Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on 𝕋³

References

[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. 10.1007/BF01896020Suche in Google Scholar

[2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), no. 1, 145–171. 10.1090/S0894-0347-99-00283-0Suche in Google Scholar

[3] J. Bourgain and C. Demeter, The proof of the l 2 decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1, 351–389. 10.4007/annals.2015.182.1.9Suche in Google Scholar

[4] A. Chabert, A weakly turbulent solution to the cubic nonlinear harmonic oscillator on ℝ 2 perturbed by a real smooth potential decaying to zero at infinity, Comm. Partial Differential Equations 49 (2024), no. 3, 185–216. 10.1080/03605302.2024.2302017Suche in Google Scholar

[5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ 3 , Ann. of Math. (2) 167 (2008), no. 3, 767–865. 10.4007/annals.2008.167.767Suche in Google Scholar

[6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113. 10.1007/s00222-010-0242-2Suche in Google Scholar

[7] B. Dodson, Global well-posedness and scattering for the defocusing, L 2 -critical nonlinear Schrödinger equation when d ≥ 3 , J. Amer. Math. Soc. 25 (2012), no. 2, 429–463. 10.1090/S0894-0347-2011-00727-3Suche in Google Scholar

[8] B. Dodson, Global well-posedness and scattering for the defocusing, L 2 critical, nonlinear Schrödinger equation when d = 1 , Amer. J. Math. 138 (2016), no. 2, 531–569. 10.1353/ajm.2016.0016Suche in Google Scholar

[9] B. Dodson, 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. 10.1215/00127094-3673888Suche in Google Scholar

[10] B. Dodson, Global well-posedness and scattering for the focusing, cubic Schrödinger equation in dimension d = 4 , Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 1, 139–180. 10.24033/asens.2385Suche in Google Scholar

[11] B. Dodson, C. Miao, J. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 3, 759–787. 10.1016/j.anihpc.2016.05.004Suche in Google Scholar

[12] C. Gao, C. Miao and J. Yang-Urbain, The interctitical defocusing nonlinear Schrödinger equations with radial initial data in dimensions four and higher, Anal. Theory Appl. 35 (2019), no. 2, 205–234. 10.4208/ata.OA-0006Suche in Google Scholar

[13] C. Gao and Z. Zhao, On scattering for the defocusing high dimensional inter-critical NLS, J. Differential Equations 267 (2019), no. 11, 6198–6215. 10.1016/j.jde.2019.06.019Suche in Google Scholar

[14] M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 3, 917–941. 10.1016/j.anihpc.2008.04.002Suche in Google Scholar

[15] S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H 1 ⁢ ( 𝕋 3 ) , Duke Math. J. 159 (2011), no. 2, 329–349. 10.1215/00127094-1415889Suche in Google Scholar

[16] S. Herr, D. Tataru and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4 ⁢ d and applications, J. Reine Angew. Math. 690 (2014), 65–78. 10.1515/crelle-2012-0013Suche in Google Scholar

[17] A. D. Ionescu and B. Pausader, Global well-posedness of the energy-critical defocusing NLS on ℝ × 𝕋 3 , Comm. Math. Phys. 312 (2012), no. 3, 781–831. 10.1007/s00220-012-1474-3Suche in Google Scholar

[18] A. D. Ionescu and B. Pausader, The energy-critical defocusing NLS on 𝕋 3 , Duke Math. J. 161 (2012), no. 8, 1581–1612. 10.1215/00127094-1593335Suche in Google Scholar

[19] 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), no. 2, 147–212. 10.1007/s11511-008-0031-6Suche in Google Scholar

[20] C. E. Kenig and F. Merle, Scattering for H ˙ 1 / 2 bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc. 362 (2010), no. 4, 1937–1962. 10.1090/S0002-9947-09-04722-9Suche in Google Scholar

[21] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), no. 2, 353–392. 10.1006/jdeq.2000.3951Suche in Google Scholar

[22] 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), no. 6, 1203–1258. 10.4171/jems/180Suche in Google Scholar

[23] R. Killip and M. Visan, Energy-supercritical NLS: Critical H ˙ s -bounds imply scattering, Comm. Partial Differential Equations 35 (2010), no. 6, 945–987. 10.1080/03605301003717084Suche in Google Scholar

[24] G. E. Lee, Local wellposedness for the critical nonlinear Schrödinger equation on 𝕋 3 , Discrete Contin. Dyn. Syst. 39 (2019), no. 5, 2763–2783. 10.3934/dcds.2019116Suche in Google Scholar

