Global Existence and Large-time Behavior of Solutions to the Cauchy Problem of One-dimensional Viscous Radiative and Reactive Gas

TL;DR

This paper establishes the global existence and asymptotic stability of smooth solutions to the one-dimensional viscous radiative and reactive gas equations in an unbounded domain, addressing a gap in existing research.

Contribution

It constructs global smooth solutions for the Cauchy problem with large initial data and analyzes their long-term behavior, which was previously unexplored in unbounded domains.

Findings

Proved uniform positive bounds on specific volume and temperature.

Demonstrated nonlinear stability of the equilibrium state.

Extended results to unbounded domains for the first time.

Abstract

Although there are many results on the global solvability and the precise description of the large time behaviors of solutions to the initial-boundary value problems of the one-dimensional viscous radiative and reactive gas in bounded domain with two typical types of boundary conditions, no result is available up to now for the corresponding problems in unbounded domain. This paper focuses on the Cauchy problem of such a system with prescribed large initial data and the main purpose is to construct its global smooth non-vacuum solutions around a non-vacuum constant equilibrium state and to study the time-asymptotically nonlinear stability of such an equilibrium state. The key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the temperature.