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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Scattering for the cubic Schrödinger equation in 3D with randomized radial initial data
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by Nicolas Camps PDF
Trans. Amer. Math. Soc. 376 (2023), 285-333 Request permission

Abstract:

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space $\mathbb {R}^3$, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in $\mathbb {R}^4$, to the energy-subcritical equation in $\mathbb {R}^3$. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the $I$-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime.
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Additional Information
  • Nicolas Camps
  • Affiliation: Université Paris-Saclay, Laboratoire de mathématiques d’Orsay, UMR 8628 du CNRS, Bâtiment 307, 91405 Orsay Cedex, France
  • ORCID: 0000-0002-7451-3576
  • Email: nicolas.camps@universite-paris-saclay.fr
  • Received by editor(s): October 25, 2021
  • Received by editor(s) in revised form: March 21, 2022
  • Published electronically: October 13, 2022
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 285-333
  • MSC (2020): Primary 35B40; Secondary 42B37, 35A01, 35Q55, 35B60, 35R60
  • DOI: https://doi.org/10.1090/tran/8737