[25] C. Lu and J. Zheng, The radial defocusing energy-supercritical NLS in dimension four, J. Differential Equations 262 (2017), no. 8, 4390–4414. 10.1016/j.jde.2017.01.005Suche in Google Scholar

[26] Y. Luo, On long time behavior of the focusing energy-critical NLS on ℝ d × 𝕋 via semivirial-vanishing geometry, J. Math. Pures Appl. (9) 177 (2023), 415–454. 10.1016/j.matpur.2023.07.006Suche in Google Scholar

[27] C. Miao, J. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal. 267 (2014), no. 6, 1662–1724. 10.1016/j.jfa.2014.06.016Suche in Google Scholar

[28] C. Miao and J. Zheng, Scattering theory for the defocusing fourth-order Schrödinger equation, Nonlinearity 29 (2016), no. 2, 692–736. 10.1088/0951-7715/29/2/692Suche in Google Scholar

[29] J. Murphy, Intercritical NLS: Critical H ˙ s -bounds imply scattering, SIAM J. Math. Anal. 46 (2014), no. 1, 939–997. 10.1137/120898280Suche in Google Scholar

[30] J. Murphy, The defocusing H ˙ 1 / 2 -critical NLS in high dimensions, Discrete Contin. Dyn. Syst. 34 (2014), no. 2, 733–748. 10.3934/dcds.2014.34.733Suche in Google Scholar

[31] J. Murphy, The radial defocusing nonlinear Schrödinger equation in three space dimensions, Comm. Partial Differential Equations 40 (2015), no. 2, 265–308. 10.1080/03605302.2014.949379Suche in Google Scholar

[32] E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ℝ 1 + 4 , Amer. J. Math. 129 (2007), no. 1, 1–60. 10.1353/ajm.2007.0004Suche in Google Scholar

[33] 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), no. 1, 165–202. 10.1215/S0012-7094-07-14015-8Suche in Google Scholar

[34] D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math. 130 (2008), no. 3, 571–634. 10.1353/ajm.0.0000Suche in Google Scholar

[35] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007), no. 2, 281–374. 10.1215/S0012-7094-07-13825-0Suche in Google Scholar

[36] K. Yang and L. Zhao, Global well-posedness and scattering for mass-critical, defocusing, infinite dimensional vector-valued resonant nonlinear Schrödinger system, SIAM J. Math. Anal. 50 (2018), no. 2, 1593–1655. 10.1137/17M1131830Suche in Google Scholar

[37] K. Yang and Z. Zhao, On scattering asymptotics for the 2D cubic resonant system, J. Differential Equations 345 (2023), 447–484. 10.1016/j.jde.2022.11.056Suche in Google Scholar

[38] X. Yu, Global well-posedness and scattering for the defocusing H ˙ 1 / 2 -critical nonlinear Schrödinger equation in ℝ 2 , Anal. PDE 14 (2021), no. 7, 2225–2268. 10.2140/apde.2021.14.2225Suche in Google Scholar

[39] X. Yu and H. Yue, On the global well-posedness for the periodic quintic nonlinear Schrödinger equation, SIAM J. Math. Anal. 56 (2024), no. 2, 1851–1902. 10.1137/22M1531063Suche in Google Scholar

[40] X. Yu, H. Yue and Z. Zhao, Global Well-posedness for the focusing cubic NLS on the product space ℝ × 𝕋 3 , SIAM J. Math. Anal. 53 (2021), no. 2, 2243–2274. 10.1137/20M1364953Suche in Google Scholar

[41] H. Yue, Global well-posedness for the energy-critical focusing nonlinear Schrödinger equation on 𝕋 4 , J. Differential Equations 280 (2021), 754–804. 10.1016/j.jde.2021.01.031Suche in Google Scholar

[42] T. F. Zhao, The defocusing energy-supercritical NLS in higher dimensions, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 7, 911–925. 10.1007/s10114-017-6499-2Suche in Google Scholar

[43] Z. Zhao, Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide ℝ 2 × 𝕋 2 , J. Hyperbolic Differ. Equ. 16 (2019), no. 1, 73–129. 10.1142/S0219891619500048Suche in Google Scholar

[44] Z. Zhao, On scattering for the defocusing nonlinear Schrödinger equation on waveguide ℝ m × 𝕋 (when m = 2 , 3 ), J. Differential Equations 275 (2021), 598–637. 10.1016/j.jde.2020.11.023Suche in Google Scholar

[45] J. Q. Zheng, The defocusing energy-supercritical Hartree equation, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 4, 547–566. 10.1007/s10114-014-3299-9Suche in Google Scholar