数理解析研究所講究録
第 1690 巻 2010 年 95-104

95

Landau solutions for incompressible Navier-Stokes
equations and applications
Hideyuki Miura
Department of Mathematics
Osaka University
Osaka 560-0043 Japan
Dedicated to Professor Kenji Nishihara on the occasion of his 60th birthday

1

Introduction

This article is based on a joint work with Tai-Peng Tsai (University of British
Columbia). We consider point singularities of very weak solutions of the $3D$ stationary Navier-Stokes equations in a finite region in. Tlie Navier-Stokes equations.
for the velocity
and pressure $p:\zeta$ ]with external force :$\Omega$$u:\Omegaarrow \mathbb{R}^{3}$$\mathbb{R}^{3}$$arrow \mathbb{R}$$f$$\Omegaarrow \mathbb{R}^{3}$are$-\triangle u+(u\cdot\nabla)u+\nabla p=f$

A very weak solution is a vector fiinction
bution sense:$u$$(livu=0,$,in$L_{loc}^{2}(\Omega)$$\int-u\cdot\triangle\varphi+u_{j}u_{i}\partial_{j}\varphi_{i}=\langle f,$ $\varphi\}$and $\int u\cdot\nabla h=0$ for any
and$h\in C_{c}^{\infty}(\Omega)$.

,$(x\in\Omega)$which satisfies (1.1)$\forall\varphi\in C_{c,\sigma}^{\infty}(\Omega)$

Here the force

$f$

(1.1)

.
$i_{I1}$

distri-,is allowed to be a distribution$C_{(.\cdot\sigma)}^{\infty}(\Omega)=\{\varphi\in C_{c,}^{\infty}(\Omega, \mathbb{R}^{3}):div\varphi=0\}$.

‘In this definition the pressure is not needed. Deiiote $B_{R}=\{x\in \mathbb{R}^{3} : |x|<R\}$ andfor $R>0$ .
We are concerned with tlie behavior of very weak solutions which solve (1.1) in the
punctured ballwitli zero force, i.e., $f=0$ . There are a lot of studies on thisproblem [3, 11, 13, 14, 2, 8]. Shapiro [13, 14] proved the removable singularity theorem
under soine assumptions on . He proved that if
for some
and$u(x)=o(|x|^{-1})(xarrow 0)$ , then $(u, p)$ can be defined atso that it is a smooth solutioii$B_{Il}^{c}=\mathbb{R}^{3}\backslash B_{R}$$B_{2}\backslash \{0\}$$u$$\tau\iota\in L^{3+\epsilon}(B_{2})$$0$$\epsilon>0$

96. Choe and Kim [2] obtained similar results by using theof (1.1) inwliole balltheories of the hydrodynamic potentials and homogencous harmonic polynomials.
Kim and Kozono [8] recently proved tltat if $u\in L^{3}(B_{2})$ or $u(x)=o(|x|^{-1})(xarrow 0)$ ,then the saine conclusion holds. As mentioned in [8], their result is optimal in thesense that if their assumption is replaced by$tI_{lC}$$B_{2}$$|u(x)$

I(1.2)$\leq C_{*}|x|^{-1}$

for $0<|x|<2$ , then the siiigularity is not removable in general, due to Landau solutions,which is the family of explicit singular solutions calculated by L. D. Landau [6].
The purpose of this article is to characterize the singularity and to identify theleading order behavior of very weak solutions satisfying the threshold assumption
is sufficiently small. We show that it is given by Landau(1.2) when tlie constantsolutions. In order to state main result, we recall Landau solutions.
Landau obtained his solutions in 1944, see [6, 7]. They can be parametrized bythere exists a unique $(-1)-$in the following way: For each
vectors
which isof (1.1) together with an associated pressure
liomogeneous solutioii
and they solve
are smooth in(-2)-homogeneous, such that
$C_{*}$$b\in \mathbb{R}^{3}$$b\in \mathbb{R}^{3}$$P^{b}$$U^{b}$$U^{b},$ $P^{b}$$\mathbb{R}^{3}\backslash \{0\}$$-\triangle\uparrow\iota+(u\cdot\nabla)c\iota+\nabla p=b\delta$(1.3)$divu=0$ .

