Existence of energy-variational solutions to hyperbolic conservation laws

Thomas Eiter; Robert Lasarzik

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arXiv · 2211.12307

Existence of energy-variational solutions to hyperbolic conservation laws

Thomas Eiter·Robert Lasarzik

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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.

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Thomas Eiter, Robert Lasarzik. 2022-11-22. Existence of energy-variational solutions to hyperbolic conservation laws. https://arxiv.org/abs/2211.12307

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