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On Admissibility Criteria for Weak Solutions of the Euler Equations

Abstract

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper, we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct, in more than one space dimension, we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique.

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Correspondence to Camillo de Lellis.

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Communicated by Y. Brenier

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de Lellis, C., Székelyhidi, L. On Admissibility Criteria for Weak Solutions of the Euler Equations. Arch Rational Mech Anal 195, 225–260 (2010). https://doi.org/10.1007/s00205-008-0201-x

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Keywords

  • Weak Solution
  • Euler Equation
  • Differential Inclusion
  • Entropy Solution
  • Energy Inequality