The Euler equations as a differential inclusion

Abstract

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 ${\mathbb{R}}^n$ with $n\ge 2$. 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.

arXiv comment16 pages; v2: corrected typos, simplified some proofs; v3: 20 pages, added a second (more direct) proof

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