数理解析研究所講究録
第 1875 巻 2014 年 19-28

19Leray’s problem on $D$ -solutions
to the stationary Navier-Stokes equations
past an obstacle
Horst Heck
Department of Engineering and Information Technology
Bern University of Applied Sciences
CH-3400 Burgdorf, Switzerland
horst. heck@bfh.ch
Hyunseok Kim
Department of Mathematics
Sogang University
Seoul, 121-742, Korea
kimh@sogang.ac.kr

Hideo KOZONO
Department of Mathematics
Waseda University
Tokyo 169-8555, Japankozono@waseda.jp

Introduction.
Let be an exterior domain inNavier-Stokes equations in :$\Omega$$\mathbb{R}^{3}$

with smooth boundary$\partial\Omega\in C^{\infty}$. We consider the stationary$\Omega$

(N- )
$S$

$\{\begin{array}{l}-\triangle u+u\cdot\nabla u+\nabla p=f in \Omega,divu=0 in \Omega,u=0 on \partial\Omega,u(x)arrow u^{\infty} as |x|arrow\infty,\end{array}$where $u=u(x)=(u_{1}(x), u_{2}(x), u_{3}(x))$ and $p=p(x)$ denote the unknown velocity vector andthe unknown pressure at $x=(x_{1}, x_{2}, x_{3})\in\Omega$ , while $f=f(x)=(f_{1}(x), f_{2}(x), f_{3}(x))$ is the given
external force, andis the prescribed constant vector in
at infinity. In
the pioneer work of Leray [14], it was shown that for every
and forevery, there exists at least one weak solution of (N- ) withsuch$u^{\infty}=(u_{1}^{\infty}, u_{2}^{\infty}, u_{3}^{\infty})$$\mathbb{R}^{3}$$f\in\dot{H}^{-1,2}(\Omega)\equiv\dot{H}_{0}^{1,2}(\Omega)^{*}$$u^{\infty}\in \mathbb{R}^{3}$

$S$

$u$

$\int_{\Omega}|\nabla u(x)|^{2}dx<\infty$

that$\int_{\Omega}|u(x)-u^{\infty}|^{6}dx<\infty.$

Here and in what follows,denotes the closure ofwith respect to the homogeneous
norm
for $1<q<\infty$ . Leray named such a weak solution $uD$ -solution of (N- ) because
it has a finite Dirichlet integral in . The asymptotic behavior of $D$ -solution at infinity had
been improved by Finn [3], Fujita [4] and Ladyzhenskaya [13] in such a way that$\dot{H}_{0}^{1,q}(\Omega)$$C_{0}^{\infty}(\Omega)$$\Vert\nabla u\Vert_{Lq}$$S$$\Omega$$u$$u(x)arrow u^{\infty}$

uniformly as$|x|arrow\infty,$

20
provided has a compact support in . In his paper [14], Leray proposed the problem whether
every -solution satisfies the energy identity$\Omega$

$f$

$D$

$u$

( $EI$ )$\int_{\Omega}\nabla u\cdot\nabla(u-a)dx+\int_{\Omega}u\cdot\nabla a\cdot(u-a)dx=\langle f,$ $u-a\rangle$

satisfying $|x|\geq R$
for allsuch that $diva=0$ in
for all. The
anddenotes the duality pairing ofwith some large $R>0$ . Here$D$ -solutions. It is still an open questionofproblem
uniquenessisaquestion
importantsecond, then the $D$ -solution ofwhether there exists a small constant such that if$D$(N- ) is unique. This is so-called a uniqueness theorem of -solutions for arbitrary small given
anddata
In this article, we shall give final affirmative answers to these two questions provided. It should be noted that the corresponding results to those in the case $u^{\infty}=0$ are still
open questions. See e.g., Nakatsuka [15]. There is another notion of physically reasonable( $PR$ )solutions introduced by Finn [2], [3]. We call the solution of (N- ) physically reasonable if it
holds$a\in C^{1}(\overline{\Omega})$$\Omega,$$\langle\cdot,$$a|_{\partial\Omega}=0,$$x\in\Omega$$a(x)\equiv u^{\infty}$$\dot{H}^{-1,2}(\Omega)$$\cdot\rangle$$\dot{H}_{0}^{1,2}(\Omega)$$u$$\Vert f\Vert_{\dot{H}^{1,2}}+|u^{\infty}|\leq\delta$$\delta$$S$$f\in\dot{H}^{-1,2}(\Omega)$$u^{\infty}\in \mathbb{R}^{3}.$$u^{\infty}\neq$

