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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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), 1-35. 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), 1493-1510. 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), 190-206. 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), 1336-1339.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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), 1-35. 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), 1493-1510. 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), 190-206. 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), 1336-1339.
[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.
[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.
[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.
[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.
[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.
[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.

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