The stokes operator with Neumann boundary conditions in Lipschitz domains

Abstract

In the first part of the paper, we give a satisfactory definition of the Stokes operator in Lipschitz domains in $ {\mathbb{R}^n} $ when boundary conditions of Neumann type are considered. We then proceed to establish optimal global Sobolev regularity results for vector fields in the domains of fractional powers of this Neumann–Stokes operator. Finally, we study the existence, regularity, and uniqueness of mild solutions of the Navier–Stokes system with Neumann boundary conditions. Bibliography: 43 titles. Illustrations: 2 figures.