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On the existence of solutions to generalized Navier--Stokes--Fourier system with dissipative heating

Anna Abbatiello, Miroslav Bulicek, Daniel Lear

Abstract

We consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier--Stokes--Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck--Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck--Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.

On the existence of solutions to generalized Navier--Stokes--Fourier system with dissipative heating

Abstract

We consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier--Stokes--Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck--Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck--Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.
Paper Structure (35 sections, 226 equations)

This paper contains 35 sections, 226 equations.

Table of Contents

  1. Introduction
  2. Beyond Oberbeck--Boussinesq system
  3. Cauchy stress tensor and heat flux
  4. Boundary and initial conditions
  5. Definition of solution and main theorem
  6. Basic definitions and function spaces
  7. Definition of weak and suitable weak solutions and main theorem
  8. Definition of approximating systems and their solutions
  9. Limit in the approximating system
  10. Uniform estimates
  11. Limit as k
  12. Solvability of the k-approximation - Lemma \ref{['mainlemma']}
  13. Galerkin approximations
  14. Estimates independent of m
  15. Estimates independent of m for the velocity:
  16. ...and 20 more sections

Theorems & Definitions (1)

  • proof