PDE Seminar Abstracts

Hideo Kozono

Tohoku University, Japan

22 Mar 2010 3-4pm, Carslaw Room 273

Tohoku University, Japan

22 Mar 2010 3-4pm, Carslaw Room 273

Consider the stationary Navier-Stokes equations in a bounded domain $\Omega \subset {\mathbb{R}}^{3}$ 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 ${\gamma}_{0}={\gamma}_{1}=\dots ={\gamma}_{L}=0$.

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