Special Session 7: 

Singular set of critical space solution to Navier-Stokes flow

hi jun choe
yonsei university
Co-Author(s):    Minsuk Yang, Joeg Wolf
We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier--Stokes equations. We present a new regularity criterion for the weak solution $u$ satisfying the condition $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ without any smallness assumption on that scale, where $L^{3,w}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space. As an application, we conclude that there are at most a finite number of blowup points at any singular time $t$. The condition that the weak Lebesgue space norm of the veclocity field $u$ is bounded in time is encompassing type I singularity and significantly weaker than the end point case of the so-called Ladyzhenskaya--Prodi--Serrin condition proved by Escauriaza--Sergin--\v{S}ver\`{a}k.