Probabilistic pointwise convergence problem of Schrödinger equations on manifolds

Wang, Junfang; Yan, Wei; Yan, Xiangqian

doi:10.1090/proc/15440

Probabilistic pointwise convergence problem of Schrödinger equations on manifolds
HTML articles powered by AMS MathViewer

by Junfang Wang, Wei Yan and Xiangqian Yan PDF

Proc. Amer. Math. Soc. 149 (2021), 3367-3378 Request permission

Abstract:

In this paper, we investigate the probabilistic pointwise convergence problem of Schrödinger equation on the manifolds. We prove probabilistic pointwise convergence of the solutions to Schrödinger equations with the initial data in $L^{2}(\mathrm {\mathbf {T}}^{n})$, where $\mathrm {\mathbf {T}}=[0,2\pi )$, which require much less regularity for the initial data than the rough data case. We also prove probabilistic pointwise convergence of the solutions to Schrödinger equation with Dirichlet boundary condition for a large set of random initial data in $\cap _{s<\frac {1}{2}}H^{s}(\Theta )$, where $\Theta$ is three dimensional unit ball, which require much less regularity for the initial data than the rough data case.

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
J. Bourgain, A remark on Schrödinger operators, Israel J. Math. 77 (1992), no. 1-2, 1–16. MR 1194782, DOI 10.1007/BF02808007
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, Some new estimates on oscillatory integrals, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 83–112. MR 1315543
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, On the Schrödinger maximal function in higher dimension, Tr. Mat. Inst. Steklova 280 (2013), no. Ortogonal′nye Ryady, Teoriya Priblizheniĭ i Smezhnye Voprosy, 53–66; English transl., Proc. Steklov Inst. Math. 280 (2013), no. 1, 46–60. MR 3241836, DOI 10.1134/s0081543813010045
J. Bourgain, A note on the Schrödinger maximal function, J. Anal. Math. 130 (2016), 393–396. MR 3574661, DOI 10.1007/s11854-016-0042-8
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
Lennart Carleson, Some analytic problems related to statistical mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 5–45. MR 576038
Yong Chen and Hongjun Gao, The Cauchy problem for the Hartree equations under random influences, J. Differential Equations 259 (2015), no. 10, 5192–5219. MR 3377523, DOI 10.1016/j.jde.2015.06.021
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
Mingjuan Chen and Shuai Zhang, Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities, Nonlinear Anal. 190 (2020), 111608, 23. MR 3997682, DOI 10.1016/j.na.2019.111608
Chu-Hee Cho, Sanghyuk Lee, and Ana Vargas, Problems on pointwise convergence of solutions to the Schrödinger equation, J. Fourier Anal. Appl. 18 (2012), no. 5, 972–994. MR 2970037, DOI 10.1007/s00041-012-9229-2
Erin Compaan, Renato Lucà, and Gigliola Staffilani, Pointwise convergence of the Schrödinger flow, Int. Math. Res. Not. IMRN 1 (2021), 599–650. MR 4198507, DOI 10.1093/imrn/rnaa036
Michael G. Cowling, Pointwise behavior of solutions to Schrödinger equations, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 83–90. MR 729347, DOI 10.1007/BFb0069152
Chao Deng and Shangbin Cui, Random-data Cauchy problem for the Navier-Stokes equations on $\Bbb T^3$, J. Differential Equations 251 (2011), no. 4-5, 902–917. MR 2812576, DOI 10.1016/j.jde.2011.05.002
Björn E. J. Dahlberg and Carlos E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 205–209. MR 654188
C. Demeter and S. Guo, Schrödinger maximal function estimates via the pseudoconformal transformation, arXiv:1608.07640, 2016.
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
Xiumin Du, Larry Guth, and Xiaochun Li, A sharp Schrödinger maximal estimate in $\Bbb R^2$, Ann. of Math. (2) 186 (2017), no. 2, 607–640. MR 3702674, DOI 10.4007/annals.2017.186.2.5
Xiumin Du and Ruixiang Zhang, Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions, Ann. of Math. (2) 189 (2019), no. 3, 837–861. MR 3961084, DOI 10.4007/annals.2019.189.3.4
Xiumin Du, Larry Guth, Xiaochun Li, and Ruixiang Zhang, Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates, Forum Math. Sigma 6 (2018), Paper No. e14, 18. MR 3842310, DOI 10.1017/fms.2018.11
D. Eceizabarrena and R. Luc$\grave {a}$, Converence over fractals for the periodic Schrödinger equation, arXiv:2005.07581, 2020.
