-
Previous Article
Global solutions of shock reflection problem for the pressure gradient system
- CPAA Home
- This Issue
-
Next Article
On the Sobolev embedding properties for compact matrix quantum groups of Kac type
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 |
$ \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*} $ |
$ P_m $ |
$ m $ |
$ H^s(\mathbb{R}^d) $ |
$ s_c-1/2 $ |
$ d\geq3 $ |
$ s_c: = d/2-2/(m-1) $ |
$ (d-1)s_c/d $ |
$ m = 2 $ |
$ s_c-\min\{1,d/4\} $ |
$ m\geq3 $ |
$ \begin{align*} (i\partial_t+\varepsilon\Delta+\Delta^2)u = P_m((\partial_x^\alpha \overline{u})_{|\alpha|\leq2}),\ m\geq2 \end{align*} $ |
$ {\mathbb{R}}^d $ |
$ d\geq2 $ |
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. |
[1] |
Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205 |
[2] |
Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 |
[3] |
Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093 |
[4] |
Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122 |
[5] |
Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2721-2757. doi: 10.3934/dcdsb.2021156 |
[6] |
Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 |
[7] |
Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure and Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 |
[8] |
Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure and Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 |
[9] |
Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 |
[10] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284 |
[11] |
Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 |
[12] |
Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527 |
[13] |
Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082 |
[14] |
Dan-Andrei Geba, Evan Witz. Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations. Electronic Research Archive, 2020, 28 (2) : 627-649. doi: 10.3934/era.2020033 |
[15] |
Shaoming Guo, Xianfeng Ren, Baoxiang Wang. Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4207-4253. doi: 10.3934/dcds.2021034 |
[16] |
Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024 |
[17] |
Huafei Di, Yadong Shang, Jiali Yu. Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electronic Research Archive, 2020, 28 (1) : 221-261. doi: 10.3934/era.2020015 |
[18] |
Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations and Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023 |
[19] |
Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations and Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 |
[20] |
Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]