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Time-asymptotic stability of composite wave of viscous shocks and viscous contact wave for Navier-Stokes-Fourier equations

Xushan Huang, Hobin Lee

TL;DR

The paper proves nonlinear time-asymptotic stability for a composite viscous wave in the 1D Navier–Stokes–Fourier system, composed of two shifted viscous shocks and a viscous contact wave. It employs the a-contraction method with time-dependent shifts and carefully constructed weight functions to handle wave interactions, establishing global existence of a strong solution and convergence to the composite profile as $t\to\infty$, up to shifts with vanishing speeds. The analysis builds a robust a priori framework, including zeroth- and higher-order energy estimates and a relative entropy approach, to control perturbations in $H^1$ and dissipation terms, while ensuring wave separation. The results extend the stability theory from single waves to complex multi-wave composites, providing insight into the long-time behavior of compressible NSF flows with multiple interacting viscous waves.

Abstract

We investigate the nonlinear time-asymptotic stability of the composite wave consisting of two viscous shocks and a viscous contact discontinuity for the one-dimensional compressible Navier-Stokes-Fourier (NSF) equations. Specifically, we establish that if the composite wave strength and the perturbations are sufficiently small, the NSF system admits a unique global-in-time strong solution, which converges uniformly in space as time tends to infinity, towards the corresponding composite wave, up to dynamical shifts in the positions of the two viscous shocks. Notably, the strengths of the two viscous shocks can be chosen independently. Our proof relies upon the $a$-contraction method with time-dependent shifts and suitable weight functions.

Time-asymptotic stability of composite wave of viscous shocks and viscous contact wave for Navier-Stokes-Fourier equations

TL;DR

The paper proves nonlinear time-asymptotic stability for a composite viscous wave in the 1D Navier–Stokes–Fourier system, composed of two shifted viscous shocks and a viscous contact wave. It employs the a-contraction method with time-dependent shifts and carefully constructed weight functions to handle wave interactions, establishing global existence of a strong solution and convergence to the composite profile as , up to shifts with vanishing speeds. The analysis builds a robust a priori framework, including zeroth- and higher-order energy estimates and a relative entropy approach, to control perturbations in and dissipation terms, while ensuring wave separation. The results extend the stability theory from single waves to complex multi-wave composites, providing insight into the long-time behavior of compressible NSF flows with multiple interacting viscous waves.

Abstract

We investigate the nonlinear time-asymptotic stability of the composite wave consisting of two viscous shocks and a viscous contact discontinuity for the one-dimensional compressible Navier-Stokes-Fourier (NSF) equations. Specifically, we establish that if the composite wave strength and the perturbations are sufficiently small, the NSF system admits a unique global-in-time strong solution, which converges uniformly in space as time tends to infinity, towards the corresponding composite wave, up to dynamical shifts in the positions of the two viscous shocks. Notably, the strengths of the two viscous shocks can be chosen independently. Our proof relies upon the -contraction method with time-dependent shifts and suitable weight functions.

Paper Structure

This paper contains 22 sections, 13 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Main ideas
  4. Preliminaries
  5. Viscous shock wave
  6. Viscous contact wave
  7. Poincaré-type inequality
  8. A priori estimate and proof of the main theorem
  9. Local existence of solutions
  10. Construction of the weight function
  11. Construction of shifts
  12. A priori estimate
  13. Conclusion
  14. Zeroth Order Estimates
  15. Wave interaction estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(v_+,u_+,\theta_+) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+$ be a given constant state. Then there exist positive constants $\delta_0$ and $\varepsilon_0$ such that the following assertions hold. Assume that the states $(v_-,u_-,\theta_-)$, $(v_*,u_*,\theta_*)$, and $(v^*,u^*,\the and are sufficiently close in the sense that Let $(\widetilde{v}_1, \widetilde{u}_1, \widetilde{\t

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3: KV21, Lemma 2.9
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • ...and 11 more