SMS scnews item created by Daniel Daners at Thu 25 Feb 2010 0851
Type: Seminar
Modified: Thu 25 Feb 2010 0936; Wed 10 Mar 2010 1236
Distribution: World
Expiry: 22 Mar 2010
Calendar1: 22 Mar 2010 1505-1600
CalLoc1: Carslaw 273
Auth: daners@p7153.pc.maths.usyd.edu.au
PDE Seminar
Leray’s inequality in multi-connected domains
Kozono
Update: Note the change of date
Hideo Kozono
Tohoku University, Japan
22 March 2010, 3-4pm, Carslaw Room 273
Abstract
Consider the stationary Navier-Stokes equations in a bounded
domain Ω ⊂ ℝ3 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.
Check also the PDE Seminar page. Enquiries to Florica Cξrstea or Daniel Daners.