{
  "id": "2104.07124",
  "version": "v1",
  "published": "2021-04-14T20:56:57.000Z",
  "updated": "2021-04-14T20:56:57.000Z",
  "title": "Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces",
  "authors": [
    "Mirela Kohr",
    "Sergey E. Mikhailov",
    "Wolfgang L. Wendland"
  ],
  "comment": "49 pages",
  "categories": [
    "math.AP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-14T20:56:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J57",
      "35Q30",
      "46E35"
    ],
    "keywords": [
      "lipschitz domain",
      "non-homogeneous dirichlet-transmission problems",
      "linear anisotropic stokes system",
      "nonlinear problems",
      "nonlinear anisotropic navier-stokes system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}