Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations

Abstract

It was proved by Karch and Pilarczyk that Landau solutions are asymptotically stable under any $L^2$-perturbation. In our earlier work with L. Li, we have classified all $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any $L^2$-perturbation.

Introduction

Consider the incompressible stationary Navier-Stokes Equations in ${\mathbb{R}}^3$,

$$\{\begin{array}{cc}\hfill & -\mathrm{\Delta }u+(u\cdot \mathrm{\nabla })u+\mathrm{\nabla }p=0,\hfill \\ \hfill & \text{div}u=0.\hfill \end{array}$$

These equations are invariant under the scaling $u(x)\to \lambda u(\lambda x)$ and $p(x)\to {\lambda }^{2}p(\lambda x)$, $\lambda >0$ and it is natural to study solutions which are invariant under this scaling. These solutions are referred to as $(-1)$-homogeneous solutions (although p is $(-2)$-homogeneous).

Let $x=(x_1,x_2,x_3)$ be Euclidean coordinates and $e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$ be the corresponding unit normal vectors. Denote $x^{\prime }=(x_1,x_2)$. Let $(r,\theta ,\varphi )$ be the spherical coordinates, where r is the radial distance from the origin, θ is the angle between the radius vector and the positive $x_3$-axis, and ϕ is the meridian angle about the $x_3$-axis. A vector field u can be written as

$$u=u_r{e}_r+u_{\theta }{e}_{\theta }+u_{\varphi }{e}_{\varphi },$$

where

$$e_r=\left(\begin{array}{c}\hfill \sin \theta \cos \varphi \hfill \\ \hfill \sin \theta \sin \varphi \hfill \\ \hfill \cos \theta \hfill \end{array}\right),e_{\theta }=\left(\begin{array}{c}\hfill \cos \theta \cos \varphi \hfill \\ \hfill \cos \theta \sin \varphi \hfill \\ \hfill -\sin \theta \hfill \end{array}\right),e_{\varphi }=\left(\begin{array}{c}\hfill -\sin \varphi \hfill \\ \hfill \cos \varphi \hfill \\ \hfill 0\hfill \end{array}\right).$$

A vector field u is called axisymmetric if $u_r$, $u_{\theta }$ and $u_{\varphi }$ are independent of ϕ, and is called no-swirl if $u_{\varphi }=0$.

In 1944, L.D. Landau [7] discovered a 3-parameter family of explicit $(-1)$-homogeneous solutions of the stationary NSE in $C^{\infty }({\mathbb{R}}^3\setminus \{0\})$. These solutions, now called Landau solutions, are axisymmetric with no-swirl and have exactly one singularity at the origin. Tian and Xin proved in [23] that all $(-1)$-homogeneous, axisymmetric nonzero solutions of (1) in $C^{\infty }({\mathbb{R}}^3\setminus \{0\})$ are Landau solutions. Šverák proved in [21] that all (-1)-homogeneous nonzero solutions of (1) in $C^{\infty }({\mathbb{R}}^3\setminus \{0\})$ are Landau solutions. There have also been works on $(-1)$-homogeneous solutions of (1), see [2], [14], [15], [16], [17], [19], [20], [24], [25]. In [10], [11], [12], the $(-1)$-homogeneous axisymmetric solutions of (1) in $C^{\infty }({\mathbb{R}}^3\setminus \{(x_1,x_2)=0\})$ with a possible singular ray $\{(x_1,x_2)=0\}$ was studied, where such solutions with no-swirl were classified in [10] and [11], and existence of such solutions with nonzero swirl was proved in [10] and [12].

There has been much work in literature on the existence of weak solutions and $L^2$-decay of weak solutions of the evolutionary Navier-Stokes equations, see e.g. [1], [3], [6], [8], [9], [13], [18], [22] and the references therein. Such $L^2$-decay of weak solutions can be viewed as the asymptotically stability of the zero stationary solution of (1). The asymptotic stability problem has been studied for other nonzero stationary solutions of (1) with some possible singularities in ${\mathbb{R}}^3$. Karch and Pilarczyk proved in [4] that small Landau solutions are asymptotically stable under $L^2$-perturbations. The $L^2$ asymptotic stability of other solutions with singularities is also studied in [5]. With special $(-1)$-homogeneous solutions which are different from Landau solutions obtained in [10], [11], [12], it is worth to explore the asymptotic stability or instability of these solutions. In this paper, we start this study for a family of solutions which are the simplest and least singular solutions among the solutions found in [10], [11], [12].

