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

TL;DR

This work examines the stability of viscous regularizations for 2-D strictly hyperbolic systems in the context of the multidimensional Gelfand problem. By linearizing the 2-D Saint-Venant (shallow water) system about a lake-at-rest state and introducing viscosity matrices $C$ and $D$, the authors derive a dispersion relation and demonstrate that axis-aligned waves can be stable while oblique modes may be linearly unstable; these predictions are corroborated by high-order numerical simulations for both linear and nonlinear shallow-water models. The paper contributes two stability conjectures that reduce the multidimensional problem to spectral properties of combinations of convective and viscous matrices, and provides explicit first-order correction formulas to test stability in low dimensions. These results extend the classical Majda–Pego instability to multiple dimensions and offer a framework for approaching the Gelfand problem in 2-D and 3-D, with potential implications for modeling viscous effects in fluid dynamics.

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.

Figures (5)

• Figure 3.1: $w$ as a function of $k$ for $\varepsilon=1$ (left) and $\varepsilon=5$ (right).
• Figure 4.1: Linear case: $h$ computed with $\varepsilon=5$ (stable regime) at different times.
• Figure 4.2: Same as in Figure \ref{['fig4']}, but for $\varepsilon=1$ (unstable regime).
• Figure 4.3: Nonlinear case: $h$ computed with $\varepsilon=5$ (stable regime) at different times.
• Figure 4.4: Same as in Figure \ref{['fig6']}, but for $\varepsilon=1$ (unstable regime).

Theorems & Definitions (3)

• Conjecture 5.1
• Remark 5.2
• Conjecture 5.3