Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

TL;DR

This paper investigates boundary value problems for anisotropic Stokes and Navier-Stokes systems with non-homogeneous conditions in Lipschitz domains, establishing well-posedness, existence, and uniqueness results using variational methods and fixed point theorems.

Contribution

It provides the first analysis of non-homogeneous Dirichlet-transmission problems for anisotropic fluid systems in Lipschitz domains with explicit conditions for solution uniqueness.

Findings

Proved well-posedness of linear anisotropic Stokes system

Established existence of weak solutions for nonlinear Navier-Stokes system

Derived explicit conditions for solution uniqueness

Abstract

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an $L^\infty$-tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in ${\mathbb R}^{n\times n}$ with zero matrix traces. We analyze, in $L^2$-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a Lipschitz interface in ${\mathbb R}^n$, $n\ge 2$ ($n=2,3$ for the nonlinear problems). The transversal interface intersects the boundary of the Lipschitz domain. First, we use a mixed variational approach to prove well-posedness results for the linear anisotropic Stokes system. Then we show the existence of a weak solution for the nonlinear anisotropic Navier-Stokes system by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.