Local well-posedness of perturbed Navier-Stokes system around Landau solutions

1.   Introduction

The initial value problem of the Navier-Stokes system is described as follows

where $u = (u_1,u_2,u_3)$ is the velocity and $p$ is the pressure.

It is well known that Leray [16] proved the global existence of weak solutions for divergence free initial data $u_0\in L^2(\mathbb{R}^3)$ and $f = 0$. The uniqueness and regularity for the solutions still remain open, see $e.g.$ [14] and references therein. For well-posedness results to the Navier-Stokes system, Kato [9] proved the local well-posedness for the general initial data in $L^n(\mathbb{R}^n)$ and the global well-posedness for the small initial data in $L^n(\mathbb{R}^n)$. Giga and Miyakawa [3] and Taylor [22] gave the same result in certain Morrey spaces. In 2001, Koch and Tataru [11] proved the global well-posedness evolving from small initial data in the space $BMO^{-1},$ in which they need $u\in L^2_{\text{loc}}(\mathbb{R}^n \times [0,\infty) )$ in order to make sense of the system. Moreover, self-similar solutions $u\notin L^3(\mathbb{R}^3)$, but belong to $L^2_{\text{uloc}}(\mathbb{R}^3)$, see [2,24,26]. The definition of $L^q_{\text{uloc}}$ will be given in (1.6) later.

For the Navier-Stokes system with $u_0\in L^2_{\text{uloc}}$, there are some results on uniformly locally square integrable solutions. Basson [1] described such solutions. Lemarié-Rieusset [14,15] gave the local existence of weak solution $u$ for when $u_0\in L^2_{\text{uloc}}$. Moreover, global weak solution exists for the decaying initial data $u_0\in E_2 = \left\{f \in L_{\text {uloc }}^{2}:\lim_{|x_{0}| \rightarrow \infty}\|f\|_{L^{2}\left(B\left(x_{0}, 1\right)\right)} = 0 \right\}$. Kikuchi and Seregin's paper [10] extend above results which include forcing terms in the equations. Very recently, Kown and Tsai [12] generalizes the global existence with non-decaying initial data whose local oscillations decay.

For the uniformly local-$L^3$ integrable functions space $L^3_{\text{uloc}}$, Lemarié-Rieusset [14] gave the applications of the space $L^3_{\text{uloc}}$ to the Navier-Stokes system. Hineman and Wang [6] obtained the local well-posedness of Nematic liquid crystal flow for any initial data $(u_0, d_0)$ with small $L^3_{\text{uloc}}$-norm of $(u_0,\nabla d_0).$

The stationary Navier-Stokes system in $\mathbb{R}^3$ has the form

When $f = (b(c)\delta_0,0,0)$ with $b(c) = \frac{8\pi c}{3(c^2-1)}\left(2+6c^2-3c(c^2-1)\ln\left(\frac{c+1}{c-1}\right)\right)$ and $\delta_0$ Dirac measure, the following formulas

with $|x| = \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$ and constant $|c|>1$ are the distributional solutions to system (1.2) in $\mathbb{R}^3$. The explicit stationary solutions (1.3) were given by Landau [13]. These solutions (1.3) are called Landau solutions. Landau solutions are in $L^2_{\text{uloc}}$ space. Tian and Xin [23] proved that all $(-1)-$homogeneous, axisymmetric nonzero solutions of system (1.2) in $C^2(\mathbb{R}^3\backslash \{0\})$ are Landau solutions. $\check {\mathrm S}$ver$\acute {\mathrm a}$k [21] proved that Landau solutions are the only $(-1)-$homogeneous solutions in $C^2(\mathbb{R}^3\backslash \{0\})$. More details can be seen in [13,17,18,19,20,23].

