Regularizing Effect of the Forward Energy Cascade in the Inviscid Dyadic Model

Abstract

We study the inviscid dyadic model of the Euler equations and prove some regularizing properties of the nonlinear term that occur due to forward energy cascade. We show every solution must have 3/5 L^2-based (or 1/10 L^3-based) regularity for all positive time. We conjecture this holds up to Onsager's scaling, where the L^2-based exponent is 5/6 and the L^3-based exponent is 1/3.

13 pages, 2 figures. Corrected misprints, added a new reference, elaborated on how the final bound for Theorem 4.2 was achieved, and specified assumption about positive solutions earlier in the paper