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Solving the 4NLS with white noise initial data

Part of: Stochastic analysis Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 November 2020

Tadahiro Oh ,
Nikolay Tzvetkov  and
Yuzhao Wang
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; E-mail: hiro.oh@ed.ac.uk
Nikolay Tzvetkov
Affiliation:
CY Cergy Paris University, Cergy-Pontoise, F-95000, UMR 8088 du CNRS, France; E-mail: nikolay.tzvetkov@cyu.fr
Yuzhao Wang
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; E-mail: hiro.oh@ed.ac.uk School of Mathematics, School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom; E-mail: y.wang.14@bham.ac.uk

Abstract

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We construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

Keywords

fourth-order nonlinear Schrrödinger equationbiharmonic nonlinear Schrrödinger equationwhite noiseinvariant measurerandom-resonant/nonlinear decompositionrandom Fourier restriction norm space

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Differential Equations
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e48
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Bényi, A. and Oh, T., ‘Modulation spaces, Wiener amalgam spaces, and Brownian motions’, Adv. Math. 228 (2011), no. 5, 2943–2981.CrossRefGoogle Scholar
Bényi, A., Oh, T., and Pocovnicu, O., ‘Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS’, in Excursions in Harmonic Analysis Vol. 4, Appl. Numer. Harmon. Anal. (Birkhäuser/Springer, Cham, 2015), 3–25.CrossRefGoogle Scholar
Bényi, Á., Oh, T., and Pocovnicu, O., ‘Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^3$ ’, Trans. Amer. Math. Soc. Ser. B 6 (2019), 114–160.CrossRefGoogle Scholar
Bényi, Á., Oh, T., and Pocovnicu, O., ‘On the probabilistic Cauchy theory for nonlinear dispersive PDEs’, in Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Anal. (Birkhäuser/Springer, Cham, 2019), 1–32.Google Scholar
Bourgain, J., ‘Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations’, Geom. Funct. Anal. 3 (1993), 107–156.CrossRefGoogle Scholar
Bourgain, J., ‘Periodic nonlinear Schrödinger equation and invariant measures’, Comm. Math. Phys. 166 (1994), no. 1, 1–26.CrossRefGoogle Scholar
Bourgain, J., ‘Invariant measures for the 2D-defocusing nonlinear Schrödinger equation’, Comm. Math. Phys. 176 (1996), no. 2, 421–445.CrossRefGoogle Scholar
Bourgain, J., ‘Invariant measures for the Gross-Piatevskii equation’, J. Math. Pures Appl. 76 (1997), no. 8, 649–702.CrossRefGoogle Scholar
Bourgain, J., ‘Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity’, Internat. Math. Res. Notices 1998, no. 5, 253–283.CrossRefGoogle Scholar
Bourgain, J., ‘Nonlinear Schrödinger equations’, in Hyperbolic Equations and Frequency Interactions (Park City, UT, 1995), IAS/Park City Math. Ser., 5 (Amer. Math. Soc., Providence, RI, 1999), 3–157.Google Scholar
Bourgain, J. and Bulut, A., ‘Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: the 2D case’, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 6, 1267–1288.CrossRefGoogle Scholar
Bourgain, J. and Bulut, A., ‘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.CrossRefGoogle Scholar
Bringmann, B., ‘Almost sure local well-posedness for a derivative nonlinear wave equation’, to appear in Int. Math. Res. Not. Google Scholar
Burq, N., Thomann, L., and Tzvetkov, N., ‘Long time dynamics for the one dimensional non linear Schrödinger equation’, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2137–2198.CrossRefGoogle Scholar
Burq, N. and Tzvetkov, N., ‘Random data Cauchy theory for supercritical wave equations. I. Local theory’, Invent. Math. 173 (2008), no. 3, 449–475.CrossRefGoogle Scholar
Burq, N. and Tzvetkov, N., ‘Random data Cauchy theory for supercritical wave equations. II. A global existence result’, Invent. Math. 173 (2008), no. 3, 477–496.CrossRefGoogle Scholar
Catellier, R. and Chouk, K., ‘Paracontrolled distributions and the 3-dimensional stochastic quantization equation’, Ann. Probab. 46 (2018), no. 5, 2621–2679.CrossRefGoogle Scholar
Choffrut, A. and Pocovnicu, O., ‘Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line’, Int. Math. Res. Not. 2018, no. 3, 699–738.Google Scholar
Christ, M., ‘Power series solution of a nonlinear Schrödinger equation’, in Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., 163 (Princeton Univ. Press, Princeton, NJ, 2007), 131–155.Google Scholar
