Abstract of Proposed Research Ioan Bejenaru
This project is to study the Schroedinger map equation. This is a nonlinear Schroedinger equation which arises from geometrical considerations and also as a version of the Heisenberg model of ferro-magnetism. The fundamental question is, under what conditions on the data, is there global existence of the solutions and when is there "blow-up" of solutions. To study this, we shall investigate the dynamics of solutions with very rough initial data and will use fine tools from harmonic analysis and Riemannian geometry as well as partial differential equations methods. It is expected that the analysis will be more difficult than the Wave map equation which has been studied extensively recently. The Schroedinger map equation has a special nonlinear term; under this project various other non-generic nonlinearities will also be studied.
Since the primary problems to be studied in this project arise in well-known physical theories, the results obtained should have some impact for theories of electromagnetism and ferromagnetic materials. They may well prove to be prototypical problems for a number of other important equations that arise in geometry and physics and for which existence and blow-up results are yet to be obtained.
