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Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations
1. | Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, United States |
References:
[1] |
T. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153.
doi: 10.1098/rspa.1972.0074. |
[2] |
J. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363.
doi: 10.1098/rspa.1975.0106. |
[3] |
J. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type,, Proc. Roy. Soc. (London) Ser. A, 411 (1987), 395.
doi: 10.1098/rspa.1987.0073. |
[4] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 107.
doi: 10.1007/BF01896020. |
[5] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation,, Geom. Funct. Anal., 3 (1993), 209.
doi: 10.1007/BF01895688. |
[6] |
M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonoical defocusing equations,, Amer. J. Math, 125 (2003), 1235.
doi: 10.1353/ajm.2003.0040. |
[7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index,, Electron. J. Differential Equations, 26 (2001).
|
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm,, Commun. Pure. Appl. Anal., 2 (2003), 33.
|
[9] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm,, Discrete Contin. Dyn. Syst., 9 (2003), 31.
|
[10] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$,, J. Amer. Math. Soc., 16 (2003), 705.
doi: 10.1090/S0894-0347-03-00421-1. |
[11] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications,, J. Funct. Anal., 211 (2004), 173.
doi: 10.1016/S0022-1236(03)00218-0. |
[12] |
J. Colliander and P. Raphaël, Rough blowup solutions to the $L^2$ critical NLS,, Math. Ann., 345 (2009), 307.
doi: 10.1007/s00208-009-0355-3. |
[13] |
L. Farah, Global rough solutions to the critical generalized KdV equation,, J. Differential Equations, 249 (2010), 1968.
doi: 10.1016/j.jde.2010.05.010. |
[14] |
L. Farah, F. Linares and A. Pastor, The supercritical generalized KdV equation: global well-posedness in the energy space and below,, Math. Res. Lett., 18 (2011), 357.
|
[15] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I.,, J. Funct. Anal., 74 (1987), 160.
doi: 10.1016/0022-1236(87)90044-9. |
[16] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II.,, J. Funct. Anal., 94 (1990), 308.
doi: 10.1016/0022-1236(90)90016-E. |
[17] |
A. Grünrock, A bilinear Airy-estimate with application to gKdV-3,, Differential Integral Equations, 18 (2005), 1333.
|
[18] |
A. Grünrock, M. Panthee and J. Silva, A remark on global well-posedness below $L^2$ for the GKDV-3 equation,, Differential Integral Equations, 20 (2007), 1229.
|
[19] |
Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$,, J. Math. Pures. Appl., 91 (2009), 583.
doi: 10.1016/j.matpur.2009.01.012. |
[20] |
C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33.
doi: 10.1512/iumj.1991.40.40003. |
[21] |
C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Comm. Pure Appl. Math., 46 (1993), 527.
doi: 10.1002/cpa.3160460405. |
[22] |
C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573.
doi: 10.1090/S0894-0347-96-00200-7. |
[23] |
N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity,, Differential Integral Equations, 22 (2009), 447.
|
[24] |
H. Koch and J. Marzuola, Small data scattering and soliton stability in $\dot H^{-1/6}$ for the quartic KdV equation,, Anal. PDE, 5 (2012), 145.
doi: 10.2140/apde.2012.5.145. |
[25] |
D. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and a new typ of long stationary waves,, Philos. Mag., 39 (1895), 422. Google Scholar |
[26] |
Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations,, Arch. Ration. Mech. Anal., 157 (2001), 219.
doi: 10.1007/s002050100138. |
[27] |
Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation,, Geom. Funct. Anal., 11 (2001), 74.
doi: 10.1007/PL00001673. |
[28] |
Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation,, J. Amer. Math. Soc., 15 (2002), 617.
doi: 10.1090/S0894-0347-02-00392-2. |
[29] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited,, Nonlinearity, 18 (2005), 391.
doi: 10.1088/0951-7715/18/1/004. |
[30] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity,, Math. Ann., 341 (2008), 391.
doi: 10.1007/s00208-007-0194-z. |
[31] |
Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation I: dynamics near the soliton, preprint,, \arXiv{1204.4625}, (). Google Scholar |
[32] |
Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation II: minimal mass dynamics, preprint,, \arXiv{1204.4624}, (). Google Scholar |
[33] |
Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation III: exotic regimes, preprint,, \arXiv{1209.2510}, (). Google Scholar |
[34] |
F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation,, J. Amer. Math. Soc., 14 (2001), 555.
doi: 10.1090/S0894-0347-01-00369-1. |
[35] |
F. Merle and L. Vega, $L^2$ stability of solitons for the KdV equation,, Int. Math. Res. Not., 13 (2003), 735.
doi: 10.1155/S1073792803208060. |
[36] |
C. Miao, S. Shao, Y. Wu and G. Xu, The low regularity global solutions for the critical generalized KdV equation,, Dyn. Partial Differ. Equ., 7 (2010), 265.
|
[37] |
R. Miura, The Korteweg-de Vries equation: a survey of results,, SIAM Rev., 18 (1976), 412.
doi: 10.1137/1018076. |
[38] |
S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation,, Commun. Pure. Appl. Anal., 2 (2003), 277.
doi: 10.3934/cpaa.2003.2.277. |
[39] |
T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839.
doi: 10.1353/ajm.2001.0035. |
[40] |
T. Tao, Scattering for the quartic generalized Korteweg-de Vries equation,, J. Differential Equations, 232 (2007), 623.
doi: 10.1016/j.jde.2006.07.019. |
[41] |
M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.
|
[42] |
M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure Appl. Math, 39 (1986), 51.
doi: 10.1002/cpa.3160390103. |
show all references
References:
[1] |
T. Benjamin, The stability of solitary waves,, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153.
doi: 10.1098/rspa.1972.0074. |
[2] |
J. Bona, On the stability theory of solitary waves,, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363.
doi: 10.1098/rspa.1975.0106. |
[3] |
J. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type,, Proc. Roy. Soc. (London) Ser. A, 411 (1987), 395.
doi: 10.1098/rspa.1987.0073. |
[4] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 107.
doi: 10.1007/BF01896020. |
[5] |
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation,, Geom. Funct. Anal., 3 (1993), 209.
doi: 10.1007/BF01895688. |
[6] |
M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonoical defocusing equations,, Amer. J. Math, 125 (2003), 1235.
doi: 10.1353/ajm.2003.0040. |
[7] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index,, Electron. J. Differential Equations, 26 (2001).
|
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm,, Commun. Pure. Appl. Anal., 2 (2003), 33.
|
[9] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm,, Discrete Contin. Dyn. Syst., 9 (2003), 31.
|
[10] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$,, J. Amer. Math. Soc., 16 (2003), 705.
doi: 10.1090/S0894-0347-03-00421-1. |
[11] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications,, J. Funct. Anal., 211 (2004), 173.
doi: 10.1016/S0022-1236(03)00218-0. |
[12] |
J. Colliander and P. Raphaël, Rough blowup solutions to the $L^2$ critical NLS,, Math. Ann., 345 (2009), 307.
doi: 10.1007/s00208-009-0355-3. |
[13] |
L. Farah, Global rough solutions to the critical generalized KdV equation,, J. Differential Equations, 249 (2010), 1968.
doi: 10.1016/j.jde.2010.05.010. |
[14] |
L. Farah, F. Linares and A. Pastor, The supercritical generalized KdV equation: global well-posedness in the energy space and below,, Math. Res. Lett., 18 (2011), 357.
|
[15] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I.,, J. Funct. Anal., 74 (1987), 160.
doi: 10.1016/0022-1236(87)90044-9. |
[16] |
M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II.,, J. Funct. Anal., 94 (1990), 308.
doi: 10.1016/0022-1236(90)90016-E. |
[17] |
A. Grünrock, A bilinear Airy-estimate with application to gKdV-3,, Differential Integral Equations, 18 (2005), 1333.
|
[18] |
A. Grünrock, M. Panthee and J. Silva, A remark on global well-posedness below $L^2$ for the GKDV-3 equation,, Differential Integral Equations, 20 (2007), 1229.
|
[19] |
Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$,, J. Math. Pures. Appl., 91 (2009), 583.
doi: 10.1016/j.matpur.2009.01.012. |
[20] |
C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33.
doi: 10.1512/iumj.1991.40.40003. |
[21] |
C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,, Comm. Pure Appl. Math., 46 (1993), 527.
doi: 10.1002/cpa.3160460405. |
[22] |
C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation,, J. Amer. Math. Soc., 9 (1996), 573.
doi: 10.1090/S0894-0347-96-00200-7. |
[23] |
N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity,, Differential Integral Equations, 22 (2009), 447.
|
[24] |
H. Koch and J. Marzuola, Small data scattering and soliton stability in $\dot H^{-1/6}$ for the quartic KdV equation,, Anal. PDE, 5 (2012), 145.
doi: 10.2140/apde.2012.5.145. |
[25] |
D. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and a new typ of long stationary waves,, Philos. Mag., 39 (1895), 422. Google Scholar |
[26] |
Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations,, Arch. Ration. Mech. Anal., 157 (2001), 219.
doi: 10.1007/s002050100138. |
[27] |
Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation,, Geom. Funct. Anal., 11 (2001), 74.
doi: 10.1007/PL00001673. |
[28] |
Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation,, J. Amer. Math. Soc., 15 (2002), 617.
doi: 10.1090/S0894-0347-02-00392-2. |
[29] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited,, Nonlinearity, 18 (2005), 391.
doi: 10.1088/0951-7715/18/1/004. |
[30] |
Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity,, Math. Ann., 341 (2008), 391.
doi: 10.1007/s00208-007-0194-z. |
[31] |
Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation I: dynamics near the soliton, preprint,, \arXiv{1204.4625}, (). Google Scholar |
[32] |
Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation II: minimal mass dynamics, preprint,, \arXiv{1204.4624}, (). Google Scholar |
[33] |
Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation III: exotic regimes, preprint,, \arXiv{1209.2510}, (). Google Scholar |
[34] |
F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation,, J. Amer. Math. Soc., 14 (2001), 555.
doi: 10.1090/S0894-0347-01-00369-1. |
[35] |
F. Merle and L. Vega, $L^2$ stability of solitons for the KdV equation,, Int. Math. Res. Not., 13 (2003), 735.
doi: 10.1155/S1073792803208060. |
[36] |
C. Miao, S. Shao, Y. Wu and G. Xu, The low regularity global solutions for the critical generalized KdV equation,, Dyn. Partial Differ. Equ., 7 (2010), 265.
|
[37] |
R. Miura, The Korteweg-de Vries equation: a survey of results,, SIAM Rev., 18 (1976), 412.
doi: 10.1137/1018076. |
[38] |
S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation,, Commun. Pure. Appl. Anal., 2 (2003), 277.
doi: 10.3934/cpaa.2003.2.277. |
[39] |
T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations,, Amer. J. Math., 123 (2001), 839.
doi: 10.1353/ajm.2001.0035. |
[40] |
T. Tao, Scattering for the quartic generalized Korteweg-de Vries equation,, J. Differential Equations, 232 (2007), 623.
doi: 10.1016/j.jde.2006.07.019. |
[41] |
M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.
|
[42] |
M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure Appl. Math, 39 (1986), 51.
doi: 10.1002/cpa.3160390103. |
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