$0$

$S$

$u$

( $PR$ )as$u(x)-u^{\infty}=O(|x|^{-\alpha})$for some $\alpha>1/2$ . If
( $WR$ )is a $PR$-solution of (N- ) with
$S$

$u$

$u(x)-u^{\infty}=O(|x|^{-1}(1+s_{x})^{-1})$,$|x|arrow\infty$, then$f\in C_{0}^{\infty}(\Omega)$$s_{x} \equiv|x|-\frac{x\cdot u^{\infty}}{|u^{\infty}|}$

$u$

as

behaves like$|x|arrow\infty,$

which exhibits a parabolic wake region behind the obstacle. It had been shown by Finn [3], every $PR$-solution becomes necessarily a $D$ -solution. The
that in the case whenconverse assertion was treated by Babenko [1] who proved that if $f\equiv 0$ , then every $D$ -solutionof (N- ) satisfies ( $PR$ ) with $\alpha=1$ . As a result, it turns out that every $D$ -solution with
$f\equiv 0$ has a parabolic wake region such as ( $WR$ ). Later on, Galdi [6], [7], [8], [9] and Farwig [5]succeeded to handle more general by introducing anisotropic weight. functions, and obtained
more precise asymptotic behavior of than ( $WR$ ) in the class of $PR$-solutions. Furthermore,in terms of the
Kobayashi-Shibata [11] showed the stability of $PR$-solutions for small andOseen semi-group in If-spaces.
$u$$f\in C_{0}^{\infty}(\Omega)$

$u$

$S$

$f$

$u$

$u^{\infty}$$f$

1

Results.

Before stating our results, let us introduce some notation and then give our definition of Dwith compact-vector functions
is the set of all
solutions of (N- ).
vector-summabelall$1<q<\infty,$$div\varphi=0$
standsfor. For
support in , such thatparingbetweendualitythe. We denote bywith the norm
functions onwith respect to the, where $1/q+1/q’=1.\dot{H}_{0}^{1,q}(\Omega)$ denotes the closure ofand$S$$C^{\infty}$$\varphi=(\varphi_{1}, \varphi_{2}, \varphi_{3})$$C_{0,\sigma}^{\infty}(\Omega)$$L^{q}$$L^{q}(\Omega)$$\Omega$$\Omega$$L^{q}(\Omega)$$(\cdot, \cdot)$$\Vert\cdot\Vert_{L^{q}}$$C_{0}^{\infty}(\Omega)$$L^{q’}(\Omega)$, where$i,j=1,2,3.$

homogeneous norm$\Vert\nabla\varphi\Vert_{L^{q}}$

, and
of
Finally, fordenotes the duality pairing betweenby, we define the space$\dot{H}_{0}^{1,q’}(\Omega)$$\langle f,$$u^{\infty}\in \mathbb{R}^{3}$$A(u^{\infty})\equiv${$\nabla\varphi=(\frac{\partial\varphi_{i}}{\partial x_{j}}),$$\phi\rangle$$a\in C^{1}(\overline{\Omega});diva=0,$$\dot{H}^{-1,q}(\Omega)$$f\in\dot{H}^{-1,q}(\Omega)$is the dual spaceand$\phi\in\dot{H}_{0}^{1,q’}(\Omega)$$A(u^{\infty})$$a|_{\partial\Omega}=0,$$a(x)\equiv u^{\infty}$

for all$x\in \mathbb{R}^{3}$

satisfying $|x|>R$ }.

