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On the continuity in time and the exponential decay of solutions to a generalized Navier--Stokes--Fourier system in 2D

Miroslav Bulíček, Petr Kaplický, Lucie Wintrová

TL;DR

The paper analyzes a 2D incompressible non-Newtonian Navier–Stokes–Fourier system with $p\ge 2$ under Dirichlet boundary conditions, proving the global existence of weak solutions that satisfy an entropy equality and, importantly, continuity in time of the temperature $\vartheta$. The authors employ Galerkin approximations, a robust entropy framework, and Minty’s method to identify the constitutive stress and establish energy–entropy balance, including a minimum principle ensuring $\vartheta\ge \mu>0$. A key novelty is the demonstration that $\vartheta$ is continuous in time in $L^1$, enabling the construction of a problem-tailored Lyapunov functional and proving nonlinear exponential convergence to a steady state, even with inhomogeneous boundary data. Together, these results advance the understanding of thermodynamically consistent weak solutions for non-Newtonian, heat-conducting fluids in two dimensions and illuminate their long-time behavior and stability.

Abstract

We consider the flow of a generalized Newtonian incompressible heat-conducting fluid in a bounded domain, subject to homogeneous Dirichlet boundary conditions for velocity and Dirichlet boundary conditions for temperature. For a fluid whose constitutive equation for the Cauchy stress follows a power law with exponent~$p$, we prove that, for~$p\ge 2$ and for initial data with finite energy, there always exists a global-in-time weak solution that additionally satisfies the entropy equality. The main novelty of this work is that we rigorously establish the continuity of temperature with respect to time, a property not previously proven in this setting. This time continuity allows us to construct a Lyapunov functional designed specifically for the problem, which in turn implies that the steady solution is nonlinearly stable and attracts all suitable weak solutions, even exponentially.

On the continuity in time and the exponential decay of solutions to a generalized Navier--Stokes--Fourier system in 2D

TL;DR

The paper analyzes a 2D incompressible non-Newtonian Navier–Stokes–Fourier system with under Dirichlet boundary conditions, proving the global existence of weak solutions that satisfy an entropy equality and, importantly, continuity in time of the temperature . The authors employ Galerkin approximations, a robust entropy framework, and Minty’s method to identify the constitutive stress and establish energy–entropy balance, including a minimum principle ensuring . A key novelty is the demonstration that is continuous in time in , enabling the construction of a problem-tailored Lyapunov functional and proving nonlinear exponential convergence to a steady state, even with inhomogeneous boundary data. Together, these results advance the understanding of thermodynamically consistent weak solutions for non-Newtonian, heat-conducting fluids in two dimensions and illuminate their long-time behavior and stability.

Abstract

We consider the flow of a generalized Newtonian incompressible heat-conducting fluid in a bounded domain, subject to homogeneous Dirichlet boundary conditions for velocity and Dirichlet boundary conditions for temperature. For a fluid whose constitutive equation for the Cauchy stress follows a power law with exponent~, we prove that, for~ and for initial data with finite energy, there always exists a global-in-time weak solution that additionally satisfies the entropy equality. The main novelty of this work is that we rigorously establish the continuity of temperature with respect to time, a property not previously proven in this setting. This time continuity allows us to construct a Lyapunov functional designed specifically for the problem, which in turn implies that the steady solution is nonlinearly stable and attracts all suitable weak solutions, even exponentially.
Paper Structure (13 sections, 1 theorem, 141 equations)

This paper contains 13 sections, 1 theorem, 141 equations.

Table of Contents

  1. Introduction
  2. Constitutive relations
  3. Notation
  4. Assumptions on data
  5. Main results
  6. Proof of the existence theorem
  7. Galerkin approximations
  8. Limit passage M
  9. Minimum principle
  10. Uniform estimates and limit passages in N
  11. Initial condition for temperature
  12. Entropy equality
  13. Continuity

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^2$ be a bounded domain with Lipschitz boundary and $T>0$. Assume that $\boldsymbol{\mathcal{S}}^\ast$ and $\kappa$ satisfy kap--tensor. Additionally, assume that $\pmb f$, $\pmb u_0$, $\vartheta_0$, and $\hat{\vartheta}$ fulfill conditions--muconditions. Then there exi and satisfying i1--ic in the following sense: Momentum equation: The Cauchy stress is of the form $

Theorems & Definitions (1)

  • Theorem 1.1: Existence of a solution fulfilling entropy equality