Existence of energy-variational solutions to hyperbolic conservation laws
- Eiter, Thomas
- Lasarzik, Robert
2020 Mathematics Subject Classification
- 35L45 35L65 35A01 35A15 35D99 35Q31 76B03 76N10
- Existence, generalized solutions, conservation laws, time discretization, weak-strong, uniqueness, Euler equations
oduce 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.