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Stability of Riemann Shocks for isothermal Euler by Inviscid limits of global-in-time large Navier-Stokes flows

Saehoon Eo, Namhyun Eun, Moon-Jin Kang, HyeonSeop Oh

TL;DR

The work addresses the stability of Riemann shocks for the one-dimensional isothermal Euler system by analyzing vanishing viscosity limits of the corresponding Navier-Stokes system with potentially degenerate viscosity. It establishes global existence of large strong solutions to the 1D isothermal NS with density bounded away from vacuum and constructs viscous shocks as the NS counterpart to Euler shocks. Central to the results is a contraction framework based on relative entropy and Bresch-Desjardins entropy, combined with a shift-coupled, weighted estimate that yields uniform-in-$\nu$ controls and stability of shocks under vanishing viscosity. Consequently, the paper provides a rigorous justification for the inviscid limit stability of Riemann shocks in the isothermal setting, including existence, uniqueness (up to shifts), and convergence in the vanishing-viscosity regime. The findings offer a framework for understanding shock stability in degenerate-viscosity isothermal flows and contribute to the broader theory of inviscid limits for hyperbolic conservation laws.

Abstract

In this paper, we study the isothermal gas dynamics. We first establish the global existence of strong solutions to the one-dimensional isothermal Navier-Stokes system for smooth initial data without any smallness conditions, assuming that the initial density has strictly positive lower bound. The existence result allows for possibly degenerate viscosity coefficients and admits different asymptotic states at the far fields. We then prove a contraction property for the strong solutions perturbed from viscous shocks, yielding uniform estimates with respect to the viscosity coefficients. This covers any large perturbations, and consequently, we establish the inviscid limits and their stability estimate. In other words, we demonstrate the stability of Riemann shocks to the one-dimensional isothermal Euler system in the class of vanishing viscosity limits of the associated Navier-Stokes system.

Stability of Riemann Shocks for isothermal Euler by Inviscid limits of global-in-time large Navier-Stokes flows

TL;DR

The work addresses the stability of Riemann shocks for the one-dimensional isothermal Euler system by analyzing vanishing viscosity limits of the corresponding Navier-Stokes system with potentially degenerate viscosity. It establishes global existence of large strong solutions to the 1D isothermal NS with density bounded away from vacuum and constructs viscous shocks as the NS counterpart to Euler shocks. Central to the results is a contraction framework based on relative entropy and Bresch-Desjardins entropy, combined with a shift-coupled, weighted estimate that yields uniform-in- controls and stability of shocks under vanishing viscosity. Consequently, the paper provides a rigorous justification for the inviscid limit stability of Riemann shocks in the isothermal setting, including existence, uniqueness (up to shifts), and convergence in the vanishing-viscosity regime. The findings offer a framework for understanding shock stability in degenerate-viscosity isothermal flows and contribute to the broader theory of inviscid limits for hyperbolic conservation laws.

Abstract

In this paper, we study the isothermal gas dynamics. We first establish the global existence of strong solutions to the one-dimensional isothermal Navier-Stokes system for smooth initial data without any smallness conditions, assuming that the initial density has strictly positive lower bound. The existence result allows for possibly degenerate viscosity coefficients and admits different asymptotic states at the far fields. We then prove a contraction property for the strong solutions perturbed from viscous shocks, yielding uniform estimates with respect to the viscosity coefficients. This covers any large perturbations, and consequently, we establish the inviscid limits and their stability estimate. In other words, we demonstrate the stability of Riemann shocks to the one-dimensional isothermal Euler system in the class of vanishing viscosity limits of the associated Navier-Stokes system.

Paper Structure

This paper contains 22 sections, 23 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Global existence of strong solutions to Navier-Stokes systems
  3. Uniqueness and stability of Riemann shocks to Euler system
  4. Proof of Theorem \ref{['thm:MV']} and \ref{['thm:KV-AP']}: Existence
  5. Proof of Theorem \ref{['thm_main']}: Contraction Property
  6. Preliminaries: the method of $a$-contraction with shifts
  7. Transformation of the system \ref{['main0']}
  8. Global and local estimates on the relative quantities
  9. Properties of small shock waves
  10. Relative entropy method
  11. Construction of the weight function
  12. Maximization in terms of $h-\tilde{h}$
  13. Main proposition
  14. Proof of Theorem \ref{['thm_main']} from Proposition \ref{['prop:main']}
  15. Proof of Proposition \ref{['prop:main']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\rho_0$ and $u_0$ be the initial data such that for some constants $\underline{\kappa}_0,\overline{\kappa}_0$, and where $\bar{\rho}$ and $\bar{u}$ are smooth monotone functions such that Let $\nu:{\mathbb R}^+\to{\mathbb R}^+$ be a function such that for some constants $C>0$ and $q\in[0,1/2)$. and Then, there exists a global-in-time unique strong solution $(\rho,u)$ of NS-Eulerian-pressu

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • ...and 37 more