Denote $U=u\cdot r\sin \theta$ and $y=\cos \theta$. By the divergence free property of u we have $u_r=\frac{1}{r}{U}_{\theta }^{\prime }$. For (-1)-homogeneous axisymmetric no-swirl solutions, (1) can be reduced to

$$(1-y^2)U_{\theta }^{\prime }+2y{U}_{\theta }+\frac{1}{2}{U}_{\theta }^2=c_1(1-y)+c_2(1+y)+c_3(1-y^2).$$

For $c_1\ge -1$ and $c_2\ge -1$, let

$${\overline{c}}_3(c_1,c_2):=-\frac{1}{2}(\sqrt{1+c_1}+\sqrt{1+c_2})(\sqrt{1+c_1}+\sqrt{1+c_2}+2),$$

where $c_1,c_2,c_3$ are real numbers. Denote $c=(c_1,c_2,c_3)$ and

$$J:=\{c\in {\mathbb{R}}^3|c_1\ge -1,c_2\ge -1,c_3\ge {\overline{c}}_3(c_1,c_2)\}.$$

In [11], it was proved that there exist ${\gamma }^{-},{\gamma }^{+}\in C^0(J,\mathbb{R})$, satisfying ${\gamma }^{-}(c)<{\gamma }^{+}(c)$ if $c_3>{\overline{c}}_3(c_1,c_2)$, and ${\gamma }^{-}(c)={\gamma }^{+}(c)$ if $c_3={\overline{c}}_3(c_1,c_2)$, such that equation (2) has a unique solution $U_{\theta }^{c,\gamma }$ in $C^{\infty }(-1,1)\cap C^0[-1,1]$ satisfying $U_{\theta }^{c,\gamma }(0)=\gamma$ for every c in J and ${\gamma }^{-}(c)\le \gamma \le {\gamma }^{+}(c)$. In particular, ${\gamma }^{+}(0)>0$ and ${\gamma }^{-}(0)<0$. Moreover, let

$$u^{c,\gamma }\equiv u_r^{c,\gamma }{e}_r+u_{\theta }^{c,\gamma }{e}_{\theta }={(U_{\theta }^{c,\gamma })}^{\prime }{e}_r+\frac{U_{\theta }^{c,\gamma }}{\sin \theta }{e}_{\theta },p^{c,\gamma }=\frac{1}{r^2}(u_r^{c,\gamma }-\frac{1}{2}{(u_{\theta }^{c,\gamma })}^2)=\frac{1}{r^2}({(U_{\theta }^{c,\gamma })}^{\prime }-\frac{{(U_{\theta }^{c,\gamma })}^2}{2{\sin }^2\theta }).$$

$\{(u^{c,\gamma },p^{c,\gamma })|c\in J,{\gamma }^{-}(c)\le \gamma \le {\gamma }^{+}(c)\}$ are all $(-1)$-homogeneous axisymmetric no-swirl solutions of (1) in $C^{\infty }({\mathbb{R}}^3\setminus \{(x_1,x_2)=0\})$. It was also obtained in [11] that

$$\begin{gathered} U_{\theta }^{c,\gamma }(-1)=\{\begin{array}{cc}2+2\sqrt{1+c_1},\hfill & \text{when }\gamma ={\gamma }^{+}(c),\hfill \\ 2-2\sqrt{1+c_1},\hfill & \text{otherwise},\hfill \end{array} \\ {U}_{\theta }^{c,\gamma }(1)=\{\begin{array}{cc}-2-2\sqrt{1+c_2},\hfill & \text{when }\gamma ={\gamma }^{-}(c),\hfill \\ -2+2\sqrt{1+c_2},\hfill & \text{otherwise}.\hfill \end{array} \end{gathered}$$

