Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems

TL;DR

This paper develops a generalized kinetic relation framework for nonconservative hyperbolic systems with entropy, enabling the formulation of shock wave solutions and their properties, supported by examples including turbulent fluid models.

Contribution

It introduces a new kinetic relation for nonconservative systems, extending previous theories and linking it to existing path-based definitions, with applications to turbulent fluid dynamics.

Findings

Kinetic relation is equivalent to path-based definitions for the considered systems.

The framework applies to various nonconservative systems in practical applications.

Detailed analysis of traveling waves in a turbulent fluid model demonstrates the theory's effectiveness.

Abstract

For a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by Abeyaratne, Knowles, and Truskinovsky for subsonic phase transitions and by LeFloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for nonconservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch, and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconservative system. We illustrate with several examples that nonconservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics, we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function.