Collection of papers in honor of Sergey Godunov (Editors: Yu. L. Trakhinin, M.A. Shishlenin)

On the Gelfand problem and viscosity matrices for two-dimensional hyperbolic systems of conservation laws

S. Chu^ab, I. Kliakhandler^c, A. Kurganov<sup>db

a</sup> Department of Mathematics, RWTH Aachen University, 52056, Aachen, Germany
^b Shenzhen International Center for Mathematics, Southern University of Science and Technology, 518055, Shenzhen, China
^c Department of Mathematics, Michigan Technological University, 49931, Houghton, MI, USA
^d Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, 518055, Shenzhen, China

Abstract: We present counter-intuitive examples of viscous regularizations of a two-dimensional strictly hyperbolic systems 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 with non-symmetric fluxes and diffusion terms. Our conjectures provide direct answer to Gelfand's problem both in one- and multi-dimensional cases.

Keywords: Viscosity matrices, hyperbolic systems of conservation laws, Saint-Venant system of shallow water equations.

UDC: 517.95

MSC: 35L65, 35B35, 76R99

Received November 1, 2024, published December 31, 2024

Language: English

DOI: 10.33048/semi.2024.21.B06