Dyadic models for fluid equations: a survey
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.
AMS Centennial Fellowship | Funder: American Mathematical Society
Mathematical Analysis of Magnetohydrodynamic Flows with Hall Effect | Funder: National Science Foundation | Grant ID: DMS-2009422
Mathematical aspects of the magneto-hydrodynamics with Hall effect | Funder: National Science Foundation | Grant ID: DMS-1815069
Von Neumann Fellowship | Funder: Institute for Advanced Study
CitationDai, M., Cheskidov, A.Friedlander, S. (2023). Dyadic models for fluid equations: a survey. Journal of Mathematical Fluid Mechanics.