Calculus of Variations and Geometric Measure Theory

C. De Lellis - L. J. Székelyhidi

On admissibility criteria for weak solutions of the Euler equations

created by delellis on 27 Dec 2007
modified on 03 May 2011


Published Paper

Inserted: 27 dec 2007
Last Updated: 3 may 2011

Journal: Arch. Ration. Mech. Anal.
Volume: 195
Pages: 225-260
Year: 2010


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 we show bounded initial data for which admissible solutions to the $p$--system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.

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Keywords: conservation laws, differential inclusions, Euler equations, energy inequality