Calculus of Variations and Geometric Measure Theory

C. De Lellis - L. J. Székelyhidi

The Euler equation as a differential inclusion

created by delellis on 04 Feb 2007
modified on 03 May 2011


Published Paper

Inserted: 4 feb 2007
Last Updated: 3 may 2011

Journal: Ann. of Math.
Volume: 170
Pages: 1417-1436
Year: 2009


In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $R^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy--decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

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Keywords: differential inclusions, Euler equations, weak solutions