# American Institute of Mathematical Sciences

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

 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.
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.

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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. 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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