I’ve just uploaded to the arXiv the paper “The cubic nonlinear Schrödinger equation in two dimensions with radial data“, joint with Rowan Killip and Monica Visan, and submitted to the Annals of Mathematics. This is a sequel of sorts to my paper with Monica and Xiaoyi Zhang, in which we established global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation (NLS) $iu_t + \Delta u = |u|^{4/d} u$ in three and higher dimensions $d \geq 3$ assuming spherically symmetric data. (This is another example of the recently active field of critical dispersive equations, in which both coarse and fine scales are (just barely) nonlinearly active, and propagate at different speeds, leading to significant technical difficulties.)

In this paper we obtain the same result for the defocusing two-dimensional mass-critical NLS $iu_t + \Delta u= |u|^2 u$, as well as in the focusing case $iu_t + \Delta u= -|u|^2 u$ under the additional assumption that the mass of the initial data is strictly less than the mass of the ground state. (When mass equals that of the ground state, there is an explicit example, built using the pseudoconformal transformation, which shows that solutions can blow up in finite time.) In fact we can show a slightly stronger statement: for spherically symmetric focusing solutions with arbitrary mass, we can show that the first singularity that forms concentrates at least as much mass as the ground state.

Like the higher-dimensional paper, the first step is to use the concentration-compactness theory of NLS to reduce matters to studying solutions which are almost periodic modulo the symmetries of the NLS. At this point, though, we have to take a slightly different tack. In higher dimensions we have the luxury (at least in principle) of using Morawetz estimates, although some truncation in space and frequency is needed before these estimates are applicable in the available regularity for these solutions (which is $L^2$). In one and two dimensions, these estimates cease to be coercive and are essentially useless for us; furthermore, even in higher dimensions, the Morawetz inequality is not well suited for the focusing problem when the mass gets too close to that of the ground state. The natural substitute for these inequalities is the virial identity, which works in all dimensions and still has good coercivity properties in the focusing case when the mass gets close to that of the ground state. However, it turns out that the virial identity does not give particularly useful results if applied immediately. Instead, we give a combinatorial argument (which works in all dimensions, both for focusing and defocusing, and does not use spherical symmetry) which allows us to reduce even further, from almost periodic solutions to three special “enemy” solutions, namely a soliton-like solution, a self-similar solution, and a solution with a double high-low frequency cascade. We then eliminate each of these enemies by regularity arguments based on Duhamel’s formula. For instance, in the self-similar case one can use Duhamel’s formula to show that these solutions have finite energy, at which point the self-similar blowup is inconsistent with energy conservation. A somewhat similar situation occurs with the double high-low frequency cascade. Finally, for the soliton-like solution, Duhamel’s formula (together with an incoming/outgoing radiation decomposition from harmonic analysis, based ultimately on Hankel functions) one also obtains finite energy, which turns out to be enough, together with the virial identity, to be inconsistent with soliton-like behavior.

The arguments in this paper seem to suggest that a reasonable strategy to establish critical global well-posedness and scattering is to pass to very special almost-periodic solutions, use Duhamel’s formula to establish as much regularity as one can, and then use all the conservation laws and monotonicity formulae that are available to derive a contradiction. It continues to be a challenge to make these arguments work in the non-radial case, when translation (and Galilean) symmetries become a significant problem.