Quantitative estimates for the Navier-Stokes equations via spatial concentration

Abstract: It remains open as to whether or not the 3D Navier-Stokes equations lose smoothness (`blow-up') in finite time. Starting from Jean Leray, many authors provided increasingly refined necessary conditions for a finite-time blow-up to occur. The majority of these blow-up behaviours are formulated in terms of critical or subcritical quantities, which are notions relating to the scaling symmetry of the Navier-Stokes equations. Very recently, Tao used a new quantitative approach to infer that certain 'slightly supercritical' quantities for the Navier-Stokes equations must become unbounded near a potential blow-up.

In this talk I'll discuss a new strategy for proving quantitative bounds for the Navier-Stokes equations, as well as applications to behaviours near a potential singularity . As a first application, we prove a new potential blow-up rate, which is optimal for a certain class of potential non-zero backward discretely self-similar solutions. As a second application, we quantify a conditional qualitative regularity result of Seregin (2012), which says that if the critical L_{3} norm of the velocity field is bounded along a sequence of times tending to time $T$ then no blow-up occurs at time $T$.

This talk is based upon joint work with Christophe Prange (CNRS, Université de Bordeaux).

analysis of PDEs

Audience: researchers in the topic

PDE seminar via Zoom