Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces
| Autor: | Kohr, Mirela, Mikhailov, Sergey E., Wendland, Wolfgang L. |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 49 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|