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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.

MSC:
 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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References:
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