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Asymptotic stability of planar viscous shock wave to three-dimensional relaxed compressible Navier-Stokes equations

Renyong Guan, Yuxi Hu

TL;DR

The paper proves nonlinear time-asymptotic stability of shifted planar viscous shocks for the 3-D relaxed compressible Navier–Stokes equations with a Maxwell-type relaxation model. It combines the relative entropy method with $a$-contraction with shifts, along with weighted energy and high-order dissipation estimates, to obtain global-in-time stability under small shock strength and perturbations. A key technical advance is closing the energy estimates uniformly in the relaxation parameter $\tau$ by introducing enhanced dissipation terms and leveraging a 3-D Poincaré inequality. Moreover, the authors establish a relaxation limit as $\tau\to0$, showing convergence to the classical 3-D Navier–Stokes system and its planar shock, thereby validating the relaxed model as a singular perturbation of the Newtonian theory in the planar shock regime.

Abstract

This paper establishes the nonlinear time-asymptotic stability of shifted planar viscous shock waves for the three-dimensional relaxed compressible Navier-Stokes equations, in which a modified Maxwell-type model replaces the classical Newtonian constitutive relation. Under the assumptions of sufficiently small shock strength and initial perturbations, we prove that planar viscous shock waves are nonlinearly stable. The main steps of our analysis are as follows. First, using the relative entropy method together with the framework of $a$-contraction with shifts, we derive energy estimates for the weighted relative entropy of perturbations. We then successively obtain high-order and dissipation estimates via direct energy arguments, which provide the required a priori bounds. Combining these estimates with a local existence result, we establish the global asymptotic stability of the shifted planar viscous shock wave. Finally, we show that as the relaxation parameter tends to zero, solutions of the relaxed system converge globally in time to those of the classical Navier-Stokes system.

Asymptotic stability of planar viscous shock wave to three-dimensional relaxed compressible Navier-Stokes equations

TL;DR

The paper proves nonlinear time-asymptotic stability of shifted planar viscous shocks for the 3-D relaxed compressible Navier–Stokes equations with a Maxwell-type relaxation model. It combines the relative entropy method with -contraction with shifts, along with weighted energy and high-order dissipation estimates, to obtain global-in-time stability under small shock strength and perturbations. A key technical advance is closing the energy estimates uniformly in the relaxation parameter by introducing enhanced dissipation terms and leveraging a 3-D Poincaré inequality. Moreover, the authors establish a relaxation limit as , showing convergence to the classical 3-D Navier–Stokes system and its planar shock, thereby validating the relaxed model as a singular perturbation of the Newtonian theory in the planar shock regime.

Abstract

This paper establishes the nonlinear time-asymptotic stability of shifted planar viscous shock waves for the three-dimensional relaxed compressible Navier-Stokes equations, in which a modified Maxwell-type model replaces the classical Newtonian constitutive relation. Under the assumptions of sufficiently small shock strength and initial perturbations, we prove that planar viscous shock waves are nonlinearly stable. The main steps of our analysis are as follows. First, using the relative entropy method together with the framework of -contraction with shifts, we derive energy estimates for the weighted relative entropy of perturbations. We then successively obtain high-order and dissipation estimates via direct energy arguments, which provide the required a priori bounds. Combining these estimates with a local existence result, we establish the global asymptotic stability of the shifted planar viscous shock wave. Finally, we show that as the relaxation parameter tends to zero, solutions of the relaxed system converge globally in time to those of the classical Navier-Stokes system.

Paper Structure

This paper contains 10 sections, 15 theorems, 342 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Planar viscous shock wave
  4. Construction of shift
  5. Local solution and a priori estimates
  6. Proof of proposition \ref{['p1']}
  7. Estimates for weighted relative entropy
  8. High-order energy estimates
  9. Dissipative estimates
  10. Proof of Theorem \ref{['th1.2']}

Key Result

Theorem 1.1

Assume the constant states $(v_\pm, u_{1\pm}) \in \mathbb{R}^+ \times \mathbb{R}$ satisfy 1.11rh-1.12, and the relaxation parameter $\tau$ satisfies where $\sigma_\ast=\sqrt{\frac{p(v_+)-p(v_-)}{v_--v_+}}$. Let $(v^s, u^s, \Pi_1^s, \Pi_2^s)$ be the planar 2-viscous shock solutions of 1.6-1.13. Then, there exist constants $\delta_0,\varepsilon_0>0$ such that for any initial data $(v_0, u_0, \Pi_{1

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Proposition 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • ...and 11 more