We denote $u(x,t)$ be the solution to the Navier-Stokes system (1.1) with the given external force $f = (b(c)\delta_0,0,0)$ and initial data $u_0 = v_c+w_0$. By a direct calculation, functions $w(x,t) = u(x,t)-v_c(x)$ and $\pi(x) = p(x,t)-p_c(x)$ satisfy the following perturbed Navier-Stokes system

The explicit formula of $\pi$ is as follows

for which detailed calculation can be seen in Appendix.

Karch and Pilarczyk [7] show that perturbed solutions of Landau solutions to the Navier-Stokes system exist globally under $L^2$- perturbations. In 2017, Karch, Pilarczyk and Schonbek [8] generalized the work of [7]. They presented a new method to show the global existence for a large class of solutions including the Landau ones. Based on these results, we are inspired to study local well-posedness of weak solutions to the perturbed Navier-Stokes system (1.4) with initial data $w_0\in L^q_{\text{uloc}}(\mathbb{R}^{3})$ in our work.

First, we give some notations used in this paper. Ball $B(x, r)$ is a ball in $\mathbb{R}^{3}$ centered at $x$ with a radius $r$,

The spaces $L_{\text {uloc}}^{q}$, $1 \leq q \leq \infty,$ and $U^{s, p}\left(t_{0}, t\right)$ for $1 \leq s, p \leq \infty$ and $0 \leq t_{0}<t \leq \infty,$ defined by

and

When $t_{0} = 0,$ we simply use $U_{T}^{s, p} = U^{s, p}(0, T) .$ Note that $U^{\infty, p}\left(t_{0}, t\right) = L^{\infty}\left(t_{0}, t ; L_{\text {uloc }}^{p}\right)$.

Set $L^2$ local energy space

where

The definition of $L^q$ local energy solution, $q\geq 2,$ is as follows

Definition 1.1. ($L^q$ local energy solution) Let $w_0\in L_{\text{uloc}}^q$, $\text{div}w_0 = 0.$ A pair of functions $(w,\pi)$ is a local energy solution to the perturbed Navier-Stokes system (1.4) with initial data $w_0$ in $\mathbb{R}^3 \times (0,T)$ for $0<T<\infty,$ if the functions satisfy the following conditions:

$(1)$ $w\in U^{\infty, q}_T$, $\nabla (|w|^{\frac q2})\in U^{2,2}_T$ and $\pi\in L_{\text{loc}}^{q} ([0,T); L_{\text{loc}}^{\frac{2q}{q+1}}(\mathbb{R}^3))$;

$(2)$ $(w,\pi)$ satisfies the perturbed Navier-Stokes system (1.4) in the sense of distributions;

$(3)$ the function $t \mapsto \int_{R^{3}} w(x, t) \cdot \varphi(x) dx$ is continuous on $[0, T]$ for any compactly supported function $\varphi \in C_c^{\infty}\left(\mathbb{R}^{3}\right)$. Furthermore, for any compact set $K \subset \mathbb{R}^{3}$,

$(4)$ $(w,\pi)$ satisfies the following local energy inequality

for any $t \in(0, T)$ and for all non-negative smooth functions $\xi \in C_c^{\infty}((0,T)\times \mathbb{R}^{3});$

$(5)$ For any $x_{0} \in \mathbb{R}^{3},$ there exists a function $c_{x_{0}}(t) \in L^{q}(0, T)$ such that

where

for $\Gamma(x) = \frac{1}{4\pi |x|}.$

Our main result is as follows

Theorem 1.2. There exist positive universal constants $c_3$, $\varepsilon_{1}$ and $C$ with the following properties,

(i) For every $|c|\geq c_3$, $w_{0} \in L_{\mathit{\text{uloc}}}^{q}$, $q\geq 2$ with div $w_{0} = 0$, if

there exists a $L^q$ local energy solution $(w, \pi)$ on $\mathbb{R}^{3} \times(0, T)$ to the perturbed Navier-Stokes system (1.4) with initial data $w_0$, satisfying

(ii) Furthermore, when $q\geq 3,$ the solution is unique.

