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

Siran Li, Haitao Wang, Jianing Yang

TL;DR

The paper proves global well-posedness for small BV∩L^1 perturbations of a 1D dynamic combustion Navier–Stokes model. It develops a BV-aware Green's-function framework to handle nonlinear coupling with the reactant fraction and to propagate BV perturbations globally in time. Local existence is achieved via an iterative scheme with BV-controlled estimates, which is then extended globally through detailed singular/regular Green’s-function analysis and a stopping-time argument, yielding optimal $t^{-1/2}$ decay. The results extend recent BV-based global well-posedness theories from 1D isentropic NS and NS–FT systems to reactive flow models, providing a rigorous foundation for the long-time behavior of BV-weak solutions in combustion dynamics.

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].

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

TL;DR

The paper proves global well-posedness for small BV∩L^1 perturbations of a 1D dynamic combustion Navier–Stokes model. It develops a BV-aware Green's-function framework to handle nonlinear coupling with the reactant fraction and to propagate BV perturbations globally in time. Local existence is achieved via an iterative scheme with BV-controlled estimates, which is then extended globally through detailed singular/regular Green’s-function analysis and a stopping-time argument, yielding optimal decay. The results extend recent BV-based global well-posedness theories from 1D isentropic NS and NS–FT systems to reactive flow models, providing a rigorous foundation for the long-time behavior of BV-weak solutions in combustion dynamics.

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 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 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].
Paper Structure (17 sections, 44 theorems, 305 equations)

This paper contains 17 sections, 44 theorems, 305 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heat kernel with constant conductivity coefficient
  4. BV functions on ${\mathbb{R}}$
  5. Heat kernel with BV-conductivity coefficient
  6. A lemma from Fourier anlaysis
  7. Local solution
  8. Iteration scheme
  9. Convergence of the iteration scheme
  10. Improved regularity in time
  11. Stability and Uniqueness
  12. Green's Function
  13. High-Frequency Analysis ($\eta\rightarrow \infty$), Singular Part
  14. Low-Frequency Analysis ($\eta\rightarrow 0$), Regular Part
  15. Global Well-posedness
  16. ...and 2 more sections

Key Result

Theorem 2

There exists a universal constant $\delta>0$ such that the following holds. Suppose that the initial data $(v_0, u_0, \theta_0, z_0)\equiv (u,v,\theta,z)|_{\{t=0\}}$ satisfy ini. Then there exists $t_{\sharp}>0$ such that Eq. PDE,2 (with Lipschitz continuous reacting rate function $\phi$) admits a w

Theorems & Definitions (78)

  • Definition 1
  • Theorem 2: Local existence and regularity
  • Theorem 3: Stability and uniqueness
  • Theorem 4: Global existence and large-time behaviour
  • Definition 1.1
  • Lemma 1.1: Lemma 2.1 ChenK2024
  • Definition 1.2: Fundamental solution
  • Lemma 1.2: Lemma 2.6 in LiuTP2022
  • Lemma 1.3: Liu--Yu LiuTP2022
  • Lemma 1.4: Lemma 2.2 in Wang--Yu--Zhang WangHT2022
  • ...and 68 more