The Euler equations as a differential inclusion

Details The Euler equations as a differential inclusion The Euler equations as a differential inclusion

2007-02-05

arxiv.org/abs/math/0702079v3

Ann. of Math. (2) 170 (2009), no. 3, 1417-1436

Mathematics

Analysis of PDEs

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

Scientific paper

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 $n\geq 2$. 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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