Duke Mathematical Journal

Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d=2

Benjamin Dodson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article we prove that the defocusing, cubic nonlinear Schrödinger initial value problem is globally well posed and scattering for u0L2(R2). The proof uses the bilinear estimates of Planchon and Vega and a frequency-localized interaction Morawetz estimate similar to the high-frequency estimate of Colliander, Keel, Staffilani, Takaoka, and Tao and especially the low-frequency estimate of Dodson.

Article information

Source
Duke Math. J., Volume 165, Number 18 (2016), 3435-3516.

Dates
Received: 11 April 2014
Revised: 31 January 2016
First available in Project Euclid: 12 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1473686054

Digital Object Identifier
doi:10.1215/00127094-3673888

Mathematical Reviews number (MathSciNet)
MR3577369

Zentralblatt MATH identifier
1361.35164

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35P25: Scattering theory [See also 47A40]

Keywords
nonlinear Schrodinger equation well-posedness scattering asymptotic completeness

Citation

Dodson, Benjamin. Global well-posedness and scattering for the defocusing, $L^{2}$ -critical, nonlinear Schrödinger equation when $d=2$. Duke Math. J. 165 (2016), no. 18, 3435--3516. doi:10.1215/00127094-3673888. https://projecteuclid.org/euclid.dmj/1473686054


