Stokes Problems in Irregular Domains with Various Boundary Conditions

Abstract

Different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in R 3 , such as Dirichlet, Neumann, Hodge or Robin boundary conditions are presented here. The situation is a little different from the case of smooth domains. The analysis of the problem involves a good comprehension of the behaviour near the boundary. The linear Stokes operator associated to the various boundary conditions is first studied. Then a classical fixed point theorem is used to show how the properties of the operator lead to local solutions or global solutions for small initial data.