As mentioned earlier, we would like to study the asymptotic stability or instability of the $(-1)$-homogeneous axisymmetric stationary solutions found in [10], [11], [12]. Different from Landau solutions, these solutions are singular at the north pole N and/or south pole S. These solutions u satisfy either $0<\underset{|x|=1,x^{\prime }\to 0}{\lim \sup }|x||x^{\prime }||\mathrm{\nabla }u(x)|<\infty$ or $\underset{|x|=1,x^{\prime }\to 0}{\lim \sup }|x^{\prime }{|}^2|\mathrm{\nabla }u(x)|>0$, while Landau solutions satisfy $\underset{|x|=1}{\sup }|x{|}^2|\mathrm{\nabla }u|<\infty$. In this paper, we study the stability of $(-1)$-homogeneous axisymmetric no-swirl solutions satisfying $0<\underset{x^{\prime }\to 0}{\lim \sup }|x||x^{\prime }||\mathrm{\nabla }u(x)|<\infty$. These solutions are the family $\{(u^{c,\gamma },p^{c,\gamma })|(c,\gamma )\in M\}$, where

$$M:=\{(c,\gamma )|c_1=c_2=0,c_3>-4,{\gamma }^{-}(c)<\gamma <{\gamma }^{+}(c)\}.$$

For any $(c,\gamma )\in M$, $U_{\theta }^{c,\gamma }$ satisfies

$$\{\begin{array}{cc}\hfill & (1-y^2){(U_{\theta }^{c,\gamma })}^{\prime }+2y{U}_{\theta }^{c,\gamma }+\frac{1}{2}{(U_{\theta }^{c,\gamma })}^2=c_3(1-y^2),-1<y<1,\hfill \\ \hfill & U_{\theta }(0)=\gamma .\hfill \end{array}$$

Proposition 1.1

Let $(c,\gamma )\in M$, then $(u^{c,\gamma }(x),p^{c,\gamma }(x))$ satisfies

$$\{\begin{array}{cc}\hfill & -\mathit{\Delta }{u}^{c,\gamma }+u^{c,\gamma }\cdot \mathrm{\nabla }{u}^{c,\gamma }+\mathrm{\nabla }{p}^{c,\gamma }=(4\pi c_3\ln |x_3|{\partial }_{x_3}{\delta }_{(0,0,x_3)}-b^{c,\gamma }{\delta }_0)e_3,x\in {\mathbb{R}}^3,\hfill \\ \hfill & \mathit{\text{div}}{u}^{c,\gamma }=0,x\in {\mathbb{R}}^3,\hfill \end{array}$$

where

$$b^{c,\gamma }=\underset{-1}{\overset{1}{\int }}(y|U_{\theta }^{\prime }{|}^2-\frac{2-y^2}{1-y^2}{U}_{\theta }-\frac{y}{1-y^2}{U}_{\theta }^2)dy.$$

Equations (6) and (7) are understood in the following distribution sense: for any $\phi \in C_c^{\infty }({\mathbb{R}}^3)$, $j=1,2,3$,

$$\underset{{\mathbb{R}}^3}{\int }(\mathrm{\nabla }{u}_j\mathrm{\nabla }\phi -u_i{u}_j{\partial }_{x_i}\phi -p{\partial }_{x_j}\phi )=[4\pi c_3\underset{-\infty }{\overset{\infty }{\int }}\ln |x_3|{\partial }_{x_3}\phi (0,0,x_3)d{x}_3-b^{c,\gamma }\phi (0)]{\delta }_{j3}{e}_3,$$

and

$$\underset{{\mathbb{R}}^3}{\int }{u}^{c,\gamma }\cdot \mathrm{\nabla }\phi =0.$$

We now study the stability of the family of solutions $\{u^{c,\gamma }|(c,\gamma )\in M\}$. Let ${\dot{H}}^1({\mathbb{R}}^3)$ denote the closure of $C_c^{\infty }({\mathbb{R}}^3,{\mathbb{R}}^3)$ under the norm ${\Vert \mathrm{\nabla }u\Vert }_{L^2({\mathbb{R}}^3)}$, and for $1\le p<\infty$,

$$L_{\sigma }^p({\mathbb{R}}^3)=\{u\in L^p({\mathbb{R}}^3)|\text{div}u=0\},{\dot{H}}_{\sigma }^1({\mathbb{R}}^3)=\{u\in {\dot{H}}^1({\mathbb{R}}^3)|\text{div}u=0\},$$

and

$${\Vert u\Vert }_{L_{\sigma }^p({\mathbb{R}}^3)}:={\Vert u\Vert }_{L^p({\mathbb{R}}^3)},{\Vert u\Vert }_{{\dot{H}}_{\sigma }^1({\mathbb{R}}^3)}={\Vert \mathrm{\nabla }u\Vert }_{L^2({\mathbb{R}}^3)}.$$