,function. Whendenotes the Dirac
in tlie sense of distributions, where$b=(0,0, \beta)$ , they have the following explicit formulas in spherical coordinates$r\cos\theta)$ :siii
with $x=$ $(r\sin\theta\cos\phi,$in$\mathbb{R}^{3}$$\delta$$\delta$$r,$$r\sin\theta$$U=\uparrow\underline{2},$$\phi$$\phi,$$( \frac{A^{2}-1}{(A-\cos\theta)^{2}}-1)e_{r}-\frac{2_{1}\sin\theta}{r(A-c_{\text{ノ}}os\theta)}e_{\theta}$

,

(1.4)

$P= \frac{-.4(A\cos\theta,-1)}{?^{2}(A-\cos\theta)^{2}}$and $e_{\theta}=(-\sin\theta\sin\phi, \sin\theta\cos\phi, \cos\theta)$ . The parameters$A\in(1, \infty]$ are related by the formula

where$\theta,$$e_{r}= \frac{x}{r}$$\beta=16\pi(A+\frac{1}{2}A^{2}\log\frac{A-1}{A+1}+\frac{4A}{3(A^{2}-1)})$$\beta\geq 0$

and.

The formulas for general can be obtained from rotation. One checks directly that). Recently) as $|b|arrow 0$ (or(oraiid
is inonotone in
Sverak [15] $ol)served$ that Laiidau solutions were the only solutions of (1.1) in, without assuming axisyminetry.
which are smooth and (-l)-homogeneous inHence Landau solutioiis can be regarded as the canonical family of the solutions for(1.1). See also [18, 1, 9] for related results.
If $u,$ is a solution of (1.1), we will denote by$b$$\Vert\uparrow\cdot U^{b}\Vert_{L}\infty$$|b|$$\Vert rU^{b}\Vert_{L^{\infty}}arrow 0$$\infty$$\infty$$\mathbb{R}^{3}\backslash \{0\}$$\mathbb{R}^{3}\backslash \{0\}$$p$$T_{ij}(u,p)=p\delta_{ij}+u_{i}u_{j}-\partial_{i}\uparrow\iota_{j}-\partial_{j^{1/}i}$97the momentum flux density tensor in the fluid, which plays an important role toat . Our main result is the following.
determine the equation for$(\tau\iota,p)$$0$

Theorem 1.1 For any $q\in(1,3)$ , there is a small $C_{*}=C_{*}(q)>0$ such that, if,
satisfying (1.2) inis a very weak solution of (1.1) with zero force in$|p(x)|\leq
C|x|^{-2}$
constant,, unique up to athere is a scalar function satisfying$(u,p)$, andwithso thatsatisfies (1.3) in$u$$B_{2}\backslash \{0\}$$B_{2}\backslash \{0\}$$tf_{l_{\text{ノ}}}en$$p$$B_{2}$$b_{i}= \int_{\iota|=1}T_{i.j}(\uparrow\iota,p)7l_{j}(x)$$\Vert u-U^{b}\Vert_{W^{1,q}(B_{1})}+\sup_{x\in B_{1}}|x|^{3’ q-1}|(u-U^{b})(x)|\leq CC_{*}$

where the constant $C$ is independent of

$q$

and

$u$

(1.5)

,

.

.
The exponent can be regarded as the degree of the approximation of by$C_{*}(q)$shrinks
is. But in our theorem,
The closer gets to 3, the less singular$u-U^{b}\inL^{\infty}$. However, it
to zero as $qarrow 3_{-}$ . Ideally, one would like to prove thatseeins quite subtle in view of the following model equation for a scalar function,
$u$

$q$

$U^{b}$$c\iota-U^{b}$$q$$-\triangle v+cv=0$ ,$c=\triangle v/v$.

