Global Smooth Solution of Nonlinear Schr\"odinger Equation

The spatially periodic initial problem and Cauchy problem for nonlinear Schr\"odinger equations are considered. The existence and uniqueness of global solution with infinite smooth initial data $u_0$, i.e. $u_0,\;|u_0|^{2p}u_0\in H^{\infty}$, are established. Moreover these two problem with initial data $u_0\in H^m$ are globally well-posed provided the Fourier frequency of $u_0$ is contained in a bounded compact set. The equations studied here cover $L^2$ and $\dot{H}^1$ critical and supercritical, defocusing and focusing nonlinear Schr$\ddot{o}$dinger equations.