On a resolvent estimate for the Stokes system with Neumann boundary condition

Abstract

We obtain the $L_p$ estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in $\mathbb R^n$. The result has been obtained by Grubb and Solonnikov [7, 8, 9, 10] by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from [7, 8, 9, 10]. The point of our proof is to use the space $W^{-1}_p$ in order to handle with the equation $\nabla\cdot u = g$ in the half-space model problem as well as in the whole space one. Our result is an extension of the paper by Farwig and Sohr [5] about the Dirichlet zero condition case to the Neumann boundary condition case.