Well-posedness, Smoothness and Blow-up for Incompressible Navier-Stokes Equations

For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on $\mathbb{R}^d$ or $\mathbb{T}^d:=\mathbb{R}^d/\mathbb{Z}^d,$ up to a time explicitly given by the initial datum and three constants coming from the upper bounds of the heat kernel and the Riesz transform. A mild well-posedness is also proved for $L^p$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.