On the equivalence of generalized solution concepts for systems of hyperbolic conservations laws in fluid dynamics

TL;DR

This paper explores the equivalence of various generalized solution concepts for fluid dynamics PDEs, demonstrating their unification under certain conditions for key models like Euler and Euler-related systems.

Contribution

It establishes the equivalence of measure-valued, dissipative weak, and energy-variational solutions for several fundamental fluid dynamics equations.

Findings

Proves equivalence of solution concepts for incompressible Euler equations.

Shows energy-variational and refined dissipative solutions are equivalent for several models.

Controls defect measures sharply by the energy defect in key fluid systems.

Abstract

We investigate the relation between several generalized solution concepts for nonlinear PDE systems from fluid dynamics. More precisely, we study measure-valued solutions, dissipative weak solutions, and energy-variational solutions. For the incompressible Euler equations, we prove the equivalence of all three concepts, provided that the energy inequality is formulated in the appropriate way. For several important examples of conservation laws arising in fluid dynamics, we establish the equivalence between energy-variational and suitably refined dissipative weak solutions, where the defect measures are controlled sharply by the energy defect. These examples comprise the compressible isentropic Euler system, the Euler--Korteweg system, and the Euler--Poisson system.