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.

2-2:50pm
3-3:50pm

Title: Kolmogorov Problem and Wiener-type Criterion for the Removability of the Fundamental Singularity for the Parabolic PDEs 

Speaker: Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan

Abstract: The major problem in the Analysis of PDEs is the characterization of singularities reflecting the natural phenomena. I will present my solution of the Kolmogorov's Problem (1928) expressed in terms of the new Wiener-type criterion for the removability of the fundamental singularity for the heat equation. The new concept of regularity or irregularity of singularity point for the parabolic (or elliptic) PDEs is defined according to whether or not the caloric (or harmonic) measure of the singularity point is null or positive. The new Wiener-type criterion precisely characterizes the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or caloric measure of the singularity point, asymptotic laws for the conditional Brownian motion, and criteria for thinness in minimal-fine topology.

The main goal of these two lectures is to sketch the proof of the new Wiener-type criterion for the removability of the fundamental singularity and, equivalently, the unique solvability of the singular Dirichlet problem for the heat equation. This work fully characterizes the removability of non-isolated boundary singularities through the fine-topological thinness of the exterior set near the singularity point. A significant tool in this characterization is the new concept of h-capacity of Borel sets, which measures thinness and establishes singularity behavior via the divergence of weighted sums of h-capacities within nested shells.