Nonlinear Schr\"odinger equations at non-conserved critical regularity
We study the critical initial-value problem for defocusing nonlinear Schr\"odinger equations.
We adapt techniques that were originally developed to treat the mass- and energy-critical equations to the case of `non-conserved' critical regularity. In particular, we follow the minimal counterexample approach to the induction on energy technique of Bourgain.
For a range of dimensions and critical regularities, we prove that any solution that remains bounded in the critical Sobolev space must exist globally in time, obey spacetime bounds, and scatter to a free solution. In certain cases, the main result applies only to radial solutions. An equivalent formulation of the main result is the statement that any solution that fails to scatter must blow up its critical Sobolev norm.