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 L-2-dense set of Holder continuous initial data in the class of Holder 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 Szekelyhidi in Arch. Rat. Mech. Anal. 224: 471-514, 2017) and generalizes (Buckmaster et al. in Comm. Pure Appl. Math. 72(2):229-274, 2018).