On the stationary Navier–Stokes problem in 3D exterior domains

ABSTRACT

The main purpose of this paper is to show that in a three-dimensional exterior Lipschitz domain $\mathrm{\Omega }={\mathbb{R}}^3\setminus {\cup }_{i=1}^m{\overline{\mathrm{\Omega }}}_i$ the stationary Navier–Stokes equations have a solution which converges at infinity to a constant vector and assumes a boundary value $a\in L^2\left(\mathrm{\partial \Omega }\right)$ (or $a\in W^{-1/3,3}\left(\mathrm{\partial \Omega }\right)$ if Ω is of class $C^{1,1}$), provided $\sum _{i=1}^m|{\int }_{\mathrm{\partial }{\mathrm{\Omega }}_i}a\cdot n|\underset{\xi \in \mathrm{\partial }{\mathrm{\Omega }}_i}{max}(1/|\xi -x_i|)<8\pi \nu$. Moreover, for large value of the viscosity ν we prove existence, uniqueness and asymptotics of a solution $u\in L^3\left(\mathrm{\Omega }\right)$ for Ω and a polar symmetric.

Keywords:

• Stationary Stokes and Navier–Stokes equations
• three-dimensional exterior domains
• boundary value problems

AMS CLASSIFICATIONS:

• Primary: 76D05, 35Q30
• Secondary: 31B10, 76D03

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We use a standard notation as, e.g. in [3]. In particular, italic light-face letters, except $o,x,y,\xi ,\zeta$, denote scalars or vectors, $(x,y)$ and $(\xi ,\zeta )$ usually stands for points running on domains and surfaces, respectively; $o\in {\mathbb{R}}^3\setminus \overline{\mathrm{\Omega }}$ is the origin of the reference frame $(o,\{e_i{\}}_{i=1,2,3})$; $r=r\left(x\right)=|x|$, $e_r=x/|x|$; n denotes the exterior (with respect to Ω) normal to $\mathrm{\partial \Omega }$; $u\cdot \mathrm{\nabla }u$ is the vector with components $u_i{\mathrm{\partial }}_i{u}_j$. $S_R\left(x\right)$ is the ball of radius R centered at x; $T_R\left(x\right)=S_{2R}\left(x\right)\setminus S_R\left(x\right)$; $S_R=S_R\left(o\right)$ (we suppose $S_R\supset \complement \mathrm{\Omega }$); $T_R=T_R\left(o\right)$; ${\mathrm{\Omega }}_R=\mathrm{\Omega }\cap S_R$. For $q\in (1,+\mathrm{\infty })$, $D^{1,q}\left(\mathrm{\Omega }\right)=\{\varphi :\parallel \mathrm{\nabla }\varphi {\parallel }_{L^q\left(\mathrm{\Omega }\right)}<+\mathrm{\infty }\}$, $D_0^{1,q}\left(\mathrm{\Omega }\right)$ is the completion of $C_0^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ with respect to the norm $\parallel \mathrm{\nabla }\varphi {\parallel }_{L^q\left(\mathrm{\Omega }\right)}$ and $D^{-1,q^{\prime }}\left(\mathrm{\Omega }\right)$ is the dual space of $D_0^{1,q}\left(\mathrm{\Omega }\right)$; the subscript σ in $K_{\sigma }$, where K is a functional space, means that the elements of K are divergence free. If $f\left(x\right)$ and $g\left(r\right)$ are functions defined in a neighborhood of the infinity, the Landau symbols $f\left(x\right)=o\left(g\right(r\left)\right)$, $f\left(x\right)=O\left(g\right(r\left)\right)$ mean respectively that $\underset{r\to +\mathrm{\infty }}{lim}\left(f\right(x\left)/g\right(r\left)\right)=0$ and $f\left(x\right)/g\left(r\right)$ is bounded at large distance.

2 On the contrary, starting from the paper by Giga [33] the problem of existence of solutions to the Stokes and Navier–Stokes problems in bounded domains with data in $L^q\left(\mathrm{\partial \Omega }\right)$ has been extensively studied (see [3] and the references therein).

3 Under suitable symmetry assumptions, the existence of solution to (1(1) $\begin{array}{rl}\nu \mathrm{\Delta }u-u\cdot \mathrm{\nabla }u-\mathrm{\nabla }p& =0\mathrm{in}\mathrm{\Omega },\\ \mathrm{div}u& =0\mathrm{in}\mathrm{\Omega },\\ u& =a\mathrm{on}\mathrm{\partial \Omega },\\ u-u_0& =o\left(1\right)\end{array}$(1) ) without any assumption on the boundary datum is proved in [10]. An analogous problem in four dimensions has been recently studied in [34].

4 See (i)–(vii) of Section 2.1.

5 The way in which the solution attains the boundary value will be clear from the context of the proof of Theorem 1.1.

6 It is worth recalling that uniqueness is lost for small viscosity (see, e.g. [3]).

7 A trivial but large class of solutions to (1(1) $\begin{array}{rl}\nu \mathrm{\Delta }u-u\cdot \mathrm{\nabla }u-\mathrm{\nabla }p& =0\mathrm{in}\mathrm{\Omega },\\ \mathrm{div}u& =0\mathrm{in}\mathrm{\Omega },\\ u& =a\mathrm{on}\mathrm{\partial \Omega },\\ u-u_0& =o\left(1\right)\end{array}$(1) )${}_{1,2}$ vanishing at infinity more rapidly than $r^{-1}$ or belonging to $L^3\left(\mathrm{\Omega }\right)$ are given by the pairs $(u,p)=(\mathrm{\nabla }\varphi ,$ $-\frac{1}{2}|\mathrm{\nabla }\varphi {|}^2)$ with $\varphi =o\left(1\right)$ harmonic functions in Ω. Clearly, $u=O\left(r^{-2}\right)$ (and $u\in L^q\left(\mathrm{\Omega }\right),q>3/2)$ is also a solution to the Stokes equations with constant pressure and must satisfy necessary compatibility conditions, for instance ${\int }_{\mathrm{\partial }{S}_R}u=0$, for every (large) R.

8 In such a case, (16(16) $\begin{array}{rl}{v}_i\left[\psi \right]\left(x\right)& ={\int }_{\mathrm{\partial \Omega }}{\mathcal{U}}_{ij}(x-\xi ){\psi }_j\left(\xi \right){\mathrm{da}}_{\xi },\\ P\left[\psi \right]\left(x\right)& ={\int }_{\mathrm{\partial \Omega }}{\mathcal{P}}_j(x-\xi ){\psi }_j\left(\xi \right){\mathrm{da}}_{\xi }.\end{array}$(16) )${}_1$ has to be understood as the value of the functional ${\psi }_j$ at ${\mathcal{U}}_{ij}$.

9 Note that this knowledge must be exact, not approximate!

10 In [28], existence is proved for $u_0=0$. The extension of this result to system (79(79) $\begin{array}{rl}-\nu \mathrm{\Delta }w+(h+w+u_{ϵ}+u_0)\cdot \mathrm{\nabla }(h+w)& \\ +(h+w)\cdot \mathrm{\nabla }{u}_{ϵ}-\nu \mathrm{\Delta }\eta +\mathrm{\nabla }Q& =0\mathrm{in}{\mathrm{\Omega }}_k,\\ \mathrm{div}w& =0\mathrm{in}{\mathrm{\Omega }}_k,\\ w& =0\mathrm{on}\mathrm{\partial }{\mathrm{\Omega }}_k\end{array}$(79) ) does not present difficulties.

11 We extend w to Ω by setting $w=0$ outside ${\mathrm{\Omega }}_k$.