Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

We study the cubic defocusing nonlinear Schrödinger equation on R with supercritical initial data. For randomized initial data in H(R), we prove almost sure local wellposedness for 1 7 < s < 1 and almost sure scattering for 11 14 < s < 1. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and – for the almost sure scattering result – an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, where we effectively combine the advantages of the different decompositions.

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