Giacomo Gigante and Fernando Soria, On the boundedness in $H^{1/4}$ of the maximal square function associated with the Schrödinger equation, J. Lond. Math. Soc. (2) 77 (2008), no. 1, 51–68. MR 2389916, DOI 10.1112/jlms/jdm087
Hiroyuki Hirayama and Mamoru Okamoto, Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst. 36 (2016), no. 12, 6943–6974. MR 3567827, DOI 10.3934/dcds.2016102
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
Joel L. Lebowitz, Harvey A. Rose, and Eugene R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys. 50 (1988), no. 3-4, 657–687. MR 939505, DOI 10.1007/BF01026495
Sanghyuk Lee, On pointwise convergence of the solutions to Schrödinger equations in $\Bbb R^2$, Int. Math. Res. Not. (2006), Art. ID 32597, 21. MR 2264734
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. Luca and M. Rogers, An improved neccessary condition for Schrödinger maximal estimate, arXiv:1506.05325, 2015.
Renato Lucà and Keith M. Rogers, Coherence on fractals versus pointwise convergence for the Schrödinger equation, Comm. Math. Phys. 351 (2017), no. 1, 341–359. MR 3613507, DOI 10.1007/s00220-016-2722-8
Changxing Miao, Jianwei Yang, and Jiqiang Zheng, An improved maximal inequality for 2D fractional order Schrödinger operators, Studia Math. 230 (2015), no. 2, 121–165. MR 3476484, DOI 10.4064/sm8292-1-2016
Changxing Miao, Junyong Zhang, and Jiqiang Zheng, Maximal estimates for Schrödinger equations with inverse-square potential, Pacific J. Math. 273 (2015), no. 1, 1–19. MR 3290442, DOI 10.2140/pjm.2015.273.1
A. Moyua, A. Vargas, and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Internat. Math. Res. Notices 16 (1996), 793–815. MR 1413873, DOI 10.1155/S1073792896000499
A. Moyua and L. Vega, Bounds for the maximal function associated to periodic solutions of one-dimensional dispersive equations, Bull. Lond. Math. Soc. 40 (2008), no. 1, 117–128. MR 2409184, DOI 10.1112/blms/bdm096
Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, and Gigliola Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1275–1330. MR 2928851, DOI 10.4171/JEMS/333
Andrea R. Nahmod, Nataša Pavlović, and Gigliola Staffilani, Almost sure existence of global weak solutions for supercritical Navier-Stokes equations, SIAM J. Math. Anal. 45 (2013), no. 6, 3431–3452. MR 3131480, DOI 10.1137/120882184
Andrea R. Nahmod and Gigliola Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1687–1759. MR 3361727, DOI 10.4171/JEMS/543
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
Tadahiro Oh and Oana Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\Bbb {R}^3$, J. Math. Pures Appl. (9) 105 (2016), no. 3, 342–366 (English, with English and French summaries). MR 3465807, DOI 10.1016/j.matpur.2015.11.003
Oana Pocovnicu, Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $\Bbb {R}^d$, $d=4$ and $5$, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 8, 2521–2575. MR 3668066, DOI 10.4171/JEMS/723
Keith M. Rogers, Ana Vargas, and Luis Vega, Pointwise convergence of solutions to the nonelliptic Schrödinger equation, Indiana Univ. Math. J. 55 (2006), no. 6, 1893–1906. MR 2284549, DOI 10.1512/iumj.2006.55.2827
Per Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), no. 3, 699–715. MR 904948, DOI 10.1215/S0012-7094-87-05535-9
S. Shao, On localization of the Schrödinger maximal operator, arXiv:1006.2787v1, 2010.
T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. MR 2033842, DOI 10.1007/s00039-003-0449-0
T. Tao and A. Vargas, A bilinear approach to cone multipliers. II. Applications, Geom. Funct. Anal. 10 (2000), no. 1, 216–258. MR 1748921, DOI 10.1007/s000390050007
Luis Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), no. 4, 874–878. MR 934859, DOI 10.1090/S0002-9939-1988-0934859-0
Xing Wang and Chunjie Zhang, Pointwise convergence of solutions to the Schrödinger equation on manifolds, Canad. J. Math. 71 (2019), no. 4, 983–995. MR 3984026, DOI 10.4153/cjm-2018-001-4
Ting Zhang and Daoyuan Fang, Random data Cauchy theory for the incompressible three dimensional Navier-Stokes equations, Proc. Amer. Math. Soc. 139 (2011), no. 8, 2827–2837. MR 2801624, DOI 10.1090/S0002-9939-2011-10762-7
Ting Zhang and Daoyuan Fang, Random data Cauchy theory for the generalized incompressible Navier-Stokes equations, J. Math. Fluid Mech. 14 (2012), no. 2, 311–324. MR 2925111, DOI 10.1007/s00021-011-0069-7