# PDE Seminar

## The structure of helicity and global smooth solutions of navier-stokes equations

### Fanghua Lin教授 (NYU Shanghai)

#### 2013年10月14日（周一）下午2:00-3:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. Part of mysteries of the 3-D incompressible Navier-Stokes equations may be lie in the physical quantity: helicity which is rather mystrious in itself. It is well-known that for 3-D incompressible Euler equations, the global integral of helicity is conserved. However, the helicity does not have a fixed sign, and hence it has not been used in analysis. In a recent joint work with Zhen Lei and Yi Zhou, we explored a structure of helicity, and from it, we derived a new energy law for regular solutions of the 3-D incompressible Navier-Stokes Equations. Unlike the classical Leray-Hopf, (which is only known one) which is supercritical with respect to natural scalings, our new energy is actually critical with respect scalings. The latter gives a hope to understand the analytical difficulties of the Navier Stokes equations. Indeed, I shall present some of our preliminary results on global existence of smooth solutions.