Young-measure solutions for multidimensional systems of conservation laws

TL;DR

This paper introduces a variational approach to Young measure solutions for multidimensional conservation laws, establishing existence results and proposing improvements to the classical concept, supported by scalar case analysis.

Contribution

It presents a novel variational method for Young measure solutions and proves their existence under specific structural conditions.

Findings

Existence of Young measure solutions established

A new variational functional measures departure from weak solutions

Proposed improvements to classical Young measure concepts

Abstract

We explore Young measure solutions of systems of conservation laws through an alternative variational method that introduces a suitable, non-negative error functional to measure departure of feasible fields from being a weak solution. Young measure solutions are then understood as being generated by minimizing sequences for such functional much in the same way as in non-convex, vector variational problems. We establish an existence result for such generalized solutions based on an appropriate structural condition on the system. We finally discuss how the classic concept of a Young measure solution can be improved, and support our arguments by considering a scalar, single equation in dimension one.