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This monograph provides a comprehensive overview on a class of nonlinear dispersive equations, such as nonlinear Schrodinger equation, nonlinear Klein-Gordon equation, KdV equation as well as the Navier-Stokes equations and the Boltzmann equation. The global wellposedness to the Cauchy problem for those equations are systematically studied by using the Harmonic analysis methods. This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects- and even ambitious undergraduate students.