.
If we choose $v=\log|x|$ , then $c(x)\in L^{3/2}$ and $\lim_{|x|arrow 0}|x|^{2}|c(x)|=0$ , butwhich hasIn equation (3.2) for the difference $w=u-U^{b}$ , there is a term
similar behavior as $cv$ above.
This work is inspired by Korolev-Sverak [9] in which they study the asymptotic as$|x|arrow\infty$ of solutions of (1.1) satisfying (1.2) in. They show that the leading
is sufficiently small. Our theorem canbehavior is also given by Landau solutions if
be considered as a dual version of their result. However, their proof is based on theis a cut-off function supported
unique existence of the difference $\varphi(u-U^{b})$ where
near infinity. If one tries the same approach for our problem, one needs to choose ashrinking to the origin, which produce very
sequencewith the supports ofsingular force terms near the origin. Instead, we prove Lemma 2.3 which defines thenear the origin,equation for $(u, p)$ at the origin. Since the equation for is same as
the -functions at the origin cancel in the equation for the difference. Then applying
the approach of Kim-Kozono [8], we prove the unique existence of the difference infor $1<r<3/2$ , where
for $3/2\leq r<3$ and uniqueness in.
in the norm
is the closure of$v\not\in L^{\infty}$$(w\cdot\nabla)U^{b}$$\mathbb{R}^{3}\backslash B_{1}$$C_{*}$$\varphi$$1-\varphi_{k}$$\varphi_{k}$$u$$U^{b}$$\delta$$W_{0}^{1,r}\cap L_{wk}^{3}(B_{2})$$W_{0}^{1,r}(B_{2})$$W_{0}^{1,r}(B_{2})$2$C_{0}^{\infty}(B_{2})$$W_{0}^{1,r}(B_{2})$Preliminaries

In this section we collect some lemmas for the proof of Theorem 1.1. The first lemmarecalls O’Neil’s inequalities [12], which are H\"older type inequalities in Lorentz spaces.

98

Sce [10, 8] for siinpler proofs in these special cases. Wc denote the Lorentz spaces by$L^{p,\prime}(1<l^{J<}\infty, 1\leq q\leq\infty)$ .
Note $L_{wk}^{3}=L’$} .
$\infty$$n\geq 2$ .
Lemma 2.1
i
Let $1<p_{1},p_{2}<\infty$ with $1/p:=1/p_{1}+1/p_{2}<1$ and let $1\leq?_{1},$
)
have
and
$LctB=B_{2}\subset \mathbb{R}^{\tau\iota},$$)$$f\in L^{p_{1},r_{1}}$$g\in L^{\rho_{2},r_{2}},$$?\iota$

Let $1<r<n$ . For

ii)

. For

$e$

$\Vert fg\Vert_{L(B)}\rho,r\leq C\Vert f\Vert_{L^{p_{11}}(B)}\Vert g\Vert_{L,(B)}\iota^{r_{2}}$$u\prime l\iota ereC=C(p_{1}, r_{1},p_{2}, r_{2})$$r_{2}\leq\infty$

for $r:= \min\{r_{1}, r_{2}\}$ ,.

$f\in M^{\gamma 1,r}(B),$$\iota ve$

have$|If\Vert_{L^{n-\tau}(B)}\Delta L_{r}\leq C\Vert f\Vert_{W^{1,r}(B)}$,where $C=C(n, r)$ .
For our application, we will let $n=3,1<r<3$ , and we have$\Vert fg\Vert_{L(B)}\leq C\Vert f\Vert_{L_{u\dagger k}^{3}}\Vert g\Vert_{L}\mu_{-r},r\leq C_{r}\Vert f\Vert_{L_{wk}^{3}(B)}\Vert g\Vert_{W^{1},(B)}$

(2.1)

.

The next lemma is on interior estimates for Stokes system with no assumption on
the pressure.
Lemma 2.2 Assume $v\in L^{1}$ is a distribution solution of the Stokes system$-Av_{i}+\partial_{i}p=\partial_{j}f_{ij}$anddent of$f\in L^{\mathfrak{l}}$$v$

for some$r\in(1, \infty)$. Then,$divv=0$$v\in W_{loc}^{1,r}$

and $R$ ,$in$$B_{2R}$

and, for some constant$\Vert\nabla v\Vert_{L^{r}(B_{R})}\leq C_{r}\Vert f\Vert_{L^{r}(\cdot)}B_{\vee}R)+C_{r}R^{-4+3/r}\Vert\uparrow)\Vert_{L^{1}(B_{2R})}$$C_{r}$

indepen-.

