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Global well-posedness and relaxation limit for relaxed compressible Navier-Stokes-Fourier equations in bounded domain

Yuxi Hu, Xiaoning Zhao

TL;DR

This work studies a one-dimensional relaxed Navier–Stokes–Fourier system in a bounded domain with Cattaneo heat conduction and Maxwell-type stress, reformulated in Lagrangian coordinates to reveal a uniform characteristic boundary. The authors develop an approximate non-characteristic boundary problem and prove local well-posedness via maximal nonnegative boundary conditions, then derive a weighted energy framework that yields uniform a priori estimates independent of the relaxation parameters. Using these estimates, they obtain global solvability for the approximate system and employ compactness to pass to the limit, obtaining a global solution for the original system and establishing a global relaxation limit to the classical Navier–Stokes–Fourier equations. A key structural cancellation of higher-order boundary terms enables closure of the energy estimates, highlighting the integrative role of the energy method in handling strongly coupled hyperbolic–parabolic dynamics. Overall, the paper provides a rigorous foundation for global solvability and relaxation behavior in 1D relaxed NS-F models and offers methods potentially extendable to similar systems.

Abstract

This paper investigates an initial boundary value problem for the relaxed one-dimensional compressible Navier-Stokes-Fourier equations. By transforming the system into Lagrangian coordinates, the resulting formulation exhibits a uniform characteristic boundary structure. We first construct an approximate system with non-characteristic boundaries and establish its local well-posedness by verifying the maximal nonnegative boundary conditions. Subsequently, through the construction of a suitable weighted energy functional and careful treatment of boundary terms, we derive uniform a priori estimates, thereby proving the global well-posedness of smooth solutions for the approximate system. Utilizing these uniform estimates and standard compactness arguments, we further obtain the existence and uniqueness of global solutions for the original system. In addition, the global relaxation limit is established. The analysis is fundamentally based on energy estimates.

Global well-posedness and relaxation limit for relaxed compressible Navier-Stokes-Fourier equations in bounded domain

TL;DR

This work studies a one-dimensional relaxed Navier–Stokes–Fourier system in a bounded domain with Cattaneo heat conduction and Maxwell-type stress, reformulated in Lagrangian coordinates to reveal a uniform characteristic boundary. The authors develop an approximate non-characteristic boundary problem and prove local well-posedness via maximal nonnegative boundary conditions, then derive a weighted energy framework that yields uniform a priori estimates independent of the relaxation parameters. Using these estimates, they obtain global solvability for the approximate system and employ compactness to pass to the limit, obtaining a global solution for the original system and establishing a global relaxation limit to the classical Navier–Stokes–Fourier equations. A key structural cancellation of higher-order boundary terms enables closure of the energy estimates, highlighting the integrative role of the energy method in handling strongly coupled hyperbolic–parabolic dynamics. Overall, the paper provides a rigorous foundation for global solvability and relaxation behavior in 1D relaxed NS-F models and offers methods potentially extendable to similar systems.

Abstract

This paper investigates an initial boundary value problem for the relaxed one-dimensional compressible Navier-Stokes-Fourier equations. By transforming the system into Lagrangian coordinates, the resulting formulation exhibits a uniform characteristic boundary structure. We first construct an approximate system with non-characteristic boundaries and establish its local well-posedness by verifying the maximal nonnegative boundary conditions. Subsequently, through the construction of a suitable weighted energy functional and careful treatment of boundary terms, we derive uniform a priori estimates, thereby proving the global well-posedness of smooth solutions for the approximate system. Utilizing these uniform estimates and standard compactness arguments, we further obtain the existence and uniqueness of global solutions for the original system. In addition, the global relaxation limit is established. The analysis is fundamentally based on energy estimates.

Paper Structure

This paper contains 4 sections, 19 theorems, 273 equations.

Table of Contents

  1. Introduction
  2. approximate system and local existence
  3. uniform a prior estimates
  4. Passing to the limit and proof of main theorems

Key Result

Theorem 1.2

Let Assumption assu be satisfied. Then, there exists a constant $\epsilon_0>0$ such that if the system yuan-boundary has a unique global solution $(v,u,\theta,q,S)\in C^1([0,\infty)\times\Omega)$ with for any $\delta_0>0$ and satisfying where $C$ is a universal constant independent of $\tau$.

Theorems & Definitions (33)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Remark 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 23 more