The Stokes Equations

by Werner Varnhorn

The present book consists of three parts. In the first part a theory of solvability for the stationary Stokes equations in exterior domains is developed. We prove existence of strong solutions in Sobolev spaces and use a localisation principle and the divergence equation to deduce further properties of the solution (uniqueness, asymptotics). The second part considers the resolvent equations with methods of potential theory. We present the explicit fundamental tensor for general space dimension and solve the boundary value problems via systems of integral equations. New a-priori norm estimates for the solution are developed independently of small resolvent parameter. This leads to the uniform boundedness of the semigroup associated with the Stokes operator, also for the case of a two-dimensional exterior domain. In the third part we approximate the nonstationary equations with help of the resolvent and prove convergence of optimal order in a scale of Sobolev spaces. This includes the nonlinear Navier-Stokes equations, which can be regularized with methods of time delay.

154 pages, Paperback

First published August 30, 1994