Tliis lemma is [17], Theorem 2.2. Although the statement in [17] assumes $v\in$, its proof only requires $v\in L^{1}$ . This lemma can be also considered as [?, Lemma
A.2] restricted to time-independent functions.
The following lemma shows the first part of Theorem 1.1, except (1.5). In particsolves (1.3).
ular, it shows that
$W_{loc}^{1,r}$$(c\iota,p)$satisfy $ing$
weak solution of (1.1) rvith
force in(illowed to be large), there is a scalar function,satisfy irig
with$|p(x)|\leq C|.r|^{-2}$ , unique up to a $co$ nstant, such that $(u, p)$ satisfies (1.3) in$S7nooth$.
Moreover,inare$b_{i}= \int_{x|=1’}T_{ij}(c\iota, p)n_{j}(x)$ .
Lemma 2.3 If
(1.2)

$u$

is a

$zer\cdot 0$$ve7^{\cdot}y$$B_{2}\backslash \{0\}$$p$$ir\iota B_{2}\backslash \{0\}(v)itl\iota C_{*}$$B_{2}$$?\iota,$$p$$B_{2}\backslash \{0\}$99. Lemina
in
Proof. For each $R\in(0,1/2],$ is a very weak solution inis smooth and. Tlic usual theory shows that
2.2 shows is a weak solution in$(u,p_{R})$
solves (1.1) in, unique up to a constant, so thattliere is a scalar function, see e.g. [5]. By the scaling argunient in Sverak-Tsai [17] using Lemma 2.2,
we have for $x\in B_{3R}-B_{2ti}$ ,$u$$W_{loc}^{1,2}$$u$$L^{\infty}$$B_{2}-\overline{B}_{R}$$\uparrow($$p_{R}$$B_{2}-\overline{B}_{R}$$| \nabla^{h}u(x)|\leq\frac{C_{h}^{t}C_{*}\prime}{|x|^{k+1}}$

for $k=1,2,$

(2.2)

,

$\ldots$

can bewhere $C_{k}=C_{k}(C_{*})$ are independent of $R\in(0,1/2]$ and its dependence on
is unique. Since
dropped if $C_{*}\in(O, 1)$ . Varying $R,$ $(2.2)$ is valid for, and define $p(x)=$up to a constant, we can fix it by requiring $p_{R}=p_{1/2}$ in$|\nablap(x)|\leq CC_{*}|x|^{-3}$ .
$p_{R}(x)$ for any
with $R=|x|/2$ . By the equation,
Integrating from $|x|=1$ we get $|p(x)|\leq CC_{*}|x|^{-2}$ . In particular$C_{*}$$x\in B_{3/2}\backslash \{0\}$$p_{R}$$B_{2}\backslash \overline{B}_{1’ 2}$$x\in B_{2}\backslash \{0\}$$|T_{ij}(u,p)(x)|\leq CC_{*}|x|^{-2}$for$x\in B_{3’ 2}\backslash \{0\}$

.

(2.3)

Denote $NS(u)=-\triangle u+(?r, \cdot\nabla)u+\nabla p$ . We have $NS(u)_{i}=\partial_{j}T_{ij}(u)$ in the sense,of distributions. Thus, by divergence theorem and $NS(u)=0$ in$B_{2}\backslash \{0\}$(2.4)$b_{i}= \int_{|x|=1}T_{ij}(u,p)n_{j}(x)=\int_{|x|=R}T_{ij},(u,p)n_{j}(x)$for any $R\in(0,2)$ . Let$\langle NS(u)_{i},$$\phi$

be any test function in$C_{c}^{\infty}(B_{1})$. For small$\epsilon>0$,$\phi\}=-/T_{ij}(u)\partial_{j}\phi$$=-.[B_{1} \backslash B_{\epsilon}T_{ij}(\iota x)\partial_{j}\phi-\int_{B_{\epsilon}}T_{ii}(u)\partial_{j}\phi$$= \int_{B}\partial_{j}T_{ij}(u)\phi+.[T_{ij}(u)\phi\uparrow z_{j}-\int_{oB_{1}}T_{ij}(u)\phi n,$ $- \int_{B_{\epsilon}}T_{ij}(u)\partial_{j}\phi$.