Export citation

References

  • [1] H. Bahouri and P. Gérard, High frequency approximations of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175.
  • [2] H. Bahouri and J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 783–789.
  • [3] P. Bégout and A. Vargas, Mass concentration phenomena for the ${L}^{2}$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), 5257–5282.
  • [4] H. Berestycki and P.-L. Lions, Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon, C. R. Math. Acad. Sci. Paris 288 (1979), A395–A398.
  • [5] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107–156.
  • [6] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II: The KdV-equation, Geom. Funct. Anal. 3 (1993), 209–262.
  • [7] J. Bourgain, Refinements of Strichartz’ inequality and applications to $2\mathrm{D}\mbox{-}\mathrm{NLS}$ with critical nonlinearity, Int. Math. Res. Not. IMRN 1998, no. 5, 253–283.
  • [8] J. Bourgain, Scattering in the energy space and below for $3\mathrm{D}$ $\mathrm{NLS}$, J. Anal. Math. 75 (1998), 267–297.
  • [9] J. Bourgain, Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145–171.
  • [10] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc., Providence, 1999.
  • [11] H. Brezis and J.-M. Coron, Convergence of solutions of $H$-systems or how to blow bubbles, Arch. Ration. Mech. Anal. 89 (1985), 21–56.
  • [12] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodes Mat. 26, Inst. Mat., Rio de Janeiro, 1993.
  • [13] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Amer. Math. Soc., Providence, 2003.
  • [14] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^{1}$, Manuscripta Math. 61 (1988), 477–494.
  • [15] T. Cazenave and F. B. Weissler, “Some remarks on the nonlinear Schrödinger equation in the subcritical case” in New Methods and Results in Nonlinear Field Equations (Bielefeld, 1987), Lecture Notes in Phys. 347, Springer, Berlin, 1989.
  • [16] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$, Nonlinear Anal. 14 (1990), 807–836.
  • [17] M. Christ, J. Colliander, and T. Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal. 254 (2008), 368–395.
  • [18] M. Christ, J. Colliander, and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), 1235–1293.
  • [19] M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 409–425.
  • [20] J. Colliander, M. Grillakis, and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\mathbf{R}^{2}$, Int. Math. Res. Not. IMRN 2007, no. 23, art. ID rnm90.
  • [21] J. Colliander, M. Grillakis, and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 62 (2009), 920–968.
  • [22] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett. 9 (2002), 659–682.
  • [23] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbf{R}^{3}$, Comm. Pure Appl. Math. 21 (2004), 987–1014.
  • [24] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on $\mathbf{R}^{2}$, Discrete Contin. Dyn. Syst. 21 (2007), 665–686.
  • [25] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation on $\mathbf{R}^{3}$, Ann. of Math. (2) 167 (2008), 767–865.
  • [26] J. Colliander and T. Roy, Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic $\mathrm{NLS}$ on $\mathbf{R}^{2}$, Commun. Pure Appl. Anal. 10 (2011), 397–414.
  • [27] B. Dodson, Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation, Commun. Pure Appl. Anal. 10 (2011), 127–140.
  • [28] B. Dodson, Global well-posedness and scattering for the defocusing $L^{2}$-critical nonlinear Schrödinger equation when $d\geq3$, J. Amer. Math. Soc. 25 (2012), 429–463.
  • [29] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285 (2015), 1589–1618.
  • [30] B. Dodson, Global well-posedness and scattering for the defocusing $L^{2}$-critical nonlinear Schrödinger equation when $d=1$, Amer. J. Math. 138 (2016), 531–569.
  • [31] P. Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233.
  • [32] P. Germain, N. Masmoudi, and J. Shatah, Global solutions for $2\mathrm{D}$ quadratic Schrödinger equations, J. Math. Pures Appl. (9) 97 (2012), 505–543.
  • [33] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Astérisque 237 (1996), 163–187, Séminaire Bourbaki 1994/1995, no. 796.
  • [34] J. Ginibre, A. Soffer, and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal. 110 (1992), 96–130.
  • [35] J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), 363–401.
  • [36] J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), 163–188.
  • [37] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys. 18 (1977), 1794–1797.
  • [38] M. G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (1990), 485–509.
  • [39] M. G. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), 749–774.
  • [40] M. G. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations 25 (2000), 1827–1844.
  • [41] M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 917–941.
  • [42] L. Kapitanski, Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett. 1 (1994), 211–223.
  • [43] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.
  • [44] C. E. Kenig and F. 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), 645–675.
  • [45] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), 147–212.
  • [46] C. E. Kenig and F. Merle, Scattering for $\dot{H}^{1/2}$ bounded solutions to the cubic, defocusing NLS in $3$ dimensions, Trans. Amer. Math. Soc. 362, no. 4 (2010), 1937–1962.
  • [47] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), 353–392.
  • [48] S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal. 235 (2006), 171–192.
  • [49] R. Killip, T. Tao, and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 (2009), 1203–1258.
  • [50] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math. 132 (2010), 361–424.
  • [51] R. Killip and M. Visan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE 5 (2012), 855–885.
  • [52] R. Killip and M. Visan, “Nonlinear Schrodinger equations at critical regularity” in Evolution Equations, Clay Math. Proc. 17, Amer. Math. Soc., Providence, 2013, 325–437.
  • [53] R. Killip, M. Visan, and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE 1 (2008), 229–266.
  • [54] H. Koch and D. Tataru, Dispersive estimates for principally normal pseudodifferential operators, Comm. Pure Appl. Math. 58 (2005), 217–284.
  • [55] H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN 2007, no 16, art. ID rnm053.
  • [56] H. Koch and D. Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in ${H}^{-1/4}$, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), 955–988.
  • [57] H. Koch, D. Tataru, and M. Visan, “Well-posedness for nonlinear dispersive equations” in Dispersive Equations and Nonlinear Waves, Oberwolfach Semin. 45, Springer, Basel, 2014, 87–109.
  • [58] J. E. Lin and W. A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal. 30 (1978), 245–263.
  • [59] F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), 427–454.
  • [60] F. Merle and L. Vega, Compactness at blow-up time for ${L}^{2}$ solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not. IMRN 1998, no. 8, 399–425.
  • [61] S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations, Duke Math. J. 91 (1998), 393–408.
  • [62] C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equation, Proc. R. Soc. Ser. A 306 (1968), 291–296.
  • [63] A. Moyua, A. Vargas, and L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in $\mathbf{R}^{3}$, Duke Math. J. 96 (1999), 547–574.
  • [64] J. Murphy, Inter-critical $\mathrm{NLS}$: Critical $\dot{H}^{s}$ bounds imply scattering, SIAM J. Math. Anal. 46 (2014), 939–997.
  • [65] J. Murphy, The defocusing $\dot{H}^{1/2}$-critical NLS in high dimensions, Discrete Contin. Dynam. Syst. 34 (2014), 733–748.
  • [66] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal. 169 (1999), 201–225.
  • [67] F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 261–290.
  • [68] E. Rhyckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{1+4}$, Amer. J. Math. 129 (2007), 1–60.
  • [69] J. Shatah and M. Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), 503–518.
  • [70] J. Shatah and M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. IMRN 1994, no. 7, 303–309.
  • [71] J. Shatah and M. Struwe, Geometric Wave Equations, Courant Lect. Notes Math. 2, Amer. Math. Soc., Providence, 1998.
  • [72] A. Stefanov, Strichartz estimates for the Schrödinger equation with radial data, Proc. Amer. Math. Soc. 129 (2001), 1395–1401.
  • [73] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math Ser. 30, Princeton Univ. Press, Princeton, 1970.
  • [74] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
  • [75] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.
  • [76] M. Struwe, Globally regular solutions to the $u^{5}$ Klein-Gordon equation, Ann. Sc. Norm. Sup. Pisa. Cl. Sci. (4) 15 (1988), 495–513.
  • [77] T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), 1471–1485.
  • [78] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359–1384.
  • [79] T. Tao, On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ. 1 (2004), 1–48.
  • [80] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math. 11 (2005), 57–80.
  • [81] T. Tao, Nonlinear Dispersive Equations, CBMS Reg. Conf. Ser. Math. 106, Amer. Math. Soc., Providence, 2006.
  • [82] T. Tao, A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations, Dyn. Partial Differ. Equ. 4 (2007), 1–53.
  • [83] T. Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math. 15 (2009), 265–282.
  • [84] T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dyn. Partial Differ. Equ. 3 (2006), 93–110.
  • [85] T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 2005, no. 118.
  • [86] T. Tao, M. Visan, and X. Zhang. The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32 (2007), 1281–1343.
  • [87] T. Tao, M. Visan, and X. Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), 165–202.
  • [88] T. Tao, M. Visan, and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math. 20 (2008), 881–919.
  • [89] D. Tataru, Local and global results for wave maps, I, Comm. Partial Differential Equations 23 (1998), 1781–1793.
  • [90] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progr. Math. 100, Birkhäuser, Boston, 1991.
  • [91] M. E. Taylor, Partial Differential Equations, I, 2nd ed., Appl. Math. Sci. 115, Springer, New York, 2011; II, Appl. Math. Sci. 116; III, Appl. Math. Sci. 117.
  • [92] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007), 281–374.
  • [93] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Ph.D. dissertation, University of California, Los Angeles, Calif., 2006.
  • [94] M. I. Weinstein, “The nonlinear Schrödinger equation—singularity formation, stability and dispersion” in The Connection between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, Col., 1987), Contemp. Math. 99, Amer. Math. Soc., Providence, 1989, 213–232.
  • [95] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), 415–426.
  • [96] V. Zakharov and E. Kuznetsov, Quasi-classical theory for three dimensional wave collapse, Sov. Phys. JETP 64 (1986), 773–780.