Leray’s inequality in multi-connected domains

Abstract

Consider the stationary Navier-Stokes equations in a bounded domain $\Omega \subset {ℝ}^3$ whose boundary $\partial \Omega$ consists of $L+1$ disjoint closed surfaces ${\Gamma }_0$, ${\Gamma }_1$, …, ${\Gamma }_L$ with ${\Gamma }_1$, …, ${\Gamma }_L$ inside of ${\Gamma }_0$. The Leray inequality of the given boundary data $\beta$ on $\partial \Omega$ plays an important role for the existence of solutions. It is known that if the flux ${\gamma }_i\equiv {\int }_{{\Gamma }_i}\beta \cdot \nu dS=0$ on ${\Gamma }_i$($\nu$: the unit outer normal to ${\Gamma }_i$) is zero for each $i=0,1,\dots ,L$, then the Leray inequality holds. We prove that if there exists a sphere $S$ in $\Omega$ separating $\partial \Omega$ in such a way that ${\Gamma }_1,\dots ,{\Gamma }_k$, $1\le k\le L$ are contained in $S$ and that ${\Gamma }_{k+1},\dots ,{\Gamma }_L$ are in the outside of $S$, then the Leray inequality necessarily implies that ${\gamma }_1+\dots +{\gamma }_k=0$. In particular, suppose that for each each $i=1,\dots ,L$ there exists a sphere $S_i$ in $\Omega$ such that $S_i$ contains only one ${\Gamma }_i$. Then the Leray inequality holds if and only if ${\gamma }_0={\gamma }_1=\dots ={\gamma }_L=0$.