21with some $R>0.$

Our definition of$D$-solutions to (N- ) reads as follows.
$S$

Definition. Let
andmeasurable function-solution of (N- ) if the following conditions (i), (ii) and (iii) are satisfied.
$f\in\dot{H}^{-1,2}(\Omega)$$D$$u^{\infty}\in \mathbb{R}^{3}.$

$A$

on

$u$

is called a$\Omega$

$S$

(i)
(ii)

$\nabla u\in L^{2}(\Omega)$with $divu=0$ in$u(\cdot)-u^{\infty}\in L^{6}(\Omega)$$\Omega$and $u=0$ on$\partial\Omega$;;(iii) it holds that(E)$(\nabla u, \nabla\varphi)+(u\cdot\nabla u, \varphi)=\langle f,$

for all$\varphi\rangle$$\varphi\in C_{0,\sigma}^{\infty}(\Omega)$.

Remark. For every $D$ -solution of (N- ), there exists a unique scalar function
to an additive constant such that
$u$

(E’)

$S$

$(\nabla u, \nabla\phi)+(u\cdot\nabla u, \phi)+(p, div\phi)=\langle f,$$\phi\rangle$

for all$\phi\in C_{0}^{\infty}(\Omega)$$p\in L_{loc}^{2}(\Omega)$up.

Our first result on the energy identity ( $EI$ ) now reads:
Theorem 1.1 Assume that$u$$f\in\dot{H}^{-1,2}(\Omega)$of (N- ) satisfies(1.1)$u^{\infty}\in \mathbb{R}^{3}$with$(\nabla u, \nabla u)-(\nabla u, \nabla a)+(u\cdot\nabla a, u-a)=\langle f,$ $u-a\rangle$

Moreover, if in addition
(1.2)

where

and

$u^{\infty}\neq 0$. Then every$D$-solution$S$$f\in\dot{H}^{-1,2}(\Omega)\cap L^{q}(\Omega)$

for all$a\in A(u^{\infty})$.

for some $1<q<2$ , then it holds that$\int_{\Omega}|\nabla u|^{2}dx+u^{\infty}\cdot\int_{\partial\Omega}T(u,p)\cdot vdS=\langle f, u-u^{\infty}\rangle,$$T(u,p)\equiv(_{\vec{\partial x_{j}}}^{\partial u}+\#_{x_{i}}^{\partial u}-\delta_{ij}p)_{1\leq i,j\leq 3}$

normal to

denotes the stress tensor and where$\nu$

is the unit outer$\partial\Omega.$

Remarks. (i) Galdi [8] and Farwig [5] showed a similar result to that of Theorem 1.1 under
the assumption that
. On the other hand, for the validity of theenergy identity (1.1), we do not need any condition on except for.
(ii) The corresponding problem for $u^{\infty}=0$ is still open. Indeed, up to the present, theenergy identity (1.1) is shown under the hypothesis that
, wheredenotes the Lorentz space on . For instance, see Kozono-Yamazaki [12].
$f\in\dot{H}^{-1,2}(\Omega)\cap L^{\frac{4}{3}}(\Omega)\cap L^{\frac{3}{2}}(\Omega)$$f\in\dot{H}^{-1,2}(\Omega)$$f$$u\in\dot{H}^{1,2}(\Omega)\cap L^{3,\infty}(\Omega)$$L^{q,r}(\Omega)$$\Omega$

Next, we consider the uniqueness of $D$ -solutions under the smallness assumption on the given
data.

Theorem 1.2 There is a constant(1.3)$\delta_{1}=\delta_{1}(\Omega)>0$

such that if$\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|\leq\delta_{1}|u^{\infty}|^{\frac{1}{2}},$$u^{\infty}\neq 0$

and$f\in\dot{H}^{-1,2}(\Omega)$satisfy22then there exists a uniquesubject to the estimate
(1.4)

$D$

-solution$u$of (N- ). Moreover, such a solution$S$$|u^{\infty}|^{\frac{1}{4}}\Vert u-u^{\infty}\Vert_{L^{4}}+\Vert\nabla u\Vert_{L^{2}}\leq C(\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|)$$u$is necessarily,

where $C=C(\Omega)$ .

