Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with <i>L</i><sub>∞</sub> tensor coefficient under relaxed ellipticity condition
Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland
Abstract
The first aim of this paper is to show well-posedness of Dirichlet and transmission problems in bounded and exterior Lipschitz domains in $ {\mathbb R}^n $, $ n\geq 3 $, for the anisotropic Stokes system with $ L_{\infty } $ viscosity tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices with zero matrix trace, with data from standard and weighted Sobolev spaces. To this end we reduce the linear problems to equivalent mixed variational formulations and show that the variational problems are well-posed. Then we use the Leray-Schauder fixed point theorem and establish the existence of a weak solution for nonlinear Dirichlet and transmission problems for the anisotropic Navier-Stokes system in bounded Lipschitz domains in $ {\mathbb R}^3 $, with general (including large) data in Sobolev spaces. For exterior domains in $ {\mathbb R}^3 $, the analysis of the nonlinear Dirichlet and transmission problems in weighted Sobolev spaces relies on the existence result for the Dirichlet problem for the anisotropic Navier-Stokes system in a family of bounded Lipschitz domains. The obtained estimates for pressure in $ {\mathbb R}^3 $ look new also for the classical isotropic case.