Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation

Kihoon Seong

doi:10.3934/cpaa.2020247

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doi: 10.3934/cpaa.2020247

Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation

Kihoon Seong ^,

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

^* Corresponding author

Received January 2020 Revised July 2020 Published October 2020

Fund Project: The author is supported by NRF-2018R1D1A1A09083345

We consider the low regularity behavior of the fourth order cubic nonlinear Schrödinger equation (4NLS)

$ \begin{align*} \begin{cases} i\partial_tu+\partial_x^4u = \pm \vert u \vert^2u, \quad(t,x)\in \mathbb{R}\times \mathbb{R}\\ u(x,0) = u_0(x)\in H^s\left(\mathbb{R}\right). \end{cases} \end{align*} $

In [29], the author showed that this equation is globally well-posed in

$ H^s(\mathbb{R}), s\geq -\frac{1}{2} $

and mildly ill-posed in the sense that the solution map fails to be locally uniformly continuous for

$ -\frac{15}{14}<s<-\frac{1}{2} $

. Therefore,

$ s = -\frac{1}{2} $

is the lowest regularity that can be handled by the contraction argument. In spite of this mild ill-posedness result, we obtain an a priori bound below

$ s<-1/2 $

. This an a priori estimate guarantees the existence of a weak solution for

$ -3/4<s<-1/2 $

. Our method is inspired by Koch-Tataru [17]. We use the

$ U^p $

and

$ V^p $

based spaces adapted to frequency dependent time intervals on which the nonlinear evolution can still be described by linear dynamics.

Keywords: fourth order nonlinear Schrödinger equation, low regularity solutions, frequency dependent time scale, U_p and V_p spaces, normal form reduction.

Mathematics Subject Classification: 35Q55.

Citation: Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020247

References:

[1]	M. Ben-Artzi, H. Koch and J. C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8. Google Scholar
[2]	M. Christ, J. Colliander and T. Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal., 254 (2008), 368-395. doi: 10.1016/j.jfa.2007.09.005. Google Scholar
[3]	M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Am. J. Math., 125 (2003), 1235-1293. Google Scholar
[4]	M. Christ, J. Holmer and D. Tataru, Low regularity a priori bounds for the modified Korteweg-de Vries equation, Lib. Math., 32 (2012), 51-75. doi: 10.14510/lm-ns.v32i1.32. Google Scholar
[5]	J. Colliander, M. Keel, G. Staffilani, T. Takaoka and H. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669. doi: 10.1137/S0036141001384387. Google Scholar
[6]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, arXiv:math/0203218. Google Scholar
[7]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, arXiv:math/0110045. Google Scholar
[8]	B. Guo and B. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$, Differ. Integral Equ., 15 (2002), 1073-1083. Google Scholar
[9]	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 Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar
[10]	S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\Bbb T^3)$, Duke Math. J., 159 (2011), 329-349. Google Scholar
[11]	S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304. doi: 10.1007/s00222-008-0115-0. Google Scholar
[12]	V. I. Karpman, Lyapunov approach to the soliton stability in highly dispersive systems. I. Fourth order nonlinear Schrödinger equations, Phys. Lett. A, 215 (1996), 254-256. Google Scholar
[13]	V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. Google Scholar
[14]	C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. Google Scholar
[15]	C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar
[16]	H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s({\Bbb R})$, Int. Math. Res. Not., (2003), 1449–1464. Google Scholar
[17]	H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., (2007), 1073–7928. doi: 10.1093/imrn/rnm053. Google Scholar
[18]	H. Koch and D. Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in $H^{-\frac14}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988. Google Scholar
[19]	H. Koch and D. Tataru, Conserved energies for the cubic nonlinear Schrödinger equation in one dimension, Duke Math. J., 167 (2018), 3207-3313. Google Scholar
[20]	H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014. Google Scholar
[21]	C. Kwak, Periodic fourth-order cubic NLS: Local well-posedness and non-squeezing property, J. Math. Anal. Appl., 461 (2018), 1327-1364. Google Scholar
[22]	B. Liu, A priori bounds for KdV equation below $H^{-\frac34}$, J. Funct. Anal., 268 (2015), 501-554. doi: 10.1016/j.jfa.2014.06.020. Google Scholar
[23]	T. Oh, P. Sosoe and N. Tzvetkov, An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, J. Éc. polytech. Math., 5 (2018), 793–841. doi: 10.5802/jep.83. Google Scholar
[24]	T. Oh and N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Rel., 169 (2017), 1121-1168. doi: 10.1007/s00440-016-0748-7. Google Scholar
[25]	T. Oh and Y. Wang, Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, arXiv:1707.02013. Google Scholar
[26]	B. Pausader, The cubic fourth-order Schrödinger equation, arXiv:0807.4916. Google Scholar
[27]	Roberto A. Capistrano-Filho, Márcio Cavalcante, Fernando A. Gallego, Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, arXiv:1812.11079. Google Scholar
[28]	J. i. Segata, Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearit, Math. Methods Appl. Sci., 29 (2006), 1785-1800. doi: 10.1002/mma.751. Google Scholar
[29]	K. Seong, Well-Posedness and Ill-Posedness for the Fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces, arXiv:1911.03253. Google Scholar
[30]	H. Takaoka and Y. Tsutsumi, Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition, Int. Math. Res. Not., 56 (2004), 3009-3040. Google Scholar
[31]	Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. Google Scholar

