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

TL;DR

This paper proves the global well-posedness of solutions and the relaxation limit for relaxed compressible Navier-Stokes-Fourier equations in bounded domains using energy estimates and boundary analysis.

Contribution

It introduces a new approach to establish global solutions and relaxation limits for the relaxed compressible Navier-Stokes-Fourier system in bounded domains.

Findings

Established global well-posedness of smooth solutions.

Proved the existence and uniqueness of solutions for the original system.

Derived the global relaxation limit.

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.