial Oseen’s equations. Properties of the fundamental solutions

In this section, our purpose is to discuss briefly the relation between the scalar equation (1.3) and the vectorial Oseen’s system (1.2). Indeed, the di¤erence between equation (1.3) and system (1.2) seems to be reinforced by the presence of the pressure p and the divergence equation divu¼h. This di¤erence also appears in terms of the fundamental solutions. However, there is a simple method for solving system (1.2) by solving equation (1.3) and the Laplace equation, in such a way that the divergence condition is automatically fulfilled. This method reveals that the fundamental solutions are intimately linked by a simple relation.

Formally, let y and s¼ ðs1;. . .;snÞ be solution of the Laplace equations Dy¼h in Rⁿ; Ds¼ f þ‘ h2 qy

qx₁

in Rⁿ:

Consider in addition a vector function F¼ ðF1;. . .;F_nÞwhose componentsF_i, 1aian, are solution of the scalar equations TFi¼si. Then the pair ðu;pÞ defined by

u¼‘yþDF‘ðdivFÞ; p¼divs;

is solution of the system (1.2). In other words, the existence of solutions of (1.2) can be obtained by treating the Laplace equation and the scalar equation (1.3). On the other hand, concerning the fundamental solution ðOi;j;e_jÞ of (1.2) we know that

Oij¼ d_ijD q² qxiqxj

! j

ej¼ q

qx_j Dþ2 q qx₁

j:

Here, i;j¼1;. . .;n andj satisfiesTj¼Ewhere E is the fundamental solution of the Laplace equation. It is well known (see for instance [11]) that the fundamental solution O of the scalar equation (1.3) in Rⁿ is given by

OðxÞ ¼ ð2pÞⁿ⁼²jxj¹ⁿ⁼²e^x1Kn=21ðjxjÞ:

In particular, if the dimension is odd, namely if n¼2mþ1 for some integer mb1, then

OðxÞ ¼1 2

1 ð2pÞ^m

1

jxj^me^x1jxjX^m1

k¼0

ðmþk1Þ!

k!ðmk1Þ!

1 ð2jxjÞ^k; Thus

OðxÞ ¼1

2e^xjxj if n¼1; OðxÞ ¼ 1 4p

e^x1jxj

jxj if n¼3:

In order to clarify the relation between ðOij;ejÞ and O it is convenient to use the notion of Riesz transform (see for instance [22]). Recall that the Riesz transforms of a function uAL^pðRⁿÞ, p>1, are defined by

c R_ju

R_juðxÞ ¼ ix_j

jxj^uu; j¼1;. . .;n;

where RRc_jjuu and ^uu denote the Fourier transform of R_ju and u respectively.

Among the properties of the operators Ri, i¼1;. . .;n, let us recall that they preserve the class L^pðRⁿÞ and satisfy

R_iR_jðDuÞ ¼ q²u qx_iqx_j

Consequently the relation between ðOij;ejÞ andOis summarized in terms of the Riesz transforms as follows

Lemma 3.1. For each i;jan,

Oij¼ ðdijIþRiRjÞO: ð3:9Þ Since Ri maps L^pðR³Þ into itself, one can easily get some properties of Oij

from those of O. Let us enumerate some of them. We state the following proposition whose proof is given in appendix A.

Proposition 3.2. We have

(a) If n¼3, then OAL^pðR³Þ, 2<p<3.

(b) If n¼3, then Og₂AL^pðR³Þ, 2<paþy. (c) If n¼4, Og2 AL^pðR⁴Þ, 2<p<4.

(d) If n¼5, Og2 AL^pðR⁵Þ, 2<p<5=2.

Assertions (b)–(d) allow us to decompose the convolution O f of the fun- damental solution O (or Oij) with any function f into the sum g₂ fþ ðOg2Þ f. The advantage of this decomposition is twofold: firstly, as stated

in Proposition 3.2, the function Og2 has a better behavior than O. The second advantage lies in the fact thatg₂ubelongs to W^2;pðRⁿÞifu belongs to L^p (see Lizorkin spaces in Section 4.1 hereafter or in [6]).

