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Spectral stability of shock profiles for hyperbolically regularized systems of conservation laws

Johannes Bärlin

Abstract

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable, if the shock amplitude is sufficiently small. This means that an associated Evans function $\mathcal{E}:Λ\rightarrow\mathbb{C}$ with $Λ\subset\mathbb{C}$ an open superset of the closed right half plane $\mathbb{H}^+\equiv\{κ\in\mathbb{C}:\text{Re}\,κ\geq 0\}$, has only one zero, namely a simple zero at $0$. The result is analogous to the one obtained in [FS02] and [PZ04] for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in [PZ04], [MZ09], [Ued09] .

Spectral stability of shock profiles for hyperbolically regularized systems of conservation laws

Abstract

We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable, if the shock amplitude is sufficiently small. This means that an associated Evans function with an open superset of the closed right half plane , has only one zero, namely a simple zero at . The result is analogous to the one obtained in [FS02] and [PZ04] for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in [PZ04], [MZ09], [Ued09] .
Paper Structure (11 sections, 29 theorems, 416 equations)

This paper contains 11 sections, 29 theorems, 416 equations.

Table of Contents

  1. Introduction
  2. Domain of consistent splitting and main theorem
  3. Resolvent kernel
  4. Semi-group theory
  5. Construction of the resolvent kernel
  6. Grassmannian flows
  7. Spectral stability: Proof of main theorem
  8. Inner regime
  9. Outer regime
  10. Outmost regime
  11. Acknowledgments

Key Result

Lemma 1.1

If A1-A4 hold then there exist $\delta > 0$ and functions $\mu_i \in C^\infty(B_\delta(0);\mathbb{R}), r_i \in C^\infty(B_\delta(0);\mathbb{R}^n\backslash \lbrace 0 \rbrace)$ for $i=1, \ldots, n$ such that for all $v \in B_\delta(0) \subset \mathbb{R}^n$ the matrices $Dg(v)$ and $Dg(v) \pm Df(v)$ ar where $\delta_{ij}$ denotes the Kronecker delta, and Furthermore, for all $v,w \in B_\delta(0)$ an

Theorems & Definitions (60)

  • Lemma 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Definition 1.7
  • ...and 50 more