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Relative Entropy Contractions for Extremal Shocks of Nonlinear Hyperbolic Systems without Genuine Nonlinearity

Jeffrey Cheng

TL;DR

The work addresses stability of extremal shocks in 1-D hyperbolic systems that lack genuine nonlinearity, focusing on $1$- and $n$-shocks in concave-convex or convex-concave characteristic fields. It extends the relative entropy framework by employing $a$-contraction with shifts to obtain $L^2$-stability up to a moving interface for shocks satisfying the Lax condition, including applications to nonlinear elastodynamics. Two principal results establish (i) $L^2$-stability up to shift for shocks under suitable field structure, and (ii) stability for shocks of moderate strength in the opposite ordering, with a scalar case allowing removal of the moderate-strength restriction. The approach hinges on constructing a dissipation functional with a shift, proving non-positivity of cont-dissipation on a convex set and for admissible shocks, and then deducing the contraction which yields stability and quantitative bounds. The findings broaden the class of systems for which classical shocks are provably stable in $L^2$, providing concrete insights for elastodynamics and scalar models with non-convex fluxes, and offering a robust tool for analyzing discontinuous solutions in non-GNL contexts.

Abstract

We study extremal shocks of $1$-d hyperbolic systems of conservation laws which fail to be genuinely nonlinear. More specifically, we consider either $1$- or $n$-shocks in characteristic fields which are either concave-convex or convex-concave in the sense of LeFloch. We show that the theory of $a$-contraction can be applied to obtain $L^2$-stability up to shift for these shocks in a class of weak solutions to the conservation law whose shocks obey the Lax entropy condition. Our results apply in particular to the $2 \times 2$ system of nonlinear elastodynamics.

Relative Entropy Contractions for Extremal Shocks of Nonlinear Hyperbolic Systems without Genuine Nonlinearity

TL;DR

The work addresses stability of extremal shocks in 1-D hyperbolic systems that lack genuine nonlinearity, focusing on - and -shocks in concave-convex or convex-concave characteristic fields. It extends the relative entropy framework by employing -contraction with shifts to obtain -stability up to a moving interface for shocks satisfying the Lax condition, including applications to nonlinear elastodynamics. Two principal results establish (i) -stability up to shift for shocks under suitable field structure, and (ii) stability for shocks of moderate strength in the opposite ordering, with a scalar case allowing removal of the moderate-strength restriction. The approach hinges on constructing a dissipation functional with a shift, proving non-positivity of cont-dissipation on a convex set and for admissible shocks, and then deducing the contraction which yields stability and quantitative bounds. The findings broaden the class of systems for which classical shocks are provably stable in , providing concrete insights for elastodynamics and scalar models with non-convex fluxes, and offering a robust tool for analyzing discontinuous solutions in non-GNL contexts.

Abstract

We study extremal shocks of -d hyperbolic systems of conservation laws which fail to be genuinely nonlinear. More specifically, we consider either - or -shocks in characteristic fields which are either concave-convex or convex-concave in the sense of LeFloch. We show that the theory of -contraction can be applied to obtain -stability up to shift for these shocks in a class of weak solutions to the conservation law whose shocks obey the Lax entropy condition. Our results apply in particular to the system of nonlinear elastodynamics.
Paper Structure (11 sections, 14 theorems, 81 equations, 3 figures)

This paper contains 11 sections, 14 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction & main results
  2. Concave-convex and convex-concave characteristic fields
  3. Main results
  4. General framework
  5. Assumptions on the system
  6. The method of $a$-contraction with shifts
  7. Proof of Theorems \ref{['main1']} & \ref{['main2']}
  8. Proof of Propositions \ref{['mainprop1']} & \ref{['mainprop2']}
  9. Applications
  10. Scalar conservation laws
  11. Nonlinear elastodynamics

Key Result

Lemma 1.1

There exists $c^*, c^{**} > 0$ such that for all $(u,v) \in \mathcal{V} \times \mathcal{V}$:

Figures (3)

  • Figure 1: Stability in the scalar setting
  • Figure 2: Stability for $1$-shocks
  • Figure 3: Stability for $2$-shocks

Theorems & Definitions (29)

  • Lemma 1.1: from MR2807139, MR2508169
  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • ...and 19 more