Smooth solutions for the dyadic model

TL;DR

This paper establishes well-posedness results for the dyadic model, a simplified framework for understanding well-posedness and blow-up phenomena in fluid dynamics equations like Navier-Stokes and Euler.

Contribution

It proves well-posedness of positive solutions for both viscous and inviscid dyadic models within relevant parameter ranges, advancing understanding of these simplified models.

Findings

Well-posedness of viscous dyadic model in Navier-Stokes scaling range

Well-posedness of inviscid dyadic model under strong transport conditions

Provides insights into blow-up and regularity issues in fluid models

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.