Remark 1.1. From (2.38) and (3.54), we could see a more detailed dependence of $c_3$.

Scheme of the proof and organization of the paper. In Section $2$, we give some results which will be used in the proof of Theorem 1.2. In Section $3$, we prove Theorem 1.2 by classical approximation theory. In Appendix, we give the details to derive the integral formula of pressure $\pi$, i.e. $(1.5).$

Let us complete this section by the notations that we shall use in this article.

Notations.

$\bullet$ We denote $\|\cdot\|_{p}$ or $\|\cdot\|_{L^p}$ the norm of the Lebesgue space $L^p(\mathbb{R}^3)$ with $p\in [1,\infty].$

$\bullet$ We denote $\|\cdot\|_{L_t^p(L_x^q)}$ the norm of the Lebesgue space $L_t^p([0,\infty); L_x^q(\mathbb{R}^3))$ with $p, q\in [1,\infty].$

$\bullet$ We use the homogeneous Sobolev space $\dot{H}^{1}\left(\mathbb{R}^{3}\right) = \left\{u \in \mathcal{S}^{\prime}\left(\mathbb{R}^{3}\right):\nabla u \in L^{2}\left(\mathbb{R}^{3}\right)\right\}$.

$\bullet$ $C_0^{\infty}(\mathbb{R}^3)$ denotes the set of smooth and compactly supported functions.

$\bullet$ The $i$th coordinate ($i = 1,2,3$) of a vector $u$ will be denoted by $u_i.$ Set $(\cdot,\cdot)$ be the $L^2(\mathbb{R}^3)$ inner product. We use notation $A\lesssim B$ to denote $A\leq CB$, where $C$ is an absolute constant.

2.   Localized-mollified system

We consider approximate solutions $(w^\epsilon,\pi^\epsilon)$ to the following localized-mollified system in $\mathbb{R}^{3} \times(0, T)$

where $\mathcal{J}_{\epsilon}(v) = v * \eta_{\epsilon}, \epsilon>0,$ the mollifier $\eta_{\epsilon}(x) = \varepsilon^{-3} \eta\left(\frac{x}{\epsilon}\right)$ with positive $\eta\in C_c^{\infty}(B(0,1)),$ $\int \eta dx = 1.$ Localization factor $\Phi_{\epsilon}(x) = \Phi(\epsilon x), \epsilon>0$ with $\Phi\in C_c^{\infty}(\mathbb{R}^3)),$

We will construct approximate solution $w^\epsilon$ directly in $\mathcal{E}_{T}$ since $w_0\in L^2_{\text{uloc}}$ has no decay. First, we give a property of Landau solution $v_c$ which can be obtained by direct calculation.

Lemma 2.1. The explicit formula of $v_c$ is (1.3), we have

Then, we give a fundamental inequality with the singular weight in Sobolev spaces: the so-called Hardy inequality which go back to the pioneering work by G.H. Hardy [4,5].

Lemma 2.2. For any $f$ in $\dot{H}^1({\mathbb{R}^{3}}),$ there holds

By the Duhamel principle, we can write the solution to system (2.1) into the following integral formulation

The following lemma give the construction of mild solution (see Chap. 5 in [25]) to system (2.1) in the space $\mathcal{E}_{T}$.

Lemma 2.3. For each $0<\epsilon<1$, $\|w_0\|_{L^2_{\mathit{\text{uloc}}}}\le B$ and $\|v_c\|_{L^2_{\mathit{\text{uloc}}}}\le2C_0 B$. If $0<T<min(1, C\epsilon^3B^{-2}),$ we can find a unique solution $w = w^\epsilon$ to the integral form of (2.1) such that

satisfying

where $C>0$ and $C_{0}>1$ are absolute constants.

