## 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.## References

- Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu,
*On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\Bbb {R}^d$, $d\geq 3$*, Trans. Amer. Math. Soc. Ser. B**2**(2015), 1–50. MR**3350022**, DOI 10.1090/btran/6 - Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu,
*Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS*, Excursions in harmonic analysis. Vol. 4, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2015, pp. 3–25. MR**3411090** - Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu,
*Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $\Bbb {R}^3$*, Trans. Amer. Math. Soc. Ser. B**6**(2019), 114–160. MR**3919013**, DOI 10.1090/btran/29 - Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu,
*On the probabilistic Cauchy theory for nonlinear dispersive PDEs*, Landscapes of time-frequency analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2019, pp. 1–32. MR**3889875** - J. Bourgain,
*Periodic nonlinear Schrödinger equation and invariant measures*, Comm. Math. Phys.**166**(1994), no. 1, 1–26. MR**1309539**, DOI 10.1007/BF02099299 - Jean Bourgain,
*Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation*, Comm. Math. Phys.**176**(1996), no. 2, 421–445. MR**1374420**, DOI 10.1007/BF02099556 - J. Bourgain,
*Refinements of Strichartz’ inequality and applications to $2$D-NLS with critical nonlinearity*, Internat. Math. Res. Notices**5**(1998), 253–283. MR**1616917**, DOI 10.1155/S1073792898000191 - J. Bourgain,
*Scattering in the energy space and below for 3D NLS*, J. Anal. Math.**75**(1998), 267–297. MR**1655835**, DOI 10.1007/BF02788703 - N. Burq and L. Thomann,
*Almost sure scattering for the one dimensional nonlinear Schrödinger equation*, Preprint, arXiv:2012.13571, 2020. - Nicolas Burq and Nikolay Tzvetkov,
*Random data Cauchy theory for supercritical wave equations. I. Local theory*, Invent. Math.**173**(2008), no. 3, 449–475. MR**2425133**, DOI 10.1007/s00222-008-0124-z - Nicolas Burq and Nikolay Tzvetkov,
*Random data Cauchy theory for supercritical wave equations. II. A global existence result*, Invent. Math.**173**(2008), no. 3, 477–496. MR**2425134**, DOI 10.1007/s00222-008-0123-0 - N. Camps and L. Gassot,
*Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations*, International Mathematics Research Notices, 2022, DOI 10.1093/imrn/rnac194. - M. Christ, J. Colliander, and T. Tao,
*Ill-posedness for nonlinear Schrödinger and wave equations*, Preprint, arXiv:0311048, 2003. - J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao,
*Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\Bbb R^3$*, Comm. Pure Appl. Math.**57**(2004), no. 8, 987–1014. MR**2053757**, DOI 10.1002/cpa.20029 - James Colliander and Tadahiro Oh,
*Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\Bbb T)$*, Duke Math. J.**161**(2012), no. 3, 367–414. MR**2881226**, DOI 10.1215/00127094-1507400 - P. Constantin and J.-C. Saut,
*Local smoothing properties of dispersive equations*, J. Amer. Math. Soc.**1**(1988), no. 2, 413–439. MR**928265**, DOI 10.1090/S0894-0347-1988-0928265-0 - Giuseppe Da Prato and Arnaud Debussche,
*Two-dimensional Navier-Stokes equations driven by a space-time white noise*, J. Funct. Anal.**196**(2002), no. 1, 180–210. MR**1941997**, DOI 10.1006/jfan.2002.3919 - Benjamin Dodson,
*Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n=3$ via a linear-nonlinear decomposition*, Discrete Contin. Dyn. Syst.**33**(2013), no. 5, 1905–1926. MR**3002734**, DOI 10.3934/dcds.2013.33.1905 - Benjamin Dodson,
*Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when $d=2,3$ and $u_0$ is radial*, Camb. J. Math.**7**(2019), no. 3, 283–318. MR**4010063**, DOI 10.4310/CJM.2019.v7.n3.a2 - Benjamin Dodson, Jonas Lührmann, and Dana Mendelson,
*Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation*, Adv. Math.**347**(2019), 619–676. MR**3920835**, DOI 10.1016/j.aim.2019.02.001 - Benjamin Dodson, Jonas Lührmann, and Dana Mendelson,
*Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data*, Amer. J. Math.**142**(2020), no. 2, 475–504. MR**4084161**, DOI 10.1353/ajm.2020.0013 - C. Fan and D. Mendelson,
*Construction of $L^2$ log-log blowup solutions for the mass critical nonlinear Schrödinger equation*, Preprint, arXiv:2010.07821, 2020. - M. Gubinelli, H. Koch, T. Oh, and L. Tolomeo,
*Global dynamics for the two-dimensional stochastic nonlinear wave equations*, Internat. Math. Res. Notices (2021). - Martin Hadac, Sebastian Herr, and Herbert Koch,
*Well-posedness and scattering for the KP-II equation in a critical space*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**26**(2009), no. 3, 917–941. MR**2526409**, DOI 10.1016/j.anihpc.2008.04.002 - Sebastian Herr, Daniel Tataru, and Nikolay Tzvetkov,
*Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\Bbb T^3)$*, Duke Math. J.**159**(2011), no. 2, 329–349. MR**2824485**, DOI 10.1215/00127094-1415889 - Carlos E. Kenig and Frank Merle,
*Scattering for $\dot H^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions*, Trans. Amer. Math. Soc.**362**(2010), no. 4, 1937–1962. MR**2574882**, DOI 10.1090/S0002-9947-09-04722-9 - Carlos E. Kenig, Gustavo Ponce, and Luis Vega,
*Global well-posedness for semi-linear wave equations*, Comm. Partial Differential Equations**25**(2000), no. 9-10, 1741–1752. MR**1778778**, DOI 10.1080/03605300008821565 - Rowan Killip, Jason Murphy, and Monica Visan,
*Almost sure scattering for the energy-critical NLS with radial data below $H^1(\Bbb R^4)$*, Comm. Partial Differential Equations**44**(2019), no. 1, 51–71. MR**3933623**, DOI 10.1080/03605302.2018.1541904 - Jonas Lührmann and Dana Mendelson,
*Random data Cauchy theory for nonlinear wave equations of power-type on $\Bbb {R}^3$*, Comm. Partial Differential Equations**39**(2014), no. 12, 2262–2283. MR**3259556**, DOI 10.1080/03605302.2014.933239 - R. Coifman and Y. Meyer,
*Commutateurs d’intégrales singulières et opérateurs multilinéaires*, Ann. Inst. Fourier (Grenoble)**28**(1978), no. 3, xi, 177–202 (French, with English summary). MR**511821**, DOI 10.5802/aif.708 - Tadahiro Oh, Mamoru Okamoto, and Oana Pocovnicu,
*On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities*, Discrete Contin. Dyn. Syst.**39**(2019), no. 6, 3479–3520. MR**3959438**, DOI 10.3934/dcds.2019144 - A. Poiret,
*Solutions globales pour l’équation de Schrödinger cubique en dimension 3*, Preprint, arXiv:1207.1578, 2012. - Aurélien Poiret, Didier Robert, and Laurent Thomann,
*Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator*, Anal. PDE**7**(2014), no. 4, 997–1026. MR**3254351**, DOI 10.2140/apde.2014.7.997 - J. Shen, A. Soffer, and Y. Wu,
*Almost sure well-posedness and scattering of 3d cubic nonlinear Schrödinger equation*, Preprint, arXiv:2110.11648, 2021. - M. Spitz,
*Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data*, arXiv preprint arXiv:2110.11051, 2021. - Qingtang Su,
*Global well posedness and scattering for the defocusing, cubic NLS in $\Bbb R^3$*, Math. Res. Lett.**19**(2012), no. 2, 431–451. MR**2955773**, DOI 10.4310/MRL.2012.v19.n2.a14 - Chenmin Sun and Nikolay Tzvetkov,
*Concerning the pathological set in the context of probabilistic well-posedness*, C. R. Math. Acad. Sci. Paris**358**(2020), no. 9-10, 989–999 (English, with English and French summaries). MR**4196770**, DOI 10.5802/crmath.102 - Chenmin Sun and Bo Xia,
*Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three*, Illinois J. Math.**60**(2016), no. 2, 481–503. MR**3680544**

## 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