Remarks. (i) Galdi [8] showed that if$u^{\infty}\neq 0$

and$f\in L^{\frac{6}{5}}(\Omega)\cap L^{\frac{3}{2}}(\Omega)$satisfy$\Vert f\Vert {}_{L}S+|u^{\infty}|\leq\delta_{1}$, our result covers that ofthen there exists a unique $D$ -solution. Since. Hence,
Galdi [8]. Furthermore, we do not need any redundant assumption such as$D$for-solutions
uniquenessofonquestion
Leray’stoanswer
Theorem 1.2 seems to be a final
small data.
(ii) The case when $u^{\infty}=0$ , such a uniqueness result as in Theorem 1.2 is known in more
restrictive situations. For instance, Nakatsuka [15] treated the case $u^{\infty}=0$ , and proved thatfor every $3<r<\infty$ there is a constant $\delta=\delta(r)>0$ such that if $\{u,p\}$ and $\{v, q\}$ withsatisfy ( ’) and if$L^{\frac{6}{5}}(\Omega)\subset\dot{H}^{-1,2}(\Omega)$$f\in L^{\frac{3}{2}}(\Omega)$$\nabla u,$$\nabla v,p,$$E$$q\in L^{\frac{3}{2},\infty}(\Omega)$$\Vert u\Vert_{L^{3,\infty}}\leq\delta, v\in L^{3}(\Omega)+L^{r}(\Omega)$,then it holds that$\{u,p\}=\{v, q\}.$and some redundant
regularity on another solution . It is still an open question whether any norm of solutions of(N- ) with $u^{\infty}=0$ can be controlled by . For details, we refer to Kim-Kozono [10].

In his result, it is necessary to assume the smallness of one solution

$u$

$u$

$v$

$f$

$S$

2

Oseen equations.

In this section, we investigate the following Oseen equations.

(Os)$\{\begin{array}{l}-\triangle v+u^{\infty}\cdot\nabla v+\nabla\pi=f in \Omega,divv=0 in \Omega,v=0 on \partial\Omega,v(x)arrow 0 as |x|arrow\infty.\end{array}$Let us introduce the two function spaces$\tilde{H}^{1,q}(\Omega)$and$\tilde{H}^{2,q}(\Omega)$

defined by$\tilde{H}^{1,q}(\Omega)\equiv\{v\in L^{\overline{4}\overline{q}}(\Omega);\nablav\in L^{q}(\Omega)\}4_{\underline{A}}, 1<q<4,$$\tilde{H}^{2,q}(\Omega)\equiv\{v\in\tilde{H}^{1,\frac{4}{4}\underline{B}_{\overline{q}}}(\Omega);\nabla^{2}v\in L^{q}(\Omega)\}, 1<q<2.$

Then we have the following results on unique solvability of (Os).
Lemma 2.1 Let$u^{\infty}\neq 0$$\tilde{H}^{1,q_{2}}(\Omega)\cross L_{loc}^{1}(\Omega)$. Assume that $1<q_{1},$ $q_{2}<4$ . The solutionof $(Os)$ is unique.