Proof. Set the map

We will do contraction mapping in the local energy space $\mathcal{E}_{T}$ which is defined in (1.7). According to Lemma 2.4 in [12], for any $T>0$, we have

for $f\in L^2_{\text{uloc}}$, and

for $F\in U_T^{2,2}$. Hence, by (2.8)-(2.10) and $T\leq 1,$ we obtain

Note that

and

We have

for some constants $C_0, C_1.$ Hence, for any $w,z\in \mathcal{E}_{T}$, there holds

By Picard contraction theorem and $\|v_c\|_{L^2_{\text{uloc}}}\le2C_0 B$, if $T$ satisfies

there exists a fixed point $w\in \mathcal{E}_{T}$ of $w = \Psi(w)$ satisfying

We will give a uniform bound of $(w^\epsilon,\pi^\epsilon)$ on a uniform time $[0,T]$ in the following lemma

Lemma 2.4. For each $0<\epsilon<1$, let $(w^\epsilon,\pi^\epsilon)$ be the solution to system (2.1) on $\mathbb{R}^{3} \times [0,T_\epsilon].$ If $|c|\geq c_3$ and $w_{0} \in L_{\mathit{\text{uloc}}}^{2}$ with div $w_{0} = 0$, there exists a small constant positive $\varepsilon_1$ independent of $\epsilon$ and $\|w_0\|_{L^2_{\mathit{\text{uloc}}}}$ such that, if $T_{\epsilon} \leq \tilde{T} = \varepsilon_{1}\left(1+\left\|w_{0}\right\|_{L_{\mathit{\text{uloc}}}^{2}}^{4}\right)^{-1}$, then

where the constant $C$ is independent of $\epsilon$ and $T_\epsilon$.

Proof. Note that we can derive an integral formula of pressure $\pi^{\epsilon}$ similar to $\pi$ for which the detailed proof can be seen in Appendix

for $\Gamma(x) = \frac{1}{4\pi |x|}.$ For any fixed point $x_0,$ we define $\widehat{\pi}^{\epsilon}_{x_{0}}(x, t)$ on $B\left(x_{0}, \frac{3}{2}\right) \times[0, T]$ by

Therefore, ${\pi^{\epsilon}}-\widehat{\pi}^{\epsilon}_{x_{0}}$ depends only on $x_0$ and t. Hence, $\nabla{\pi^{\epsilon}} = \nabla \widehat{\pi}^{\epsilon}_{x_{0}}$ on $B\left(x_{0}, \frac{3}{2}\right) \times[0, T]$. Hence, $(w^{\epsilon},\widehat{\pi}^{\epsilon}_{x_{0}})$ is another solution to system (2.1). We will replace $\pi^{\epsilon}$ by $\widehat{\pi}^{\epsilon}_{x_{0}}$ in the following procedure.

Take $\psi(x,s) = \phi^2(x)\theta(s)$ with $\text{supp} \phi(x)\in B(x_0, \frac32),$ $\theta(s)\in C_c^\infty$ on $[0,T_\epsilon)$ and $\theta(s) = 1$ on $[0,t].$ Using $2w^\epsilon\psi$ as a text function in $(2.1)_1$, we have

for any $0<t<T_\epsilon.$ Then we have

By Hölder's inequality and $\left|\nabla \Phi_{\epsilon}\right| \lesssim \epsilon \leq 1$, we obtain

and

By Hölder's inequality, we have

According to (2.20), we have

Moreover, by Calderon-Zygmund theorem, there holds

Since $x \in B\left(x_{0}, \frac{3}{2}\right)$ and $y \in B\left(x_{0}, 2\right)^{c},$ we have

Hence,

where we take $B\left(x_{0}, 2^{k+1}\right) \subset \cup_{j = 1}^{a_{k}} B\left(x_{j}^{k}, 1\right)$ with $a_{k} \leq 2^{3 k}$. Therefore

for $i = 1,2,3.$ By interpolation and Young's inequality, we have

By Calderon-Zygmund theorem, there holds

Similar to (2.29), we have

Combining with (2.25) and (2.30)-(2.33), we obtain

Similar to (2.31), we have

For $J_6 = 2\int_{0}^{t} \int w^{\epsilon} \cdot \nabla w^{\epsilon} \cdot v_{c} \Phi_{\epsilon} \phi^{2} d x d s$, we have

where the first inequality holds because of Hardy inequality and Hölder's inequality.

