In this seminar, we give some brief introduction to the theory of dispersive partial differential equations:

 

        - We give some basic well-posedness theory of 3D cubic nonlinear Schödinger equation (NLS) using Strichartz inequality.

        - We introduce the notion of ground state which is the critical element of Gargliardo-Nirenberg inequality. We study the concentration-compactness technique in the formulation of the bubble decomposition.

        - As a final goal, we introduce Holmer-Roudenko's (08, CMP) global existence versus finite time blowup dichotomy of 3D focusing cubic NLS (without scattering part).

 

This kind of approach can be extended to many other cases, for example, a work of Hong-Kwon-Y. (19, JMPA), which studies a system of NLS with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal.