# Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data

## TL;DR

This paper demonstrates that for the two-dimensional isentropic Euler equations, there exist smooth initial conditions leading to infinitely many bounded admissible weak solutions, highlighting non-uniqueness in solutions to these equations.

## Contribution

It constructs smooth initial data that admits infinitely many solutions using a novel approach involving generalized fan subsolutions and the relation to Burgers equation.

## Key findings

- Existence of smooth initial data with multiple solutions
- Construction of solutions via generalized fan subsolutions
- Application of De Lellis and Székelyhidi's theory to non-constant data

## Abstract

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Sz\'ekelyhidi and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.