Existence of energy-variational solutions to hyperbolic conservation laws

TL;DR

This paper introduces energy-variational solutions for hyperbolic conservation laws, establishing their existence and properties, and demonstrating their equivalence to dissipative weak solutions for Euler equations.

Contribution

It defines a new class of solutions called energy-variational solutions and proves their existence for key fluid dynamics equations, extending the theoretical framework.

Findings

Energy-variational solutions satisfy weak-strong uniqueness.

Existence is proven via a time-discretization scheme.

Solutions coincide with dissipative weak solutions for Euler equations.

Abstract

We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions.