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Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $
Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea |
$ \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*} $ |
$ H^s(\mathbb{R}), s\\\geq -\frac{1}{2} $ |
$ -\frac{15}{14}<s<-\frac{1}{2} $ |
$ s = -\frac{1}{2} $ |
$ s<-1/2 $ |
$ -3/4<s<-1/2 $ |
$ U^p $ |
$ V^p $ |
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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