The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities

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June 2020, 19(6): 3367-3385. doi: 10.3934/cpaa.2020149

The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities

Shuai Zhang ^1,, and Shaopeng Xu ^2,

1.	School of Mathematical Sciences, Peking University, Beijing 100871, China
2.	School of Science, Hainan University, Haikou 570228, China

^* Corresponding author

Received October 2019 Revised January 2020 Published March 2020

Fund Project: Shaopeng Xu is supported by Hainan Provincial Natural Science Foundation of China (No.2019RC168)

Chen and Zhang [7] consider the probabilistic Cauchy problem of the fourth order Schrödinger equation

$ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha u)_{|\alpha|\leq2},(\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq3, \end{align*} $

where

$ P_m $

is a homogeneous polynomial of degree

$ m $

. The almost sure local well-posedness and small data global existence were obtained in

$ H^s(\mathbb{R}^d) $

with the regularity threshold

$ s_c-1/2 $

when

$ d\geq3 $

, where

$ s_c: = d/2-2/(m-1) $

is the scaling critical regularity. For the lower regularity threshold

$ (d-1)s_c/d $

with

$ m = 2 $

and

$ s_c-\min\{1,d/4\} $

with

$ m\geq3 $

, we get the corresponding well-posedness of the following fourth order nonlinear Schrödinger equation

$ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq2 \end{align*} $

on

$ {\mathbb{R}}^d $

(

$ d\geq2 $

) with random initial data.

Keywords: Random initial data, fourth order Schrödinger equation, supercritical well-posedness, decay estimates, truncated bilinear estimates.

Mathematics Subject Classification: Primary: 35Q41, 35Q55; Secondary: 35R60, 60H15.

Citation: Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149

References:

[1]	M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8.
[2]	Á. 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, Vol. 4, Birkhäuser/Springer, Cham, (2015), 3–25.
[3]	J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., 166 (1994), 1-26.
[4]	J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.
[5]	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.
[6]	N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. Ⅱ. A global existence result, Invent. Math., 173 (2008), 477-496. doi: 10.1007/s00222-008-0123-0.
[7]	J. 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, 23. doi: 10.1016/j.na.2019.111608.
[8]	J. Colliander, J. Delort, C. Kenig and G. Staffilani, Bilinear estimates and applications to 2{D} NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325. doi: 10.1090/S0002-9947-01-02760-X.
[9]	V. D. Dinh, Well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437.
[10]	B. Dodson, J. Lührmann and D. Mendelson, Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data, Adv. Math., 347 (2019), 619-676. doi: 10.1016/j.aim.2019.02.001.
[11]	K. B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. A., 369 (1979), 105-114.
[12]	M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. Henri Poincare Anal. Non Lineaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.
[13]	C. C. Hao, L. Hsiao and B. X. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265. doi: 10.1016/j.jmaa.2005.06.091.
[14]	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.
[15]	H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, preprint, arXiv: 1505.06497. doi: 10.3934/dcds.2016102.
[16]	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.
[17]	Z. H. Huo and Y. L. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Commun. Partial Differ. Equ., 32 (2007), 1493-1510. doi: 10.1080/03605300701629385.
[18]	B. Ilan, G. Fibich and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math, 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.
[19]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), R1336. doi: 10.1016/0375-9601(95)00752-0.
[20]	V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Physica D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.
[21]	J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on {$\mathbb{R}^3$}, Commun. Partial Differ. Equ., 39 (2014), 2262-2283. doi: 10.1080/03605302.2014.933239.
[22]	J. Lührmann and D. Mendelson, On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on {$\Bbb R^3$}, New York J. Math., 22 (2016), 209-227.
[23]	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.
[24]	B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.
[25]	J. Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976.
[26]	M. Ruzhansky, B. X. Wang and H. Zhang, Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl., 105 (2016), 31-65. doi: 10.1016/j.matpur.2015.09.005.
[27]	Y. Z. Wang, Global well-posedness for the generalised fourth-order Schrödinger equation, Bull. Aust. Math. Soc., 85 (2012), 371-379. doi: 10.1017/S0004972711003327.
[28]	B. X. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differ. Equ., 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004.
[29]	B. X. Wang, L. F. Zhao and B. L. Guo, Isometric decomposition operators, function spaces {$E^\lambda_{p, q}$} and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.

show all references

References:

[1]	M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8.
[2]	Á. 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, Vol. 4, Birkhäuser/Springer, Cham, (2015), 3–25.
[3]	J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., 166 (1994), 1-26.
[4]	J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys., 176 (1996), 421-445.
[5]	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.
[6]	N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. Ⅱ. A global existence result, Invent. Math., 173 (2008), 477-496. doi: 10.1007/s00222-008-0123-0.
[7]	J. 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, 23. doi: 10.1016/j.na.2019.111608.
[8]	J. Colliander, J. Delort, C. Kenig and G. Staffilani, Bilinear estimates and applications to 2{D} NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325. doi: 10.1090/S0002-9947-01-02760-X.
[9]	V. D. Dinh, Well-posedness, regularity and ill-posedness for the nonlinear fourth-order Schrödinger equation, Bull. Belg. Math. Soc. Simon Stevin, 25 (2018), 415-437.
[10]	B. Dodson, J. Lührmann and D. Mendelson, Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data, Adv. Math., 347 (2019), 619-676. doi: 10.1016/j.aim.2019.02.001.
[11]	K. B. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. A., 369 (1979), 105-114.
[12]	M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. Henri Poincare Anal. Non Lineaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.
[13]	C. C. Hao, L. Hsiao and B. X. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265. doi: 10.1016/j.jmaa.2005.06.091.
[14]	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.
[15]	H. Hirayama and M. Okamoto, Random data Cauchy theory for the fourth order nonlinear Schrödinger equation with cubic nonlinearity, preprint, arXiv: 1505.06497. doi: 10.3934/dcds.2016102.
[16]	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.
[17]	Z. H. Huo and Y. L. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Commun. Partial Differ. Equ., 32 (2007), 1493-1510. doi: 10.1080/03605300701629385.
[18]	B. Ilan, G. Fibich and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math, 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.
[19]	V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), R1336. doi: 10.1016/0375-9601(95)00752-0.
[20]	V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Physica D, 144 (2000), 194-210. doi: 10.1016/S0167-2789(00)00078-6.
[21]	J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on {$\mathbb{R}^3$}, Commun. Partial Differ. Equ., 39 (2014), 2262-2283. doi: 10.1080/03605302.2014.933239.
[22]	J. Lührmann and D. Mendelson, On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on {$\Bbb R^3$}, New York J. Math., 22 (2016), 209-227.
[23]	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.
[24]	B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.
[25]	J. Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976.
[26]	M. Ruzhansky, B. X. Wang and H. Zhang, Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl., 105 (2016), 31-65. doi: 10.1016/j.matpur.2015.09.005.
[27]	Y. Z. Wang, Global well-posedness for the generalised fourth-order Schrödinger equation, Bull. Aust. Math. Soc., 85 (2012), 371-379. doi: 10.1017/S0004972711003327.
[28]	B. X. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differ. Equ., 232 (2007), 36-73. doi: 10.1016/j.jde.2006.09.004.
[29]	B. X. Wang, L. F. Zhao and B. L. Guo, Isometric decomposition operators, function spaces {$E^\lambda_{p, q}$} and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.

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