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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., 166 (1994), 1-26.   Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[24]

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

[25]

J. Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., 166 (1994), 1-26.   Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[24]

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

[25]

J. Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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