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

Abstract

For any divergence free initial datum $u_0$ with $\Vert u_0{\Vert }_{\infty }+\Vert \nabla u_0{\Vert }_{L^p}+\Vert {\nabla }^2{u}_0{\Vert }_{L^p}<\infty$ 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.

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