The Euler equations as a differential inclusion


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.


Camillo De Lellis

Institut für Mathematik
Universität Zürich
CH-8057 Zürich

László Székelyhidi Jr.

Departement Mathematik
ETH Zürich
CH-8092 Zürich