Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Van Duong Dinh

doi:10.3934/cpaa.2020284

Article Contents

2021, Volume 20, Issue 2: 651-680. Doi: 10.3934/cpaa.2020284

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Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Van Duong Dinh

Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France, Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam

Received: April 2020

Revised: September 2020

Early access: December 2020

Published: February 2021

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)

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Abstract

We consider the cubic nonlinear fourth-order Schrödinger equation

$ i \partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 $

on $ \mathbb R^N, N\geq 5 $ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

Keywords:

Fourth-order nonlinear Schrödinger equation,
almost sure well-posedness,
wiener randomization,
probabilistic Strichartz estimates.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01.

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References

[1]	M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth-order Schrödinger equations, C. R. Acad. Sci., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8.
[2]	A. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, in Excursions in Harmonic Analysis, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-20188-7_1.
[3]	A. Bényi, T. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb R^d, d\geq 3$, T. Am. Math. Soc. Ser. B, 2 (2015), 1-50. doi: 10.1090/btran/6.
[4]	A. Bényi, T. Oh and O. Pocovnicu, Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb R^3$, T. Am. Math. Soc. B, 6 (2019), 114-160. doi: 10.1090/btran/29.
[5]	T. Boulenger and E. Lenzmann, Blowup for biharmonic NL4S, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544. doi: 10.24033/asens.2326.
[6]	J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.
[7]	N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., 173 (2008), 449-475. doi: 10.1007/s00222-008-0124-z.
[8]	N. Burq, L. Thomann and N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, 63 (2013), 2137-2198.
[9]	M. Chen and S. Zhang, Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities, Nonlinear Anal., 190 (2020), 111608. doi: 10.1016/j.na.2019.111608.
[10]	Y. Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE, 5 (2012), 913-960. doi: 10.2140/apde.2012.5.913.
[11]	V. D. Dinh, On well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437.
[12]	V. D. Dinh, Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space, Nonlinear Anal., 172 (2018), 115-140. doi: 10.1016/j.na.2018.03.003.
[13]	V. D. Dinh, On blowup solutions to the focusing intercritical nonlinear fourth-order Schrödinger equation, J. Dynam. Differ. Equ., 31 (2019), 1793-1823. doi: 10.1007/s10884-018-9690-y.
[14]	V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations, arXiv: 2001.03022.
[15]	B. Dodson, J. Lührmann and D. Mendelson, Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation, Adv. Math., 347 (2019), 619-676. doi: 10.1016/j.aim.2019.02.001.
[16]	Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Commun. Partial Differ. Equ., 41 (2016), 185-207. doi: 10.1080/03605302.2015.1116556.
[17]	M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.
[18]	S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb{T}^3)$, Duke Math. J., 159 (2011), 329-349. doi: 10.1215/00127094-1415889.
[19]	H. Hirayama and M. Okamoto, Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst., 36 (2016), 6943-6974. doi: 10.3934/dcds.2016102.
[20]	H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, arXiv: 1505.06497.
[21]	V. I. Karpman, Stabiliztion of solition instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
[22]	V. I. Karpman and A. G. Shagalov, Stability of solition described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. doi: 10.1016/S0375-9601(97)00063-7.
[23]	M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.
[24]	R. Killip, J. Murphy and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb R^4)$, Commun. Partial Differ. Equ., 44 (2019), 51-71. doi: 10.1080/03605302.2018.1541904.
[25]	H. Koch, D. Tataru and M. Visan, Dispersive equations and nonlinear waves, Birkhäuse 45, Springer Basel, 2014.
[26]	J. Lührmann and D. Mendelson, Random data Cauchy theory for the nonlinear wave equations of power-type on $ \mathbb R^3$, Commun. Partial Differ. Equ., 39 (2014), 2262-2283. doi: 10.1080/03605302.2014.933239.
[27]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing energy critical nonlinear Schrödinger equation of fourth order in the radial case, J. Differ. Equ., 246 (2009), 3715-3749. doi: 10.1016/j.jde.2008.11.011.
[28]	C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energy critical nonlinear Schrödinger equations of fourth order in dimensions $d\geq 9$, J. Differ. Equ., 251 (2011), 3381-3402. doi: 10.1016/j.jde.2008.11.011.
[29]	C. Miao, H. Wu and J. Zhang, Scattering theory below energy for the cubic fourth-order Schrödinger equation, Math. Narchr., 288 (2015), 798-823. doi: 10.1002/mana.201400012.
[30]	T. Oh, M. Okamoto and O. Pocovnicu, On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities, Discrete Contin. Dyn. Syst., 39 (2019), 3479-3520. doi: 10.3934/dcds.2019144.
[31]	B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.
[32]	B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292. doi: 10.3934/dcds.2009.24.1275.
[33]	B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.
[34]	B. Pausader and S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J. Hyper. Differ. Equ., 7 (2010), 651-705. doi: 10.1142/S0219891610002256.
[35]	B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191. doi: 10.1088/0951-7715/26/8/2175.
[36]	S. Zhang and S. Xu, The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities, Commun. Pure Appl. Anal., 19 (2020), 3367-3385. doi: 10.3934/cpaa.2020149.
[37]	S. Zhu, H. Yang and J. Zhang, Limiting profile of the blow-up solutions for the fourth-order nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 7 (2010), 187-205. doi: 10.4310/DPDE.2010.v7.n2.a4.