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

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.