$\{v, \pi\}\in\tilde{H}^{1,q_{1}}(\Omega)+$23
Lemma 2.2 (i) For

with $\frac{3}{2}<q<4$ , there exists a unique solution $\{v, \pi\}\in$. Moreover, for every $\frac{3}{2}<q<3$ and every $M>0$ there is a constantof$C=C(q, M, \Omega)$ such thatis a solution of $(Os)$ with, thenif
it holds that$f\in\dot{H}^{-1,q}(\Omega)$$(Os)$$\tilde{H}^{1,q}(\Omega)\cross L^{q}(\Omega)$$\{v, \pi\}\in\tilde{H}^{1,q}(\Omega)\cross L^{q}(\Omega)$$|u^{\infty}|\leq M$$k_{1}\Vert v\Vert_{L^{\frac{4q}{4-q}}}+\Vert\nabla v\Vert_{Lq}+\Vert\pi\Vert_{Lq}\leq C\Vert f\Vert_{\dot{H}^{-1,q}},$where(ii) For every $f\in L^{q}(\Omega)$ with $1<q<2$ , there exists a unique solution, where. Moreover, for every $1<q< \frac{3}{2}$ and everyof $(Os)$ with
$M>0$ there is a constant $C=C(q, M, \Omega)$ such that if
is a solution$(Os)$ with
,
then
it
holds
that
of
$k_{1} \equiv\min.\{1, |u^{\infty}|^{\frac{1}{4}}\}.$$\{v, \pi\}\in\tilde{H}^{2,q}(\Omega)\cross$$L^{q*}(\Omega)$$\nabla\pi\in L^{q}(\Omega)$$\frac{1}{q_{*}}=\frac{1}{q}-\frac{1}{3}$$\{v, \pi\}\in\tilde{H}^{1,q}(\Omega)\cross L^{q}(\Omega)$$|u^{\infty}|\leq M$$k_{2}\Vert v\Vert_{L}\mu_{-q}^{2}+k_{1}\Vert\nabla v\Vert_{L^{\frac{4q}{4-q}}}+\Vert\nabla^{2}v\Vert_{Lq}+\Vert\pi\Vert_{Lq*}+\Vert\pi\Vert_{Lq}\leq C\Vert f\Vert_{Lq},$

where

3

$k_{2}=k_{1}^{2} \equiv\min.\{1, |u^{\infty}|^{\frac{1}{2}}\}.$

Proof of Theorems.

The following lemma is based on Lemma 2.2 and plays a key role for the proof of Theorem 1.1.
Lemma 3.1 Let(i) If in addition$u^{\infty}\neq 0$

and$f\in\dot{H}^{-1,2}(\Omega)$. Let be a -solution of (N- ).
, then it holds that
for
$D$

$u$

$f\in\dot{H}^{-1,2}(\Omega)\cap\dot{H}^{-1,q}(\Omega)$$S$$\frac{4}{3}<q<4$,$u-u^{\infty}\in L^{\overline{4}-\overline{q}}(\Omega)4\Delta, u^{\infty}\cdot\nabla u\in\dot{H}^{-1,q}(\Omega)$$\nabla u\in L^{q}(\Omega)$(ii) If in addition,

for some constant$p-p_{\infty}\in L^{q}(\Omega)$$f\in\dot{H}^{-1,2}(\Omega)\cap L^{q}(\Omega)$$p_{\infty}.$

for $1<q<2$ , then it holds that,$u-u^{\infty}\in L^{\overline{2}-\overline{q}}(\Omega)2A, \nabla u\in L^{\frac{4}{4}\underline{B}^{3_{B}}}\overline{q}(\Omega)\cap L^{\overline{3}-\overline{q}}(\Omega)$$p-p_{\infty}\in L^{\frac{3}{3}g_{\overline{q}}}-(\Omega)$

for some constant$\nabla^{2}u, \nabla p, u^{\infty}\cdot\nabla u\in L^{q}(\Omega)$$p_{\infty},$.

By taking $q=2$ in this lemma, we have
Corollary 3.1 Every $D$ -solution$u$of (N- ) with$S$$u^{\infty}\neq 0$

and$f\in\dot{H}^{-1,2}(\Omega)$$u-u^{\infty}\in L^{4}(\Omega) , u^{\infty}\cdot\nabla u\in\dot{H}^{-1,2}(\Omega) , p-p_{\infty}\in L^{2}(\Omega)$

for some constant$p_{\infty}.$

To deal with the nonlinear term, we need

Proposition 3.1 Let(i) Ifwith $divu=0$ in$v,$$w\in\dot{H}_{0}^{1,2}(\Omega)\cap L^{4}(\Omega)$$u\in L^{4}(\Omega)$$\Omega$.

, then it holds that$(u\cdot\nabla v, w)=-(u\cdot\nabla w, v)$(ii) If$u^{\infty}\cdot\nabla v\in\dot{H}^{-1,2}(\Omega)$and$u^{\infty}\cdot\nabla w\in\dot{H}^{-1,2}(\Omega)$.

