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On the problem of stability of viscous shocks

Sergey Bolotin, Dmitry Treschev

TL;DR

The paper investigates spectral stability of viscous shocks in systems of viscous conservation laws by introducing a tractable class of piecewise-linear vector fields and a regularization framework that reduces the stability problem to linear-algebraic computations. It proves that spectral problems for these low-regularity shocks are well-posed via a jump operator $S$ and a determinant condition $D(\lambda)$ (or $\Theta$) that governs eigenvalues, and it derives explicit stability/instability scenarios for shocks with a single discontinuity as well as in overcompressive regimes. A principal result demonstrates a bifurcation-driven instability in a two-discontinuity construction, with an eigenvalue $\lambda(\varepsilon)=c\varepsilon+o(\varepsilon)$ emerging under mild sign conditions, illustrating a Hopf-type mechanism for stability loss. The combination of exact piecewise-linear models and careful regularization yields transparent, analyzable insights into stability loss mechanisms that can be smoothed to produce smooth counterexamples, informing broader expectations for stability in more general vector fields. These results provide concrete, algebraic criteria and bifurcation pictures that complement numerical Evans-function analyses in higher-dimensional settings.

Abstract

We consider the problem of spectral stability of traveling wave solutions $u=γ(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $γ$ of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields $F$ for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.

On the problem of stability of viscous shocks

TL;DR

The paper investigates spectral stability of viscous shocks in systems of viscous conservation laws by introducing a tractable class of piecewise-linear vector fields and a regularization framework that reduces the stability problem to linear-algebraic computations. It proves that spectral problems for these low-regularity shocks are well-posed via a jump operator and a determinant condition (or ) that governs eigenvalues, and it derives explicit stability/instability scenarios for shocks with a single discontinuity as well as in overcompressive regimes. A principal result demonstrates a bifurcation-driven instability in a two-discontinuity construction, with an eigenvalue emerging under mild sign conditions, illustrating a Hopf-type mechanism for stability loss. The combination of exact piecewise-linear models and careful regularization yields transparent, analyzable insights into stability loss mechanisms that can be smoothed to produce smooth counterexamples, informing broader expectations for stability in more general vector fields. These results provide concrete, algebraic criteria and bifurcation pictures that complement numerical Evans-function analyses in higher-dimensional settings.

Abstract

We consider the problem of spectral stability of traveling wave solutions for a system of viscous conservation laws . Such solutions correspond to heteroclinic trajectories of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.

Paper Structure

This paper contains 19 sections, 27 theorems, 147 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Regularization of discontinuities
  3. Examples
  4. Shocks with a single discontinuity
  5. Proof of Theorem \ref{['theo:simple']}
  6. Overcompressive shocks
  7. Proof of Theorem \ref{['theo:over']}
  8. Proof of Theorem \ref{['theo:main']}
  9. The curves $\gamma^\pm$ and $\gamma^\varepsilon$
  10. Operator $\mathbf{L}$
  11. Eigenfunction of $\mathbf{L}$
  12. The function ${\bf v}$ for $\xi < \xi^-$ and $\xi > \xi^+$
  13. The function ${\bf v}$ for $\xi = \xi^- + 0$ and $\xi = \xi^+ - 0$
  14. The function ${\bf v}$ for $\xi^- < \xi < \xi^+$
  15. Computation of $\Theta$
  16. ...and 4 more sections

Key Result

Proposition 2.1

As $\mu\to 0$, $z^\mu(\xi)$ converges uniformly on $[-T,T]$ to $z(\xi)$, while $v^\mu(\xi)$ is uniformly bounded and converges pointwise on $[-T,T]\setminus \{0\}$ to $v(\xi)$. The functions $z(\xi),v(\xi)$ are smooth on $[-T,0]$ and $[0,T]$ and are solutions of the initial value problems where

Figures (2)

  • Figure 1: Overcompressive shock
  • Figure 2: Splitting of a saddle connection

Theorems & Definitions (33)

  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • Remark 2.1
  • Corollary 2.1
  • Proposition 2.2
  • Corollary 2.2
  • Proposition 2.3
  • Theorem 1
  • Theorem 2
  • ...and 23 more