Existence and selection of solutions in the energy-variational framework with applications in fluid dynamics

TL;DR

This paper introduces a new existence theory for energy-variational solutions to a broad class of PDEs, relaxing previous assumptions and applying to fluid dynamics models like Euler--Korteweg and binormal curvature flow.

Contribution

It generalizes the existence results for energy-variational solutions by relaxing regularity and energy growth assumptions, with applications to fluid dynamics systems.

Findings

Established existence of solutions under weaker assumptions.

Applied theory to Euler--Korteweg and binormal curvature flow systems.

Discussed criteria for selecting specific solutions in multi-valued sets.

Abstract

We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation of the assumptions on the regularity weight and the admissibility of energies with merely linear growth. We apply the abstract theory to the Euler--Korteweg system and to the equation for binormal curvature flow, which serve as examples that require the first and second generalization, respectively. Moreover, we discuss criteria that are suitable for the selection of particular energy-variational solutions in the possibly multi-valued solution set.