Global well‐posedness for one‐dimensional compressible Navier–Stokes system in dynamic combustion with small BV∩L1 initial data

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 partial differential equation (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 in the ‐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 linearized system. The large‐time behavior of the global BV weak solutions is also characterized. 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.