Non-uniqueness for the Euler Equations up to Onsager’s Critical Exponent

In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical 1/3. Along the way, and more importantly, we identify a natural condition on “blow-up” of the associated subsolution, which acts as the signature of the non-uniqueness mechanism. This improves previous results on non-uniqueness obtained in (Daneri in Comm. Math. Phys. 329(2):745–786, 2014; Daneri and Székelyhidi in Arch. Rat. Mech. Anal. 224: 471–514, 2017) and generalizes (Buckmaster et al. in Comm. Pure Appl. Math. 72(2):229–274, 2018).