2021/04/14 by Mirela Kohr, Kohr, Mirela, Sergey E. Mikhailov +3
Computer Science · Engineering · Mathematics · #31C #35J57 #35Q30 #46E35 #76D #76M #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2104.07124
openalex publication_date 2021/04/14 · openalex created_date 2022/08/24 · openalex updated_date 2026/07/28
This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^∞-tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in \mathbb Rn× n with zero matrix traces. We analyze, in L2-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 Rn, n≥ 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.