A Note on Measure-Valued Solutions to the Full Euler System

TL;DR

This paper constructs specific measure-valued solutions to the full Euler system from identical initial data, demonstrating that not all measure-valued solutions can be approximated by weak solutions, highlighting a key difference from incompressible Euler equations.

Contribution

It shows that the weak* closure of weak solutions does not encompass all measure-valued solutions for the full Euler system, revealing fundamental differences from incompressible flows.

Findings

Constructed measure-valued solutions from the same initial data.

Demonstrated measure-valued solutions not approximable by weak solutions.

Highlighted contrast with incompressible Euler equations.

Abstract

We construct two particular solutions of the full Euler system which emanate from the same initial data. Our aim is to show that the convex combination of these two solutions form a measure-valued solution which may not be approximated by a sequence of weak solutions. As a result, the weak* closure of the set of all weak solutions, considered as parametrized measures, is not equal to the space of all measure-valued solutions. This is in stark contrast with the incompressible Euler equations.