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;
2017-01-01

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.
2017
convex integration, Euler equations, Mikado flows
File in questo prodotto:
File Dimensione Formato  
2017_ArchRationMechAnal_224_Daneri.pdf

non disponibili

Descrizione: Articolo principale
Tipologia: Versione Editoriale (PDF)
Licenza: Non pubblico
Dimensione 753 kB
Formato Adobe PDF
753 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/7297
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 87
  • ???jsp.display-item.citation.isi??? 84
social impact