On the weak solutions to the equations of a compressible heat conducting gas

TL;DR

This paper demonstrates that the Euler-Fourier system for a compressible heat conducting gas admits infinitely many weak solutions for any smooth initial data, highlighting non-uniqueness and complexity in such systems.

Contribution

It applies convex integration to prove the existence of infinitely many weak solutions for the Euler-Fourier system, even with fixed initial data.

Findings

Existence of infinitely many global weak solutions for any smooth initial data.

Multiple solutions can conserve total energy for the same initial density and temperature.

Convex integration is effective in analyzing compressible heat conducting gases.

Abstract

We consider the weak solutions to the Euler-Fourier system describing the motion of a compressible heat conducting gas. Employing the method of convex integration, we show that the problem admits infinitely many global-in-time weak solutions for any choice of smooth initial data. We also show that for any initial distribution of the density and temperature, there exists an initial velocity such that the associated initial-value problem possesses infinitely many solutions that conserve the total energy.