Weighted $L^p - L^q$ estimates of the Stokes semigroup in some unbounded domains

Abstract

We consider the Navier-Stokes equations in half-space and a perturbed half-space in $L^p$ space with Muckenhoupt weight. As the first step, we shall describe the Helmholtz decompostion of $L^p$ space with Muckenhoupt weight and the weighted resolvent estimates for the Stokes equations. Next we shall show the $L^p - L^q$ estimates of Stokes semigroup with $\langle{x}\rangle^s$ weight. Finally as the application of the weighted $L^p - L^q$ estimates, we shall obtain the weighted asymptotic behavior of the solution to the Navier-Stokes equations.