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 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 L2-based exponent is 5 and the L 3-based exponent is 1 .