- Series
- PDE Seminar
- Time
- Tuesday, January 23, 2024 - 2:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Zoran Grujic – University of Alabama Birmingham – zgrujic@uab.edu – https://www.uab.edu/cas/mathematics/people/faculty-directory/zoran-grujic
- Organizer
- Gong Chen
It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system exhibits global-in-time regularity as long as the hyper-diffusion exponent is greater or equal to 5/4. One should note that at 5/4, the system is critical, i.e., the energy level and the scaling -invariant level coincide. What happens in the super-critical regime, the hyper-diffusion exponent being strictly between 1 and 5/4 remained a mystery.
The goal of this talk is to demonstrate that as soon as the hyper-diffusion exponent is greater than 1, a class of monotone blow-up scenarios consistent with the analytic structure of the flow (prior to the possible singular time) can be ruled out (a sort of 'runaway train' scenario). The argument is in the spirit of the regularity theory of the 3D HD NS system in 'turbulent scenario' (in the super-critical regime) developed by Grujic and Xu, relying on 'dynamic interpolation' – however, it is much shorter, tailored to the class of blow-up profiles in view. This is a joint work with Aseel Farhat.