Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ³

Bényi, Árpád; Oh, Tadahiro; Pocovnicu, Oana

doi:10.1090/btran/29

Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb {R}^3$
HTML articles powered by AMS MathViewer

by Árpád Bényi, Tadahiro Oh and Oana Pocovnicu HTML | PDF

Trans. Amer. Math. Soc. Ser. B 6 (2019), 114-160

Abstract:

We consider the cubic nonlinear Schrödinger equation (NLS) on $\mathbb {R}^3$ with randomized initial data. In particular, we study an iterative approach based on a partial power series expansion in terms of the random initial data. By performing a fixed point argument around the second order expansion, we improve the regularity threshold for almost sure local well-posedness from our previous work. We further investigate a limitation of this iterative procedure. Finally, we introduce an alternative iterative approach, based on a modified expansion of arbitrary length, and prove almost sure local well-posedness of the cubic NLS in an almost optimal regularity range with respect to the original iterative approach based on a power series expansion.

References

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, 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
Á. Bényi, T. Oh, and O. Pocovnicu, On the probabilistic Cauchy theory for nonlinear dispersive PDEs, Landscapes of Time-Frequency Analysis, 1–32, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, 2019.
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
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. Brereton, Almost sure local well-posedness for the supercritical quintic NLS, Tunisian J. Math. 1 (2019), no. 3, 427–453.
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, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1, 1–30. MR 3141727, DOI 10.4171/JEMS/426
Thierry Cazenave and Fred B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987) Lecture Notes in Math., vol. 1394, Springer, Berlin, 1989, pp. 18–29. MR 1021011, DOI 10.1007/BFb0086749
M. Christ, Power series solution of a nonlinear Schrödinger equation, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 131–155. MR 2333210
M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arXiv:math/0311048 [math.AP].
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
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2) 167 (2008), no. 3, 767–865. MR 2415387, DOI 10.4007/annals.2008.167.767
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
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
Giuseppe Da Prato and Arnaud Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab. 31 (2003), no. 4, 1900–1916. MR 2016604, DOI 10.1214/aop/1068646370
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
Peter K. Friz and Martin Hairer, A course on rough paths, Universitext, Springer, Cham, 2014. With an introduction to regularity structures. MR 3289027, DOI 10.1007/978-3-319-08332-2
J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), no. 1, 163–188. MR 1151250, DOI 10.1007/BF02099195
Massimiliano Gubinelli and Nicolas Perkowski, Lectures on singular stochastic PDEs, Ensaios Matemáticos [Mathematical Surveys], vol. 29, Sociedade Brasileira de Matemática, Rio de Janeiro, 2015. MR 3445609
Martin Hadac, Sebastian Herr, and Herbert Koch, Erratum to “Well-posedness and scattering for the KP-II equation in a critical space” [Ann. I. H. Poincaré—AN 26 (3) (2009) 917–941] [MR2526409], Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 3, 971–972. MR 2629889, DOI 10.1016/j.anihpc.2010.01.006
Martin Hairer, Introduction to regularity structures, Braz. J. Probab. Stat. 29 (2015), no. 2, 175–210. MR 3336866, DOI 10.1214/14-BJPS241
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
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
Justin Holmer and Svetlana Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), no. 2, 435–467. MR 2421484, DOI 10.1007/s00220-008-0529-y
Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. MR 833073
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
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
R. Killip, J. Murphy, and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb {R}^4)$, arXiv:1707.09051 [math.AP], 2017.
Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm053, 36. MR 2353092, DOI 10.1093/imrn/rnm053
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
Jonas Lührmann and Dana Mendelson, On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $\Bbb R^3$, New York J. Math. 22 (2016), 209–227. MR 3484682
H. P. McKean, Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Comm. Math. Phys. 168 (1995), no. 3, 479–491. MR 1328250, DOI 10.1007/BF02101840
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, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac. 60 (2017), no. 2, 259–277. MR 3702002, DOI 10.1619/fesi.60.259
T. Oh, M. Okamoto, and O. Pocovnicu, On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities, arXiv:1708.01568 [math.AP], Discrete Contin. Dyn. Syst. A., to appear.
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
T. Oh, O. Pocovnicu, and Y. Wang, On the stochastic nonlinear Schrödinger equations with non-smooth additive noise, arXiv:1805.08412 [math.AP], Kyoto J. Math, to appear.
T. Oh, N. Tzvetkov, and Y. Wang, Solving the 4NLS with white noise initial data, preprint.
T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201–222. MR 1741843
R. E. A. C. Paley and A. Zygmund, On some series of functions (1), (2), (3), Proc. Camb. Philos. Soc. 26 (1930), 337–357, 458–474; 28 (1932), 190–205.
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
Oana Pocovnicu and Yuzhao Wang, An $L^p$-theory for almost sure local well-posedness of the nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris 356 (2018), no. 6, 637–643 (English, with English and French summaries). MR 3806892, DOI 10.1016/j.crma.2018.04.009
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
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
Laurent Thomann, Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 6, 2385–2402 (English, with English and French summaries). MR 2569900, DOI 10.1016/j.anihpc.2009.06.001
Norbert Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. MR 1503035, DOI 10.2307/1968102
Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945, DOI 10.1007/BF01212420
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