On strong continuity of weak 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.