The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability

Nahmod, Andrea

doi:10.1090/bull/1516

The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability
HTML articles powered by AMS MathViewer

by Andrea R. Nahmod PDF

Bull. Amer. Math. Soc. 53 (2016), 57-91 Request permission

Abstract:

The field of nonlinear dispersive and wave equations has undergone significant progress in the last twenty years thanks to the influx of tools and ideas from nonlinear Fourier and harmonic analysis, geometry, analytic number theory and most recently probability, into the existing functional analytic methods. In these lectures we concentrate on the semilinear Schrödinger equation defined on tori and discuss the most important developments in the analysis of these equations. In particular, we discuss in some detail recent work by J. Bourgain and C. Demeter proving the $\ell ^2$ decoupling conjecture and as a consequence the full range of Strichartz estimates on either rational or irrational tori, thus settling an important earlier conjecture by Bourgain.

References

Mark J. Ablowitz, Nonlinear dispersive waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2011. Asymptotic analysis and solitons. MR 2848561, DOI 10.1017/CBO9780511998324
Jonathan Bennett, Anthony Carbery, and Terence Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261–302. MR 2275834, DOI 10.1007/s11511-006-0006-4
S. N. Bose, Plancks Gesetz und Lichtquantenhypothese, Zeitschrift für Physik, 26 (1924), 178.
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
J. Bourgain, Eigenfunction bounds for the Laplacian on the $n$-torus, Internat. Math. Res. Notices 3 (1993), 61–66. MR 1208826, DOI 10.1155/S1073792893000066
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
Jean Bourgain, Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions (Park City, UT, 1995) IAS/Park City Math. Ser., vol. 5, Amer. Math. Soc., Providence, RI, 1999, pp. 3–157. MR 1662829, DOI 10.1090/coll/046
J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications, vol. 46, American Mathematical Society, Providence, RI, 1999. MR 1691575, DOI 10.1090/coll/046
J. Bourgain, On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 1–20. MR 2331676
J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math. 193 (2013), no. 1, 441–458. MR 3038558, DOI 10.1007/s11856-012-0077-1
Jean Bourgain and Ciprian Demeter, Improved estimates for the discrete Fourier restriction to the higher dimensional sphere, Illinois J. Math. 57 (2013), no. 1, 213–227. MR 3224568
J. Bourgain and C. Demeter, New bounds for the discrete Fourier restriction to the sphere in four and five dimensions, to apper in Int. Math. Res. Notices (2014). Preprint at http://arxiv.org/pdf/1310.5244.pdf
J. Bourgain and C. Demeter, The proof of the $\ell ^2$ decoupling conjecture. Annals of Math. (2) 182 (2015), no. 1, 351–389.
J. Bourgain, C. Demeter, Decouplings for curves and hypersurfaces with nonzero Gaussian curvature, Preprint at http://arxiv.org/abs/1409.1634.pdf (2014)
Jean Bourgain and Larry Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295. MR 2860188, DOI 10.1007/s00039-011-0140-9
F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), no. 2, 483–491. MR 2600446, DOI 10.3934/cpaa.2010.9.483
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
T. Chen, C. Hainzl, N. Pavlović, R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, preprint (2013), http://arxiv.org/abs/1307.3168.pdf.
Thomas Chen and Nataša Pavlović, Recent results on the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Math. Model. Nat. Phenom. 5 (2010), no. 4, 54–72. MR 2662450, DOI 10.1051/mmnp/20105403
Thomas Chen and Nataša Pavlović, The quintic NLS as the mean field limit of a boson gas with three-body interactions, J. Funct. Anal. 260 (2011), no. 4, 959–997. MR 2747009, DOI 10.1016/j.jfa.2010.11.003
Thomas Chen and Nataša Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d=3$ based on spacetime norms, Ann. Henri Poincaré 15 (2014), no. 3, 543–588. MR 3165917, DOI 10.1007/s00023-013-0248-6
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113. MR 2651381, DOI 10.1007/s00222-010-0242-2
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observations of Bose–Einstein condensation in a dilute atomic vapor, Science 269 (1995), 198–201.
W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497–522. MR 1158383
Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, and Nikolaos Tzirakis, Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D, Discrete Contin. Dyn. Syst. 19 (2007), no. 1, 37–65. MR 2318273, DOI 10.3934/dcds.2007.19.37
C. Demeter, Incidence theory and discrete restriction estimates, Preprint at http://arxiv.org/pdf/1401.1873v1.pdf (2014).
S. Demirbas, Local Well-posedness for 2-D Schrödinger Equation on Irrational Tori and Bounds on Sobolev Norms, preprint (2013), http://arxiv.org/abs/1307.0051.
A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften 1: 3. (1925).
Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265–283. MR 2209131, DOI 10.1007/s00205-005-0388-z
László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math. 59 (2006), no. 12, 1659–1741. MR 2257859, DOI 10.1002/cpa.20123
László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515–614. MR 2276262, DOI 10.1007/s00222-006-0022-1
L. Erdős, B. Schlein, H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation, Phys. Rev. Lett. 98 (2007), no.4, 040404.
