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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

 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.
Citation: Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure and 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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [31] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation I: dynamics near the soliton, preprint,, \arXiv{1204.4625}, (). [32] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation II: minimal mass dynamics, preprint,, \arXiv{1204.4624}, (). [33] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation III: exotic regimes, preprint,, \arXiv{1209.2510}, (). [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. [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. [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. [37] R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev., 18 (1976), 412-459. 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), 277-296. 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), 839-908. doi: 10.1353/ajm.2001.0035. [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. [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), 51-67. doi: 10.1002/cpa.3160390103.

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##### 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. [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. [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. [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. [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. [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. [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). [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. [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. [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. [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. [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. [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. [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. [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. [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. [17] A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential Integral Equations, 18 (2005), 1333-1339. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [31] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation I: dynamics near the soliton, preprint,, \arXiv{1204.4625}, (). [32] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation II: minimal mass dynamics, preprint,, \arXiv{1204.4624}, (). [33] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical gKdV equation III: exotic regimes, preprint,, \arXiv{1209.2510}, (). [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. [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. [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. [37] R. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev., 18 (1976), 412-459. 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), 277-296. 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), 839-908. doi: 10.1353/ajm.2001.0035. [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. [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), 51-67. doi: 10.1002/cpa.3160390103.
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