Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition

Abstract

In this paper, we obtain local energy decay estimates and $L_p$-$L_q$ estimates of the solutions to the Stokes equations with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equations. Comparing with the non-slip boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null force at the boundary.