Dyadic models for fluid equations: a survey

TL;DR

This survey reviews dyadic models as simplified systems that mimic key features of fluid PDEs, highlighting recent advances in understanding their solutions' existence, uniqueness, and regularity.

Contribution

It provides a comprehensive overview of dyadic models and summarizes recent research findings on their mathematical properties.

Findings

Results on existence of solutions

Results on uniqueness of solutions

Results on regularity of solutions

Abstract

Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years, including the groundbreaking work of Ladyzhenskaya on the Navier-Stokes equations. However crucial questions such as the existence, uniqueness and regularity of the three dimensional Navier-Stokes equations remain open. Partly because of this mathematical challenge and partly motivated by the phenomena of turbulence, insights into the full PDEs have been sought via the study of simpler approximating systems that retain some of the original nonlinear features. One such simpler system is an infinite dimensional coupled set of nonlinear ordinary differential equations referred to a dyadic model. In this survey we provide a brief overview of dyadic models and describe recent results. In particular, we discuss results for certain dyadic models in the context of existence, uniqueness and regularity of solutions.