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Singular periodic solutions for the p-laplacian ina punctured domain
Long-term stability for kdv solitons in weighted Hs spaces
| 1. | Department of Mathematics Wofford College, 429 North Church Street, Spartanburg, SC 29303 |
| 2. | Department of Mathematics and Statistics, Wake Forest University, P.O. Box 7388, Winston Salem, NC 27109 |
In this work, we consider the stability of solitons for the KdV equation below the energy space, using spatially-exponentially-weighted norms. Using a combination of the I-method and spectral analysis following Pego and Weinstein, we are able to show that, in the exponentially weighted space, the perturbation of a soliton decays exponentially for arbitrarily long times. The finite time restriction is due to a lack of global control of the unweighted perturbation.
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] |
T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 |
| [5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Fund. Anal., 211 (2004), 173-218. |
| [6] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrodinger equation below the energy norm, Commun. Pure. Appl. Anal., 2 (2003), 33-50. |
| [7] |
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. |
| [8] |
Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901. |
| [9] |
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. |
| [10] |
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. |
| [11] |
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. |
| [12] |
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. |
| [13] |
F. Merle and L. Vega, L2 stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753.
doi: 10.1155/S1073792803208060. |
| [14] |
T. Mizumachi, Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., 32 (2001), 1050-1080. |
| [15] |
T. Mizumachi and N. Tzvetkov, L2-stability of solitary waves for the KdV equation via Pego and Weinstein's method, preprint, arXiv: 1403.5321. |
| [16] |
L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776. |
| [17] |
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005. |
| [18] |
R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349. |
| [19] |
B. Pigott, Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations, Commun. Pure. Appl. Anal., 13 (2014), 389-418.
doi: 10.3934/cpaa.2014.13.389. |
| [20] |
B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). |
| [21] |
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. |
| [22] |
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. |
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. |
| [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] |
T. Buckmaster and H. Koch, The Korteweg-de Vries equation at H-1 regularity, arXiv: 1112.4657 |
| [5] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Fund. Anal., 211 (2004), 173-218. |
| [6] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the instability of the nonlinear Schrodinger equation below the energy norm, Commun. Pure. Appl. Anal., 2 (2003), 33-50. |
| [7] |
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. |
| [8] |
Z. Guo and B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Differential Equations, 246 (2009), 3864-3901. |
| [9] |
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. |
| [10] |
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. |
| [11] |
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. |
| [12] |
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. |
| [13] |
F. Merle and L. Vega, L2 stability of solitons for the KdV equation, Int. Math. Res. Not., 13 (2003), 735-753.
doi: 10.1155/S1073792803208060. |
| [14] |
T. Mizumachi, Large time asymptotics of solutions around solitary waves to the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., 32 (2001), 1050-1080. |
| [15] |
T. Mizumachi and N. Tzvetkov, L2-stability of solitary waves for the KdV equation via Pego and Weinstein's method, preprint, arXiv: 1403.5321. |
| [16] |
L. Molinet and F. Ribaud, The Cauchy problem for dissipative Korteweg de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J., 50 (2001), 1745-1776. |
| [17] |
L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, Int. Math. Res. Not., 37 (2002), 1979-2005. |
| [18] |
R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 2 (1994), 305-349. |
| [19] |
B. Pigott, Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations, Commun. Pure. Appl. Anal., 13 (2014), 389-418.
doi: 10.3934/cpaa.2014.13.389. |
| [20] |
B. Pigott and S. Raynor, Asymptotic stability for KdV solitons in weighted spaces via iteration, Submitted, (2013). |
| [21] |
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. |
| [22] |
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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