In the last line, the first integral is zero since $NS(u)=0$ and the third integral is zero.
since $\phi=0$ . By the pointwise estimate (2.3), the last integral is bounded by
On the other hand, by (2.4),$C\epsilon^{3-2}$$1_{\partial B_{\in}}^{T_{ij}(u)\phi n_{j}}arrow b_{i}\phi(0)$as$\epsilonarrow 0$.

Thus $(u,p)$ solves (1.3) and we have proved the lemma.
It follows from the proof tliat $|b|\leq CC_{*}$, for $C_{*}<1$ . With tliis lemma, we havecompletely proved Theorem 1.1 in the case $q<3/2$ . In the case $3/2\leq q<3$ , itremains to prove 1.5 .
$\square$

$($

$)$

100

Proof of main theorem

3

In tliis section, we present the proof of Theorem 1.1. We first prove that solutions. Wc next apply this result to obtain thc pointwise estimate. For what
belong tofollows, denote(3.1)$u)=u-U$, $U=U^{b}$ .
$I\eta_{!}^{r1,q}$

By Lemma 2.3, there is a function$\tilde{p}$such that$(w,\tilde{p})$satisfies in$-\triangle w+U\cdot\nabla w+\tau v\cdot\nabla(U+w)+\nabla\tilde{p}=0$$|w(x)| \leq\frac{CC}{|x|}*$,,$B_{2}$

that$divw=0$ ,

(3.2)

.

$| \tilde{p}(x)|\leq\frac{c,c}{|x|^{2}}*$

Note that the -functions at the origin cancel.
$\delta$

3.1

regularity$W^{1,q}$In this subsection we will show $w\in W^{1,q}(B_{1})$ . Fix a cut off function. We localize by introducing
and $\varphi=0$ in$B_{9/8}$$\varphi$with $\varphi=1$ in$w$$B_{11/8}^{c}$$v=\varphi w+\zeta$

where is a solution of the problem $div\zeta=-\nabla\varphi\cdot w$ . By Galdi [4, Ch.3] Theorem
3.1, tlrere exists such a satisfying$\zeta$$\zeta$$supp\zeta\subset B_{3\prime 2}\backslash B_{1}$The vector$v$

is supported in,$\overline{B}_{3’ 2}$$\Vert\nabla\zeta\Vert_{L^{100}}\leq C\Vert\nabla\varphi\cdot w\Vert_{L^{100}}\leq CC_{*}$and satisfies $v\in W^{1,r}\cap L_{wk}^{3}$ for $r<3/2$ ,$-\triangle v+U\cdot\nabla u+v\cdot\nabla(U+u)+\nabla\pi=f$,where$\pi=\varphi\tilde{p}$

.

(3.3)

$divv=0$ ,,$f=-2(\nabla\varphi\cdot\nabla)w-(\triangle\varphi)c\{)+(U\cdot\nabla\varphi)w+(\varphi^{2}-\varphi)w\cdot\nabla w+(w\cdot\nabla\varphi)w+\tilde{p}\nabla\varphi$$-\triangle\zeta+(U\cdot\nabla)\zeta+\zeta\cdot\nabla(U+\varphi w+\zeta)+\varphi w\cdot\nabla\zeta$

is supportcd in the annulus$\overline{B}_{3’ 2}\backslash B_{1}$. One verifies directly that, for some

$1\leq’\cdot\leq 100s\iota\iota 1)\Vert f\Vert_{\nu V_{0}^{-1}’(B_{2})}\leq C_{1}C_{*}$

Our proof is based on tlie following lemmas.

.

