WIAS Preprint No. 2974, (2022)

Existence of energy-variational solutions to hyperbolic conservation laws


  • Eiter, Thomas
    ORCID: 0000-0002-7807-1349
  • Lasarzik, Robert
    ORCID: 0000-0002-1677-6925

2020 Mathematics Subject Classification

  • 35L45 35L65 35A01 35A15 35D99 35Q31 76B03 76N10


  • Existence, generalized solutions, conservation laws, time discretization, weak-strong, uniqueness, Euler equations




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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