On the Stokes and Navier-Stokes equations in a perturbed half-space

Abstract

We discuss $L^p$-$L^q$ type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed half-space. Especially, we have the $L^p$-$L^q$ type estimate of the gradient of the Stokes semigroup for any $p$ and $q$ with $1 < p \leqq q < \infty$, while the same estimate holds only for the exponents $p$ and $q$ with $1 < p\leqq q \leqq n$ in the exterior domain case, where $n$ denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the $L^p$-$L^q$ type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.