Scattering for cubic fourth order nonlinear Schrödinger equation with radial data in dimension six
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- by Zuyu Ma and Yilin Song;
- Proc. Amer. Math. Soc. 153 (2025), 2539-2554
- DOI: https://doi.org/10.1090/proc/17170
- Published electronically: April 3, 2025
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Abstract:
In this article, we study the global well-posedness and scattering for the cubic fourth-order Schrödinger equation which is $\dot {H}^{1}$-critical, \begin{align*} \begin {cases} i\partial _tu+\Delta ^2u=-|u|^2u,&(t,x)\in \mathbb {R}\times \mathbb {R}^6,\\ u(0,x)=u_0(x). \end{cases} \end{align*} Inspired by Dodson [Camb. J. Math. 7 (2019), pp. 283–318] and Miao, Xu, and Yang [Commun. Contemp. 22 (2020), p. 1950004], we established the long-time Strichartz estimates in $U_{\Delta ^2}^2$ spaces by using the local smoothing estimate and radial Sobolev embedding. Together with the standard I-method, we proved the improved estimate of modified energy increment to derive the global well-posedness and scattering for radial data in $\dot {H}^{s}$ with $s > \frac {8}{7}$, which extends the previous results of Miao, Wu, and Zhang [Math. Nachr. 288 (2015), pp. 798–823].References
- J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math. 75 (1998), 267–297. MR 1655835, DOI 10.1007/BF02788703
- Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Hokkaido Univ. Preprint Ser. Math. 859 (2007), 1–8.
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal. 34 (2002), no. 1, 64–86. MR 1950826, DOI 10.1137/S0036141001394541
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett. 9 (2002), no. 5-6, 659–682. MR 1906069, DOI 10.4310/MRL.2002.v9.n5.a9
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\Bbb R^3$, Comm. Pure Appl. Math. 57 (2004), no. 8, 987–1014. MR 2053757, DOI 10.1002/cpa.20029
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\Bbb R^2$, Discrete Contin. Dyn. Syst. 21 (2008), no. 3, 665–686. MR 2399431, DOI 10.3934/dcds.2008.21.665
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2) 167 (2008), no. 3, 767–865. MR 2415387, DOI 10.4007/annals.2008.167.767
- Van Duong Dinh, Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space, Nonlinear Anal. 172 (2018), 115–140. MR 3790370, DOI 10.1016/j.na.2018.03.003
- Benjamin Dodson, Global well-posedness and scattering for the defocusing, $L^{2}$-critical nonlinear Schrödinger equation when $d\geq 3$, J. Amer. Math. Soc. 25 (2012), no. 2, 429–463. MR 2869023, DOI 10.1090/S0894-0347-2011-00727-3
- Benjamin Dodson, Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n=3$ via a linear-nonlinear decomposition, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 1905–1926. MR 3002734, DOI 10.3934/dcds.2013.33.1905
- Benjamin 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. MR 3483476, DOI 10.1353/ajm.2016.0016
- Benjamin 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. MR 3577369, DOI 10.1215/00127094-3673888
- Benjamin Dodson, Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Ann. PDE 3 (2017), no. 1, Paper No. 5, 35. MR 3625190, DOI 10.1007/s40818-017-0025-9
- Benjamin Dodson, Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when $d=2,3$ and $u_0$ is radial, Camb. J. Math. 7 (2019), no. 3, 283–318. MR 4010063, DOI 10.4310/CJM.2019.v7.n3.a2
- Benjamin Dodson, Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space, Rev. Mat. Iberoam. 38 (2022), no. 4, 1087–1100. MR 4445908, DOI 10.4171/rmi/1295
- Carlos E. Kenig and Frank Merle, Scattering for $\dot H^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc. 362 (2010), no. 4, 1937–1962. MR 2574882, DOI 10.1090/S0002-9947-09-04722-9
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
- Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm053, 36. MR 2353092, DOI 10.1093/imrn/rnm053
- Herbert Koch and Daniel Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in $H^{-\frac 14}$, Ann. Inst. H. Poincaré C Anal. Non Linéaire 29 (2012), no. 6, 955–988. MR 2995102, DOI 10.1016/j.anihpc.2012.05.006
- H. Koch, Nonlinear dispersive equations, Oberwolfach Seminars, vol. 45, Birkhäuser/Springer, Basel, 2004, pp. 41–72.
- Changxing Miao, Haigen Wu, and Junyong Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Nachr. 288 (2015), no. 7, 798–823. MR 3345105, DOI 10.1002/mana.201400012
- Changxing Miao, Guixiang Xu, and Jianwei-Urbain Yang, Global well-posedness for the defocusing Hartree equation with radial data in $\Bbb R^4$, Commun. Contemp. Math. 22 (2020), no. 2, 1950004, 35. MR 4077090, DOI 10.1142/S0219199719500044
- Changxing Miao and Jiqiang Zheng, Scattering theory for the defocusing fourth-order Schrödinger equation, Nonlinearity 29 (2016), no. 2, 692–736. MR 3461612, DOI 10.1088/0951-7715/29/2/692
- Jason Murphy, Intercritical NLS: critical $\dot {H}{}^s$-bounds imply scattering, SIAM J. Math. Anal. 46 (2014), no. 1, 939–997. MR 3166962, DOI 10.1137/120898280
- Benoit Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ. 4 (2007), no. 3, 197–225. MR 2353631, DOI 10.4310/DPDE.2007.v4.n3.a1
- Qingtang Su, Global well posedness and scattering for the defocusing, cubic NLS in $\Bbb R^3$, Math. Res. Lett. 19 (2012), no. 2, 431–451. MR 2955773, DOI 10.4310/MRL.2012.v19.n2.a14
- Daniel Tataru, Local and global results for wave maps. I, Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781–1793. MR 1641721, DOI 10.1080/03605309808821400
Bibliographic Information
- Zuyu Ma
- Affiliation: The Graduate School of China Academy of Engineering Physics, Beijing 100088, People’s Republic of China
- Email: mazuyu23@gscaep.ac.cn
- Yilin Song
- Affiliation: The Graduate School of China Academy of Engineering Physics, Beijing 100088, People’s Republic of China
- Email: songyilin21@gscaep.ac.cn
- Received by editor(s): July 21, 2024
- Received by editor(s) in revised form: November 28, 2024
- Published electronically: April 3, 2025
- Communicated by: Benoit Pausader
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 153 (2025), 2539-2554
- MSC (2020): Primary 35Q55
- DOI: https://doi.org/10.1090/proc/17170
- MathSciNet review: 4892626