American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry
  • CPAA Home
  • This Issue
  • Next Article
    Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $
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  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, 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.  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, 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.  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, 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.  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

[1]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284
[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[3]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[4]

Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $ p $-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034
[5]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002
[6]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403
[7]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[8]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127
[9]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[10]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[11]

Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123
[12]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[13]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[14]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103
[15]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[16]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[17]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[18]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[19]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
[20]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (46)
  • HTML views (48)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences