The Nash–Kuiper Theorem and the Onsager Conjecture

We give an account of the analogies between the Nash–Kuiper $C^1$ solutions of the isometric embedding problem and the weak solutions of the incompressible Euler equations which violate the energy conservation. Such analogies have lead to the recent resolution of a well-known conjecture of Lars Onsager in the theory of fully developed turbulence.

Keywords

Onsager conjecture, Euler equations, isometric embedding, $h$-principle, Nash–Kuiper theorem

2010 Mathematics Subject Classification

35A01, 35D30, 35Q31, 53A99, 53C21, 76F02

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The research of C.D.L. has been supported by the grant 200021_159403 of the Swiss National Foundation.

Published 22 October 2020