Non-uniqueness and h-principle for H\"older-continuous weak solutions of the Euler equations

TL;DR

This paper demonstrates the density of wild initial data with infinitely many weak solutions for the Euler equations, using novel stationary flows to recover Reynolds stresses, advancing understanding of non-uniqueness and the h-principle.

Contribution

It introduces a new set of stationary flows as perturbation profiles, replacing Beltrami flows, to establish the density of wild initial data in the context of Euler equations.

Findings

Wild initial data are dense in L^2 for Euler equations.

Infinitely many admissible weak solutions exist for these wild data.

New stationary flows effectively recover arbitrary Reynolds stresses.

Abstract

In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. Based on estimates developed in [Buckmaster-De Lellis-Isett-Sz\'ekelyhidi], we prove that the set of H\"older $1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial datum wild if it admits infinitely many admissible H\"older $1\slash 5-\eps$ weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds stresses.