For a given solution $(u^{c,\gamma },p^{c,\gamma })$ of (1), let $u=u(x,t)$ denote a solution of

$$\{\begin{array}{cc}\hfill & u_t-\mathrm{\Delta }u+(u\cdot \mathrm{\nabla })u+\mathrm{\nabla }p\hfill \\ \hfill & =(4\pi c_3\ln |x_3|{\partial }_{x_3}{\delta }_{(0,0,x_3)}-b^{c,\gamma }{\delta }_0)e_3,(x,t)\in {\mathbb{R}}^3\times (0,\infty ),\hfill \\ \hfill & \text{div}u=0,(x,t)\in {\mathbb{R}}^3\times (0,\infty ),\hfill \\ \hfill & u(x,0)=u^{c,\gamma }+w_0,\hfill \end{array}$$

where $w_0\in L_{\sigma }^2({\mathbb{R}}^3)$ and $b^{c,\gamma }$ is given by (7). Then $w(x,t)=u(x,t)-u^{c,\gamma }$ and $\pi (x)=p(x)-p^{c,\gamma }(x)$ satisfy the initial value problem

$$\{\begin{array}{cc}\hfill & w_t-\mathrm{\Delta }w+(w\cdot \mathrm{\nabla })w+(w\cdot \mathrm{\nabla })u^{c,\gamma }+(u^{c,\gamma }\cdot \mathrm{\nabla })w+\mathrm{\nabla }\pi =0,(x,t)\in {\mathbb{R}}^3\times (0,\infty ),\hfill \\ \hfill & \text{div}w=0,(x,t)\in {\mathbb{R}}^3\times (0,\infty ),\hfill \\ \hfill & w(x,0)=w_0(x).\hfill \end{array}$$

We study the existence and asymptotic behavior of global-in-time weak solutions of (11). Let the energy space

$$X:=L^{\infty }([0,\infty ),L_{\sigma }^2({\mathbb{R}}^3))\cap L^2([0,\infty ),{\dot{H}}_{\sigma }^1),$$

and for w in X

$${\Vert w\Vert }_X:={\Vert w\Vert }_{L^{\infty }([0,\infty ),L_{\sigma }^2({\mathbb{R}}^3))}+{\Vert w\Vert }_{L^2([0,\infty ),{\dot{H}}_{\sigma }^1)}.$$

Let $(\cdot ,\cdot )$ denote the $L^2$-inner product, i.e. $(f,g)={\int }_{{\mathbb{R}}^3}fgdx$. A vector $w\in X$ is a weak solution of (11) if for any $0\le s\le t<\infty$ and $\phi \in C([0,\infty ),H_{\sigma }^1({\mathbb{R}}^3)\cap C^1([0,\infty ),L_{\sigma }^2({\mathbb{R}}^3))$,

$$(w(t),\phi (t))+\underset{s}{\overset{t}{\int }}[(\mathrm{\nabla }w,\mathrm{\nabla }\phi )+(w\cdot \mathrm{\nabla }w,\phi )+(w\cdot \mathrm{\nabla }{u}^{c,\gamma },\phi )+(u^{c,\gamma }\cdot \mathrm{\nabla }w,\phi )]d\tau =(w(s),\phi (s))+\underset{s}{\overset{t}{\int }}(w,{\phi }_{\tau })d\tau .$$

Theorem 1.1

There exists some ${\mu }_0>0$, such that for any $c=(0,0,c_3)$, $|(c,\gamma )|<{\mu }_0$, $w_0\in L_{\sigma }^2({\mathbb{R}}^3)$, there exists a weak solution w of (11) in the energy space X. Moreover, w is weakly continuous from $[0,\infty )$ to $L_{\sigma }^2({\mathbb{R}}^3)$, and satisfies that

$${\Vert w(t)\Vert }_2^2+\underset{s}{\overset{t}{\int }}{\Vert \mathrm{\nabla }\otimes w(\tau )\Vert }_2^{2}d\tau \le {\Vert w(s)\Vert }_2^2$$

for almost all $s\ge 0$, including $s=0$ and all $t\ge s$.

