# On strong continuity of weak solutions to the compressible Euler system

## TL;DR

This paper demonstrates that for the compressible Euler system, there exist infinitely many weak solutions that lack strong continuity at a dense set of times, highlighting non-uniqueness and irregularity in solutions.

## Contribution

It introduces a refined oscillatory lemma allowing for discontinuous coefficients, enabling the construction of weak solutions with prescribed discontinuities in time.

## Key findings

- Existence of infinitely many weak solutions not strongly continuous at specified times.
- Extension of oscillatory lemma to handle discontinuous coefficients.
- Demonstration of non-uniqueness in solutions to the compressible Euler system.

## Abstract

Let $\mathcal{S} = \{ \tau_n \}_{n=1}^\infty \subset (0,T)$ be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions that are not strongly continuous at each $\tau_n$, $n=1,2,\dots$. The proof is based on a refined version of the oscillatory lemma of De Lellis and Sz\' ekelyhidi with coefficients that may be discontinuous on a set of zero Lebesgue measure.