## Analysis and Applied Mathematics Seminar

Trevor Leslie

University of Illinois at Chicago

New Energy Balance Criteria for the Navier-Stokes Equations

**Abstract:**When a Leray-Hopf weak solution to the Navier-Stokes equations has a singularity set $S$ of dimension $d$ less than 3--for example, a suitable weak solution--we find a family of $L^q L^p$ conditions that guarantee validity of the energy balance relation. Our conditions surpass the classical Lions-Ladyzhenskaya $L^4 L^4$ result in the case $d<1$. In this talk, we focus on the special case when $S$ belongs to a single time-slice. Besides allowing more flexibility in the relevant analysis (and accordingly, stronger results), the time-slice case is the one which is most relevant for the blowup problem. If time allows, we will also discuss extensions to the fractional Navier-Stokes equations.

Monday April 3, 2017 at 4:00 PM in SEO 636