Recall that ${\gamma }^{+}(0)>0$ and ${\gamma }^{-}(0)<0$. So there is some ${\mu }_0^{\prime }$, such that $\{(c,\gamma )|c_1=c_2=0,|(c_3,\gamma )|\le {\mu }_0^{\prime }\}\subset M$. We also have

Theorem 1.2

There exists some ${\mu }_0>0$, such that for any $c=(0,0,c_3)$, $|(c,\gamma )|<{\mu }_0$ and weak solution $w\in X$ of (11) satisfying (12),

$$\underset{t\to \infty }{\lim }{\Vert w(t)\Vert }_2=0.$$

Moreover, if $w_0\in L^p({\mathbb{R}}^3)\cap L_{\sigma }^2({\mathbb{R}}^3)$ for some $\frac{6}{5}<p<2$, then there exists some constant $C>0$, depending only on $(c,\gamma )$, $n,p$ and ${\Vert w_0\Vert }_p$, such that ${\Vert w(t)\Vert }_2\le C{t}^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{2})}$, for all $t>0$.

Theorem 1.1 and Theorem 1.2 can be established using the same arguments as [4], as long as the special stationary solutions $u^{c,\gamma }$ satisfy the following condition

$$|\underset{{\mathbb{R}}^3}{\int }(v\cdot \mathrm{\nabla }{u}^{c,\gamma })\cdot wdx|\le K{\Vert \mathrm{\nabla }w\Vert }_{L^2}{\Vert \mathrm{\nabla }v\Vert }_{L^2},$$

for some constant K small enough, for any divergence free $v,w\in C_c^{\infty }({\mathbb{R}}^3)$. In [4], (13) is proved by Hardy's inequality when $u^{c,\gamma }$ is replaced by small Landau solutions. In this paper, we analyze the solutions $u^{c,\gamma }$ where $(c,\gamma )\in M$, and obtain $|\mathrm{\nabla }{u}^{c,\gamma }|\le C(|c|+|\gamma |)/(|x||x^{\prime }|)$. So (13) is true if we have

$$\underset{{\mathbb{R}}^3}{\int }\frac{|v{|}^2}{|x||x^{\prime }|}dx\le K{\Vert \mathrm{\nabla }v\Vert }_{L^2}^2,$$

for any $v\in C_c^{\infty }({\mathbb{R}}^3)$. Notice (14) cannot be proved by the classical Hardy's inequality. In Section 4, we prove the following extended Hardy-type inequality, which includes (14).

Theorem 1.3

Let $n\ge 2$, $1\le p<n$, $u\in C_c^1({\mathbb{R}}^n)$, $\alpha p>1-n$, $(\alpha +\beta )p>-n$, then there exists some constant C, depending on p, α and β, such that

$${\Vert |x{|}^{\beta }|x^{\prime }{|}^{\alpha }u\Vert }_{L^p({\mathbb{R}}^n)}\le C{\Vert |x{|}^{\beta +\alpha -{\alpha }^{\prime }}|x^{\prime }{|}^{{\alpha }^{\prime }+1}\mathrm{\nabla }u\Vert }_{L^p({\mathbb{R}}^n)},$$

for all ${\alpha }^{\prime }\le \alpha$. Moreover, for any ${\alpha }^{\prime }>\alpha$ and any $C>0$, (15) fails in general.

Estimate (14) is the special case of (15) with $p=2$, $\alpha ={\alpha }^{\prime }=\beta =-\frac{1}{2}$. Then we also have (13). Given (13), Theorem 1.1 and Theorem 1.2 can be proved by the same arguments used in [4], see also [5]. So in this paper we will only prove Theorem 1.3 and (13).

Remark 1.1

In [5], Karch, Pilarczyk and Schonbek proved the asymptotic stability of a class of general time-dependent solutions u of (10) using Fourier analysis, where (13) with $u^{c,\gamma }$ replaced by u is an essential assumption. A list of spaces were given in [5] where (13) is true if $u^{c,\gamma }$ is in one of those spaces. But the solutions $u^{c,\gamma }$ we discuss here are not in those spaces.

We will analyze in Section 2 the singular behaviors of $u^{c,\gamma }$, $(c,\gamma )\in M$. In Section 3 we study the force of $u^{c,\gamma }$, $(c,\gamma )\in M$. Theorem 1.3 will be proved in Section 4. Then as stated above, Theorem 1.1 and Theorem 1.2 follow with the same arguments as in [4].

Acknowledgment. We thank Vladimír Šverák for bringing to our attention the work [4] of Karch and Pilarczyk.