Energy dissipation admissibility condition for conservation law systems admitting singular solutions

TL;DR

This paper introduces a backward energy condition to select unique solutions for conservation law systems with singular initial data, ensuring maximal energy dissipation and consistency with existing admissibility criteria.

Contribution

It defines a new backward energy condition for conservation laws with singular initial data, providing a criterion for selecting physically relevant solutions.

Findings

The backward energy condition aligns with existing admissibility conditions in gas dynamics.

It effectively handles solutions with delta measures in initial data.

The approach ensures maximal energy dissipation for singular solutions.

Abstract

The main goal of the paper is to define and use a condition sufficient to choose a unique solution to conservation law systems with a singular measure in initial data. Different approximations can lead to solutions with different distributional limits. The new notion called backward energy condition is then to single out a proper approximation of the distributional initial data. The definition is based on the maximal energy dissipation defined in \cite{CD_1973}. Suppose that a conservation law system admits a supplementary law in space--time divergent form where the time component is a (strictly or not) convex function. It could be an energy density or a mathematical entropy in gas dynamic models, for example. One of the admissibility conditions is that a proper weak solution should maximally dissipate the energy or the mathematical entropy. We show that it is consistent with other admissibility conditions in the case of Riemann problems for systems of isentropic gas dynamics with non-positive pressure in the first part of the paper. Singular solutions to these systems are described by shadow waves, nets of piecewise constant approximations with respect to the time variable. In the second part, we define and apply the backward energy condition for those systems when the initial data contains a delta measure approximated by piecewise constant functions.