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 Székelyhidi with coefficients that may be discontinuous on a set of zero Lebesgue measure.
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Notes
\({\mathbb {R}}^{d \times d}_{0,\mathrm{sym}}\) denotes the space of real symmetric matrices with zero trace, while \(\lambda _\mathrm{max}[\cdot ]\) is the maximum eigenvalue.
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Communicated by Eliot Fried.
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Anna Abbatiello: The research of A.A. is supported by Einstein Foundation, Berlin. Eduard Feireisl: The research of E.F. leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 18-05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. The stay of E.F. at TU Berlin is supported by Einstein Foundation, Berlin.
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Abbatiello, A., Feireisl, E. On Strong Continuity of Weak Solutions to the Compressible Euler System. J Nonlinear Sci 31, 33 (2021). https://doi.org/10.1007/s00332-021-09694-5
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DOI: https://doi.org/10.1007/s00332-021-09694-5