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Generalized dissipative solutions to free boundary compressible viscous models

Anna Abbatiello, Donatella Donatelli

Abstract

We study free boundary compressible viscous models that may include nonlinear viscosities. These are compressible/incompressible Navier-Stokes type systems for a non-Newtonian stress tensor. They describe the motion of a possibly non-Newtonian fluid in free flow and in congested regions. In the congested regions it appears the pressure that is the Lagrange multiplier associated with the incompressibility constraint, while in free flows it is a pressureless gas system. We establish the existence of generalized dissipative solutions in the case of in/out-flow boundary conditions and we also prove that if these solutions are smooth then they are classical solutions.

Generalized dissipative solutions to free boundary compressible viscous models

Abstract

We study free boundary compressible viscous models that may include nonlinear viscosities. These are compressible/incompressible Navier-Stokes type systems for a non-Newtonian stress tensor. They describe the motion of a possibly non-Newtonian fluid in free flow and in congested regions. In the congested regions it appears the pressure that is the Lagrange multiplier associated with the incompressibility constraint, while in free flows it is a pressureless gas system. We establish the existence of generalized dissipative solutions in the case of in/out-flow boundary conditions and we also prove that if these solutions are smooth then they are classical solutions.

Paper Structure

This paper contains 31 sections, 3 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem and main results
  3. Viscous stress
  4. Initial Data
  5. Boundary Data
  6. Definition of generalized dissipative solutions
  7. Main results
  8. The multilevel approximating problem
  9. Regularization of the potential
  10. Parabolic approximation of the equation of continuity
  11. Regularization of the pressure
  12. Galerkin approximations of the momentum balance
  13. Approximated initial and boundary data
  14. Existence of dissipative solutions
  15. Limit as $\delta \to 0$
  16. ...and 16 more sections

Key Result

Theorem 2.5

Let the assumptions P7--hyp-rb be satisfied and let $T>0$ then problem problem admits at least one generalized dissipative solution in the sense of Definition def.

Theorems & Definitions (8)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 4.1
  • Remark 4.2