## 15. ÖMG-Kongress

Jahrestagung der Deutschen Mathematikervereinigung

#### 16. bis 22. September 2001 in Wien

**Sektion 10 - Angewandte Mathematik, Industrie- und Finanzmathematik**

Dienstag, 18. September 2001, 18.30, Hörsaal 42

**The nonstationary Stokes and Navier-Stokes equations in aperture domains**

**Toshiaki Hishida**,

**TU Darmstadt**

We study the initial value problems for the Stokes and Navier-Stokes equations
in an aperture domain
,
which consists of two disjoint half spaces separated by a wall but connected
by an aperture in the wall.
This class of unbounded domains with noncompact boundaries is interesting
because of the following remarkable feature (Heywood, 1976):
either a prescribed flux of the velocity field through the aperture or
a prescribed pressure drop at infinity may be required as an additional
boundary condition in order to get a unique solution.
We consider the Stokes equation with zero flux through the aperture, which
generates a bounded analytic semigroup in
(Farwig and Sohr, 1996),
and derive some decay estimates (- estimates) of the semigroup:

for and
, where
if ; and
if .
This result improves a known result (Abels, 2000), especially, the -decay
property of
is now proved.
Using the estimates of the semigroup, we construct a unique global strong solution
to the Navier-Stokes equation (with zero flux through the aperture) for
small initial value in
.
Such a global existence theorem is well known in the cases of
whole spaces, half spaces, bounded and exterior domains.

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