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Viscous Destabilization for Large Shocks of Conservation Laws

Paul Blochas, Jeffrey Cheng

TL;DR

This work uncovers a viscous destabilization phenomenon for large shocks in 1D conservation laws by showing that the a-contraction with shifts can fail uniformly in shock amplitude for certain scalar fluxes and for a barotropic Navier-Stokes system. The authors deploy the relative entropy method with a weighted contraction and a shift X(t), constructing carefully chosen perturbations that render the time derivative of the weighted relative entropy positive at t=0, then extend the argument to Lipschitz shifts through a robust approximation framework. The key contribution is demonstrating that viscous stabilization observed for small shocks does not extend uniformly to large shocks, highlighting that a-contraction is a stronger requirement than nonlinear stability. The results thereby separate the regimes of inviscid and viscous stability and suggest fundamental limitations of a-contraction as a universal tool for large-shock stability in viscous systems. The findings have implications for understanding the precise mechanisms by which viscosity interacts with nonlinear wave structures in 1D conservation laws.

Abstract

The recent theory of $a-$contraction with shifts provides $L^2$-stability for shock waves of $1-$D hyperbolic systems of conservation laws. The theory has been established at the inviscid level uniformly in the shock amplitude, and at the viscous level for small shocks. In this work, we investigate whether the $a-$contraction property holds uniformly in the shock amplitude for some specific systems with viscosity. We show that in some cases, the $a-$contraction fails for sufficiently large shocks. This showcases a "viscous destabilization" effect in the sense that the $a$-contraction property is verified for the inviscid model, but can fail for the viscous one. This also shows that the $a$-contraction property, even among small perturbations, is stronger than the classical notion of nonlinear stability, which is known to hold regardless of shock amplitude for viscous scalar conservation laws.

Viscous Destabilization for Large Shocks of Conservation Laws

TL;DR

This work uncovers a viscous destabilization phenomenon for large shocks in 1D conservation laws by showing that the a-contraction with shifts can fail uniformly in shock amplitude for certain scalar fluxes and for a barotropic Navier-Stokes system. The authors deploy the relative entropy method with a weighted contraction and a shift X(t), constructing carefully chosen perturbations that render the time derivative of the weighted relative entropy positive at t=0, then extend the argument to Lipschitz shifts through a robust approximation framework. The key contribution is demonstrating that viscous stabilization observed for small shocks does not extend uniformly to large shocks, highlighting that a-contraction is a stronger requirement than nonlinear stability. The results thereby separate the regimes of inviscid and viscous stability and suggest fundamental limitations of a-contraction as a universal tool for large-shock stability in viscous systems. The findings have implications for understanding the precise mechanisms by which viscosity interacts with nonlinear wave structures in 1D conservation laws.

Abstract

The recent theory of contraction with shifts provides -stability for shock waves of D hyperbolic systems of conservation laws. The theory has been established at the inviscid level uniformly in the shock amplitude, and at the viscous level for small shocks. In this work, we investigate whether the contraction property holds uniformly in the shock amplitude for some specific systems with viscosity. We show that in some cases, the contraction fails for sufficiently large shocks. This showcases a "viscous destabilization" effect in the sense that the -contraction property is verified for the inviscid model, but can fail for the viscous one. This also shows that the -contraction property, even among small perturbations, is stronger than the classical notion of nonlinear stability, which is known to hold regardless of shock amplitude for viscous scalar conservation laws.
Paper Structure (38 sections, 17 theorems, 152 equations)

This paper contains 38 sections, 17 theorems, 152 equations.

Table of Contents

  1. Introduction & main results
  2. Introduction
  3. Previous work
  4. The relative entropy method
  5. The theory of $a$-contraction
  6. Large shock waves
  7. Related work
  8. Results
  9. Proof outlines
  10. Preliminaries for scalar case
  11. Existence and uniqueness of perturbations
  12. Relative entropy structure
  13. Reduction of the shift condition
  14. Estimates for $A$
  15. Proof of Theorem \ref{['maintheorem']}
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $A:\mathbb{R} \to \mathbb{R}$ be smooth, strictly convex, such that $A'(0)=0$ and such that for every real valued polynomial $P$ there exists $x$ in $\mathbb{R}_-$ such that $P(x)>A'(x)$. Fix $\tilde{a} \in W^{2,\infty}([0,1])$. Let $u_-=0$. There exists a sequence of positive numbers $(K_n)_{n where $u$ is the solution to (viscous) with initial data $u_0$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.3
  • ...and 28 more