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December  2020, 19(12): 5437-5473. 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

Received  January 2020 Revised  July 2020 Published  December 2020 Early access  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 and Applied Analysis, 2020, 19 (12) : 5437-5473. 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.

[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.

[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. 

[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.

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.

[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.

[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.

[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. 

[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.

[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. 

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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. 

[19]

H. Koch and D. Tataru, Conserved energies for the cubic nonlinear Schrödinger equation in one dimension, Duke Math. J., 167 (2018), 3207-3313.

[20]

H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014.

[21]

C. Kwak, Periodic fourth-order cubic NLS: Local well-posedness and non-squeezing property, J. Math. Anal. Appl., 461 (2018), 1327-1364. 

[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.

[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.

[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.

[25]

T. Oh and Y. Wang, Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, arXiv: 1707.02013.

[26]

B. Pausader, The cubic fourth-order Schrödinger equation, arXiv: 0807.4916.

[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.

[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.

[29]

K. Seong, Well-Posedness and Ill-Posedness for the Fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces, arXiv: 1911.03253.

[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. 

[31]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. 

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.

[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.

[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. 

[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.

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.

[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.

[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.

[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. 

[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.

[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. 

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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. 

[19]

H. Koch and D. Tataru, Conserved energies for the cubic nonlinear Schrödinger equation in one dimension, Duke Math. J., 167 (2018), 3207-3313.

[20]

H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars, Birkhäuser/Springer, Basel, 2014.

[21]

C. Kwak, Periodic fourth-order cubic NLS: Local well-posedness and non-squeezing property, J. Math. Anal. Appl., 461 (2018), 1327-1364. 

[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.

[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.

[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.

[25]

T. Oh and Y. Wang, Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, arXiv: 1707.02013.

[26]

B. Pausader, The cubic fourth-order Schrödinger equation, arXiv: 0807.4916.

[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.

[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.

[29]

K. Seong, Well-Posedness and Ill-Posedness for the Fourth order cubic nonlinear Schrödinger equation in negative Sobolev spaces, arXiv: 1911.03253.

[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. 

[31]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. 

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