Existence and Uniqueness of Global Smooth Solution of Incompressible Navier-Stokes Equation

The Cauchy problem and spatially periodic problem of incompressible Navier-Stokes equation are considered. The existence and uniqueness of global solution for these two problem with infinite smooth initial data $u_0$, i.e. $u_0,\;(u_0\cdot\nab)u_0\in H^{\infty}$, are established. Moreover these two problem with initial data $u_0\in H^m$ ($m\ge1$) are globally well-posed provided the Fourier frequency of $u_0$ is contained in a bounded compact set.