Random data final-state problem for the mass-subcritical NLS in $L^2$
HTML articles powered by AMS MathViewer
- by Jason Murphy
- Proc. Amer. Math. Soc. 147 (2019), 339-350
- DOI: https://doi.org/10.1090/proc/14275
- Published electronically: October 18, 2018
- PDF | Request permission
Abstract:
We study the final-state problem for the mass-subcritical NLS above the Strauss exponent. For $u_+\in L^2$, we perform a physical-space randomization, yielding random final states $u_+^\omega \in L^2$. We show that for almost every $\omega$, there exists a unique, global solution to NLS that scatters to $u_+^\omega$. This complements the deterministic result of Nakanishi, which proved the existence (but not necessarily uniqueness) of solutions scattering to prescribed $L^2$ final states.References
- Jacqueline E. Barab, Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys. 25 (1984), no. 11, 3270–3273. MR 761850, DOI 10.1063/1.526074
- Árpád Bényi, Tadahiro Oh, and Oana Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\Bbb {R}^d$, $d\geq 3$, Trans. Amer. Math. Soc. Ser. B 2 (2015), 1–50. MR 3350022, DOI 10.1090/btran/6
- J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), no. 1, 1–26. MR 1309539
- Jean Bourgain, Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445. MR 1374420
- Nicolas Burq, Laurent Thomann, and Nikolay Tzvetkov, Long time dynamics for the one dimensional non linear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2137–2198 (English, with English and French summaries). MR 3237443
- Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475. MR 2425133, DOI 10.1007/s00222-008-0124-z
- Nicolas Burq and Nikolay Tzvetkov, Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math. 173 (2008), no. 3, 477–496. MR 2425134, DOI 10.1007/s00222-008-0123-0
- 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
- Thierry Cazenave and Fred B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), no. 1, 75–100. MR 1171761
- James Colliander and Tadahiro Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\Bbb T)$, Duke Math. J. 161 (2012), no. 3, 367–414. MR 2881226, DOI 10.1215/00127094-1507400
- Yu Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE 5 (2012), no. 5, 913–960. MR 3022846, DOI 10.2140/apde.2012.5.913
- Anne-Sophie de Suzzoni, Large data low regularity scattering results for the wave equation on the Euclidean space, Comm. Partial Differential Equations 38 (2013), no. 1, 1–49. MR 3005545, DOI 10.1080/03605302.2012.736910
- Anne-Sophie de Suzzoni, Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space, Commun. Pure Appl. Anal. 13 (2014), no. 3, 991–1015. MR 3177685, DOI 10.3934/cpaa.2014.13.991
- B. Dodson, J. Lürhmann, and D. Mendelson, Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data, preprint, arXiv:1703.09655, 2017.
- B. Dodson, J. Lührmann, and D. Mendelson, Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation, preprint, arXiv:1802.03795, 2018.
- Damiano Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005), no. 1, 1–24. MR 2134950, DOI 10.1142/S0219891605000361
- J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), no. 1, 163–188. MR 1151250
- Justin Holmer and Nikolaos Tzirakis, Asymptotically linear solutions in $H^1$ of the 2-D defocusing nonlinear Schrödinger and Hartree equations, J. Hyperbolic Differ. Equ. 7 (2010), no. 1, 117–138. MR 2646800, DOI 10.1142/S0219891610002049
- Tosio Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 223–238. MR 1275405, DOI 10.2969/aspm/02310223
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
- R. Killip, J. Murphy, and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb {R}^4)$, preprint, arXiv:1707.09051, 2017.
- Jonas Lührmann and Dana Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $\Bbb {R}^3$, Comm. Partial Differential Equations 39 (2014), no. 12, 2262–2283. MR 3259556, DOI 10.1080/03605302.2014.933239
- Satoshi Masaki, A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation, Commun. Pure Appl. Anal. 14 (2015), no. 4, 1481–1531. MR 3359531, DOI 10.3934/cpaa.2015.14.1481
- S. Masaki, Two minimization problems on non-scattering solutions to mass-subcritical nonlinear Schrödinger equation, preprint, arXiv:1605.09234, 2016.
- Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, and Gigliola Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1275–1330. MR 2928851, DOI 10.4171/JEMS/333
- Kenji Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation, SIAM J. Math. Anal. 32 (2001), no. 6, 1265–1271. MR 1856248, DOI 10.1137/S0036141000369083
- Kenji Nakanishi and Tohru Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 1, 45–68. MR 1891695, DOI 10.1007/s00030-002-8118-9
- Aurélien Poiret, Didier Robert, and Laurent Thomann, Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator, Anal. PDE 7 (2014), no. 4, 997–1026. MR 3254351, DOI 10.2140/apde.2014.7.997
- Yoshio Tsutsumi and Kenji Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 1, 186–188. MR 741737, DOI 10.1090/S0273-0979-1984-15263-7
- 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
- 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
Bibliographic Information
- Jason Murphy
- Affiliation: Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, Missouri 65409
- MR Author ID: 1034475
- Email: jason.murphy@mst.edu
- Received by editor(s): February 23, 2018
- Received by editor(s) in revised form: June 7, 2018
- Published electronically: October 18, 2018
- Additional Notes: The author was supported by the NSF Postdoctoral Fellowship DMS-1400706 at the University of California, Berkeley.
- Communicated by: Joachim Krieger
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 339-350
- MSC (2010): Primary 35Q55
- DOI: https://doi.org/10.1090/proc/14275
- MathSciNet review: 3876753