$C_{1}$

,

(3.4)

101

Lemma 3.1 (Unique existence) For any $3/2\leq r<3_{f}$ for sufficie ntly mall $C_{*}=$$C_{*}(r)>0_{f}$ there is a unique solutionof $(3.3)-(3.4)$ in the set
$s$

$v$

$V=\{v\in W_{0}^{1,\tau}(B_{2}), \Vert v\Vert_{V} :=\Vert v\Vert_{W_{0}^{1}’(B_{2})}\leq C_{2}C_{*}\}$

for some $C_{2}>0$ independent of $r\in[3/2,3)$ .
$\cdot$Lemma 3.2 (Uniqueness) Let 1 $<r<3/2$ .
$(3.3)-(3.4)$ in
and$W_{0}^{1,r}\cap L_{wk}^{3}$

are solutions ofandis sufficiently small, then $v_{1}=v_{2}$ .

If$C_{*}+\Vert v_{1}\Vert_{L_{w\Lambda}^{3}}+\Vert v_{2}\Vert_{L_{wk}^{3}}$$botl\iota v_{1}$$v_{2}$

Assuming the above lemmas, we getregularity as follows. First we have asolution of (3.3) inby Lemma 3.1. On the other hand, both $v=\varphi?v+\zeta$byand are small solutions of (3.3) infor $r=5/4$ , and thus$v\in
W_{0}^{1,q}(B_{2})$$w\inW^{1,q}(B_{1})$Lemma 3.2. Thus.
and$W^{1,q}$$W_{0}^{1,q}(B_{2})$$\tilde{v}$$v=\tilde{v}$$W_{0}^{1}""\cap L_{wk}^{3}(B_{2})$$\tilde{v}$Proof of Lemma 3.1. Consider the following mapping : For each $v\in V$ , let
be the unique solution inof the Stokes system$\Phi$$W_{0}^{1,r}(B_{2})$$\overline{v}=\Phi v$$-\triangle\overline{v}+\nabla\overline{\pi}=f-\nabla\cdot(U\otimes v+v\otimes(U+v))$$div\overline{v}=0$.

By estimates for the Stokes system, see Galdi [4, Ch.4] Theorem 6.1, in particular(6.9), for $1<r<3$ , we have$\Vert\overline{v}\Vert_{\nu V_{0}^{1,r}(B_{2})}\leq C_{r}\Vert f\Vert_{W_{0}^{-1,r}}+C_{r}\Vert\nabla\cdot(U\otimes v+v\otimes(U+v))\Vert_{W_{0}^{-1,r}}$$\leq c_{m_{l}}..C_{1}C_{*}+C_{r}\Vert U\otimes v+u\otimes(U+v)\Vert_{L^{r}}$.

By Lemma 2.1, in particular (2.1), for $1<r<3$ ,$\Vert\overline{v}\Vert_{W_{0}^{1,?}(B_{2})}\leq C_{r}C_{1}C_{*}+C_{7}.\tilde{C}_{r}(\Vert U\Vert_{L_{u\iota k}^{3}}+\Vert v\Vert_{L_{wk}^{3}})\Vert v\Vert_{V}$We now choose $C_{2}=2C_{r}C_{1}$ . Since
if $r\geq 3/2$ , we get
sufficiently small.
We next consider the difference estimate. Let
Then$V\subset L_{wk}^{3}$$v_{1},$.

$\overline{v}=\Phi v\in V$$v_{2}\in V,\overline{v}_{1}=\Phi v_{1}$

, and

if

$C_{*}$

is$\overline{v}_{2}=\Phi v_{2}$.

$\Vert\Phi v_{1}-\Phi v_{2}\Vert_{lV^{1,r}}\leq C(\Vert U\Vert_{I_{u}^{3}}wk\cdot+\Vert v_{1}\Vert_{T_{\lrcorner}^{3}}+\Vert v_{2}\Vert_{L_{wk}^{3}})\Vert v_{1}-v_{2}\Vert_{W^{1,r}}wk$

.

