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

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

2005-02-05

arxiv.org/abs/math/0502104v1

Mathematics

Analysis of PDEs

14 pages

Scientific paper

We consider the Cauchy problem for incompressible Navier-Stokes equations $u_t+u\nabla_xu-\Delta u+\nabla p=0, div u=0 in R^d \times R^+$ 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 $t^{m+n/2}D^m_tD^{n}_x u\in L^{d+2}(R^d\times (0,T))$, as long as the $\|u\|_{L^{d+2}_{x,t}(R^d\times (0,T))}$ stays finite.

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