  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.
