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Stabilizability in optimization problems with unbounded data
Orbital and asymptotic stability of a train of peakons for the Novikov equation
Institut Denis Poisson, Université de Tours, Université d'Orleans, CNRS, Parc Grandmont 37200, Tours, France |
The Novikov equation is an integrable Camassa-Holm type equation with cubic nonlinearity. One of the most important features of this equation is the existence of peakon and multi-peakon solutions, i.e. peaked traveling waves behaving as solitons. This paper aims to prove both, the orbital and asymptotic stability of peakon trains solutions, i.e. multi-peakon solutions such that their initial configuration is increasingly ordered. Furthermore, we give an improvement of the orbital stability of a single peakon so that we can drop the non-negativity hypothesis on the momentum density. The same result also holds for the orbital stability for peakon trains, i.e. in this latter case we can also avoid assuming non-negativity of the initial momentum density. Finally, as a corollary of these results together with some asymptotic formulas for the position and momenta vectors for multi-peakon solutions, we obtain the orbital and asymptotic stability for initially not well-ordered multipeakons.
References:
[1] |
B. Alvarez-Samaniego and D. Lannes,
Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.
doi: 10.1007/s00222-007-0088-4. |
[2] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[3] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[4] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
R. M. Chen, F. Guo, Y. Liu and C. Qu,
Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.
doi: 10.1016/j.jfa.2016.01.017. |
[6] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[7] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.
|
[8] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[9] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[10] |
A. Constantin and L. Molinet,
Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89.
doi: 10.1016/S0167-2789(01)00298-6. |
[11] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[12] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
[13] |
A. Degasperis, D. D. Kholm and A. N. I. Khone,
A new integrable equation with peakon solution, Theor. Math. Phys., 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[14] |
A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, (1999), 23–37. |
[15] |
K. El Dika and L. Molinet,
Exponential decay of $H^1$-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2313-2331.
doi: 10.1098/rsta.2007.2011. |
[16] |
K. El Dika and L. Molinet,
Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1517-1532.
doi: 10.1016/j.anihpc.2009.02.002. |
[17] |
B. Fuchssteiner and A. S. Fokas,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[18] |
A. A. Himonas and C. Holliman,
The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.
doi: 10.1088/0951-7715/25/2/449. |
[19] |
H. Holden and X. Raynaud,
Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differ. Equ., 4 (2007), 39-64.
doi: 10.1142/S0219891607001045. |
[20] |
A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002, 10 pp.
doi: 10.1088/1751-8113/41/37/372002. |
[21] |
A. N. W. Hone, H. Lundmark and J. Szmigielski,
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.
doi: 10.4310/DPDE.2009.v6.n3.a3. |
[22] |
R. S. Johnson,
Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[23] |
Z. Lin and Y. Liu,
Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
doi: 10.1002/cpa.20239. |
[24] |
X. Liu, Y. Liu and C. Qu,
Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.
doi: 10.1016/j.matpur.2013.05.007. |
[25] |
Y. Martel, F. Merle and T.-P. Tsai,
Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys., 231 (2002), 347-373.
doi: 10.1007/s00220-002-0723-2. |
[26] |
A. V. Mikhailov and V. S. Novikov,
Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[27] |
L. Molinet,
A Liouville property with application to asymptotic stability for the Camassa-Holm equation, Arch. Ration. Mech. Anal., 230 (2018), 185-230.
doi: 10.1007/s00205-018-1243-3. |
[28] |
L. Molinet,
A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon, Nonlinear Anal. Real World Appl., 50 (2019), 675-705.
doi: 10.1016/j.nonrwa.2019.06.004. |
[29] |
L. Molinet, Asymptotic stability for some non positive perturbations of the Camassa-Holm peakon with application to the antipeakon-peakon profile, arXiv: 1804.06230v2 Google Scholar |
[30] |
L. Molinet,
On well-posedness results for Camassa-Holm equation on the line: A survey., J. Nonlinear Math. Phys., 11 (2004), 521-533.
doi: 10.2991/jnmp.2004.11.4.8. |
[31] |
V. Novikov, Generalizations of the Camassa-Holm type equation, J. Phys. A, 42 (2009), 342002, 14 pp.
doi: 10.1088/1751-8113/42/34/342002. |
[32] |
J. M. Palacios,
Asymptotic stability of peakons for the Novikov equation, J. Differential Equations, 269 (2020), 7750-7791.