Therefore, we obtain

where the last inequality holds because of the assumption that

Using the interpolation inequality and Young's inequality,

Combining with (2.37), we have

Hence, there exists a small constant $\varepsilon_{1}>0$ such that, if $w^{\epsilon}$ exists on $\left[0, T_{\epsilon}\right]$ for $T_{\epsilon} \leq \tilde{T} = \varepsilon_{1}\left(1+\left\|w_{0}\right\|_{L_{u l o c}^{2}}^{4}\right)^{-1}$, then we have

Combining with (2.40), we have (2.18).

Then, we can obtain the following lemma easily. We omit the details.

Lemma 2.5. The distribution solutions $\{(w^{\epsilon},{\pi}^{\epsilon})\}_{0<\epsilon <1}$ of (2.1) can be extended to the uniform time interval $[0,\tilde{T}],$ where $\tilde{T}$ is as in Lemma 2.4.

3.   Proof of Theorem 1.2

First, when $q = 2$, we give the following existence result

Proposition 3.1. Let $|c|\geq c_3$ and $w_{0} \in L_{\mathit{\text{uloc}}}^{2}$ with div $w_{0} = 0 .$ If

for some small positive constant $\varepsilon_{1}$ independent of $\epsilon$ and $\|w_0\|_{L^2_{\mathit{\text{uloc}}}}$, there exists a $L^2$ local energy solution $(w, \pi)$ on $\mathbb{R}^{3} \times(0, T)$ to the perturbed Navier-Stokes system (1.4) with initial data $w_0$, satisfying

Proof of Proposition 3.1. Our method is inspired by Theorem 3.2 in [12]. We will prove our result in the following four steps.

Step 1. Construct $\{(w^{\epsilon}, \pi^{\epsilon})\}$ on $[0,T^{\epsilon}].$

Let $(w^{\epsilon},\pi^{\epsilon})$ be the solution to the localized-mollified system (2.1). According to Lemmas 2.3 and 2.4, we construct $w^{\epsilon}\in \mathcal{E}_{T^{\epsilon}}$ on $\mathbb{R}^{3} \times [0,T^{\epsilon}]$, where $T^{\epsilon}\leq T = \varepsilon_{1}\left(1+\left\|w_{0}\right\|_{L_{\text{uloc}}^{2}}^{4}\right)^{-1}$ with constant $\varepsilon_{1}$ independent of $\epsilon$ and $\left\|w_{0}\right\|_{L_{\text{uloc}}^{2}}$. By Lemma 2.5, time interval can be extended to $[0,\tilde{T}].$ We construct pressure $\pi^{\epsilon}$ as follows

for $\Gamma(x) = \frac{1}{4 \pi|x|} .$ It is easy to check ${\pi}^{\epsilon}\in L_{\mathrm{loc}}^{2}\left([0, T) ; L_{\mathrm{loc}}^{\frac{4}{3}}\left(\mathbb{R}^{3}\right)\right)$.

Step 2. Prove that $\|w^{\epsilon}\|_{{\mathcal{E}_{T}}}$ and $\|\pi^{\epsilon}\|_{L^2\left(0,T;L^\frac43_{\text{uloc}}\right)}$ is uniformly bounded.

