Global well-posedness of NLS equations

The thesis is a survey of the I-method. After introducing the method, we discuss the implementation of this method to cubic, defocusing nonlinear Schr ̈odinger equation in the spatial dimension n = 2 and quintic, defocusing nonlinear Schr ̈odinger equation in the spatial dimension n = 1 with detailed calculations. We will find out on which type of equations one can use the I-method. We then mention our joint work with Engin Ba ̧sako ̆glu on the cubic defocusing fourth order nonlinear Schr ̈odinger equation in the spatial dimension n = 4. Lastly, we discuss advantage and disadvantage of the method and share our idea of a study plan of the same equation in the spatial dimensions n = 2, 3 as future work.