The proof of the following corollary stems from assertion (b) of Prop- osition 3.2.

Corollary 3.3. If n¼3, 1ap<3 and 2<qaþy, then O;OijA L^pðR³Þ þL^qðR³Þ, i;j¼1;2;3.

In appendix B, a proof of the following corollary is given

Corollary 3.4. Let f AL^pðR³Þ, 1ap<2 and n¼3. We set

p₁¼ 2p

2p; p₂¼ 3p

32p if p<3=2 þy if 3=2ap<2 8>

<

>: ; p₃¼ p

p1 if 1< p<2 þy if p¼1 8<

: :

Then, the following assertions are true

(a) O f AL^rðR³Þ for each r, p₁<r<p₂. Moreover, if p03=2 and p01, then O f AL^rðR³Þ with r¼p₂. In all the cases,

kO fk_LrðR³Þkkfk_LpðR³Þ:

(b) O f ¼h1þh2 with h1 AL^rðR³Þ, p₁ <rap₃, h2AW^2;pðR³Þ and kh1k_LrðR³Þþ kh2k_W2;pðR³Þkkfk_LpðR³Þ:

(c) If 1<p<3=2, then O f ¼h₁þh₂ with h₁AL^qðR³Þ and h₂AW^2;pðR³Þ with

q¼ðpþ3Þp 32p :

The same assertions remain true if O is replaced by Oij, 1ai;ja3.

4. The scalar equation

Here we deal with the scalar equation Duþ2 qu

qx1

¼ f in Rⁿ: ð4:1Þ

A first approach for treating this equation is based on a simple but e‰cient idea. It consists of rewriting the equation in term of the new unknown wðxÞ ¼e^x1uðxÞ. More precisely, let us consider the more general equation

Duþ‘f:‘u¼ f in Rⁿ;

withDf¼0. Setting wðxÞ ¼e^fðxÞ=2uðxÞ, wsatisfies the usual elliptic equation:

DwþaðxÞw¼F1 in Rⁿ; ð4:2Þ with aðxÞ ¼¹₄j‘fj²ðxÞ and F1ðxÞ ¼e^fðxÞ=2fðxÞ. Notice that equation (4.1) corresponds to the particular case f¼2x₁. In this case, equation (4.2) writes

Dwþw¼F1 in Rⁿ: ð4:3Þ

The main advantage of this new formulation is that the anisotropic charac- ter of equation (4.1) has disappeared. However, as we shall see, solutions obtained by this method could be di¤erent from those obtained by dealing directly with equation (4.1). This di¤erence is mainly due to the fact that the space of tempered distribution S⁰ðRⁿÞis not preserved under multiplication by e^x1.

These considerations lead one to distinguish two kinds of solutions of (4.1); tempered solutions, which are tempered distributions, and quasi-tempered solutions. A solutionuof (4.1) will be called quasi-tempered ife^x1uAS⁰ðRⁿÞ.

The former solutions are obtained by solving (4.1) directly and may not be unique. On the contrary, the latter solutions are unique in general. It is worth nothing that only tempered solutions of the equation (4.1) turn out to be useful in the treatment of the vectorial Oseen’s system (1.2).