(3.5)

,
Takingsufficiently small for $32\leq\uparrow’<3$ , we getwhich shows that is a contractioii mapping in $V$ and thus has a unique fixed point.
We have proved the unique existence of the solution for $(3.3)-(3.4)$ in V.
$\Vert\Phi v_{1}-\Phi v_{2}\Vert_{V}\leq\frac{1}{2}\Vert v_{1}-v_{2}\Vert_{V}$$C_{*}$$\Phi$$\square$102

Since the constant . for the Stokes estimate can bc taken the samefrom Lemma 2.1
for $\uparrow\in[3/2,3],$
is independent of . However, the constant$rarrow
3_{-}$$rarrow
3_{-}$.
, thus(ii) blows up as
has to shrink to zero as
$Re$

mark.

$C_{7}$$C_{2}$$C_{r}$$r$$C_{*}$

Proof of Lemma 3.2.

By the difference estimate (3.5), we have$\Vert\iota’\iota-v_{2}\Vert_{W^{1,r}}\leq C(\Vert U\Vert_{L_{wk}^{3}}+\Vert v_{1}\Vert_{L_{wk}^{3}}+\Vert v_{2}\Vert_{L_{wk}^{3}})\Vert v_{1}-v_{2}\Vert_{W^{1,r}}$Thus, if

3.2

$C(\Vert U\Vert_{L_{u’ k}^{3}}+\Vert v_{1}\Vert_{L_{wk}^{3}}+\Vert v_{2}\Vert_{L_{wk}^{3}})<1$.

, we conclude $v_{1}=v_{2}$ .

$\square$

Pointwise bound.
In this subsection, we will prove pointwise bound of using, let $R=|x_{0}|/4$ and $E_{k}=B(x_{0}, kR),$ $k=1,2$ .
For any fixed$1/s+1/q^{*}=1$ . We have
Note $q^{*}\in(3, \infty)$ . Let be tlie dual exponent of$w$$\Vert w\Vert_{W^{1,q}}\sim<C_{*}$$x_{0}\in B_{1/2}\backslash \{0\}$$q^{*},$$s$.

$\Vert w\Vert_{L^{1}(E_{2})}\sim<\Vert u)\Vert_{L(b_{2})}q^{*}\Vert 1\Vert_{L^{s}(E_{2})}\leq C_{*}R^{4-3\prime q}$

By the interior estimate Lemma 2.2,$\Vert\nabla_{\mathfrak{l}\ell 1}’\Vert_{L(E_{1})}q^{v}\sim<\Vert f\Vert_{L(E_{2})}q^{*}+R^{-4+3\prime q^{*}}\Vert w\Vert_{L^{1}(F_{\lrcorner 2})}$where $f=U\otimes w+rv\otimes(U+co)$ . Since $|U|+|w|_{\sim}<C_{*}|x|_{\sim}^{-1}<C_{*}R^{-1}$ in$\Vert f\Vert_{L(E_{2})}q^{*}\sim<C_{*}R^{-1}\Vert w\Vert_{L(E_{2})}q^{*}\sim<C_{*}^{2}R^{-1}$We also have$R^{-4+3’ q^{t}}\Vert w\Vert_{\Gamma_{\text{ノ}}^{1}(F_{2})}\sim<R^{-4+3/q^{*}}C_{*}R^{4-3\prime q}=C_{*}R^{-1}$$E_{2}$,.
. Thus$\Vert\nabla w\Vert_{L(E_{1})}q^{*}\sim<C_{*}R^{-1}$

By Gagliardo-Nirenberg incquality in$\Vert w\Vert_{L^{\infty}(E_{1})\sim}<$I$E_{1}$,$w\Vert_{L(E_{1})}^{1-\theta}q^{*}\Vert\nabla w||_{L^{q^{*}}(E_{1})}^{\theta}+R^{-3}||w\Vert_{L^{1}(E_{1})}$,

where $1/\infty=(1-\theta)/q^{*}+\theta(1/q_{*}-1/3)$ and thus $\theta=3/q-1$ . We conclude. Sinceis arbitrary, we have proved the pointwise bound,and completed the proof of Tlieorem 1.1.
$\Vert tl)\Vert_{L^{\infty}(E_{1})}\leq C_{*}R^{-\theta}$$x_{0}$

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