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January  2014, 13(1): 389-418. doi: 10.3934/cpaa.2014.13.389

Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations

Brian Pigott 1,

1. 

Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, United States

Received  April 2013 Revised  May 2013 Published  July 2013

We prove polynomial-in-time upper bounds for the orbital instability of solitons for subcritical generalized Korteweg-de Vries equations in $H_x^s R$ with $s < 1$. By combining coercivity estimates of Weinstein with the $I$-method as developed by Colliander, Keel, Staffilani, Takaoka, and Tao, we construct a modified energy functional which is shown to be almost conserved while providing us with an estimate of the deviation of the solution from the ground state curve. The iteration of the almost conservation law for the modified energy functional over time intervals of uniform length yields the polynomial upper bound.
Keywords: KdV equation, orbital stablity, growth of Sobolev norms, solitons..
Mathematics Subject Classification: Primary: 35Q53, 42B35; Secondary: 37K1.
Citation: Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389
References:
[1]

T. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.  Google Scholar

[2]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363-374. doi: 10.1098/rspa.1975.0106.  Google Scholar

[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), 395-412. doi: 10.1098/rspa.1987.0073.  Google Scholar

[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), 107-156. doi: 10.1007/BF01896020.  Google Scholar

[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) 209-262. doi: 10.1007/BF01895688.  Google Scholar

[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), 1235-1293. doi: 10.1353/ajm.2003.0040.  Google Scholar

[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), 7 pp. (electronic).  Google Scholar

[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), 33-50.  Google Scholar

[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), 31-54.  Google Scholar

[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), 705-749. doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[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), 173-218. doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[12]

J. Colliander and P. Raphaël, Rough blowup solutions to the $L^2$ critical NLS, Math. Ann., 345 (2009), 307-366. doi: 10.1007/s00208-009-0355-3.  Google Scholar

[13]

L. Farah, Global rough solutions to the critical generalized KdV equation, J. Differential Equations, 249 (2010), 1968-1985. doi: 10.1016/j.jde.2010.05.010.  Google Scholar

[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), 357-377.  Google Scholar

[15]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I., J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[16]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II., J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[17]

A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[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), 1229-1236.  Google Scholar

[19]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$, J. Math. Pures. Appl., 91 (2009), 583-597. doi: 10.1016/j.matpur.2009.01.012.  Google Scholar

[20]

C. 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

[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), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[22]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[23]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.  Google Scholar

[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), 145-198. doi: 10.2140/apde.2012.5.145.  Google Scholar

[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), 422-443. Google Scholar

[26]

Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219-254. doi: 10.1007/s002050100138.  Google Scholar

[27]

Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123. doi: 10.1007/PL00001673.  Google Scholar

[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), 617-664. doi: 10.1090/S0894-0347-02-00392-2.  Google Scholar

[29]

Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited, Nonlinearity, 18 (2005), 391-427. doi: 10.1088/0951-7715/18/1/004.  Google Scholar

[30]

Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391-427. doi: 10.1007/s00208-007-0194-z.  Google Scholar

[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), 555-578. doi: 10.1090/S0894-0347-01-00369-1.  Google Scholar

[35]

F. Merle and L. Vega, $L^2$ stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753. doi: 10.1155/S1073792803208060.  Google Scholar

[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), 265-288.  Google Scholar

[37]

R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev., 18 (1976), 412-459. doi: 10.1137/1018076.  Google Scholar

[38]

S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation, Commun. Pure. Appl. Anal., 2 (2003), 277-296. doi: 10.3934/cpaa.2003.2.277.  Google Scholar

[39]

T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.  Google Scholar

[40]

T. Tao, Scattering for the quartic generalized Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[41]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar

[42]

M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math, 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

show all references

References:
[1]

T. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A, 328 (1972), 153-183. doi: 10.1098/rspa.1972.0074.  Google Scholar

[2]

J. Bona, On the stability theory of solitary waves, Proc. Roy. Soc. (London) Ser. A, 344 (1975), 363-374. doi: 10.1098/rspa.1975.0106.  Google Scholar

[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), 395-412. doi: 10.1098/rspa.1987.0073.  Google Scholar

[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), 107-156. doi: 10.1007/BF01896020.  Google Scholar

[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) 209-262. doi: 10.1007/BF01895688.  Google Scholar

[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), 1235-1293. doi: 10.1353/ajm.2003.0040.  Google Scholar

[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), 7 pp. (electronic).  Google Scholar

[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), 33-50.  Google Scholar

[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), 31-54.  Google Scholar

[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), 705-749. doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[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), 173-218. doi: 10.1016/S0022-1236(03)00218-0.  Google Scholar

[12]

J. Colliander and P. Raphaël, Rough blowup solutions to the $L^2$ critical NLS, Math. Ann., 345 (2009), 307-366. doi: 10.1007/s00208-009-0355-3.  Google Scholar

[13]

L. Farah, Global rough solutions to the critical generalized KdV equation, J. Differential Equations, 249 (2010), 1968-1985. doi: 10.1016/j.jde.2010.05.010.  Google Scholar

[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), 357-377.  Google Scholar

[15]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I., J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[16]

M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II., J. Funct. Anal., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[17]

A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[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), 1229-1236.  Google Scholar

[19]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(R)$, J. Math. Pures. Appl., 91 (2009), 583-597. doi: 10.1016/j.matpur.2009.01.012.  Google Scholar

[20]

C. 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

[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), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[22]

C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[23]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.  Google Scholar

[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), 145-198. doi: 10.2140/apde.2012.5.145.  Google Scholar

[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), 422-443. Google Scholar

[26]

Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal., 157 (2001), 219-254. doi: 10.1007/s002050100138.  Google Scholar

[27]

Y. Martel and F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal., 11 (2001), 74-123. doi: 10.1007/PL00001673.  Google Scholar

[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), 617-664. doi: 10.1090/S0894-0347-02-00392-2.  Google Scholar

[29]

Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations, revisited, Nonlinearity, 18 (2005), 391-427. doi: 10.1088/0951-7715/18/1/004.  Google Scholar

[30]

Y. Martel and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann., 341 (2008), 391-427. doi: 10.1007/s00208-007-0194-z.  Google Scholar

[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), 555-578. doi: 10.1090/S0894-0347-01-00369-1.  Google Scholar

[35]

F. Merle and L. Vega, $L^2$ stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753. doi: 10.1155/S1073792803208060.  Google Scholar

[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), 265-288.  Google Scholar

[37]

R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev., 18 (1976), 412-459. doi: 10.1137/1018076.  Google Scholar

[38]

S. Raynor and G. Staffilani, Low regularity stability of solitons for the KdV equation, Commun. Pure. Appl. Anal., 2 (2003), 277-296. doi: 10.3934/cpaa.2003.2.277.  Google Scholar

[39]

T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.  Google Scholar

[40]

T. Tao, Scattering for the quartic generalized Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[41]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.   Google Scholar

[42]

M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math, 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.  Google Scholar

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