Karen Zaya : On Regularity Properties for Fluid Equations

Posted by Christof Sparber , part of the Analysis and Applied Mathematics Seminar.

At

Nov. 23, 2015, 4 p.m.

In

SEO 636

Abstract

Fundamental mathematical questions about the 3D Navier-Stokes remain unanswered, such as the question of the regularity of solutions to the equations. Thus it is natural to ask: If we assume a smooth solution to the 3D Navier-Stokes equations $u$ loses regularity at time $T^*$, what is the rate of blow-up? In this talk, we discuss blow-up rates of solutions in the homogeneous Sobolev spaces, in particular the new result in $\dot{H}^\frac{3}{2}$. We will also discuss a newly developed regularity criterion for the 3D Boussinesq equations, which only imposes a condition on the low modes of the velocity $u$. The key tool in the development of this weaker regularity criterion is linked to the dissipation wave number.