Remark (Relation with the Laplace equation). Let uAS₁⁰ðRⁿÞ be a so- lution of (4.1) and set w¼e^x1u. Consider the finite measure m₂ defined by m₂¼ ð2pÞⁿ⁼²d0 ð2pÞⁿ⁼²g2ðxÞdx, with d0 the Dirac measure at the origin. Let m^

m₂ be the Fourier transform of m₂. We have mm^₂ðxÞ ¼ jxj²ð1þ jxj²Þ¹¼ 1 ð1þ jxj²Þ¹. Next, let vAS⁰ðRⁿÞ be a solution of the Laplace equation Dv¼e^x1f ¼F in Rⁿ. Then, on the one hand, we have jxj²^vv¼FF^. On the other hand, since Dwþw¼F, we get wwðxÞ ¼ ð1^ þ jxj²Þ¹FF^. Thus, wwðxÞ ¼^ m^

m₂ðxÞ^vv, and wðxÞ ¼ ð2pÞⁿ⁼²m₂vðxÞ ¼ ð2pÞⁿ⁼²Ð

RⁿvðxyÞdm₂ðyÞ. Namely, w¼vg2v and u¼e^x1ðvg2vÞ ¼ ð2pÞⁿ⁼²e^x1m₂v.

4.1 Quasi-tempered solutions of scalar Oseen equation

Our aim here is to show existence of quasi-tempered solutions of the scalar Oseen equation (4.1). The main result of this paragraph is the following

Theorem 4.1. Let m;kAZ be two integers and p>1 a real. Then, the operator

T:H_k^mþ2;pðRⁿÞ !H_k^m;pðRⁿÞ is an isomorphism.

Proof of Theorem 4.1

Firstly, observe that the mapping v7!w¼e^x1v

is an isomorphism between H_k^m;pðRⁿÞ and V_k^m;pðRⁿÞ. Moreover, in the sense of distributions one can easily prove that

Tv¼e^x1ðIDÞe^x1v:

This remark allows one to deal only with the operator ID. We start with the following lemma.

Lemma 4.2. Let kb0 be an integer and p>1 be a real. Then, the operator

ID:W^kþ2;pðRⁿÞ !W^k;pðRⁿÞ is an isomorphism.

Proof. We need the following identity (see [6]) L^m;pðRⁿÞ ¼W^m;pðRⁿÞ; EmAN;

where L^m;pðRⁿÞ is the Lizorkin space defined by

L^m;pðRⁿÞ ¼ fuAS⁰ðRⁿÞ;u¼gmv;vAL^pðRⁿÞg:

In terms of Fourier transform, the unique solution of the equation ðIDÞw¼h;

is given by w¼F¹ðð1þ jxj²Þ¹FðhÞÞ ¼F¹ð^gg₂FðhÞÞ. Hence, hAW^m;pðRⁿÞ ,hAL^m;pðRⁿÞ

, ð^gg_mÞ¹^hhAFðL^pðRⁿÞÞ , ðgg^_mÞ¹ðgg^₂Þ¹ww^AFðL^pðRⁿÞÞ , ðgg^_mþ2Þ¹ww^AFðL^pðRⁿÞÞ ,wAW^mþ2;pðRⁿÞ: 9

Lemma 4.3. Let kAZand mAZbe two integers and p>1 a real. Then, the operator

ID:V_k^mþ2;pðRⁿÞ !V_k^m;pðRⁿÞ is an isomorphism.

Proof. We know that the operatorIDis one to one fromS⁰ðRⁿÞ into S⁰ðRⁿÞ. Hence, for F AV_k^m;pðRⁿÞ, there exists a unique wAS⁰ðRⁿÞ such that

Dwþw¼F in Rⁿ:

Suppose first that kb0. Let us prove by induction on k the following F AV_k^0;pðRⁿÞ ) ðwAV_k^2;pðRⁿÞ and kwk_V^2;p

k ðRⁿÞkkFk_V^0;p

k ðRⁿÞÞ: ð4:4Þ This follows immediately from Lemma 4.2 when k¼0. Suppose that (4.4) holds for 0;. . .;k and suppose that F AV_kþ1^0;pðRⁿÞ. Necessarily wAV_k^2;pðRⁿÞ.

Setting ww~¼hxi^kþ1w, one proves easily that

Dww~þww~¼hxi^kþ1F2‘hxi^kþ1:‘w ðDhxi^kþ1Þw:

The right hand side belongs to L^pðRⁿÞ since wAV_k^2;pðRⁿÞ and jD^ahxi^kjk hxi^kjajfor any multi-indexa. It follows that ww~AV₀^2;pðRⁿÞ, and, consequently, wAV_kþ1^2;pðRⁿÞ since

Ejaja2; jD^awjk X

jnja2;naa

jDⁿhxi^k1D^anwwj~ k X

jnja2;naa

jhxi^k1jnjD^anwwj:~ This completes the proof of (4.4). Similarly, let us prove by induction on kb0 the following

FAV_k^0;pðRⁿÞ ) ðwAV_k^2;pðRⁿÞ and kwk_V^2;p

kðRⁿÞkkfk_V^0;p

k ðRⁿÞÞ: ð4:5Þ This holds clearly for k¼0. Suppose that it holds for 0;. . .;k and let F A V_k1^0;p ðRⁿÞ. Setting FF~¼hxi^k1F AL^pðRⁿÞ, ww~¼ ðIDÞ¹FF~AV₀^2;pðRⁿÞ and h¼whxi^k1ww, one gets after a few calculation~

Dhþh¼ ðDhxi^kþ1Þww~þ‘hxi^kþ1:‘ww:~

The right hand side of the last identity belongs to V_k^0;pðRⁿÞ. From induc- tion hypothesis we deduce that hAV_k^2;pðRⁿÞ,!V_k1^2;p ðRⁿÞ. It follows that wAV_k1^2;p ðRⁿÞ since hxi^kþ1ww~AV_k1^2;p ðRⁿÞ. This completes the proof of (4.5).

At this stage, the assertion of Lemma 4.3 is proved when m¼0 and kAZ. Now, suppose that mb1, and let wAS⁰ðRⁿÞ be the unique solution of Dwþw¼F with F AV_k^m;pðRⁿÞ. Then, for each multi-index a, jajam, D^aw satisfies DðD^awÞ þD^aw¼D^aF AV_k^0;pðRⁿÞ. Hence, D^awAV_k^2;pðRⁿÞ for each a, jajam. We conclude that wAV_k^mþ2;pðRⁿÞ which provides the proof of Lemma 4.3 formb0. The proof forma2 is based on a classical duality argument. It remains to treat the case m¼ 1. Let F AV_k^1;pðRⁿÞ for somekAZ. Then,FAV_k^2;pðRⁿÞ. Hence, there existswAV_k^0;pðRⁿÞsuch that Dwþw¼F (here we used the result of Lemma 4.3 for m¼ 2). Since in addition, q_iFAV_k^2;pðRⁿÞ, i¼1;. . .;n, we deduce that q_iwAV_k^0;pðRⁿÞ for i¼1;. . .;n since DðqiwÞ þqiw¼qiF. Thus, wAV_k^1;pðRⁿÞ. 9

4.2 Tempered solutions of the scalar Oseen equation

Our aim here is to look for solutions of the scalar equation (4.1) which are tempered distributions. This is a first step toward the resolution of the vectorial Oseen’s equations (1.2). In the sequel, for each integer mAZ, we consider the space

W~

W_k^m;pðRⁿÞ ¼ uAW_k^m;pðRⁿÞ; qu

qx₁ AW_k^m2;pðRⁿÞ

;

equipped with the norm

kukWW~_k^m;pðRⁿÞ¼ kuk_W^p m;p

k ðRⁿÞþ qu qx₁

p W_k^m2;pðRⁿÞ

( )1=p

:

When mb0 and kAZ, we set WW~_k^m;p0ðRⁿÞ the dual space of WW~_k^m;pðRⁿÞ.

Clearly, we have the embedding W~

W_k^m;p0ðRⁿÞ,!W_k^m;p0ðRⁿÞ,!WW~_k^m;p0ðRⁿÞ:

In what follows, Q_k denotes the sum H_k⁰ þP_k1, where H_k⁰ is the space of homogeneous polynomials of degree k and depending only on x2;. . .;xn. We denote by Q^þ_l (resp. Q_l) the subspace of all the polynomials pAQ_l satisfy- ing Tp¼0 (resp. Tp¼0). Notice that the mapping pðx1;x2;. . .;xnÞ !

pðx1;x₂;. . .;x_nÞ is one to one from Q^þ_l into Q_l. We have the lemma Lemma 4.4. Let mb0, 1<p<þy and kAZ such that kþn=pB f0;. . .;mg. A function uAWW~_k^m;pðRⁿÞsatisfies Tu¼0 if and only if uAQ^þ_l with l¼ ½kþn=pm 1.

Proof. If u is a tempered distribution such that Tu¼0, then ðjxj²ix₁Þuu^¼0. Since jxj²ix₁ vanishes only at x¼0, we deduce that u is polynomial. If in addition uAWW~_k^m;pðRⁿÞ then it belongs necessarily to Q_l. In the following theorems, we give some isomorphism results concerning the operator T, which in consequence yield existence results for the scalar Oseen equation. In the first theorem, we look for a solution that belongs to W~

W₀^1;pðRⁿÞ. Theorems 4.6, 4.7 and 4.8 concern the regularity of solutions.

Theorem 4.5. Assume that n=p01 and n=p⁰01. Then, the operator T : ~WW₀^1;pðRⁿÞ=P½1n=p!W₀^1;pðRⁿÞ ?P_½1n=p⁰

is an isomorphism.

Theorem4.6. Let mb2 be an integer, and suppose that n=p0f1;. . .;mg, then the operator

T : ~WW₀^m;pðRⁿÞ=Q^þ_½mn=p!W₀^m2;pðRⁿÞ is an isomorphism.

Theorem4.7. Let mb2be an integer, and suppose that n=p⁰0f1;. . .;mg and n=p0f1;2g if m is even and n=p01. Let WW~₀^mþ2;pðRⁿÞ ??P_½mn=p0 be the space of all the functions uAWW~₀^mþ2;pðRⁿÞ satisfying the conditions

EpAP_½m2n=p⁰;hu;pi¼0;

EpAP_½mn=p0; qu qx₁;p

¼0:

Then, the operator

T: ~WW₀^mþ2;pðRⁿÞ ??P_½mn=p⁰!W₀^m;pðRⁿÞ ?P_½mn=p⁰ is an isomorphism.

By duality and transposition, Theorem 4.6 yields

Theorem 4.8. Suppose that mb2, 1< p<þy and n=p⁰0f1;. . .;mg, then the operator

T :W₀^mþ2;pðRⁿÞ !WW~₀^m;pðRⁿÞ ?Q_½mn=p0

is an isomorphism.

The following proposition plays a prominent role in the proof of Theorems 4.5–4.6,

Proposition 4.9. Let 1<p<þy, mb2 such that n=pBf1;. . .;mg, and set l¼minðm1;½mn=pÞ. Then,

qAinfQ_lkuþqkWW~₀^m;pðRⁿÞkjuj_W^m;p

0 ðRⁿÞþ qu

qx1 W₀^m2;pðRⁿÞ

for each uAWW~₀^m;pðRⁿÞ.

Proof of Proposition 4.9

Observe first that the semi-norm ½u_m;p¼ juj_W^m;p

0 ðRⁿÞþ qu

qx1 W₀^m2;pðRⁿÞ

is a norm on WW~₀^m;pðRⁿÞ=Q_l. Indeed, if ½u_m;p¼0, then juj_W^m;p

0 ðRⁿÞ¼0 and u is polynomial. Since u belongs to W₀^m;pðRⁿÞ, its degree is necessarily less or equal to l. Moreover, since qu

qx1

belongs to W₀^m2;pðRⁿÞ, the degree of qu qx1

is less or equal to l2¼minðm3;½m2n=pÞ. Hence, uAQ_l.

