February  2021, 20(2): 651-680. doi: 10.3934/cpaa.2020284

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

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 Published  December 2020

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

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.
Citation: Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284
References:
[1]

M. Ben-ArtziH. 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.  Google Scholar

[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.  Google Scholar

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A. BényiT. 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.  Google Scholar

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T. Boulenger and E. Lenzmann, Blowup for biharmonic NL4S, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

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J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.   Google Scholar

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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.  Google Scholar

[8]

N. BurqL. Thomann and N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, 63 (2013), 2137-2198.   Google Scholar

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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.  Google Scholar

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Y. Deng, Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE, 5 (2012), 913-960.  doi: 10.2140/apde.2012.5.913.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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B. DodsonJ. 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.  Google Scholar

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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.  Google Scholar

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M. HadacS. 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.  Google Scholar

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S. HerrD. 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.  Google Scholar

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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.  Google Scholar

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H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, arXiv: 1505.06497. Google Scholar

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V. I. Karpman, Stabiliztion of solition instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.   Google Scholar

[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.  Google Scholar

[23]

M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.   Google Scholar

[24]

R. KillipJ. 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.  Google Scholar

[25]

H. Koch, D. Tataru and M. Visan, Dispersive equations and nonlinear waves, Birkhäuse 45, Springer Basel, 2014.  Google Scholar

[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.  Google Scholar

[27]

C. MiaoG. 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.  Google Scholar

[28]

C. MiaoG. 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.  Google Scholar

[29]

C. MiaoH. 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.  Google Scholar

[30]

T. OhM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. ZhuH. 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.  Google Scholar

show all references

References:
[1]

M. Ben-ArtziH. 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.  Google Scholar

[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.  Google Scholar

[3]

A. BényiT. 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.  Google Scholar

[4]

A. BényiT. 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.  Google Scholar

[5]

T. Boulenger and E. Lenzmann, Blowup for biharmonic NL4S, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[6]

J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.   Google Scholar

[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.  Google Scholar

[8]

N. BurqL. Thomann and N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, 63 (2013), 2137-2198.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations, arXiv: 2001.03022. Google Scholar

[15]

B. DodsonJ. 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.  Google Scholar

[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.  Google Scholar

[17]

M. HadacS. 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.  Google Scholar

[18]

S. HerrD. 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.  Google Scholar

[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.  Google Scholar

[20]

H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, arXiv: 1505.06497. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

M. Keel and T. Tao, Endpoint Strichartz estimates, Am. J. Math., 120 (1998), 955-980.   Google Scholar

[24]

R. KillipJ. 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.  Google Scholar

[25]

H. Koch, D. Tataru and M. Visan, Dispersive equations and nonlinear waves, Birkhäuse 45, Springer Basel, 2014.  Google Scholar

[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.  Google Scholar

[27]

C. MiaoG. 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.  Google Scholar

[28]

C. MiaoG. 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.  Google Scholar

[29]

C. MiaoH. 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.  Google Scholar

[30]

T. OhM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. ZhuH. 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.  Google Scholar

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