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

Abstract

We discuss $L^p - L^q$ type estimate of the Stokes semigroup and its application to the Navier-Stokes equations in a perturbed half-space and an aperture domain. Especially, we have the $L^p$-$L^q$ type estimate of the gradient of the Stokes semigroup for any $p$ and $q$ with $1 \le p \le q \lt \infty$, while the same estimate holds only for the exponents $p$ and $q$ with $1 \lt p \le q \le n$ in the exterior domain case, where $n$ denotes the space dimension. And therefore, we can get better results concerning the asymptotic behavior 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 is also obtained in a perturbed half-space and an aperture domain by the perturbation argument. This is one of the reason why the result in our case is essentially better compared with the exterior domain case.