Global Solutions for Small Nonlinear Long Range Perturbations of Two Dimensional Schrödinger Equations
In the book, a global existence of solutions to the two dimensional Schrödinger equations is proved. The equation allows for a quadratic nonlinearity containing the space derivatives. The initial data are small. The setting is ‘critical’ in the sense that the nonlinearity becomes non-integrable when applied to the fundamental solution of the Schrödinger operator. To overcome this difficulty, a suitably chosen function is subtracted from the solution to cancel out the ‘worst’ terms. The main technique of the book is classical harmonic analysis. Symbolic calculus, dyadic decomposition, carefully developed linear theory, together with some nonlinear estimates (e.g., for products) are the keystones of the final success, and the solution is found in a suitable weighted space. The book is rather technical but is mostly self-contained.