The Stokes and Navier-Stokes equations in an aperture domain

Abstract

We consider the nonstationary Stokes and Navier-Stokes equations in an aperture domain $\mathrm{\Omega }\subset {\mathbf{R}}^n$, $n\ge 2$. For this purpose, we prove $L^p$-$L^q$ type estimate of the Stokes semigroup in the aperture domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin. We apply them to the Navier-Stokes initial value problem in the aperture domain. As a result, we can prove the global existence of a unique solution to the Navier-Stokes problem with the vanishing flux condition and some decay properties as $t\to \mathrm{\infty }$, when the initial velocity is sufficiently small in the $L^n$ space. Moreover we can prove the time-local existence of a unique solution to the Navier-Stokes problem with the non-trivial flux condition.