Now, let us prove that this semi-norm is equivalent to the norm of W~

W₀^m;pðRⁿÞ=Q_l. We need the following lemma (see [2]).

Lemma 4.10. Let kAZ and sb1 be two integers such that n=pþk0 f1;. . .;sg. Let q¼minðs1;½skn=pÞ. Then, the semi-norm jvj_W^s;p

k ðRⁿÞ

defines on W_k^s;pðRⁿÞ=Pq a norm which is equivalent to the quotient norm.

Suppose first that lb0, then from Lemma 4.10 it follows that EvAW₀^ml;pðRⁿÞ; inf

cARkvþck_W^ml;p

0 ðRⁿÞkjvj_W^ml;p

0 ðRⁿÞ; ð4:6Þ EvAW₀^ml1;pðRⁿÞ; kvk_W^ml1;p

0 ðRⁿÞkjvj_W^ml1;p

0 ðRⁿÞ; ð4:7Þ EvAW_lm^l;pðRⁿÞ; inf

qAPl1

kvqk_W^l;p

lmðRⁿÞkjvj_W^l;p

lmðRⁿÞ; ð4:8Þ Now, let P_k⁰, k being an integer, be the space of all the polynomials of degree less than or equal toland depending only onx2;. . .;xn. Namely, ifkb0, then

P_k⁰ ¼H₀⁰þ þH_k⁰:

If k<0, P_k⁰ ¼ f0g. We shall use the following Lemma.

Lemma 4.11. Let m, l and p be as in Proposition 4.9. Then, inf

qAP_l⁰kuþqk_W^m;p

0 ðRⁿÞkjuj_W^m;p

0 ðRⁿÞþ qu qx₁

W₀^m1;pðRⁿÞ

for each uAW₀^m;pðRⁿÞ.

Proof [of lemma 4.11]. For each kb0, we set

Lk ¼ fm¼ ð0;m₂;. . .;m_nÞ;jmj ¼m₂þ þm_n¼kg:

The proof of Lemma 4.11 rests on an identification between the space H_k, kb0, and R^cardðLkÞ by means of the mapping

qAH_k! ðD^mqÞ_mALkAR^cardðLkÞ: Next, from Lemma 4.10, we can write

inf

qAP_l⁰kuþqk_W^m;p

0 ðRⁿÞ¼ inf

plAH_l⁰ inf

pl1AH_l1⁰

. . . inf

p0AH₀⁰ku ðp_lþpl1þ þp0Þk_W^m;p

0 ðRⁿÞ

k inf

plAH_l⁰

. . . inf

p1AH₁⁰

X

mAL1

kD^muD^mðplþ þp₁Þk_W^m1;p

0 ðRⁿÞþ qu

qx₁

W₀^m1ðRⁿÞ

( )

:

Since the polynomial p₁AH₁⁰ can be identified to the constantsðD^mp₁;jmj ¼1Þ, it follows that

inf

plAH_l⁰

. . . inf

p1AH₁⁰

X

mAL1

kD^muD^mðp_lþ þp₁Þk_W^m1;p

0 ðRⁿÞþ qu

qx1

W₀^m1ðRⁿÞ

( )

k inf

plAH_l⁰

. . . inf

p2AH₂⁰

X

mAL2

kD^muD^mðp_lþ þp2Þk_W^m1;p

0 ðRⁿÞþ qu

qx₁

W₀^m1ðRⁿÞ

( )

:

More generally each polynomial function p_k AH_k⁰ can be identified to the constants ðD^mp_k;jmj ¼kÞ. Then, repeating the argument gives

inf

qAP_l⁰kuþqk_W^m;p

0 ðRⁿÞkjuj_W^m;p

0 ðRⁿÞþ qu qx₁

W₀^m1;pðRⁿÞ

;

which completes the proof of Lemma 4.11.

