On the global solutions of quasilinear dispersive equations

Abstract

This thesis mainly focuses on certain nonlinear dispersive equations where the classical

Picard's fix-point argument fails in obtaining the desired local or global solutions.

More specifically, Chapter two proves the local well-posedness of the KP-I initial

value problem on the torus T^2 with initial data in the Besov space B^1_{2,1} through a

short-time estimate approach. Chapter three constructs global solutions to a modified

ionic Euler-Poisson system in two dimensions, given the initial data is small smooth

irrotational perturbation of the constant background. The main ingredients in the

proof is a quasi-linear I-method approach, along with the Fourier transform method

analyzing its space-time resonance feature.