A BKM-type criterion for the 3D incompressible Euler equations (Mustafa Aydin, USC)

Abstract: The three-dimensional incompressible Euler equations describe the motion of an ideal fluid, yet the mechanisms that govern the possible loss of regularity of smooth solutions remain only partially understood. A classical result of Beale, Kato, and Majda shows that if a smooth solution breaks down in finite time, then the time integral of the vorticity’s supremum norm must diverge, providing a sharp conditional criterion for regularity.

In this talk, I will present a new blow-up criterion in the spirit of the Beale–Kato–Majda theorem that emphasizes a different form of control. Instead of requiring bounds on the full vorticity, the criterion involves tangential derivatives of the velocity field, and shows that smooth solutions persist as long as these derivatives remain appropriately bounded in time. The result holds in a variety of settings, including the whole space, periodic domains, and domains with boundaries.