Chung, J., Guo, Z., Kwon, S., and Oh, T., ‘Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle’, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 1273–1297.CrossRefGoogle Scholar
Colliander, J., and Oh, T., ‘Almost sure well-posedness of the cubic nonlinear Schrödinger equation below ${L}^2(T)$ ’, Duke Math. J . 161 (2012), no. 3, 367–414.CrossRefGoogle Scholar
Da Prato, G. and Debussche, A., ‘Two-dimensional Navier-Stokes equations driven by a space-time white noise’, J. Funct. Anal. 196 (2002), no. 1, 180–210.CrossRefGoogle Scholar
Da Prato, G. and Debussche, A., ‘Strong solutions to the stochastic quantization equations’, Ann. Probab. 31 (2003), no. 4, 1900–1916.Google Scholar
de Bouard, A. and Debussche, A., ‘The Korteweg-de Vries equation with multiplicative homogeneous noise’, in Stochastic Differential Equations: Theory and Applications, Interdiscip. Math. Sci., 2 (World Sci. Publ., Hackensack, NJ, 2007), 113–133.CrossRefGoogle Scholar
Forlano, J., Oh, T., and Wang, Y., ‘Stochastic cubic nonlinear Schrödinger equation with almost space-time white noise’, J. Aust. Math. Soc. 1 09 (2020), no. 1, 44–67.CrossRefGoogle Scholar
Ginibre, J., Tsutsumi, Y., and Velo, G., ‘On the Cauchy problem for the Zakharov system’, J. Funct. Anal. 151 (1997), no. 2, 384–436.CrossRefGoogle Scholar
Grafakos, L., Modern Fourier Analysis, 2e, Graduate Texts in Mathematics, 250 (Springer, New York, 2009).Google Scholar
Gross, L., ‘Abstract Wiener spaces’, in Proc. 5th Berkeley Sym. Math. Stat. Prob. 2 (1965), 31–42.Google Scholar
Grünrock, A., and Herr, S., ‘Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data’, SIAM J. Math. Anal. 39 (2008), no. 6, 1890–1920.CrossRefGoogle Scholar
Gubinelli, M., Imkeller, P., and Perkowski, P., ‘Paracontrolled distributions and singular PDEs’, Forum Math. Pi 3 (2015), e6, 75 pp.CrossRefGoogle Scholar
Gubinelli, M., Koch, H., and Oh, T., ‘Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity’, arXiv:1811.07808 [math.AP].Google Scholar
Guo, Z., Kwon, S., and Oh, T., ‘Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS’, Comm. Math. Phys. 322 (2013), no. 1, 19–48.CrossRefGoogle Scholar
Guo, Z. and Oh, T., ‘Non-existence of solutions for the periodic cubic nonlinear Schrödinger equation below ${L}^2$ ’, Internat. Math. Res. Not . 2018, no. 6, 1656–1729.Google Scholar
Hairer, M., ‘A theory of regularity structures’, Invent. Math. 198 (2014), 269–504.CrossRefGoogle Scholar
Hairer, M., ‘Singular stochastic PDEs’, in Proceedings of the International Congress of Mathematicians–Seoul 2014 , Vol. IV (Kyung Moon Sa, Seoul, 2014), 49–73.Google Scholar
Hani, Z., Pausader, B., Tzvetkov, N., and Visciglia, N., ‘Modified scattering for the cubic Schrödinger equation on product spaces and applications’, Forum Math. Pi 3 (2015), e4.Google Scholar
Hayashi, N. and Naumkin, P., ‘Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations’, Amer. J. Math. 120 (1998), no. 2, 369–389.CrossRefGoogle Scholar
Ivanov, B.A. and Kosevich, A.M., ‘Stable three-dimensional small-amplitude soliton in magnetic materials’, So . J. Low Temp. Phys. 9 (1983), 439–442.Google Scholar
Kappeler, T. and Topalov, P., ‘Global wellposedness of KdV in ${H}^{-1}\left(T,R\right)$ , Duke Math. J . 135 (2006), no. 2, 327–360.CrossRefGoogle Scholar
Kuo, H., ‘Gaussian measures in Banach spaces’, in Lecture Notes in Mathematics, Vol. 463 (Springer-Verlag, Berlin-New York, 1975).Google Scholar
Kwak, C., ‘Periodic fourth-order cubic NLS: Local well-posedness and Non-squeezing property’, J. Math. Anal. Appl. 461 (2018), no. 2, 1327–1364.CrossRefGoogle Scholar
McKean, H.P., ‘Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger’, Comm. Math. Phys. 168 (1995), no. 3, 479–491. ‘Erratum: Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger’, Comm. Math. Phys. 173 (1995), no. 3, 675.Google Scholar
Mourrat, J.-C. and Weber, H., ‘The dynamic ${\varPhi}_3^4$ model comes down from infinity’, Comm. Math. Phys . 356 (2017), no. 3, 673–753.Google Scholar
Nakanishi, K., Takaoka, H., and Tsutsumi, Y., ‘Local well-posedness in low regularity of the mKdV equation with periodic boundary condition’, Discrete Contin. Dyn. Syst. 28 (2010), no. 4, 1635–1654.CrossRefGoogle Scholar
Nahmod, A. and Staffilani, G., ‘Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space’, J. Eur. Math. Soc. 17 (2015), 1687–1759.CrossRefGoogle Scholar
