AbstractsMathematics

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.