On the local Smoothness of Solutions of the Navier-Stokes Equations

Abstract

We consider the Cauchy problem for incompressible Navier-Stokes equations $u_t+u{\nabla }_{x}u-\Delta u+\nabla p=0,divu=0in{R}^d\times R^{+}$ with initial data $a\in L^d(R^d)$, and study in some detail the smoothing effect of the equation. We prove that for $T<\infty$ and for any positive integers $n$ and $m$ we have $t^{m+n/2}{D}_t^m{D}_x^{n}u\in L^{d+2}(R^d\times (0,T))$, as long as the $\Vert u{\Vert }_{L_{x,t}^{d+2}(R^d\times (0,T))}$ stays finite.

14 pages