The 2-d isentropic compressible Euler equations may have infinitely many solutions which conserve energy

TL;DR

This paper demonstrates that the 2D isentropic compressible Euler equations can have infinitely many energy-conserving weak solutions for certain initial data, challenging uniqueness expectations in fluid dynamics.

Contribution

It proves the existence of infinitely many weak solutions that conserve energy for specific Riemann and Lipschitz initial data, extending prior results on non-uniqueness.

Findings

Existence of infinitely many energy-conserving solutions for Riemann initial data.

Existence of infinitely many energy-conserving solutions for Lipschitz initial data.

Challenges the uniqueness of solutions in the 2D isentropic Euler equations.

Abstract

We consider the 2-d isentropic compressible Euler equations. It was shown in by E. Chiodaroli, C. De Lellis and O. Kreml that there exist Riemann initial data as well as Lipschitz initial data for which there exist infinitely many weak solutions that fulfill an energy inequality. In this note we will prove that there is Riemann initial data for which there exist infinitely many weak solutions that conserve energy, i.e. they fulfill an energy equality. As in the aforementioned paper we will also show that there even exists Lipschitz initial data with the same property.