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Linear damping estimates for periodic roll wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws

L. Miguel Rodrigues, Kevin Zumbrun

TL;DR

This work proves that for periodic roll waves in the inviscid Saint-Venant framework and related hyperbolic balance laws, high-frequency spectral stability implies a Lyapunov-type linear damping estimate, provided the system size satisfies $n\le 6$; a new dimension-dependent linear-algebraic lemma underpins this threshold. The approach combines diagonalization into sonic and transverse scalar modes with a Kawashima-type energy method and carefully chosen weights to translate spectral gaps into exponential decay of $H^s$ energies, including the handling of sonic points. While the results lay a solid foundation for nonlinear stability analyses, extending to full nonlinear stability remains challenging due to degeneracies from phase shifts and an infinite-dimensional family of nearby traveling waves. The findings clarify when high-frequency stability yields energy damping and delineate a precise dimensional boundary, offering a robust tool for stability analyses in hyperbolic balance laws with discontinuities.

Abstract

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions 7 and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov's Lemma for ODE that is to our knowledge new.

Linear damping estimates for periodic roll wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws

TL;DR

This work proves that for periodic roll waves in the inviscid Saint-Venant framework and related hyperbolic balance laws, high-frequency spectral stability implies a Lyapunov-type linear damping estimate, provided the system size satisfies ; a new dimension-dependent linear-algebraic lemma underpins this threshold. The approach combines diagonalization into sonic and transverse scalar modes with a Kawashima-type energy method and carefully chosen weights to translate spectral gaps into exponential decay of energies, including the handling of sonic points. While the results lay a solid foundation for nonlinear stability analyses, extending to full nonlinear stability remains challenging due to degeneracies from phase shifts and an infinite-dimensional family of nearby traveling waves. The findings clarify when high-frequency stability yields energy damping and delineate a precise dimensional boundary, offering a robust tool for stability analyses in hyperbolic balance laws with discontinuities.

Abstract

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions 7 and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov's Lemma for ODE that is to our knowledge new.

Paper Structure

This paper contains 17 sections, 8 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Reader's guide
  3. Plan of the paper
  4. A linear algebraic lemma
  5. The Saint-Venant equations
  6. Spectral analysis
  7. Linear damping estimate
  8. Towards nonlinear stability
  9. General systems
  10. Spectral analysis
  11. Linear damping estimate
  12. Discussion and open problems
  13. Counterexample in ${\mathcal{M}}_4(\mathbb C)$, and extension to ${\mathcal{M}}_5(\mathbb R)$
  14. Extension to ${\mathcal{M}}_5(\mathbb R)$
  15. Counterexample in ${\mathcal{M}}_4(\mathbb C)$
  16. ...and 2 more sections

Key Result

Lemma 2.1

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • proof
  • Remark 4.1
  • Lemma 4.2
  • proof
  • ...and 10 more