On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb {R}^d$, $d \geq 3$
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- by Árpád Bényi, Tadahiro Oh and Oana Pocovnicu HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 2 (2015), 1-50
Abstract:
We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) $: i \partial _t u + \Delta u = \pm |u|^{2}u$ on $\mathbb {R}^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity $s_\mathrm {crit} = \frac {d-2}{2}$. More precisely, given a function on $\mathbb {R}^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove ‘conditional’ almost sure global well-posedness for $d = 4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $d \ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.References
- Thomas Alazard and Rémi Carles, Loss of regularity for supercritical nonlinear Schrödinger equations, Math. Ann. 343 (2009), no. 2, 397–420. MR 2461259, DOI 10.1007/s00208-008-0276-6
- Antoine Ayache and Nikolay Tzvetkov, $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4425–4439. MR 2395179, DOI 10.1090/S0002-9947-08-04456-5
- Árpád Bényi and Tadahiro Oh, Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math. 228 (2011), no. 5, 2943–2981. MR 2838066, DOI 10.1016/j.aim.2011.07.023
- Árpad Bényi, Tadahiro Oh, and Oana Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, to appear in Excursions in Harmonic Analysis.
- 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, Invariant measures for the Gross-Piatevskii equation, J. Math. Pures Appl. (9) 76 (1997), no. 8, 649–702. MR 1470880, DOI 10.1016/S0021-7824(97)89965-5
- 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
- Jean Bourgain and Aynur Bulut, Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 6, 1289–1325. MR 3226743, DOI 10.4171/JEMS/461
- Jean Bourgain and Aynur Bulut, Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball, J. Funct. Anal. 266 (2014), no. 4, 2319–2340. MR 3150162, DOI 10.1016/j.jfa.2013.06.002
- Nicolas Burq, Patrick Gérard, and Nikolay Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4) 38 (2005), no. 2, 255–301 (English, with English and French summaries). MR 2144988, DOI 10.1016/j.ansens.2004.11.003
- Nicolas Burq, Laurent Thomann, and Nikolay Tzvetkov, Long time dynamics for the one dimensional non linear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2137–2198 (English, with English and French summaries). MR 3237443, DOI 10.5802/aif.2825
- Nicolas Burq, Laurent Thomann, and Nikolay Tzvetkov, Global infinite energy solutions for the cubic wave equation, to appear in Bull. Soc. Math. France.
- 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
- Rémi Carles, Geometric optics and instability for semi-classical Schrödinger equations, Arch. Ration. Mech. Anal. 183 (2007), no. 3, 525–553. MR 2278414, DOI 10.1007/s00205-006-0017-5
- 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
- Michael Christ, James Colliander, and Terrence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), no. 6, 1235–1293. MR 2018661, DOI 10.1353/ajm.2003.0040
- 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
- Yu Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE 5 (2012), no. 5, 913–960. MR 3022846, DOI 10.2140/apde.2012.5.913
- Anne-Sophie de Suzzoni, Invariant measure for the cubic wave equation on the unit ball of $\Bbb R^3$, Dyn. Partial Differ. Equ. 8 (2011), no. 2, 127–147. MR 2857361, DOI 10.4310/DPDE.2011.v8.n2.a4
- Anne-Sophie de Suzzoni, Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space, Commun. Pure Appl. Anal. 13 (2014), no. 3, 991–1015. MR 3177685, DOI 10.3934/cpaa.2014.13.991
- Hans Feichtinger, Modulation spaces of locally compact Abelian groups, Technical report, University of Vienna (1983). in Proc. Internat. Conf. on Wavelets and Applications (Chennai, 2002), R. Radha, M. Krishna, S. Thangavelu (eds.), New Delhi Allied Publishers (2003), 1–56.
- Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989), no. 2, 307–340. MR 1021139, DOI 10.1016/0022-1236(89)90055-4
- Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), no. 2-3, 129–148. MR 1026614, DOI 10.1007/BF01308667
- 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
- Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
- 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
- Alexandru D. Ionescu and Benoit Pausader, Global well-posedness of the energy-critical defocusing NLS on $\Bbb R\times \Bbb T^3$, Comm. Math. Phys. 312 (2012), no. 3, 781–831. MR 2925134, DOI 10.1007/s00220-012-1474-3
- 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, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675. MR 2257393, DOI 10.1007/s00222-006-0011-4
- 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 and Frank Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, Amer. J. Math. 133 (2011), no. 4, 1029–1065. MR 2823870, DOI 10.1353/ajm.2011.0029
- Rowan Killip, Tadahiro Oh, Oana Pocovnicu, and Monica Vişan, Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions, Math. Res. Lett. 19 (2012), no. 5, 969–986. MR 3039823, DOI 10.4310/MRL.2012.v19.n5.a1
- Rowan Killip and Monica Vişan, Nonlinear Schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437. MR 3098643, DOI 10.1007/s00208-013-0960-z
- Rowan Killip and Monica Visan, Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering, Comm. Partial Differential Equations 35 (2010), no. 6, 945–987. MR 2753625, DOI 10.1080/03605301003717084
- Masaharu Kobayashi and Mitsuru Sugimoto, The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal. 260 (2011), no. 11, 3189–3208. MR 2776566, DOI 10.1016/j.jfa.2011.02.015
- 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
- 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, to appear in J. Eur. Math. Soc.
- Tadahiro Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegő equation, Funkcial. Ekvac. 54 (2011), no. 3, 335–365. MR 2918143, DOI 10.1619/fesi.54.335
- Tadahiro Oh and Oana Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $\mathbb {R}^3$, arXiv:1502.00575 [math.AP].
- Kasso A. Okoudjou, Embedding of some classical Banach spaces into modulation spaces, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1639–1647. MR 2051124, DOI 10.1090/S0002-9939-04-07401-5
- 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. Cambridge Philos. Soc. 26 (1930), 337–357, 458–474; 28 (1932), 190–205.
- Oana Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing cubic nonlinear wave equations on $\mathbb {R}^4$, to appear in J. Eur. Math. Soc. (JEMS).
- 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
- G. Richards, Invariance of the Gibbs measure for the periodic quartic gKdV, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire.
- E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\Bbb R^{1+4}$, Amer. J. Math. 129 (2007), no. 1, 1–60. MR 2288737, DOI 10.1353/ajm.2007.0004
- 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
- Mitsuru Sugimoto and Naohito Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal. 248 (2007), no. 1, 79–106. MR 2329683, DOI 10.1016/j.jfa.2007.03.015
- Terence Tao and Monica Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations (2005), No. 118, 28. MR 2174550
- Terence Tao, Monica Visan, and Xiaoyi Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1281–1343. MR 2354495, DOI 10.1080/03605300701588805
- 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
- Joachim Toft, Convolutions and embeddings for weighted modulation spaces, Advances in pseudo-differential operators, Oper. Theory Adv. Appl., vol. 155, Birkhäuser, Basel, 2004, pp. 165–186. MR 2090373
- Monica Vişan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions, Int. Math. Res. Not. IMRN 5 (2012), 1037–1067. MR 2899959, DOI 10.1093/imrn/rnr051
- 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
Additional Information
- Árpád Bényi
- Affiliation: Department of Mathematics, Western Washington University, 516 High Street, Bellingham, Washington 98225
- MR Author ID: 672886
- Email: arpad.benyi@wwu.edu
- Tadahiro Oh
- Affiliation: School of Mathematics, The University of Edinburgh – and – The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
- MR Author ID: 782317
- Email: hiro.oh@ed.ac.uk
- Oana Pocovnicu
- Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 – and – Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, New Jersey 08544
- MR Author ID: 948569
- Email: opocovnicu@math.princeton.edu
- Received by editor(s): October 27, 2014
- Received by editor(s) in revised form: April 20, 2015
- Published electronically: May 26, 2015
- Additional Notes: This work was partially supported by a grant from the Simons Foundation (No. 246024 to the first author). The second author was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). The third author was supported by the NSF grant under agreement No. DMS-1128155. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF
- © Copyright 2015 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 2 (2015), 1-50
- MSC (2010): Primary 35Q55
- DOI: https://doi.org/10.1090/btran/6
- MathSciNet review: 3350022