doi: 10.1016/j.jde.2020.05.039. |
[33] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
doi: 10.12775/TMNA.1996.001. |
[34] |
X. Wu and Z. Yin, Global weak solutions for the Novikov equation., J. Phys. A, 44 (2011), 055202, 17 pp.
doi: 10.1088/1751-8113/44/5/055202. |
show all references
References:
[1] |
B. Alvarez-Samaniego and D. Lannes,
Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541.
doi: 10.1007/s00222-007-0088-4. |
[2] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[3] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[4] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
R. M. Chen, F. Guo, Y. Liu and C. Qu,
Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374.
doi: 10.1016/j.jfa.2016.01.017. |
[6] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[7] |
A. Constantin and J. Escher,
Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.
|
[8] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[9] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[10] |
A. Constantin and L. Molinet,
Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89.
doi: 10.1016/S0167-2789(01)00298-6. |
[11] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[12] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
[13] |
A. Degasperis, D. D. Kholm and A. N. I. Khone,
A new integrable equation with peakon solution, Theor. Math. Phys., 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[14] |
A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, (1999), 23–37. |
[15] |
K. El Dika and L. Molinet,
Exponential decay of $H^1$-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2313-2331.
doi: 10.1098/rsta.2007.2011. |
[16] |
K. El Dika and L. Molinet,
Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1517-1532.
doi: 10.1016/j.anihpc.2009.02.002. |
[17] |
B. Fuchssteiner and A. S. Fokas,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[18] |
A. A. Himonas and C. Holliman,
The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479.
doi: 10.1088/0951-7715/25/2/449. |
[19] |
H. Holden and X. Raynaud,
Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differ. Equ., 4 (2007), 39-64.
doi: 10.1142/S0219891607001045. |
[20] |
A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002, 10 pp.
doi: 10.1088/1751-8113/41/37/372002. |
[21] |
A. N. W. Hone, H. Lundmark and J. Szmigielski,
Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289.
doi: 10.4310/DPDE.2009.v6.n3.a3. |
[22] |
R. S. Johnson,
Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.
doi: 10.1017/S0022112001007224. |
[23] |
Z. Lin and Y. Liu,
Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.
doi: 10.1002/cpa.20239. |
[24] |
X. Liu, Y. Liu and C. Qu,
Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187.
doi: 10.1016/j.matpur.2013.05.007. |
[25] |
Y. Martel, F. Merle and T.-P. Tsai,
Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys., 231 (2002), 347-373.
doi: 10.1007/s00220-002-0723-2. |
[26] |
A. V. Mikhailov and V. S. Novikov,
Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[27] |
L. Molinet,
A Liouville property with application to asymptotic stability for the Camassa-Holm equation, Arch. Ration. Mech. Anal., 230 (2018), 185-230.
doi: 10.1007/s00205-018-1243-3. |
[28] |
L. Molinet,
A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon, Nonlinear Anal. Real World Appl., 50 (2019), 675-705.
doi: 10.1016/j.nonrwa.2019.06.004. |
[29] |
L. Molinet, Asymptotic stability for some non positive perturbations of the Camassa-Holm peakon with application to the antipeakon-peakon profile, arXiv: 1804.06230v2 Google Scholar |
[30] |
L. Molinet,
On well-posedness results for Camassa-Holm equation on the line: A survey., J. Nonlinear Math. Phys., 11 (2004), 521-533.
doi: 10.2991/jnmp.2004.11.4.8. |
[31] |
V. Novikov, Generalizations of the Camassa-Holm type equation, J. Phys. A, 42 (2009), 342002, 14 pp.
doi: 10.1088/1751-8113/42/34/342002. |
[32] |
J. M. Palacios,
Asymptotic stability of peakons for the Novikov equation, J. Differential Equations, 269 (2020), 7750-7791.
doi: 10.1016/j.jde.2020.05.039. |
[33] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
doi: 10.12775/TMNA.1996.001. |
[34] |
X. Wu and Z. Yin, Global weak solutions for the Novikov equation., J. Phys. A, 44 (2011), 055202, 17 pp.
doi: 10.1088/1751-8113/44/5/055202. |
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