Variational approach for layer potentials of the Stokes system with $L_{\infty }$ symmetrically elliptic coefficient tensor and applications to Stokes and Navier-Stokes boundary problems

Abstract

The first aim of this paper is to develop a layer potential theory in $L_2$-based weighted Sobolev spaces on Lipschitz bounded and exterior domains of ${\mathbb R}^n$, $n\geq 3$, for the anisotropic Stokes system with $L_{\infty }$ viscosity coefficient tensor satisfying an ellipticity condition for symmetric matrices. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of ${\mathbb R}^n$, with the given data in $L_2$-based weighted Sobolev spaces. These results are used to define the Newtonian and layer potentials and to obtain their properties. Then we analyze well-posedness of the exterior Dirichlet, Neumann and mixed problems for the Stokes system with $L_{\infty }$ symmetrically elliptic coefficient tensor. Solutions of some of these problems are also represented in terms of the anisotropic Stokes Newtonian and layer potentials. Finally, we prove the existence of a weak solution for a transmission problem in complementary Lipschitz domains in ${\mathbb R}^3$ for the anisotropic Navier-Stokes system with general data in $L_2$-based weighted Sobolev spaces. The analysis relies on an existence result for a Dirichlet problem for the anisotropic Navier-Stokes system in a family of bounded domains, and on the Leray-Schauder fixed point theorem.