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

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.
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. doi: 10.1016/0022-1236(91)90103-C.

[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.

[3]

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

[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. doi: 10.1016/j.anihpc.2008.04.002.

[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. doi: 10.1016/j.anihpc.2010.01.006.

[6]

C. Hao, L. Hsiao, and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations,, J. Math. Anal. Appl., 320 (2006), 246. doi: 10.1016/j.jmaa.2005.06.091.

[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. doi: 10.1016/j.jmaa.2006.05.031.

[8]

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

[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. doi: 10.1016/j.na.2014.12.024.

[10]

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

[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. doi: 10.1016/j.jde.2004.09.005.

[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. doi: 10.1080/03605300701629385.

[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. doi: 10.1016/j.matpur.2011.01.002.

[14]

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

[15]

V. Karpman and A. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion,, Physica D, 144 (2000), 194. doi: 10.1016/S0167-2789(00)00078-6.

[16]

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

[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.

[18]

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

[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. doi: 10.3934/dcds.2010.27.1093.

[20]

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

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. doi: 10.1016/0022-1236(91)90103-C.

[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.

[3]

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

[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. doi: 10.1016/j.anihpc.2008.04.002.

[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. doi: 10.1016/j.anihpc.2010.01.006.

[6]

C. Hao, L. Hsiao, and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations,, J. Math. Anal. Appl., 320 (2006), 246. doi: 10.1016/j.jmaa.2005.06.091.

[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. doi: 10.1016/j.jmaa.2006.05.031.

[8]

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

[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. doi: 10.1016/j.na.2014.12.024.

[10]

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

[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. doi: 10.1016/j.jde.2004.09.005.

[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. doi: 10.1080/03605300701629385.

[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. doi: 10.1016/j.matpur.2011.01.002.

[14]

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

[15]

V. Karpman and A. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion,, Physica D, 144 (2000), 194. doi: 10.1016/S0167-2789(00)00078-6.

[16]

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

[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.

[18]

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

[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. doi: 10.3934/dcds.2010.27.1093.

[20]

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

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