On the Gelfand Problem and Viscosity Matrices for Two-Dimensional Hyperbolic Systems of Conservation Laws

TL;DR

This paper investigates viscous regularizations of two-dimensional hyperbolic conservation laws, revealing that certain oblique waves can be unstable despite stability in principal directions, with numerical simulations supporting the analytical findings.

Contribution

It provides the first nontrivial multidimensional Gelfand problem results and explores stability properties of viscous regularizations with different viscosity matrices.

Findings

Oblique waves can be linearly unstable in 2D hyperbolic systems.

Numerical simulations confirm analytical stability and instability results.

First nontrivial multidimensional Gelfand problem analysis.

Abstract

We present counter-intuitive examples of a viscous regularizations of a two-dimensional strictly hyperbolic system of conservation laws. The regularizations are obtained using two different viscosity matrices. While for both of the constructed ``viscous'' systems waves propagating in either $x$- or $y$-directions are stable, oblique waves may be linearly unstable. Numerical simulations fully corroborate these analytical results. To the best of our knowledge, this is the first nontrivial result related to the multidimensional Gelfand problem. Our conjectures provide direct answer to Gelfand's problem both in one- and multi-dimensional cases.