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Unconditional stability of equilibria in thermally driven compressible fluids

Eduard Feireisl, Yong Lu, Yongzhong Sun

TL;DR

The paper proves unconditional stability of equilibria for thermally driven compressible viscous fluids modeled by the Navier–Stokes–Fourier system under boundary-driven heating. It employs a weak-solution framework with entropy inequalities and a relative-energy (Bregman) distance to a stationary state, together with a novel density-damping argument based on Bogovskii operators to compensate lack of explicit Lyapunov structure. By carefully bounding the temperature, velocity, and density dissipation and absorbing remainder terms through small-data assumptions, it establishes that the relative energy $ rac{d}{dt}\int_ ext{Ω} E( ho, heta,oldsymbol{u}ig| ho_s, heta_s,oldsymbol{u}_s) ext{d}x$ decays to zero, implying convergence $ ho o ho_s$, $ heta o heta_s$, and $oldsymbol{u} ooldsymbol{u}_s$ as $t o ty$ for all global weak solutions. The results apply to Rayleigh–Bénard convection and extend the unconditional convergence framework to compressible flows, offering a global-in-time stability perspective with potential implications for turbulence and convection problems.

Abstract

We show that small perturbations of the spatially homogeneous equilibrium of a thermally driven compressible viscous fluid are globally stable. Specifically, any weak solution of the evolutionary Navier--Stokes--Fourier system driven by thermal convection converges to an equilibrium as time goes to infinity. The main difficulty to overcome is the fact the problem does not admit any obvious Lyapunov function. The result applies, in particular, to the Rayleigh--B\' enard convection problem.

Unconditional stability of equilibria in thermally driven compressible fluids

TL;DR

The paper proves unconditional stability of equilibria for thermally driven compressible viscous fluids modeled by the Navier–Stokes–Fourier system under boundary-driven heating. It employs a weak-solution framework with entropy inequalities and a relative-energy (Bregman) distance to a stationary state, together with a novel density-damping argument based on Bogovskii operators to compensate lack of explicit Lyapunov structure. By carefully bounding the temperature, velocity, and density dissipation and absorbing remainder terms through small-data assumptions, it establishes that the relative energy decays to zero, implying convergence , , and as for all global weak solutions. The results apply to Rayleigh–Bénard convection and extend the unconditional convergence framework to compressible flows, offering a global-in-time stability perspective with potential implications for turbulence and convection problems.

Abstract

We show that small perturbations of the spatially homogeneous equilibrium of a thermally driven compressible viscous fluid are globally stable. Specifically, any weak solution of the evolutionary Navier--Stokes--Fourier system driven by thermal convection converges to an equilibrium as time goes to infinity. The main difficulty to overcome is the fact the problem does not admit any obvious Lyapunov function. The result applies, in particular, to the Rayleigh--B\' enard convection problem.
Paper Structure (16 sections, 3 theorems, 93 equations)

This paper contains 16 sections, 3 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Weak solutions to the Navier--Stokes--Fourier system
  3. Structural hypotheses concerning constitutive relations
  4. Equation of state
  5. Transport coefficients
  6. Weak solutions
  7. Dissipativity in the sense of Levinson
  8. Main result -- unconditional stability
  9. Proof of Theorem \ref{['TM1']}
  10. Relative entropy inequality
  11. Estimates involving temperature dissipation
  12. Estimates involving viscosity
  13. Pressure estimates and density damping
  14. Estimates of the remainders
  15. Conclusion, unconditional convergence
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $\Omega \subset R^3$ be a bounded domain of class $C^{2 + \nu}, \ \nu > 0$. Suppose the thermodynamic functions $p$, $e$, and $s$ satisfy the hypotheses S1--convex-pm, while the transport coefficients $\mu$, $\eta$, and $\kappa$ comply with S10--S12. Then there exists a positive quantity $\mathc for any weak solution of the NSF system defined on $(T, \infty) \times \Omega$ in the sense of Defi

Theorems & Definitions (6)

  • Remark 2.1
  • Definition 2.2
  • Proposition 3.1: Bounded absorbing set
  • Theorem 4.1: Unconditional stability
  • Remark 4.2
  • Theorem 6.1: Rayleigh--Bénard problem -- unconditional stability