Relative entropy method for measure-valued solutions in natural sciences

TL;DR

This paper discusses the relative entropy framework's applications in proving uniqueness and measure-valued solutions for various physical systems, including fluid dynamics and elastodynamics, highlighting recent advances and long-term behavior analysis.

Contribution

It introduces the relative entropy method for measure-valued solutions and surveys recent results on measure-valued-strong uniqueness across multiple physical models.

Findings

Proven uniqueness of entropy solutions for scalar conservation laws.

Survey of measure-valued-strong uniqueness results in fluid dynamics and elastodynamics.

Analysis of long-time asymptotics for the McKendrick-Von Foerster equation.

Abstract

In this article we describe the applications of the relative entropy framework. In particular uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey recent results concerning measure-valued-strong uniqueness for a number of physical systems - incompressible and compressible Euler equations, compressible Navier-Stokes, polyconvex elastodynamics and general hyperbolic conservation laws, as well as long-time asymptotics of the McKendrick-Von Foerster equation.