PDE Seminar Abstracts

Leray’s inequality in multi-connected domains

Hideo Kozono
Tohoku University, Japan
22 Mar 2010 3-4pm, Carslaw Room 273

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 ${\gamma }_{0}={\gamma }_{1}=\dots ={\gamma }_{L}=0$.