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2019

On strong continuity of weak solutions to the compressible Euler system

Abbatiello, Anna, Feireisl, Eduard

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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.

On strong continuity of weak solutions to the compressible Euler system · Around

Built on

Estimates near the boundary for solutions of elliptic partial differential equations
S. Agmon, A. Douglis, and L. Nirenberg · 1959
Earlier work this paper cites.
Singular integrals and differential properties of functions
E.M. Stein · 1970
Earlier work this paper cites.
Onsager’s conjecture on the energy conservation for solutions of Euler’s equation
P. Constantin, W. E, and E. S. Titi · 1994
Earlier work this paper cites.
On admissibility criteria for weak solutions of the Euler equations
C. De Lellis and L. Székelyhidi, Jr · 2010
Earlier work this paper cites.

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