Global well-posedness and scattering for the energy-critical Schrödinger equation in $\mathbb R^3$


We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$ spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation in $\mathbb{R}^{1+3}$, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain [4] and Grillakis [20], which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain [4], but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant (first used in [12], [13]). The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of $L^2$ mass in frequency space, rules out the possibility of energy concentration.


James Colliander

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4

Markus Keel

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Gigiola Staffilani

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States

Hideo Takaoka

Department of Mathematics, Kobe University, Kobe 657-8501, Japan

Terence Tao

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States and Mathematical Sciences Institute, The Australian National University, Canberra 0200, Australia