László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc. 22 (2009), no. 4, 1099–1156. MR 2525781, DOI 10.1090/S0894-0347-09-00635-3
László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math. (2) 172 (2010), no. 1, 291–370. MR 2680421, DOI 10.4007/annals.2010.172.291
László Erdős and Horng-Tzer Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205. MR 1926667, DOI 10.4310/ATMP.2001.v5.n6.a6
Gustavo Garrigós and Andreas Seeger, A mixed norm variant of Wolff’s inequality for paraboloids, Harmonic analysis and partial differential equations, Contemp. Math., vol. 505, Amer. Math. Soc., Providence, RI, 2010, pp. 179–197. MR 2664568, DOI 10.1090/conm/505/09923
Philip Gressman, Vedran Sohinger, and Gigliola Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal. 266 (2014), no. 7, 4705–4764. MR 3170216, DOI 10.1016/j.jfa.2014.02.006
E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento (10) 20 (1961), 454–477 (English, with Italian summary). MR 128907
M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 1, 71–149. MR 3312404, DOI 10.4171/JEMS/499
Zihua Guo, Tadahiro Oh, and Yuzhao Wang, Strichartz estimates for Schrödinger equations on irrational tori, Proc. Lond. Math. Soc. (3) 109 (2014), no. 4, 975–1013. MR 3273490, DOI 10.1112/plms/pdu025
Larry Guth, A short proof of the multilinear Kakeya inequality, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 1, 147–153. MR 3300318, DOI 10.1017/S0305004114000589
L. Guth, Mini-course Notes on Decoupling and multilinear estimates in harmonic analysis, MIT (2014) http://math.mit.edu/~lguth/decouplingseminar.html.
Larry Guth, The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture, Acta Math. 205 (2010), no. 2, 263–286. MR 2746348, DOI 10.1007/s11511-010-0055-6
Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Preprint at http://arxiv.org/pdf/1311.2275v3.pdf (2013.
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
Yi Hu and Xiaochun Li, Discrete Fourier restriction associated with Schrödinger equations, Rev. Mat. Iberoam. 30 (2014), no. 4, 1281–1300. MR 3293434, DOI 10.4171/RMI/815
Alexandru D. Ionescu and Benoit Pausader, The energy-critical defocusing NLS on $\Bbb T^3$, Duke Math. J. 161 (2012), no. 8, 1581–1612. MR 2931275, DOI 10.1215/00127094-1593335
K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose–Einstein Condensation in a Gas of Sodium Atoms, Phys. Rev. Let. 75 (1995), no. 22, 3969–3973.
W. Ketterle, When Atoms behave like Waves: Bose–Einstein Condensation and the Atom Laser, Nobel Lecture (2001). Available at http://www.nobelprize.org/nobel_prizes/physics/laureates/2001/ketterle-lecture.pdf
W. Ketterle, When Atoms behave like Waves: Bose–Einstein Condensation and the Atom Laser, Video of Nobel Lecture (2001). Available at http://www.nobelprize.org/nobel_prizes/physics/laureates/2001/ketterle-lecture.html
R. Killip, M. Vişan, Scale invariant Strichartz estimates on tori and applications, preprint (2014), http://arxiv.org/abs/1409.3603.
Sergiu Klainerman and Matei Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys. 279 (2008), no. 1, 169–185. MR 2377632, DOI 10.1007/s00220-008-0426-4
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc 55 (1949), 961–962. MR 0031538, DOI 10.1090/S0002-9904-1949-09320-5
Jerome Moloney and Alan Newell, Nonlinear optics, Westview Press. Advanced Book Program, Boulder, CO, 2004. MR 2018083
Camil Muscalu and Wilhelm Schlag, Classical and multilinear harmonic analysis. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 137, Cambridge University Press, Cambridge, 2013. MR 3052498
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
L. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 (1961), 451–454.
V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbb {T}^3$ from the dynamics of many-body quantum systems, preprint (2014), http://arxiv.org/abs/1405.3003, to appear in Ann. Inst. H. Poincaré (C), Analyse Non Linéaire.
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
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
Nils Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions, J. Evol. Equ. 14 (2014), no. 4-5, 829–839. MR 3276862, DOI 10.1007/s00028-014-0240-8
H. Takaoka and N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on ${\Bbb R}\times \Bbb T$, J. Funct. Anal. 182 (2001), no. 2, 427–442. MR 1828800, DOI 10.1006/jfan.2000.3732
T. Tao, Recent Progress on the Restriction Conjecture, IAS/Park City Mathematics Series, 2003. Preprint at http://arxiv.org/pdf/math/0311181.pdf.
Terence Tao, Some recent progress on the restriction conjecture, Fourier analysis and convexity, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004, pp. 217–243. MR 2087245, DOI 10.1198/106186003321335099
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
Laurent Thomann and Nikolay Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity 23 (2010), no. 11, 2771–2791. MR 2727169, DOI 10.1088/0951-7715/23/11/003
Nathan Totz and Sijue Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem, Comm. Math. Phys. 310 (2012), no. 3, 817–883. MR 2891875, DOI 10.1007/s00220-012-1422-2
Nathan Totz, A justification of the modulation approximation to the 3D full water wave problem, Comm. Math. Phys. 335 (2015), no. 1, 369–443. MR 3314508, DOI 10.1007/s00220-014-2259-7
N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields 146 (2010), no. 3-4, 481–514. MR 2574736, DOI 10.1007/s00440-008-0197-z
Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254, DOI 10.1090/ulect/029