According to Lemma 2.4, we have

where the constant $C$ is independent of $\epsilon$ and $T.$ We consider $\pi^{\epsilon}$ in $B_{2^n}$ for each $n \in \mathbb{N}$. We rewrite (3.3) as follows

For $\pi_1^{\epsilon}$, we have

For $\pi_2^{\epsilon}$, by Calderon-Zygmund theorem, there holds

For the third term, we have

Using Calderon-Zygmund theorem, we have

On the other hand

For $\pi_4^{\epsilon}$, since $x \in B_{2^n}$ and $y \in B_{2^{n+1}}^{c},$ we have

Similar to (2.29)-(2.30), we obtain

Similar to (2.31), we have

Similar to (3.10), there holds

For the last term $\pi_8^{\epsilon}$, since $x \in B_{2^n}$ and $y \in B_{2^{n+1}}^{c},$ we have

Therefore, we deduce

Combining with above estimates, we conclude

Step 3. Find subsequence of $(w^{\epsilon}, \pi^{\epsilon})$, then show the subsequence converge to $(w,\pi)$. Similar method has been used in [10,12]. For each $n \in \mathbb{N}$, we find a limit solution of $\left(w^{\epsilon}, \pi^{\epsilon}\right)$ up to subsequence on each $[0, T] \times B_{2^{n}}$. First, we construct $w$ on the compact set $[0, T] \times B_{2}$. By uniform bounds on $w^{\epsilon}$ and the compactness argument, we can find sequences $w^{1,k}$ form ${w^{\epsilon}}$ such that

for any $\delta_1<2,$ as $k \rightarrow \infty$. Let $w = w^{1}$ on $[0, T] \times B_{2}$.

Then, we extend $w$ to $[0, T] \times B_{4}$. By the same arguments as above, we can find sequences $w^{2,k}$ form $w^{1,k}$ such that

for any $\delta_2<4,$ as $k \rightarrow \infty$.

Continuing this process, we can construct sequence $w^{n,k}$ and its limit $w$. By diagonal argument, we have

satisfying

for any $\delta_n<2^n,$ as $k \rightarrow \infty$. Furthermore,

Next, we will prove

for each $n \in \mathbb{N}$. According to formula (3.3) of $\pi^{\epsilon}$, we define $\pi^{(k)}$ as follows

And pressure

Hence, for any $m>n$

Set

and

Note that for fixed $R>0,$ we have

By (3.21) and Lebesgue dominated convergence theorem, we have

as $k \rightarrow \infty.$ Similar to estimates in Step 3, we have

and

Combined with (3.28), these four terms become very small for sufficiently large $k.$

Note that

as $k \rightarrow \infty.$

and

Combing with (3.31), we have $p_6,p_7,p_8,p_9\rightarrow 0$ as $k \rightarrow \infty.$

For $p_5$, there holds

Also

Take $m$ large enough, we can make $p_5$ and $p_{10}$ very small in the space $L^{2}\left(0, T ; L^{\frac43}\left(B_{2^n}\right)\right).$ These give the convergence (3.23).

Step 4. Check $(w,\pi)$ is a local energy solution. Proof in this step is very similar to the proof of Theorem 3.2 in [12]. For simplicity, we omit the details.

For $q>2,$ $w_0\in L_{\text{uloc}}^{q}$ implies that $w_0\in L_{\text{uloc}}^2$. By the existence results for $q = 2$ in Proposition 3.1, we have $w^\epsilon\in {\mathcal{E}_{T}}$ with initial data $w_0\in L_{\text{uloc}}^{q}$. Then, we will prove

For simplicity, we only give crucial $a$-$priori$ estimates.

Similar to (2.30), we have

for $i = 1,2,3.$ By interpolation and Young's inequality, similar to (2.31), there holds

By Calderon-Zygmund theorem, there holds

Similar to (2.33), we have

Combining with (2.25) and (3.36)-(3.39), we obtain

Similar to (3.37), we have

For $J_6 = 2\int_{0}^{t} \int w^{\epsilon} \cdot \nabla w^{\epsilon} \cdot v_{c} \Phi_{\epsilon} \phi^{2} d x d s$, we have

where the first inequality holds because of Hardy inequality and Hölder's inequality.

