The existence of singularities in the super-critical regime of the 3D hyper-dissipative (HD) Navier-Stokes equations (NSE), where the hyper-dissipation is generated by a fractional power of the Laplacian within the interval (1, 5/4), has been a longstanding question in the field of mathematical fluid dynamics since J.L. Lions' foundational work. In this talk, we address this question by establishing global regularity of the 3D HD-NSE under a condition that represents turbulent regimes. Specifically, we demonstrate that the equations remain globally regular for any power on the Laplacian greater than 1, within a suitable functional framework that incorporates intermittency.
Global Solutions in Turbulent Regimes of the Hyper-Dissipative 3D NSE
Aseel Farhat, Florida State UniversityAuthors: Aseel Farhat, Zoran Grujic
2023 AWM Research Symposium
Recent Developments in Control, Optimization, and the Analysis of Partial Differential Equations [Organized by Lorena Bociu and Pelin Guven Geredeli]