Generic Structural Stability for $2 \times 2$ Systems of Hyperbolic Conservation Laws

TL;DR

This paper proves that Riemann solutions to generic 2x2 hyperbolic conservation laws are structurally stable under perturbations, using differential topology concepts to establish the genericity of stability for a broad class of systems.

Contribution

It introduces a novel proof of generic structural stability for 2x2 hyperbolic systems using transversality and differential topology techniques.

Findings

Structural stability is related to transversality of Hugoniot loci and rarefaction curves.

The proof applies to the p-system and particle-laden thin film systems.

The assumptions include hyperbolicity, non-linearity, and a manifold regularity condition.

Abstract

This paper presents a proof of generic structural stability for Riemann solutions to $2 \times 2$ system of hyperbolic conservation laws in one spatial variable, without diffusive terms. This means that for almost every left and right state, shocks and rarefaction solutions of the same type are preserved via perturbations of the flux functions, the left state, and the right state. The main assumptions for this proof involve standard assumptions on strict hyperbolicity and genuine non-linearity, a technical assumption on directionality of rarefaction curves, and the regular manifold (submersion) assumption motivated by concepts in differential topology. We show that the structural stability of the Riemann solutions is related to the transversality of the Hugoniot loci and rarefaction curves in the state space. The regular manifold assumption is required to invoke a variant of a theorem from differential topology, Thom's parametric transversality theorem, to show the genericity of transversality of these curves. This in turn implies the genericity of structural stability. We then apply this theorem to two examples: the p-system and a $2 \times 2$ system governing the evolution of gravity-driven monodisperse particle-laden thin films. In particular, we illustrate how one can verify all the above assumptions for the former, and apply the theorem to different numerical and physical aspects of the system governing the latter.