Systems of hyperbolic conservation laws with prescribed eigencurves

TL;DR

This paper investigates the construction of hyperbolic conservation law systems with specified eigencurves, analyzing integrability conditions and providing classifications, including applications to gas dynamics and rich systems.

Contribution

It formulates and analyzes the overdetermined system for eigenvalues with prescribed eigencurves, offering a complete classification for three-equation systems and rich systems of any size.

Findings

Complete analysis of three-equation systems with prescribed eigencurves.

Characterization of systems sharing eigencurves with the Euler system.

Resolution of the general rich systems case, including examples.

Abstract

We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan-Kahler). We give a complete analysis of the possible scenarios, including examples, for systems of three equations. As an application we characterize conservative systems with the same eigencurves as the Euler system for 1-dimensional compressible gas dynamics. The case of general rich systems of any size (i.e. when the given eigenvector fields are pairwise in involution; this includes all systems of two equations) is completely resolved and we consider various examples in this class.