Smooth solutions for the dyadic model

Abstract

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.

Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system

“…A simple version of this conjecture, when reformulated on a toy model, has been proved for the dyadic model in [BMR14]. Actually, for that model one could prove regularity in full supercritical regime, with m(k) = |k|, as was done in [BMR11], but it was natural to develop there some of the main ideas on which also this paper is based. In fact here we prove that the equations for the velocity can be reduced to a suitable dyadic-like model, with infinitely many interactions though.…”

A dyadic model on a tree

“…The viscous dyadic model is studied in [5], where it is proven that for β ∈ ( 3 2 , 5 2 ] the stationary solution is unique and is a global attractor. In [4] it is proven that for the viscous case it is possible, dropping the Y n ≥ 0 condition, to explicitly provide examples of non-uniqueness of the stationary solution. In this paper we prove the existence and uniqueness of stationary solutions in l 2 for every positive value of the β and γ parameters both in viscous and inviscid dyadic models.…”

Structure Function and Fractal Dissipation for an Intermittent Inviscid Dyadic Model

“…With the latter we intend that the non-linear, formally conservative term, "fires" lumps of energy to smaller and smaller scales, making them actually disappear. In passing from Euler to Navier-Stokes, the introduction of a term corresponding to the viscosity of the fluid may sometimes be enough to brake this phenomenon (as was proved for the dyadic model with viscosity in [8]), but the non-linear term can be tailored to overcome thermal dissipation, in fact Tao in [54] used a shell model to prove that some averaged versions of three-dimensional Navier-Stokes equation have blow-up.…”