Low Dispersion Equation (group),

In this dissertation, We introduce some new methods on the local and global well-posedness theories of the dispersive equations in recent years.First, We systematically introduce the general framework of the Strichartz-based local wellposedness theory and the X_φ^s,b -based local wellposedness theory in Chapter 1. Since they are based on Banach fixed point theorem to construct the local solution. we have to establish some homogeneous, inhomogeneous linear and nonlinear estimates. Some useful estimates of Strichartz-based method and an concrete application are introduced in Section 1. The X_φ^s,b-based method is introduced in Section 2. Since it's homogeneous, inhomogeneous linear estimates are applicable to the general dispersive equation, we reduce the local well-posedness to the nonlinear estimates.Next, together the X_φ^s,b -based local wellposedness theory with the High-low frequency truncation method in Chapter 2, we consider the Klein-Gordon-Schro|¨dinger system with quadratic (Yukawa) coupling and cubic autointeractions in R²⁺¹, and prove the existence and uniqueness of global solution for rough data. It is just the low regularity problem.The Klein-Gordon-Schro|¨dinger system with quadratic (Yukawa) coupling is similar in the structure of the Zakharov system. There are many works on their low regularity. Pecher made use of the High-low frequency truncation method [86] and I-Method [88] to consider the low regularity of the Zakharov system in R¹⁺¹. At the same time, he also used the High-low frequency truncation method and studied the Klein-Gordon-Schro|¨dinger system with quadratic (Yukawa) coupling in [87]. Tzirakis used I-method to consider the Klein-Gordon-Schro|¨dinger system with quadratic (Yukawa) coupling in R^d+1,d = 1,2,3 in [109]. It was remarkable that Colliander et at. used another iteration process to obtain the almost optimal global well-posedness on 1D-Zakharov system and 3D-Klein-Gordon-Schrodinger system with quadratic (Yukawa) coupling in [24]. Their method strongly relies on the particular nonlinear structure, it isn't applicable to the Klein-Gordon-Schro|¨dinger system with quadratic (Yukawa) coupling and cubic autointeractions.Physically, data which is only in H^s for s