53rd Midwest PDE Seminar March 19-21, 2004 Department of Mathematics University of Missouri

Markus Keel University of Minnesota
A Regularity Result for Nonlinear Schröedinger Equations

Abstract

We describe recent work with J. Colliander, G. Staffilani, H. Takaoka, and T. Tao on the following nonlinear Schröedinger equation in three space dimensions,

where we start the evolution from given data at time zero. This equation is "critical" in the sense that the natural scaling enjoyed by solutions leaves the conserved energy norm of solutions unchanged. (So for example, we can't reduce to the case of small energy data simply by scaling.)

Our result is that if the initial data has finite (and possibly large) energy, then there is a unique solution that exists for all time and asymptotically approaches a linear solution.