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Dyadic Models for Fluid Equations: A Survey

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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.

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Acknowledgements

A. Cheskidov is partially supported by the NSF grant DMS-1909849. M. Dai is partially supported by the NSF grants DMS-1815069 and DMS-2009422, and the AMS Centennial Fellowship. S. Friedlander is partially supported by the NSF grant DMS-1613135. A. Cheskidov and M. Dai are grateful to IAS for its hospitality in 2021–2022. They are also grateful for the hospitality of Princeton University in 2022–2023.

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  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Alexey Cheskidov & Mimi Dai

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Susan Friedlander

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Communicated by L. Kapitanski.

Dedicated to Olga Aleksandrovna Ladyzhenskaya.

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Cheskidov, A., Dai, M. & Friedlander, S. Dyadic Models for Fluid Equations: A Survey. J. Math. Fluid Mech. 25, 62 (2023). https://doi.org/10.1007/s00021-023-00799-3

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