, then it holds that$\langle u^{\infty}\cdot\nabla v, w\rangle=-\langle u^{\infty}\cdot\nabla w, v\rangle,$$\langle a\cdot\nabla v,$$w\rangle=-\langle a\cdot\nabla w,$$v\rangle$

for all$a\in A(u^{\infty})$.

satisfies

24
3.1

Proof of Theorem 1.1.

By Definition of $D$ -solutions, we have$\langle f, \phi\rangle = (\nabla u, \nabla\phi)+(u\cdot\nabla u, \phi)-(p, div\phi)$(3.1)

for all$= (\nabla u, \nabla\phi)+((u-a)\cdot\nabla u, \phi)+\langle a\cdot\nabla u, \phi\rangle-(p-p_{\infty}, div\phi)$$\phi\in C_{0}^{\infty}(\Omega)$(3.2)

for allSince$C_{0}^{\infty}(\Omega)$

is dense in$\dot{H}_{0}^{1,2}(\Omega)\cap L^{4}(\Omega)$, we have$\langle f, \phi\rangle=(\nabla u, \nabla\phi)+((u-a)\cdot\nabla u, \phi)+\langle a\cdot\nabla u, \phi\rangle-(p-p_{\infty}, div\phi)$. By Corollary 3.1 it holds that $u-a=u-u^{\infty}+u^{\infty}-a\in$. Hence, taking $\phi=u-a$ in (3.2), we have$\phi\in\dot{H}_{0}^{1,2}(\Omega)\cap L^{4}(\Omega)$$\dot{H}_{0}^{1,2}(\Omega)\cap L^{4}(\Omega)$(3.3)$\langle f, u-a\rangle=(\nabla u, \nabla(u-a))+((u-a)\cdot\nabla u, u-a)+\langlea\cdot\nabla u, u-a\rangle.$

Furthermore by Proposition 3.1, it holds that$((u-a)\cdot\nabla u, u-a)+\langle a\cdot\nabla u, u-a\rangle$$= ((u-a)\cdot\nabla(u-a), u-a)+\langle a\cdot\nabla(u-a), u-a\rangle$$+((u-a)\cdot\nabla a, u-a)+\langle a\cdot\nabla a, u-a\rangle$$= (u\cdot\nabla a, u-a)$,from which and (3.3) we obtain$\Vert\nabla u\Vert_{L^{2}}^{2}-(\nabla u, \nabla a)+(u\cdot\nabla a, u-a)=\langle f, u-a\rangle.$

This proves (1.1).
Assume in addition that$f\in\dot{H}^{-1,2}(\Omega)\cap L^{q}(\Omega)$

for some $1<q<2$ . By Lemma 3.1 (ii), we

have$-\Delta u+u\cdot\nabla u+\nabla p=f$a.e. in$\Omega.$

Note that$a-u^{\infty}\in C_{0,\sigma}^{\infty}(\mathbb{R}^{3})$,$a-u^{\infty}=0$on$\partial\Omega.$

By integration by parts, we have$(f, a-u^{\infty}) = (-\triangle u+u\cdot\nabla u+\nabla p, a-u^{\infty})$$= (-div(T(u,p), a-u^{\infty})+(u\cdot\nabla u, a-u^{\infty})$(3.4)$= ( \nabla u, \nabla a)+u^{\infty}\cdot\int_{\partial\Omega}T(u,p)\cdot\nu dS-(u\cdot\nabla a, u)$.

Addition of (3.4) and (1.1) yields that
(3.5)$\Vert\nabla u\Vert_{L^{2}}^{2}+u^{\infty}\cdot\int_{\partial\Omega}T(u,p)\cdot\nu dS-(u\cdot\nabla a, a)=\langle f, u-u^{\infty}\rangle.$

Since $supp\nabla a$ is compact, we see easily$(u\cdot\nabla a, a)=0,$

from which and (3.5) we obtain the desired identity (1.2). This proves Theorem 1.1.

25
3.2

Proof of Theorem 1.2.

