On ruling out a class of type II blow-up scenarios in the hyper-dissipative Navier-Stokes equations

Workshop "Singularities and Partial Differential Equations representing Natural Phenomena: from potential theory to fluid mechanics" 23 July, 2025

10:30-11:20am

11:30-12:20am

Title: On ruling out a class of type II blow-up scenarios in the hyper-dissipative Navier-Stokes equations

Speaker: Zoran Grujic, University of Alabama at Birmingham, USA

Abstract: It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system does not permit formation of singularities as long as the hyper-dissipation exponent, say beta, is greater or equal to 5/4. Recall that at 5/4 the system is in the critical regime — the energy level and the scaling-invariant levels coincide — while for beta greater than 5/4 the system is in the sub-critical regime. The question of global-in-time regularity in the super-critical regime, beta strictly between 1 and 5/4, has remained a fundamental open problem in mathematical fluid dynamics, intimately related to the problem of global-in-time regularity for the NS system per se.

The main goal of the two lectures is to present a mathematical framework — built around a suitably defined scale of sparseness of the super-level sets of the components of the higher-order velocity derivatives — in which a class of `turbulent' blow-up scenarios can be ruled out as soon as the hyper-dissipation exponent is greater than 1. In particular, a class of type II generalized self-similar blow-ups is ruled out which — in turn — rules out approximately self-similar blow-ups, a prime candidate for singularity formation, in all 3D HD NS systems indicating criticality of the Laplacian/NS diffusion. A sketch of the proof — the main ideas and some key technical realizations — will also be presented. This is a joint work with L. Xu.