Mathematics Seminar - 04/25/24

Abstract:  We investigate the existence of strong solutions to the initial value problem for the incompressible Navier-Stokes equations in $\mathbb{R}^N, N\geq 3$. Our investigation shows that local in-time classical solutions do not develop singularity as long as the initial velocity lies in $(L^2(\mathbb{R}^N))^N\cap (L^\infty(\mathbb{R}^N))^N$.  This work is closely related to one of the seven millennium problems proposed by the Clay Mathematical Institute.