In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. We prove that the set of Hölder 1 / 5 - ε wild initial data is dense in L 2 , where we call an initial datum wild if it admits infinitely many admissible Hölder 1 / 5 - ε weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows in order to show that a general form of the h-principle applies to Hölder-continuous weak solutions of the Euler equations. Our result indicates that in a deterministic theory of three dimensional turbulence the Reynolds stress tensor can be arbitrary and need not satisfy any additional closure relation.

Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations

Daneri S;
2017

Abstract

In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. We prove that the set of Hölder 1 / 5 - ε wild initial data is dense in L 2 , where we call an initial datum wild if it admits infinitely many admissible Hölder 1 / 5 - ε weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows in order to show that a general form of the h-principle applies to Hölder-continuous weak solutions of the Euler equations. Our result indicates that in a deterministic theory of three dimensional turbulence the Reynolds stress tensor can be arbitrary and need not satisfy any additional closure relation.
convex integration, Euler equations, Mikado flows
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/7297
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