The Euler equations as a differential inclusion. (English) Zbl 1350.35146

Summary: 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.


35Q31 Euler equations
35D30 Weak solutions to PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
34A60 Ordinary differential inclusions
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