Duke Mathematical Journal

Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2}(\mathbb{T})$

James Colliander and Tadahiro Oh

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

We consider the Cauchy problem for the 1-dimensional periodic cubic nonlinear Schrödinger (NLS) equation with initial data below $L^{2}$. In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well-posedness of the NLS equation almost surely for the initial data in the support of the canonical Gaussian measures on $H^{s}(\mathbb{T})$ for each $s\textgreater -\frac{1}{3}$, and global well-posedness for each $s\textgreater -\frac{1}{12}$.

Article information

Source
Duke Math. J. Volume 161, Number 3 (2012), 367-414.

Dates
First available in Project Euclid: 1 February 2012

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1328105284

Digital Object Identifier
doi:10.1215/00127094-1507400

Mathematical Reviews number (MathSciNet)
MR2881226

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 37L50: Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems 37L40: Invariant measures

Citation

Colliander, James; Oh, Tadahiro. Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2 ( T ) . Duke Math. J. 161 (2012), no. 3, 367--414. doi:10.1215/00127094-1507400. http://projecteuclid.org/euclid.dmj/1328105284.


Export citation

References

  • [1] Á. Bényi and T. Oh, Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math. 228 (2011), 2943–2981.
  • [2] 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.
  • [3] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76 (1994), 175–202.
  • [4] J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), 1–26.
  • [5] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), 421–445.
  • [6] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not. IMRN 1998, no. 5, 253–283.
  • [7] J. Bourgain, “Nonlinear Schrödinger equations” in Hyperbolic Equations and Frequency Interactions (Park City, Utah, 1995), IAS/Park City Math. Ser. 5, Amer. Math. Soc., Providence, 1999, 3–157.
  • [8] N. Burq, P. Gérard, and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on Sd, Math. Res. Lett. 9 (2002), 323–335.
  • [9] N. Burq and N. Tzvetkov, Invariant measure for a three-dimensional nonlinear wave equation, Int. Math. Res. Not. IMRN 2007, no. 22, art. ID rnm108.
  • [10] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations, I: Local theory, Invent. Math. 173 (2008), 449–475.
  • [11] N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations, II: A global existence result, Invent. Math. 173 (2008), 477–496.
  • [12] M. Christ, “Power series solution of a nonlinear Schrödinger equation” in Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, 2007, 131–155.
  • [13] 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.
  • [14] M. Christ, J. Colliander, and T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint, arXiv:math/0311227v1 [math.AP]
  • [15] J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), 384–436.
  • [16] A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal. 39 (2008), 1890–1920.
  • [17] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., The Clarendon Press, Oxford Univ. Press, Oxford, 1979.
  • [18] S. Janson, Gaussian Hilbert Spaces, Cambridge Tracts in Math. 129, Cambridge Univ. Press, Cambridge, 1997.
  • [19] C. E. Kenig, G. Ponce, and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), 617–633.
  • [20] S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math. 4 (2002), 223–295.
  • [21] 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.
  • [22] H.-H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. 463, Springer, Berlin, 1975.
  • [23] J. L. Lebowitz, H. A. Rose, and E. R. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys. 50 (1988), 657–687.
  • [24] M. Ledoux and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Ergeb. Math. Grenzgeb. (3) 23, Springer, Berlin, 1991.
  • [25] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett. 16 (2009), 111–120.
  • [26] T. Oh, Invariant Gibbs measures and a.s. global well posedness for coupled KdV systems, Differential Integral Equations 22 (2009), 637–668.
  • [27] T. Oh, Invariance of the Gibbs measure for the Schrödinger–Benjamin–Ono system, SIAM J. Math. Anal. 41 (2009/10), 2207–2225.
  • [28] T. Oh, Invariance of the white noise for KdV, Comm. Math. Phys. 292 (2009), 217–236. Also, see Erratum: “Invariance of the white noise for KdV”, in preparation.
  • [29] T. Oh, “White noise for KdV and mKdV on the circle” in Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu B18 (2010), 99–124.
  • [30] T. Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac. 54 (2011), 335–365.
  • [31] T. Oh, J. Quastel, and B. Valkó, Interpolation of Gibbs measures with white noise for Hamiltonian PDE, preprint, to appear in J. Math. Pures Appl. (9), arXiv:1005.3957v1 [math.PR]
  • [32] T. Oh and C. Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below L2, Kyoto J. Math. 52 (2012), 99–115.
  • [33] R. Paley and A. Zygmund, On some series of functions (1), (2), (3), Proc. Camb. Phil. Soc. 26 (1930), 337–357; 26 (1930), 458–474; 28 (1933), 190–205.
  • [34] J. Quastel and B. Valkó, KdV preserves white noise, Comm. Math. Phys. 277 (2008), 707–714.
  • [35] L. Thomann, Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 2385–2402.
  • [36] Y. Tsutsumi, L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115–125.
  • [37] N. Tzvetkov, Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ. 3 (2006), 111–160.
  • [38] N. Tzvetkov, Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 58 (2008), 2543–2604.
  • [39] N. Tzvetkov, Construction of a Gibbs measure associated to the periodic Benjamin–Ono equation, Probab. Theory Related Fields 146 (2010), 481–514.
  • [40] P. E. Zhidkov, An invariant measure for a nonlinear wave equation, Nonlinear Anal. 22 (1994), 319–325.
  • [41] P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Math. 1756, Springer, Berlin, 2001.