By interpolation inequality and Young's inequality, we have

Therefore, we have

Hence, there exists a small constant $\varepsilon_{1}>0$ such that, if $w^{\epsilon}$ exists on $\left[0, T_{\epsilon}\right]$ for $T_{\epsilon} \leq \tilde{T} = \varepsilon_{1}\left(1+\left\|w_{0}\right\|_{L_{u l o c}^{q}}^{2q}\right)^{-1}$, then we have

Following the procedure in the proof of Lemma 3.1, we have the existence results when $q\geq 2$.

Then, we will prove the uniqueness when $q\geq 3.$ Let $u,$ $v$ be two solutions to the perturbed Navier-Stokes system (1.4) on $\mathbb{R}^{3} \times(0, T)$ with the same initial data $w_0.$ The uniqueness can be proved by the method in the proof of Theorem 4.4 in Tsai [25]. We sketch it here.

From (1.14), using interpolation theory, we have

with $\frac 2p+\frac 3s = \frac 3q.$ Then, there exists $t\in(0,1)$ sufficient small such that

where $C_4$ is given in (3.54). Set $g = v-u$, we have

Using $2g\psi$ with $\psi\in C_c^\infty((0,T)\times \mathbb{R}^3)$ as a test function, multiplying the equation $(3.48)_1$ by $2g\psi$, then integrating it, we have

Crucial part is to estimate

Denote $E(t) = \operatorname{ess} \sup _{s<t}\|g(s)\|_{L^2_{\text{uloc}}}^{2}+\int_{0}^{t}\|\nabla g\|_{L^2_{\text{uloc}}}^{2}d \tau$. Since

we have

By Hölder inequality, Hardy inequality and Lemma 2.1, we have

For term $\int_{0}^{t} \int\pi g\cdot \nabla \psi d x d s$, we use the similar decomposition as (2.20) and obtain

There holds

Combining with (3.47) and

we have $E(t)\leq0$, and finish the proof of Theorem 1.2.

4.   Appendix

Integral formula of the pressure $\pi$. Our goal is to derive the integral formula of the pressure $\pi$, i.e. (1.5). Our method is inspired by [25] and [27]. According to the perturbed Navier-Stokes system (1.4), we have

Fix $x$, take a smooth compact supported function $\xi\in C_c^\infty(B(x,2R))$ such that

Therefore, we have $|\nabla \xi|\leq\frac{C}{R}$ and $|\nabla ^2 \xi|\leq\frac{C}{R^2}$. Since

we obtain

Therefore,

Note that

where $\nabla \cdot u = 0$ and $n_i = \frac{y_i-x_i}{|x-y|}$ denotes the $i$th component of the outer normal vector of Ball $B_{2R}.$ Since $\xi = 0$ on $\partial B_{2R},$ we have

For term $IV,$ we have the following estimate

For term $I,$ integration by parts yields

where

and

for $\xi = 0$ on $\partial B_{2R}.$ The last term $\lim _{\varepsilon \rightarrow 0} \int_{\partial B_{\varepsilon}} \partial_{i}\Gamma(x-y) \left(u_{i} v_{j}\right) n_{j} \xi d s$ can be dealt as follows

By the mean value inequality, we have

Hence, we obtain $J_1 = 0.$ For $J_2$, when $i = j,$

When $i\neq j$, according to the symmetry

Combining with (4.15)-(4.18), we have

Therefore, (4.14) holds. Combining with (4.6)-(4.13), we have

Take $u_iv_j = w_iw_j, v_{ci}w_j, w_iv_{cj}$, separately, we obtain

Setting $R\rightarrow \infty$, combining with (4.5), we obtain (1.5).

Acknowledgments

The author is grateful to Prof. Yanyan Li for bringing to our attention the question studied in this paper and much useful advice. This work is partially supported by the National Natural Science Foundation of China 11771389, 11931010 and 11621101. We sincerely thank the anonymous reviewers for their constructive revision suggestions.