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Almost Sure Well-Posedness and Scattering of the 3D Cubic Nonlinear Schrödinger Equation

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We study the random data problem for 3D, defocusing, cubic nonlinear Schrödinger equation in \(H_x^s({{\mathbb {R}}}^3)\) with \(s<\frac{1}{2}\). First, we prove that the almost sure local well-posedness holds when \(\frac{1}{6}\leqslant s<\frac{1}{2}\) in the sense that the Duhamel term belongs to \(H_x^{1/2}({{\mathbb {R}}}^3)\). Furthermore, we prove that the global well-posedness and scattering hold for randomized, radial, large data \(f\in H_x^{s}({{\mathbb {R}}}^3)\) when \(\frac{17}{40}< s<\frac{1}{2}\). The key ingredient is to control the energy increment including the terms where the first order derivative acts on the linear flow, and our argument can lower down the order of derivative more than \(\frac{1}{2}\). To our best knowledge, this is the first almost sure large data global result for this model.

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Acknowledgements

J. Shen and Y. Wu are partially supported by NSFC 12171356 and 11771325. A. Soffer is partially supported by the Simons’ Foundation (No. 395767). The authors would like to thank the editor and anonymous referees for valuable comments and suggestions.

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  1. Center for Applied Mathematics, Tian** University, Tian**, 300072, China

    Jia Shen & Yifei Wu

  2. Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ, 08854, USA

    Avy Soffer

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Shen, J., Soffer, A. & Wu, Y. Almost Sure Well-Posedness and Scattering of the 3D Cubic Nonlinear Schrödinger Equation. Commun. Math. Phys. 397, 547–605 (2023). https://doi.org/10.1007/s00220-022-04500-z

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