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 Sér. I Math., 330 (2000), 87-92. doi: 10.1016/S0764-4442(00)00120-8. Google Scholar
[2]	M. Christ, J. Colliander and T. Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal., 254 (2008), 368-395. doi: 10.1016/j.jfa.2007.09.005. Google Scholar
[3]	M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Am. J. Math., 125 (2003), 1235-1293. Google Scholar
[4]	M. Christ, J. Holmer and D. Tataru, Low regularity a priori bounds for the modified Korteweg-de Vries equation, Lib. Math., 32 (2012), 51-75. doi: 10.14510/lm-ns.v32i1.32. Google Scholar
[5]	J. Colliander, M. Keel, G. Staffilani, T. Takaoka and H. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669. doi: 10.1137/S0036141001384387. Google Scholar
[6]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, arXiv:math/0203218. Google Scholar
[7]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, arXiv:math/0110045. Google Scholar
[8]	B. Guo and B. Wang, The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in $H^s$, Differ. Integral Equ., 15 (2002), 1073-1083. Google Scholar
[9]	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 Linéaire, 26 (2009), 917-941. doi: 10.1016/j.anihpc.2008.04.002. Google Scholar
[10]	S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\Bbb T^3)$, Duke Math. J., 159 (2011), 329-349. Google Scholar
[11]	S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304. doi: 10.1007/s00222-008-0115-0. Google Scholar
[12]	V. I. Karpman, Lyapunov approach to the soliton stability in highly dispersive systems. I. Fourth order nonlinear Schrödinger equations, Phys. Lett. A, 215 (1996), 254-256. Google Scholar
[13]	V. I. Karpman and A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210. Google Scholar
[14]	C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. Google Scholar
[15]	C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar
[16]	H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in $H^s({\Bbb R})$, Int. Math. Res. Not., (2003), 1449–1464. Google Scholar
[17]	H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., (2007), 1073–7928. doi: 10.1093/imrn/rnm053. Google Scholar
[18]	H. Koch and D. Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in $H^{-\frac14}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988. Google Scholar
[19]	H. Koch and D. Tataru, Conserved energies for the cubic nonlinear Schrödinger equation in one dimension, Duke Math. J., 167 (2018), 3207-3313. Google Scholar
[20]	H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014. Google Scholar
[21]	C. Kwak, Periodic fourth-order cubic NLS: Local well-posedness and non-squeezing property, J. Math. Anal. Appl., 461 (2018), 1327-1364. Google Scholar
[22]	B. Liu, A priori bounds for KdV equation below $H^{-\frac34}$, J. Funct. Anal., 268 (2015), 501-554. doi: 10.1016/j.jfa.2014.06.020. Google Scholar
[23]	T. Oh, P. Sosoe and N. Tzvetkov, An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, J. Éc. polytech. Math., 5 (2018), 793–841. doi: 10.5802/jep.83. Google Scholar
[24]	T. Oh and N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Rel., 169 (2017), 1121-1168. doi: 10.1007/s00440-016-0748-7. Google Scholar
[25]	T. Oh and Y. Wang, Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, arXiv:1707.02013. Google Scholar
[26]	B. Pausader, The cubic fourth-order Schrödinger equation, arXiv:0807.4916. Google Scholar
[27]	Roberto A. Capistrano-Filho, Márcio Cavalcante, Fernando A. Gallego, Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, arXiv:1812.11079. Google Scholar
[28]	J. i. Segata, Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearit, Math. Methods Appl. Sci., 29 (2006), 1785-1800. doi: 10.1002/mma.751. Google Scholar
[29]	K. Seong, Well-Posedness and Ill-Posedness for the Fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces, arXiv:1911.03253. Google Scholar
[30]	H. Takaoka and Y. Tsutsumi, Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition, Int. Math. Res. Not., 56 (2004), 3009-3040. Google Scholar
[31]	Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. Google Scholar

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Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation

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