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 $ \Omega\subset R^n (n\geq 3)$, 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 $ e^{-tA}$ in $ L^q_\sigma(\Omega), 1<q<\infty$ (Farwig and Sohr, 1996), and derive some decay estimates ($ L^q$-$ L^r$ estimates) of the semigroup:

$\displaystyle \Vert\nabla^j e^{-tA}f\Vert_{L^r(\Omega)}\leq Ct^{-(n/q-n/r)/2-j/2}\Vert
f\Vert_{L^q(\Omega)} $

for $ t>0$ and $ f\in L^q_\sigma(\Omega)$, where $ 1<q<r\leq\infty$ if $ j=0$; and $ 1<q\leq r\leq n$ if $ j=1$. This result improves a known result (Abels, 2000), especially, the $ L^n$-decay property of $ \nabla e^{-tA}f$ 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 $ L^n_\sigma(\Omega)$. Such a global existence theorem is well known in the cases of whole spaces, half spaces, bounded and exterior domains.

E-Mail: hishida@mathematik.tu-darmstadt.de

Zeitplan der Sektion   Tagesübersicht   Liste der Vortragenden