Analysis of energy-variational solutions for hyperbolic conservation laws

The goal of this project is to further develop the analysis of energy-variational solutions for general hyperbolic conservation laws. This is a new generalized solvability concept, which relaxes the evolution equation to a variational inequality taking into account possible energy defects during the evolution. This framework provides a rich structure and comes along with several improved properties in comparison to previous solvability concepts:

• existence of energy-variational solutions for a large class of models,
• constructive existence proof via a minimizing-movement scheme,
• semi-flow property,
• convexity and weak-star closedness of the solution set,
• in special cases: continuous dependence of solution set from the initial value.

We aim at further developing and establishing the analytic framework. In particular, we plan to exploit the above properties in order to de ne appropriate selection criteria for a unique physically reasonable candidate within the set of energy-variational solutions and to prove additional properties of the selected solutions. Moreover, we aim to use these selection criteria in the construction of approximation schemes. Overall, we plan to investigate the following topics:

1. Analysis of energy-variational solutions in the context of hyperbolic conservation laws, comparison to other solution concepts, and investigation of appropriate selection criteria.
2. Approximation of energy-variational solutions based on a sequential minimization scheme in time, respecting the introduced selection criteria.