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The Oberbeck--Boussinesq approximation and Rayleigh--Benard convection revisited

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna

Abstract

We consider the Oberbeck--Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non--local term arising in the incompressible limit of the Navier--Stokes--Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.

The Oberbeck--Boussinesq approximation and Rayleigh--Benard convection revisited

Abstract

We consider the Oberbeck--Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non--local term arising in the incompressible limit of the Navier--Stokes--Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.
Paper Structure (13 sections, 8 theorems, 112 equations)

This paper contains 13 sections, 8 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminary material, weak formulation
  3. Energy balance for the temperature
  4. Higher regularity of the temperature
  5. Validity of the maximum principle for $\Theta$
  6. Uniform bounds on the temperature
  7. Levinson dissipativity
  8. Stationary statistical solutions
  9. Convergence to equilibria
  10. Bounds on equilibrium solutions
  11. Relative energy inequality
  12. Stability of equilibrium solutions
  13. Convergence for the classical Bénard problem with temperature gradient aligned to gravitation

Key Result

Proposition 2.2

Let the velocity ${\bf u}$, and the initial distribution of the temperature be given. Then the problem w5, w6 admits at most one solution $\Theta$, $\Theta(0, \cdot) = \Theta_0$, in the class

Theorems & Definitions (14)

  • Remark 2.1
  • Proposition 2.2: Uniqueness of $\Theta$
  • proof
  • Proposition 2.3: Maximum/minimum principle
  • Theorem 3.1: Uniform bounds on the temperature
  • proof
  • Theorem 4.1: Levinson dissipativity
  • Remark 4.2
  • proof
  • Corollary 4.3: Bounded absorbing set
  • ...and 4 more