Le equazioni di Eulero dal punto di vista delle inclusioni differenziali

De Lellis, Camillo

Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 873-879

Résumé

In a recent joint paper with L. Székelyhidi we have proposed 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.