The Euler equations as a differential inclusion

Abstract

We propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in n with n 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.