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Dissipative solutions to the model of a general compressible viscous fluid with the Coulomb friction law boundary condition

Sarka Necasova, Justyna Ogorzaly, Jan Scherz

Abstract

We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this result consists of a three-level approximation method: A Galerkin approximation, the classical parabolic regularization of the continuity equation as well as convex regularizations of the potential generating the viscous stress and the boundary terms incorporating the Coulomb friction law into the dissipative formulation. This approach combines the techniques already known from the proof of the existence of dissipative solutions to a model of general compressible viscous fluids under inflow-outflow boundary conditions as well as the proof of the existence of a weak solution to the incompressible Navier-Stokes equations under the Coulomb friction law boundary condition. It is the first time that this type of boundary condition is considered in the case of compressible flow.

Dissipative solutions to the model of a general compressible viscous fluid with the Coulomb friction law boundary condition

Abstract

We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this result consists of a three-level approximation method: A Galerkin approximation, the classical parabolic regularization of the continuity equation as well as convex regularizations of the potential generating the viscous stress and the boundary terms incorporating the Coulomb friction law into the dissipative formulation. This approach combines the techniques already known from the proof of the existence of dissipative solutions to a model of general compressible viscous fluids under inflow-outflow boundary conditions as well as the proof of the existence of a weak solution to the incompressible Navier-Stokes equations under the Coulomb friction law boundary condition. It is the first time that this type of boundary condition is considered in the case of compressible flow.
Paper Structure (10 sections, 5 theorems, 159 equations)

This paper contains 10 sections, 5 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Dissipative solutions and main result
  4. Approximate system
  5. Existence of the approximate solutions
  6. Limit passage with respect to
  7. Limit passage with respect to
  8. Limit passage with respect to
  9. Proof of the main result
  10. Appendix

Key Result

Theorem 3.1

Let $T > 0$ and let $\Omega \subset \mathbb{R}^3$ be a bounded domain with boundary $\Gamma = \partial \Omega$ of class $C^{2,\eta}\bigcup C^{0,1}$ for some $\eta>0$. Let the data satisfy the conditions fcond1, fcond2 and pcond1--pcond3. Moreover, consider initial data where $P$ denotes the pressure potential defined in pcond2 and $\gamma>1$ is chosen according to the estimate coercP. Then the p

Theorems & Definitions (9)

  • Remark 3.1
  • Remark 3.2
  • Definition 3.1
  • Theorem 3.1
  • Proposition 5.1
  • Remark 6.1
  • Lemma 9.1
  • Lemma 9.2
  • Lemma 9.3