**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 ⊂ ℝ^{3} whose boundary ∂Ω consists of L + 1 disjoint closed surfaces
Γ_{0}, Γ_{1},
, Γ_{L} with Γ_{1},
, Γ_{L} inside of Γ_{0}. 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 γ_{i} ≡∫
_{Γi}β ⋅ νdS = 0 on Γ_{i}(ν: the
unit outer normal to Γ_{i}) 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 Γ_{1},
, Γ_{k}, 1 ≤ k ≤ L are contained in S and that
Γ_{k+1},
, Γ_{L} are in the outside of S, then the Leray inequality necessarily
implies that γ_{1} +
+ γ_{k} = 0. In particular, suppose that for each each
i = 1,
,L there exists a sphere S_{i} in Ω such that S_{i} contains only one Γ_{i}.
Then the Leray inequality holds if and only if γ_{0} = γ_{1} =
= γ_{L} = 0.
Check also the PDE Seminar page. Enquiries to Florica Cξrstea or Daniel Daners.