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Asymptotic stability of composite waves of two viscous shocks for relaxed compressible Navier-Stokes equations

Renyong Guan, Yuxi Hu

TL;DR

The paper analyzes the time-asymptotic stability of composite waves formed by two viscous shocks in the 1D relaxed compressible Navier–Stokes equations with Maxwell-type relaxation. It develops a framework combining relative entropy and the $a$-contraction with shifts to prove nonlinear stability under small, independent shock strengths and perturbations, and demonstrates that solutions converge to the classical Navier–Stokes system as the relaxation parameter $ au$ vanishes. Central to the result are energy methods: L^2 relative-entropy estimates, higher-order control, and dissipative bounds that manage the weaker dissipation of the relaxed system. The work also establishes a vanishing-relaxation limit in which the relaxed system converges to the classical isentropic NS equations, with the stress converging to $ ilde au^0= rac{ u}{v^0}(u^0_x)$. Overall, the results extend stability of composite viscous shocks to relaxed models and provide a rigorous link to the classical theory via the relaxation limit.

Abstract

This paper investigates the time asymptotic stability of composite waves formed by two shock waves within the context of one-dimensional relaxed compressible Navier-Stokes equations. We demonstrate that the composite waves consisting of two viscous shocks achieve asymptotic nonlinear stability under the condition of having two small, independent wave strengths and the presence of minor initial perturbations. Furthermore, the solutions of the relaxed system are observed to globally converge over time to those of the classical system as the relaxation parameter approaches zero. The methodologies are grounded in relative entropy, the $a$-contraction with shifts theory and fundamental energy estimates.

Asymptotic stability of composite waves of two viscous shocks for relaxed compressible Navier-Stokes equations

TL;DR

The paper analyzes the time-asymptotic stability of composite waves formed by two viscous shocks in the 1D relaxed compressible Navier–Stokes equations with Maxwell-type relaxation. It develops a framework combining relative entropy and the -contraction with shifts to prove nonlinear stability under small, independent shock strengths and perturbations, and demonstrates that solutions converge to the classical Navier–Stokes system as the relaxation parameter vanishes. Central to the result are energy methods: L^2 relative-entropy estimates, higher-order control, and dissipative bounds that manage the weaker dissipation of the relaxed system. The work also establishes a vanishing-relaxation limit in which the relaxed system converges to the classical isentropic NS equations, with the stress converging to . Overall, the results extend stability of composite viscous shocks to relaxed models and provide a rigorous link to the classical theory via the relaxation limit.

Abstract

This paper investigates the time asymptotic stability of composite waves formed by two shock waves within the context of one-dimensional relaxed compressible Navier-Stokes equations. We demonstrate that the composite waves consisting of two viscous shocks achieve asymptotic nonlinear stability under the condition of having two small, independent wave strengths and the presence of minor initial perturbations. Furthermore, the solutions of the relaxed system are observed to globally converge over time to those of the classical system as the relaxation parameter approaches zero. The methodologies are grounded in relative entropy, the -contraction with shifts theory and fundamental energy estimates.
Paper Structure (12 sections, 15 theorems, 301 equations)

This paper contains 12 sections, 15 theorems, 301 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Traveling wave
  4. Construction of shifts
  5. Local solution and a priori estimates
  6. Proof of the Proposition \ref{['p1']}
  7. $L^2$ energy estimates
  8. High-order energy estimates
  9. Dissipative estimates
  10. Proof of Theorem \ref{['th1.2']}
  11. proof of Lemma \ref{['le3.10']}
  12. proof of Lemma \ref{['le4.3']}

Key Result

Theorem 1.1

Let the relaxation parameter $\tau$ satisfy For a given constant state $(v_+,u_+)\in \mathbb{R}_+\times \mathbb{R}$, there exist constants $\delta_0,\varepsilon_0>0$ such that the following holds true. For any constant states $(v_-,u_-)$ and $(v_m,u_m)$ satisfying 3.15-1 with Denote $(\widetilde{v}_i,\widetilde{u}_i,\widetilde{\Pi}_i)(x-\sigma_i t)$ the i-viscous shock solution that are the trav

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 3.1
  • Proposition 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 10 more