Stationary Navier–Stokes equations under inhomogeneous boundary conditions in 3D exterior domains

Abstract

In an exterior domain $\Omega \subset \mathbb {R}^3$ having compact boundary $\partial \Omega = \bigcup _{j=1}^L\Gamma _j$ with L disjoint smooth closed surfaces $\Gamma _1, \ldots , \Gamma _L$, we consider the problem on the existence of weak solutions $\varvec{v}$ of the stationary Navier–Stokes equations in $\Omega$ satisfying $\varvec{v}|_{\Gamma _j}= \varvec{\beta _j}$, $j = 1, \ldots , L$ and $\varvec{v}\rightarrow \varvec{0}$ as $|x| \rightarrow \infty$, where $\varvec{\beta }_j$ are the given data on the boundary component $\Gamma _j$, $j=1, \ldots , L$. Our first task is to find an appropriate solenoidal extension $\varvec{b}$ into $\Omega$, i.e., $\mathrm{div}\,\varvec{b}=0$ satisfying $\varvec{b}|_{\Gamma _j} = \varvec{\beta }_j$, $j =1, \ldots , L$. By our previous result [8] on the $L^r$-Helmholtz-Weyl decomposition, $\varvec{b}$ is expressed as $\varvec{b} = \varvec{h} + \mathrm{rot}\, \varvec{w}$, where $\varvec{h}$ is a harmonic vector field depending only on the flux $\int _{\Gamma _j}\varvec{\beta }_j\cdot \varvec{\nu }dS$ through $\Gamma _j$, $j =1, \ldots , L$. Next, we prove that if $\varvec{h}$ is small in $L^3(\Omega )$, then there exists a weak solution $\varvec{v}$ with $\nabla \varvec{v} \in L^2(\Omega )$.