Global well-posedness for one-dimensional compressible Navier--Stokes system in dynamic combustion with small $BV\cap L^1$ initial data

TL;DR

This paper proves the global existence, uniqueness, and stability of small BV weak solutions to a one-dimensional compressible Navier--Stokes system modeling dynamic combustion, extending previous results to reactive gas mixtures.

Contribution

It establishes the global well-posedness for small BV initial data in a reactive gas model, including stability and long-time behavior analysis.

Findings

Global existence of weak solutions for small BV initial data

Uniqueness and stability of solutions

Characterization of large-time behavior

Abstract

We establish the global well-posedness theory of small BV weak solutions to a one-dimensional compressible Navier--Stokes model for reacting gas mixtures in dynamic combustion. The unknowns of the PDE system consist of the specific volume, velocity, temperature, and mass fraction of the reactant. For initial data that are small perturbations around the constant equilibrium state $(1, 0, 1, 0)$ in the $L^1(\mathbb{R}) \cap {\rm BV}(\mathbb{R})$-norm, we establish the local-in-time existence of weak solutions via an iterative scheme, show the stability and uniqueness of local weak solutions, and prove the global-in-time existence of solutions for initial data with small BV-norm via an analysis of the Green's function of the linearised system. The large-time behaviour of the global BV weak solutions is also characterised. This work is motivated by and extends the recent global well-posedness theory for BV weak solutions to the one-dimensional isentropic Navier--Stokes and Navier--Stokes--Fourier systems developed in [T.-P. Liu, S.-H. Yu, Commun. Pure Appl. Math. 75 (2022), 223--348] and [H. Wang, S.-H. Yu, X. Zhang, Arch. Ration. Mech. Anal. 245 (2022), 375--477].