Now, since the space Q_l can be identified to the productP_l⁰x1P_l2, for each uAWW~₀^m;pðRⁿÞ and for each p¼p₁þp₂, with ðp₁;p₂ÞAP_l⁰x₁P_l2, one can write

kupkWW~₀^m;pðRⁿÞ

¼ ku ðp1þp₂ÞkWW~₀^m;pðRⁿÞ

¼ ku ðp1þp2Þk_W^m;p

0 ðRⁿÞþ qu qx1

qp2

qx1

W₀^m2;pðRⁿÞ

: ð4:9Þ

Next, since the degree of p2 is lower than l1, it is clear that p2 A W₀^m;pðRⁿÞ. Hence, using Lemma 4.11, we have

inf

p1AP_l⁰kðup₂Þ p₁k_W^m;p

0 ðRⁿÞkjuj_W^m;p

0 ðRⁿÞþ qu qx1

qp₂ qx1

_Wm1;p

0 ðRⁿÞ

: ð4:10Þ

Now, combining (4.9), (4.10) and using the fact that the mapping p! qp qx₁ is one to one from x1P_l2 into P_l2, we get

pinfAQ_lkupkWW~₀^m;pðRⁿÞ

¼ inf

ðp1;p2ÞAP_l⁰x1Pl2

ku ðp1þp2ÞkWW~₀^m;pðRⁿÞ

k juj_W^m;p

0 ðRⁿÞþ inf

p2Ax1Pl2

qu qx1

qp2

qx1

W₀^m1;pðRⁿÞ

þ qu qx1

qp2

qx1

W₀^m2;pðRⁿÞ

( )!

k juj_W^m;p

0 ðRⁿÞþ inf

pAPl2

qu qx₁p

W₀^m1;pðRⁿÞ

þ qu qx₁p

W₀^m2;pðRⁿÞ

( )!

:

Observe that qu qx₁p

W₀^m1;pðRⁿÞ

¼ qu qx₁p

W₁^m2;pðRⁿÞ

þ qu qx_{1 W}^m1;p

0 ðRⁿÞ

k qu qx₁p

W₀^m2;pðRⁿÞ

þ qu qx_{1 W}^m1;p

0 ðRⁿÞ

ð4:11Þ

Together withe inequality (see Lemma 4.10)

pAinfP_l2

qu qx₁p

W₀^m2;pðRⁿÞ

k qu qx_{1 W}^m2;p

0 ðRⁿÞ

;

this yields therefore

qinfAQ_lkuþqkWW~₀^m;pðRⁿÞkjuj_W^m;p

0 ðRⁿÞþ qu qx_1W^m2;p

0 ðRⁿÞ

;

completing the proof. 9

Proof of Theorem 4.5

T is clearly continuous from WW~₀^1;pðRⁿÞ=P½1n=p into W₀^1;pðRⁿÞ ? P_½1n=p⁰. It is also injective, thanks to Lemma 4.4. Let us prove that it is onto. Let f AW₀^1;pðRⁿÞ ?P_½1n=p⁰. According to Proposition 4.1 of [2] and since n=p⁰01, there exists a vector function w¼ ðw1;. . .;w_nÞAL^pðRⁿÞⁿ such that divw¼ f. We set

z¼F¹ððjxj²þix₁Þ¹FfÞ ¼Xⁿ

k¼1

F¹ ix_k

jxj²þ2ix₁Fwk

! :

We need the following multiplier theorem due to Lizorkin [18] (see also [11], Lemma 4.2 Ch. VII)

Lemma 4.12. Let j;kAf1;. . .;ng. The operators h!F¹ x_kx_j

jxj²þ2ix₁Fh

!

; h!F¹ x₁

jxj²þ2ix₁Fh

!

;

are continuous from L^pðRⁿÞ into L^pðRⁿÞ, 1< p<þy. Hence, for each jan, we have

qz

qx_j ¼ Xⁿ

k¼1

F¹ x_jx_k

jxj²þ2ix₁Fw_k

!

AL^pðRⁿÞ;