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Stability and instability in scalar balance laws: fronts and periodic waves

Vincent Duchêne, Luis Miguel Rodrigues

Abstract

We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is the elucidation of the prominent role of characteristic points in the determination of both the spectra of the linearized operators and the phase dynamics involved in the nonlinear large-time evolution. For a generic class of equations an upshot of our analysis is a dramatic reduction from a tremendously wide variety of entropic traveling waves to a relatively small range of stable entropic traveling waves.

Stability and instability in scalar balance laws: fronts and periodic waves

Abstract

We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is the elucidation of the prominent role of characteristic points in the determination of both the spectra of the linearized operators and the phase dynamics involved in the nonlinear large-time evolution. For a generic class of equations an upshot of our analysis is a dramatic reduction from a tremendously wide variety of entropic traveling waves to a relatively small range of stable entropic traveling waves.

Paper Structure

This paper contains 21 sections, 34 theorems, 214 equations, 2 figures.

Table of Contents

  1. Preliminaries on traveling waves
  2. Structure of profiles
  3. Notions of stability
  4. Nonlinear stability
  5. Spectral stability
  6. Instability mechanisms
  7. Instabilities at infinity
  8. Instabilities at characteristic points
  9. Instabilities of shock positions
  10. Specialization to periodic traveling waves
  11. Classification of traveling waves
  12. Stable continuous fronts
  13. Detailed structure
  14. Spectral stability
  15. Linear estimates
  16. ...and 6 more sections

Key Result

Proposition 1.2

Let $({\underline U},\sigma,D)$ define a piecewise regular traveling-wave solution to eq-u. Let $X$ be a connected component of $\mathbb{R}\setminus D$ such that and such that $F_\sigma:{\underline U}(X)\to\mathbb{R}$ defined by is locally Lipschitz near any of its zeroes. Then ${\underline U}$ is either constant or strictly monotonic on $X$. In particular, if this is true for any connected comp

Figures (2)

  • Figure 1: Classes of possibly stable non-degenerate piecewise regular traveling wave profiles, ${\underline U}$ (constant states being omitted). The profiles represented in Figures \ref{['F.class3']}, \ref{['F.class4']} and \ref{['F.class5']} pass through the characteristic value ${\underline u}_\star=0$ at the characteristic point $x_\star=0$. All the traveling waves represented have speed $\sigma=0$ and are spectrally and nonlinearly stable by Theorem \ref{['T.classification']}.
  • Figure 2: Stable waves of class \ref{['class4']} or \ref{['class5']} in Theorem \ref{['T.classification']}. The functions $f$ and $g$ used to trace the profiles are as in Figure \ref{['F.classes']}, specifically, $f(u) = -\cos(\tfrac{7}{4}\,u)$ and $g(u) = \sin(\pi\,u)$.

Theorems & Definitions (65)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Proposition 2.1
  • ...and 55 more