Boundary Value Problems for the Stokes System

Abstract

In this chapter we present a preliminary information on the boundary value problems for the Stokes system: the statement of the boundary value problems, the Green formulas, and the Green matrices in the half-space and half-plane. The main part of the chapter is devoted to the solvability of the boundary value problems in weighted Sobolev and Holder spaces. Theorems on the unique solvability in the strong, generalized, and weak sense are proved. These results enable us to study local properties of solutions to the boundary value problems near the boundary of an arbitrary domain, i.e. to study the smoothness properties of solutions with respect to those of the data and the boundary of the domain, as well as to obtain local estimates of solutions with respect to Holder and L s -norms.