Nelson, E., ‘A quartic interaction in two dimensions’, in Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965 ) (M.I.T. Press, Cambridge, Mass., 1966), 69–73.Google Scholar
Nualart, D., The Malliavin Calculus and Related Topics, 2e, Probability and Its Applications (Springer-Verlag, Berlin, 2006).Google Scholar
Oh, T., ‘Invariant Gibbs measures and a.s. global well-posedness for coupled KdV systems’, Differential Integral Equations 22 (2009), no. 7-8, 637–668.Google Scholar
Oh, T., ‘Invariance of the white noise for KdV’, Comm. Math. Phys. 292 (2009), no. 1, 217–236.CrossRefGoogle Scholar
Oh, T., ‘Periodic stochastic Korteweg-de Vries equation with additive space-time white noise’, Anal. PDE 2 (2009), no. 3, 281–304.CrossRefGoogle Scholar
Oh, T., ‘White noise for KdV and mKdV on the circle’, in Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, B18 (Res. Inst. Math. Sci. (RIMS), Kyoto, 2010), 99–124.Google Scholar
Oh, T., ‘Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation’, Funkcial. Ekvac. 54 (2011), no. 3, 335–365.Google Scholar
Oh, T., Pocovnicu, O., and Tzvetkov, N., ‘Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces’, to appear in Ann. Inst. Fourier (Grenoble).Google Scholar
Oh, T. and Sulem, C., ‘On the one-dimensional cubic nonlinear Schrödinger equation below ${L}^2$ ’, Kyoto J. Math . 52 (2012), no. 1, 99–115.CrossRefGoogle Scholar
Oh, T. and Thomann, L., ‘A pedestrian approach to the invariant Gibbs measure for the 2- $d$ defocusing nonlinear Schrödinger equations’, Stoch. Partial Differ. Equ. Anal. Comput. 6 (2018), 397–445.Google Scholar
Oh, T., Tsutsumi, Y., and Tzvetkov, N., ‘Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third order dispersion’, C. R. Math. Acad. Sci. Paris 357 (2019), no. 4, 366–381.CrossRefGoogle Scholar
Oh, T. and Tzvetkov, N., ‘Quasi-invariant Gaussian measures for the cubic fourth-order nonlinear Schrödinger equation’, Probab. Theory Related Fields 169 (2017), 1121–1168.CrossRefGoogle ScholarPubMed
Oh, T. and Wang, Y., ‘On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle’, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 64 (2018), no. 1, 53–84.Google Scholar
Oh, T. and Wang, Y., ‘Global well-posedness of the periodic cubic fourth-order NLS in negative Sobolev spaces’, Forum Math. Sigma 6 (2018), e5, 80 pp.CrossRefGoogle Scholar
Oh, T. and Wang, Y., ‘Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces’, to appear in J. Anal. Math.Google Scholar
Ozawa, T., ‘Long range scattering for nonlinear Schrödinger equations in one space dimension’, Comm. Math. Phys. 139 (1991), no. 3, 479–493.CrossRefGoogle Scholar
Ozawa, T. and Tsutsumi, Y., ‘Space-time estimates for null gauge forms and nonlinear Schrödinger equations’, Differential Integral Equations 11 (1998), no. 2, 201–222.Google Scholar
Quastel, J. and Valkó, B., ‘KdV preserves white noise’, Comm. Math. Phys. 277 (2008), no. 3, 707–714.CrossRefGoogle Scholar
Richards, G., ‘Maximal-in-time behavior of deterministic and stochastic dispersive partial differential equations’, PhD thesis, University of Toronto (Canada), 2012.Google Scholar
Richards, G., ‘Invariance of the Gibbs measure for the periodic quartic gKdV’, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 3, 699–766.CrossRefGoogle Scholar
Simon, B., The $P{\left(\varphi \right)}_2$ Euclidean (Quantum) Field Theory, Princeton Series in Physics (Princeton University Press, Princeton, NJ, 1974).Google Scholar
Takaoka, H. and Tsutsumi, Y., ‘Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition’, Int. Math. Res. Not. 2004, no. 56, 3009–3040.CrossRefGoogle Scholar
Thomann, L., ‘Random data Cauchy problem for supercritical Schrödinger equations’, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 6, 2385–2402.CrossRefGoogle Scholar
Thomann, L. and Tzvetkov, N., ‘Gibbs measure for the periodic derivative nonlinear Schrödinger equation’, Nonlinearity 23 (2010), no. 11, 2771–2791.CrossRefGoogle Scholar
Turitsyn, S. K., ‘Three-dimensional dispersion of nonlinearity and stability of multidimensional solitons’, Teoret. Mat. Fiz. 64 (1985), 226–232 (in Russian).Google Scholar
Tzvetkov, N., ‘Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation’, Probab. Theory Related Fields 146 (2010), no. 3-4, 481–514.CrossRefGoogle Scholar