PDE Seminar Abstracts

Leray’s inequality in multi-connected domains

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


Consider the stationary Navier-Stokes equations in a bounded domain Ω 3 whose boundary Ω consists of L + 1 disjoint closed surfaces Γ0, Γ1, …, ΓL with Γ1, …, ΓL inside of Γ0. The Leray inequality of the given boundary data β on Ω plays an important role for the existence of solutions. It is known that if the flux γi Γiβ νdS = 0 on Γi(ν: the unit outer normal to Γi) is zero for each i = 0, 1,,L, then the Leray inequality holds. We prove that if there exists a sphere S in Ω separating Ω in such a way that Γ1,, Γk, 1 k L are contained in S and that Γk+1,, ΓL are in the outside of S, then the Leray inequality necessarily implies that γ1 + + γk = 0. In particular, suppose that for each each i = 1,,L there exists a sphere Si in Ω such that Si contains only one Γi. Then the Leray inequality holds if and only if γ0 = γ1 = = γL = 0.