Trevor Leslie : Energy Balance Criteria for the Navier-Stokes Equations

Posted by Trevor Leslie , part of the Graduate Analysis Seminar.

At

Oct. 31, 2016, noon

In

SEO 1227

Abstract

Let $u$ be a solution to the classical 3D Navier-Stokes equations. If $u$ is smooth, then we know that $u$ satisfies a certain energy balance relation. But what if $u$ is not smooth? It has been known since the 1960s (by a result of Lions) that $u$ satisfies the energy balance relation if $u$ is $L^4$ integrable in both time and space. We give new regularity conditions on $u$ that guarantee energy balance; our conditions depend on both the integrability of $u$ and the Hausdorff dimension $d$ of the singularity set. We recover Lions' result when $d=1$ and improve upon it when $d<1$ or when the singularity set is confined to a single time-slice. This is joint work with Roman Shvydkoy.