On oscillatory solutions to the complete Euler system

TL;DR

This paper demonstrates the ill-posedness of the Euler system for certain initial data, revealing infinitely many solutions and analyzing a singular limit using measure-valued solutions and relative energy.

Contribution

It shows the Euler system's Cauchy problem is ill-posed for $L^ abla$-initial data and explores measure-valued solutions in a singular limit context.

Findings

The Cauchy problem is ill-posed for $L^ abla$-initial data.

Existence of infinitely many measure-valued solutions.

Analysis of singular limit with relative energy.

Abstract

The Euler system in fluid dynamics is a model of a compressible inviscid fluid incorporating the three basic physical principles: Conservation of mass, momentum, and energy. We show that the Cauchy problem is basically ill-posed for the $L^\infty$-initial data in the class of weak entropy solutions. As a consequence, there are infinitely many measure-valued solutions for a vast set of initial data. Finally, using the concept of relative energy, we discuss a singular limit problem for the measure-valued solutions, where the Mach and Froude number are proportional to a small parameter.