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Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws

A. P. Chugainova, D. V. Treschev

TL;DR

This work analyzes a two-component, 2×2 hyperbolic system with potential $Q(u)$ and its viscous regularization to understand shock structure. It develops a detailed classification of shocks (fast, slow, undercompressive, overcompressive) and Jouguet waves via the traveling-wave framework, derives the Hugoniot locus, and constructs the Riemann problem solution as sequences of structured waves. A key finding is that the viscosity ratio $\mu_1/\mu_2$ governs the existence of nonclassical shocks, with undercompressive shocks possible only when $0<\mu_2<\mu_1$, and the equal-viscosity case $\mu_1=\mu_2$ yields complete stability results by reducing to decoupled Burgers equations. The results provide a rigorous pathway to predict structured shock interactions in nonlinear elastic media and related systems, highlighting how anisotropic dissipation selects physically meaningful discontinuities.

Abstract

A system of hyperbolic conservation laws $$ \partial_t u + \partial_x \partial_u Q = 0, \quad Q = u_1^3 / 3 + u_1 u_2^2, \qquad u = u(x,t) \in\mR^2, $$ as well as its viscous regularization $$ \partial_t u + \partial_x \partial_u Q = \calM \partial_x^2 u, \qquad \calM = \diag (μ_1,μ_2), \quad μ_1>0,\, μ_2>0, $$ are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann problem is constructed that consists of rarefaction waves and shocks with structure. Depending on the conditions imposed at $\pm\infty$, the solution also contains undercompressive shocks and Jouguet waves.

Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws

TL;DR

This work analyzes a two-component, 2×2 hyperbolic system with potential and its viscous regularization to understand shock structure. It develops a detailed classification of shocks (fast, slow, undercompressive, overcompressive) and Jouguet waves via the traveling-wave framework, derives the Hugoniot locus, and constructs the Riemann problem solution as sequences of structured waves. A key finding is that the viscosity ratio governs the existence of nonclassical shocks, with undercompressive shocks possible only when , and the equal-viscosity case yields complete stability results by reducing to decoupled Burgers equations. The results provide a rigorous pathway to predict structured shock interactions in nonlinear elastic media and related systems, highlighting how anisotropic dissipation selects physically meaningful discontinuities.

Abstract

A system of hyperbolic conservation laws as well as its viscous regularization are studied. It is assumed that admissible shocks are those that satisfy the requirement of existence of a structure (the traveling wave criterion). A solution of the Riemann problem is constructed that consists of rarefaction waves and shocks with structure. Depending on the conditions imposed at , the solution also contains undercompressive shocks and Jouguet waves.

Paper Structure

This paper contains 19 sections, 10 theorems, 105 equations, 7 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. The Hugoniot locus
  4. Evolutionary conditions for discontinuities. Shocks
  5. Rarefaction waves
  6. Undercompressive shocks and saddle connections
  7. Overcompressive shocks
  8. Jouguet wave
  9. Special rarefaction waves
  10. Existence of a structure of a slow shock
  11. Existence of a structure of a fast shock
  12. Stability of the structures of shocks
  13. The case of $\mu_1=\mu_2$
  14. Structures of fast and slow shocks
  15. Structures of overcompressive shocks
  16. ...and 4 more sections

Key Result

Proposition 2.1

If all points (4cp) are different, then $u^+$ and $u^\times$ are of the same type (an extremum or a saddle). The points $u^a$ and $u^b$ are also of the same (opposite) type (a saddle or an extremum).

Figures (7)

  • Figure 1: Hugoniot locus for the parameters $u_1^+ = 2$, $u_2^+ = 3$, and $p=1$.
  • Figure 2: (a) The structure of (node--node) overcompressive shocks, and (b) the saddle-node--saddle-node heteroclinic connection
  • Figure 3: The phase portraits of system (\ref{['ode']}) for $\mu_2<\mu_1$: (a) parameters lie in ${\cal P}_\infty$, and (b) parameters lie in ${\cal P}_{\hat{u}}$.
  • Figure 4: The phase portraits of system (\ref{['ode']}) for $\mu_2<\mu_1$ lying in ${\cal P}_{saddle}$: (a) in region (\ref{['set2']}) and (b) in region (\ref{['set3']}).
  • Figure 5: The phase portraits of system (\ref{['ode']}) for $\mu_2<\mu_1$ for the parameters (a) from region (\ref{['<W<']})) and (b) from region (\ref{['<W']})).
  • ...and 2 more figures

Theorems & Definitions (10)

  • Proposition 2.1
  • Proposition 3.1
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 9.1
  • Proposition 9.2
  • Proposition 10.1
  • Proposition 11.1
  • Theorem 1
  • Theorem 2