Stability of Peakons for the Degasperis-Procesi Equation

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provides us more information about stability via (1.10). For the CH equation, even if the orbital stability is proved by a simpler construction [11], our approach can also give the additional stability information. More specifically, for the CH equation (1.3) with y 0 ≥ 0, by refining the integrals of [11, Lemma 2] to each monotonic interval of u, one can obtain F 3 (u) − 4 3 B3 n ≤ M 1 ( F2 (u) − 2A 2 n) , where F 2 and F 3 are defined in (1.7), and A n and B n in (3.12) with M i and m i being the maxima and minima of u, respectively. Hence, estimate (1.10) may be obtained by following the proof of Lemma 3.6. Acknowledgements The work of Zhiwu Lin is supported partly by the NSF grants DMS-0505460 and DMS-0707397. References [1] A. Bressan and A. Constantin, Global conservative solutions of the Camassa- Holm equation, preprint, www.math.ntnu.no/conservation/2005/016.html. [2] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661–1664. [3] G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis- Procesi equation, J. Funct. Anal., 233 (2006), 60–91. [4] G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, preprint. [5] A. Constantin, Global existence of solutions and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321–362. [6] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506, 4 pp. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303–328. [8] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229–243. [9] A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787–804. 19
[10] A. Constantin and L. Molinet, Obtital stability of solitary waves for a shallow water equation, Physica D, 157 (2001), 75–89. [11] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603–610. [12] A. Degasperis, D. D. Holm, and A. N. W. Hone, A New Integral Equation with Peakon Solutions, Theoretical and Mathematical Physics, 133 (2002), 1463- 1474. [13] A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory, edited by A. Degasperis and G. Gaeta, World Scientific (1999), 23–37. [14] H. R. Dullin, G. A. Gottwald, and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Letters, 87 (2001), 4501–4504. [15] J. Escher, Y. Liu, and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. [16] J. Escher, Y. Liu, and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117. [17] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47–66. [18] D. Henry, Infinite propagation speed for the Degasperis-Procesi equation. J. Math. Anal. Appl., 311 (2005), 755–759. [19] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst. (electronic), 2 (2003), 323–380. [20] D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE, Phys. Lett. A, 308 (2003), 437-444. [21] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63–82. [22] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25–70. [23] Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis- Procesi equation, Comm. Math. Phys., 267 (2006), 801-820. [24] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, to appear in J. Nonlinear Science, (2006). 20
Page 1 and 2: Stability of Peakons for the Degasp
Page 3 and 4: These three cases exhaust in the co
Page 5 and 6: Let us denote E 2 (u) = ‖u‖ 2 X
Page 7 and 8: introduce a new idea. By constructi
Page 9 and 10: Lemma 2.3 [23] Assume u 0 ∈ H s (
Page 11 and 12: 3 Proof of stability In this primar
Page 13 and 14: To estimate the first term, by inte
Page 15 and 16: □ Without changing the integral i
Page 17 and 18: A similar argument also shows that
Page 19: (iii) can be obtained from (i) and

Stability of Peakons for the Degasperis-Procesi Equation

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