Step 1. We first show that there are constants(3.6)$\delta_{*}=\delta_{*}(\Omega)$

and $C_{*}(\Omega)>0$ such that if$\Vert f||_{H^{-1,2}}+|u^{\infty}|\leq\delta_{*}|u^{\infty}|^{\frac{1}{2}},$then every $D$ -solution

(3.7)

$u$

of (N- ) satisfies$S$$|u^{\infty}|^{\frac{1}{4}}\Vert u-a\Vert_{L^{4}}+\Vert\nabla u\Vert_{L^{2}}\leqC_{*}(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|)$for some $a\in A(u^{\infty})$ . Indeed, taking $0<R_{0}<R_{1}<\infty$ and $a\in A(u^{\infty})$ in such a way that$\Omega^{c}=\mathbb{R}^{3}\backslash \Omega\subset B_{R_{0}}(0) , supp\nabla a\subset\{R_{0}<|x|<R_{1}\}.$

we have(3.8)$\Vert a\Vert_{L\infty}+\Vert\nabla a\Vert_{L^{1}\cap L\infty}\leq C|u^{\infty}|$with $C=C(\Omega)$ .

By (1.1), we see that$\Vert\nabla u\Vert_{L^{2}}^{2}=\langle f, u-a\rangle+(\nabla u, \nabla a)+(u\cdot\nabla a, u-a)$,from which and (3.8) with the aid of the Young inequality it follows that$\Vert\nabla u\Vert_{L^{2}}^{2}\leq(\frac{1}{2}+C|u^{\infty}|)\Vert\nabla u\Vert_{L^{2}}^{2}+C\Vert f\Vert_{H}^{2_{-1,2}}+C(|u^{\infty}|^{2}+|u^{\infty}|^{4})$Hence, under the assumption
(3.9)

.

$|u^{\infty}| \leq\delta_{*}^{(1)}\equiv\min.\{1, \frac{1}{4C}\},$

we have$\frac{1}{4}\Vert\nabla u\Vert_{L^{2}}^{2} \leq C\Vert f\Vert_{H^{-1,2}}^{2}+C(|u^{\infty}|^{2}+|u^{\infty}|^{4})$$\leq C(\Vert f\Vert_{H^{-1,2}}^{2}+|u^{\infty}|^{2})$which yields that
(3.10)$\Vert\nabla u\Vert_{L^{2}}\leq C(\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|)$Next, we show the bound of(3.11),$\Vert u-a\Vert_{L^{4}}$.

. Define $v=u-a$ , and we have by (3.8) and (3.9) that$v\in\dot{H}_{0}^{1,2}(\Omega) , \Vert\nabla v\Vert_{L^{2}}\leq C(\Vert f\Vert_{\dot{H}}-1,2+|u^{\infty}|)$,and that$\{\begin{array}{l}-\triangle v+u^{\infty}\cdot\nabla v+\nabla\pi=f-Q(v) in \Omega,divv=0 in \Omega,v=0 on \partial\Omega,v(x)arrow 0 as |x|arrow\infty,\end{array}$where$Q(v)\equiv v\cdot\nabla v+(a-u^{\infty})\cdot\nabla v+v\cdot\nabla a-\triangle a+a\cdot\nabla a.$

26
By (3.8) and (3.11), it holds that$\Vert v\cdot\nabla v\Vert_{L3}4\leq\Vert v\Vert_{L^{4}}\Vert\nabla v\Vert_{L^{2}}\leq C(\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|)\Vert v\Vert_{L^{4}}$$\Vert Q(v)-v\cdot\nabla v\Vert_{\dot{H}^{-1,2}}$$=\Vert(a-u^{\infty})\cdot\nabla v+v\cdot\nabla a-\triangle a+a\cdot\nabla a\Vert_{H^{-1,2}}$$\leq C(\Vert\nabla v\Vert_{L^{2}}+|u^{\infty}|)$$\leq C(\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|)$.

Hence, it follows from Lemma 2.1 and Lemma 2.2 with $q=2$ in (i) and with

$q= \frac{4}{3}$in (ii) that$\Vert v\Vert_{L^{4}} \leq C(\frac{1}{k_{1}}\Vert f-Q(v)-v\cdot\nabla v\Vert_{\dot{H}^{-1,2}}+\frac{1}{k_{2}}\Vert v\cdot\nabla v\Vert_{L3}4)$(3.12)$\leq C(\frac{1}{k_{1}}(\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|)+\frac{1}{k_{2}}(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|)\Vert v\Vert_{L^{4}})$.

