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The Euler equations as a differential inclusion

Camillo De Lellis·László Székelyhidi Jr
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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 $n\geq 2$. 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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Camillo De Lellis, László Székelyhidi Jr. 2007-11-27. The Euler equations as a differential inclusion. https://arxiv.org/abs/math/0702079

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