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Global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction

Fucai Li, Houzhi Tang, Shuxing Zhang

TL;DR

The study proves global well-posedness for the compressible Navier–Stokes equations with Cattaneo heat conduction on $\mathbb{R}^3$ for small perturbations of the constant state and establishes optimal decay rates for the solution and heat flux. By reformulating around the steady state and employing a spectral Green-function analysis for the linearized system, the authors derive precise low-frequency decay while handling high-frequency damping through uniform energy estimates. The nonlinear analysis, using Duhamel's principle and a low–high frequency decomposition, yields sharp time-decay rates: $\|\nabla^k(n,w,\phi)\|_{L^2} \lesssim (1+t)^{-3/4-k/2}$ and $\|\nabla^k\psi\|_{L^2} \lesssim (1+t)^{-5/4-k/2}$, with corresponding lower bounds validating optimality. The results illustrate a fundamental difference from Fourier's law by showing a faster decay for the highest derivatives of the heat flux due to Cattaneo's damping, and they provide rigorous insight into the long-time behavior of hyperbolic heat conduction in compressible flows.

Abstract

The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is that whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida [Proc. Japan Acad. Ser. A Math. Sci., 55(9):337-342, 1979]. In this sense, our results first reveal the essential differences between the two laws.

Global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction

TL;DR

The study proves global well-posedness for the compressible Navier–Stokes equations with Cattaneo heat conduction on for small perturbations of the constant state and establishes optimal decay rates for the solution and heat flux. By reformulating around the steady state and employing a spectral Green-function analysis for the linearized system, the authors derive precise low-frequency decay while handling high-frequency damping through uniform energy estimates. The nonlinear analysis, using Duhamel's principle and a low–high frequency decomposition, yields sharp time-decay rates: and , with corresponding lower bounds validating optimality. The results illustrate a fundamental difference from Fourier's law by showing a faster decay for the highest derivatives of the heat flux due to Cattaneo's damping, and they provide rigorous insight into the long-time behavior of hyperbolic heat conduction in compressible flows.

Abstract

The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is that whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida [Proc. Japan Acad. Ser. A Math. Sci., 55(9):337-342, 1979]. In this sense, our results first reveal the essential differences between the two laws.
Paper Structure (15 sections, 18 theorems, 244 equations)

This paper contains 15 sections, 18 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Related Literature
  4. Main Motivation
  5. Primary challenge and ideas of the proof
  6. Preliminaries
  7. Notation
  8. Auxiliary lemmas
  9. Global existence of the classical solution
  10. Reformulated problem
  11. Time-independent energy estimates
  12. Large-time behavior of the classical solution
  13. Time-decay rates for the linearized problem
  14. Nonlinear time-decay estimates
  15. The Proofs of Theorems \ref{['thm1']} and \ref{['thm3']}

Key Result

Theorem 1.1

Let $s\geq 3$ be an integer. Assume that the initial data $(\rho_0-\rho_*,u_0,\theta_0-\theta_*,q_0)\in H^s(\mathbb{R}^3)$ satisfying for some sufficiently small constant $\delta_0>0$, then the Cauchy problem NSC-NSI-2 admits a unique global classical solution $(\rho,u,\theta,q)$ enjoying Moreover, if the initial perturbation $(\rho_0-\rho_*,u_0,\theta_0-\theta_*,q_0) \in L^1(\mathbb{R}^3)$, the

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 24 more