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May  2016, 15(3): 831-851. doi: 10.3934/cpaa.2016.15.831

Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity

Hiroyuki Hirayama 1, and  Mamoru Okamoto 2,

1. 

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602
2. 

Department of Mathematics, Institute of Engineering, Academic Assembly, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan

Received  May 2015 Revised  December 2015 Published  February 2016

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with derivative nonlinearity. In one dimensional case, the small data global well-posedness and scattering for the fourth order nonlinear Schrödinger equation with the nonlinear term $\partial _x (\overline{u}^4)$ are shown in the scaling invariant space $\dot{H}^{-1/2}$. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$, where $d$ denotes the space dimension.
Keywords: scaling critical, Cauchy problem, bounded $p$-variation., Fourth order Schrödinger equation, well-posedness.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B6.
Citation: Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
References:
[1]

F. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[2]

K. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A, 369 (1979), 105-114. Google Scholar

[3]

Y. Fukumoto, Motion of a curved vortex filament: higher-order asymptotics, in Proc. IUTAM Symp. Geom. Stat. Turbul., 2001, 211-216. doi: 10.1007/978-94-015-9638-1_25.  Google Scholar

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M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non linéaie., 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

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C. Hao, L. Hsiao, and B. 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

[7]

C. Hao, L. Hsiao, and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi dimensional spaces, J. Math. Anal. Appl., 328 (2007), 58-83. doi: 10.1016/j.jmaa.2006.05.031.  Google Scholar

[8]

N. Hayashi and P. I. Naumkin, Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905. doi: 10.1016/j.jde.2014.10.007.  Google Scholar

[9]

N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131. doi: 10.1016/j.na.2014.12.024.  Google Scholar

[10]

H. Hirayama, Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity,, FUNKCIALAJ EKVACIOJ, ().   Google Scholar

[11]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differential Equations, 214 (2005), 1-35. doi: 10.1016/j.jde.2004.09.005.  Google Scholar

[12]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510. doi: 10.1080/03605300701629385.  Google Scholar

[13]

Z. Huo and Y. Jia, Well-posedness for the fourth-order nonlinear derivative Schrödinger equation in higher dimension, J. Math. Pures Appl., 96 (2011), 190-206. doi: 10.1016/j.matpur.2011.01.002.  Google Scholar

[14]

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

[15]

V. Karpman and A. Shagalov, Stability of soliton 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

[16]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[17]

J. Segata, Well-posedness for the fourth order nonlinear Schrödinger type equation related to the vortex filament, Diff. and Integral Eqs., 16 (2003), 841-864.  Google Scholar

[18]

J. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568. doi: 10.1090/S0002-9939-04-07620-8.  Google Scholar

[19]

J. Segata, Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 27 (2010), 1093-1105. doi: 10.3934/dcds.2010.27.1093.  Google Scholar

[20]

Y. Wang, Global well-posedness for the generalized fourth-order Schrödingier equation, Bull. Aust. Math. Soc., 85 (2012), 371-379. doi: 10.1017/S0004972711003327.  Google Scholar

show all references

References:
[1]

F. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[2]

K. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A, 369 (1979), 105-114. Google Scholar

[3]

Y. Fukumoto, Motion of a curved vortex filament: higher-order asymptotics, in Proc. IUTAM Symp. Geom. Stat. Turbul., 2001, 211-216. doi: 10.1007/978-94-015-9638-1_25.  Google Scholar

[4]

M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non linéaie., 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[5]

M. Hadac, S. Herr, and H. Koch, Errantum to "Well-posedness and scattering for the KP-II equation in a critical space'' [Ann. I. H. Poincaré-AN26 (3) (2009) 917-941], Ann. Inst. H. Poincaré Anal. Non linéaie., 27 (2010), 971-972. doi: 10.1016/j.anihpc.2010.01.006.  Google Scholar

[6]

C. Hao, L. Hsiao, and B. 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

[7]

C. Hao, L. Hsiao, and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi dimensional spaces, J. Math. Anal. Appl., 328 (2007), 58-83. doi: 10.1016/j.jmaa.2006.05.031.  Google Scholar

[8]

N. Hayashi and P. I. Naumkin, Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905. doi: 10.1016/j.jde.2014.10.007.  Google Scholar

[9]

N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131. doi: 10.1016/j.na.2014.12.024.  Google Scholar

[10]

H. Hirayama, Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity,, FUNKCIALAJ EKVACIOJ, ().   Google Scholar

[11]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differential Equations, 214 (2005), 1-35. doi: 10.1016/j.jde.2004.09.005.  Google Scholar

[12]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510. doi: 10.1080/03605300701629385.  Google Scholar

[13]

Z. Huo and Y. Jia, Well-posedness for the fourth-order nonlinear derivative Schrödinger equation in higher dimension, J. Math. Pures Appl., 96 (2011), 190-206. doi: 10.1016/j.matpur.2011.01.002.  Google Scholar

[14]

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

[15]

V. Karpman and A. Shagalov, Stability of soliton 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

[16]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225. doi: 10.4310/DPDE.2007.v4.n3.a1.  Google Scholar

[17]

J. Segata, Well-posedness for the fourth order nonlinear Schrödinger type equation related to the vortex filament, Diff. and Integral Eqs., 16 (2003), 841-864.  Google Scholar

[18]

J. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568. doi: 10.1090/S0002-9939-04-07620-8.  Google Scholar

[19]

J. Segata, Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 27 (2010), 1093-1105. doi: 10.3934/dcds.2010.27.1093.  Google Scholar

[20]

Y. Wang, Global well-posedness for the generalized fourth-order Schrödingier equation, Bull. Aust. Math. Soc., 85 (2012), 371-379. doi: 10.1017/S0004972711003327.  Google Scholar

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