Hence, under the assumption(3.13)

we have(3.14)$\frac{1}{k_{2}}(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|)\leq\delta_{*}\equiv\min.\{\delta_{*}^{(1)}, \frac{1}{2C}\},$.

$\Vert u-a\Vert_{L^{4}}=\Vert v\Vert_{L^{4}}\leq\frac{C}{k_{1}}(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|)$Since the assumption (3.13) necessarily implies the assumption (3.9), we see by (3.10) and (3.14)
that if$\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|\leq\delta_{*}|u^{\infty}|^{\frac{1}{2}},$then it holds that,$|u^{\infty}|^{\frac{1}{4}}\Vert u-a\Vert_{L^{4}}+\Vert\nabla u\Vert_{L^{4}}\leq(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|)$which implies (3.7)
andbe two $D$-solutions of (N- ). Define
Step 2. We next show uniqueness. Let$v_{1}=u_{1}-a$ and $v_{2}=u_{2}-a$ with $a\in A(u^{\infty})$ as in Stepl. Then $v\equiv v_{1}-v_{2}=u_{1}-u_{2}$ fulfills$S$$u_{2}$$u_{1}$$\{\begin{array}{l}-\Delta v+u^{\infty}\cdot\nabla v+\nabla\pi=-v_{1}\cdot\nabla v-v\cdot\nabla u_{2} in \Omega,divv=0 in \Omega,v=0 on \partial\Omega,v(x)arrow 0 as |x|arrow\infty,\end{array}$Hence it follows from Lemmata 2.1 and 2.1 with
$f=-v_{1}\cdot\nabla v=div(v_{1}\otimes v)$$f=-v\cdot\nabla u_{2}$for$q= \frac{4}{3}$for $q=2$ in (i),in (ii)

27
that$\Vert v\Vert_{L^{4}} \leq C(\frac{1}{k_{1}}\Vert div(v_{1}\otimes v)\Vert_{H^{-1,2}}+\frac{1}{k_{2}}\Vert v\cdot\nabla u_{2}\Vert_{L3}4)$$\leq C(\frac{1}{k_{1}}\Vert v_{1}\otimes v\Vert_{L^{2}}+\frac{1}{k_{2}}\Vert v\Vert_{L^{4}}\Vert\nabla u_{2}\Vert_{L^{2}})$(3.15)$\leq C(\frac{1}{k_{1}}\Vert v_{1}\Vert_{L^{4}}+\frac{1}{k_{2}}\Vert\nabla u_{2}\Vert_{L^{2}})\Vert v\Vert_{L^{4}}.$

By Stepl, under the assumption$\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|\leq\delta_{*}|u^{\infty}|^{\frac{1}{2}},$

we have,$\Vert v_{1}\Vert_{L^{4}}\leq\frac{C}{k_{1}}(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|) , \Vert\nabla u_{2}\Vert_{L^{2}}\leq C(\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|)$from which and (3.15) with

(3.16)
Now, define$k_{1}^{2}=k_{2}$it follows that$\Vert v\Vert_{L^{4}}\leq\frac{C}{k_{2}}(\Vert f\Vert_{\dot{H}^{-1,2}}+|u^{\infty}|)\Vert v\Vert_{L^{4}}.$$\delta_{1}=\delta_{1}(\Omega)$so that$\delta_{1}\equiv\min.\{\delta_{*}, \frac{1}{2C}\}.$

Then under the assumption$\Vert f\Vert_{H^{-1,2}}+|u^{\infty}|\leq\delta_{1}|u^{\infty}|^{\frac{1}{2}},$

it follows from (3.16) with the aid of the relation$k_{2}= \min.\{1, |u^{\infty}|^{\frac{1}{2}}\}$

that$\Vert v\Vert_{L^{4}}\leq 0,$

which yields the desired uniqueness result. This completes the proof of Theorem 1.2.

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