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arXiv · 2110.11051

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schr\"odinger equation with supercritical data

Martin Spitz
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Abstract

We study the cubic defocusing nonlinear Schr\"odinger equation on $\mathbb{R}^4$ with supercritical initial data. For randomized initial data in $H^s(\mathbb{R}^4)$, we prove almost sure local wellposedness for $\frac{1}{7} < s < 1$ and almost sure scattering for $\frac{5}{7} < s < 1$. 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 Schr\"odinger flow with randomized data, where we effectively combine the advantages of the different decompositions.

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Martin Spitz. 2021-10-21. Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schr\"odinger equation with supercritical data. https://arxiv.org/abs/2110.11051

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