Well-posedness and global existence for the Novikov equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 3, pp. 707-727.

In this paper, we mainly study the Cauchy problem of the Novikov equation. We first establish the local well-posedness and give the precise blow-up scenario for the equation. Then we show that the equation has smooth solutions which exist globally in time. Finally we prove that peakon solutions to the equation are global weak solutions.

Publié le :
Classification : 35G25, 35L05
Wu, Xinglong 1 ; Yin, Zhaoyang 1

1 Department of Mathematics Sun Yat-sen University 510275 Guangzhou, China
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Wu, Xinglong; Yin, Zhaoyang. Well-posedness and global existence  for the Novikov equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 3, pp. 707-727. http://archive.numdam.org/item/ASNSP_2012_5_11_3_707_0/

[1] H. Amann, “Ordinary Differential Equation”, W. de Gruyter, Berlin, 1990. | MR | Zbl

[2] R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems 15 (1999), 1–4. | MR | Zbl

[3] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters 71 (1993), 1661–1664. | MR | Zbl

[4] R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech. 31 (1994), 1–33. | Zbl

[5] G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numberical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal. 28 (2008), 80–105. | MR | Zbl

[6] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London, Ser. A, Math. Phys. Eng. Sci. 457 (2001), 953–970. | MR | Zbl

[7] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equation, Acta Math. 181 (1998), 229–243. | MR | Zbl

[8] A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential Integral Equations 15 (2002), 1061–1072. | MR | Zbl

[9] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solution, Theoret. Math. Phys. 133 (2002), 1463–1474. | MR

[10] J. Dieudonne, “Foundations of Morden Analysis”, Academic Press, New York, 1969. | MR | Zbl

[11] 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. | MR | Zbl

[12] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D 4 (1981), 47–66. | MR | Zbl

[13] A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor. 41 (2008), 372002. | MR | Zbl

[14] T. Kato, Quasi-linear equation of evolution, with applications to partical differential equations, In: “Spectral Theorey and Differential Equation”, Lecture Notes in Math., Vol. 488, Spring-Verlag, Berlin, 1975, 25–70. | MR | Zbl

[15] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math. 28 (1979), 89–99. | EuDML | MR | Zbl

[16] T. Kato, On the Cauchy problem for the generalized Korteweg-de Vries equation, In: “Studies in Applied Mathematics”, Adv. Math. Suppl. Stu., Vol. 8, Academic Press, New York, 1983, 93–128. | MR | Zbl

[17] T. Kato and G. Ponce, Commutator estimation and the Euler and Navier–Stokes equation, Comm. Pure Appl. Math. 41 (1998) 891–907. | MR | Zbl

[18] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci. 17 (2007), 169–198. | MR | Zbl

[19] V. S. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A: Math. Gen. 35 (2002), 4775–4790. | MR | Zbl

[20] A. Pazy, “Semigroup of Linear Operators and Applications to Partial Differential Equations”, Spring Verlag, New York, 1986. | MR | Zbl

[21] V. S. Novikov, Generalizations of the Camassa–Holm equation, J. Phys. A: Math. Theor. 42 (2009), 342002. | MR | Zbl

[22] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutins, Illinois J. Math. 47 (2003), 649–666. | MR | Zbl

[23] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal. 212 (2004), 182–194. | MR | Zbl