## Duke Mathematical Journal

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

#### 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

http://projecteuclid.org/euclid.dmj/1328105284

Digital Object Identifier
doi:10.1215/00127094-1507400

Mathematical Reviews number (MathSciNet)
MR2881226

#### 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.

#### 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.