Global well-posedness for the periodic Novikov equation with cubic nonlinearity

References

• Hone ANW, Wang J. Integrable peakon equations with cubic nonlinearity. J. Phys. A: Math. Theor. 2008;41:10 p.
   Web of Science ®Google Scholar
• Mikhailov VS, Novikov VS. Perturbative symmetry approach. J. Phys. A: Math. Gen. 2002;35:4775–4790.
   Google Scholar
• Novikov VS. Generalizations of the Camassa--Holm equation. J. Phys. A: Math. Theor. 2009;42:14 p.
   Web of Science ®Google Scholar
• Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993;71:1661–1664.
   PubMed Web of Science ®Google Scholar
• Fokas A, Fuchssteiner B. Symplectic structures, their Bäcklund transformation and hereditary symmetries. Physica D. 1981;4:47–66.
   Web of Science ®Google Scholar
• Constantin A. On the scattering problem for the Camassa--Holm equation. Proc. R. Soc. London A. 2001;457:953–970.
   Web of Science ®Google Scholar
• Camassa R, Holm D, Hyman J. An integrable shallow water equation. Adv. Appl. Mech. 1994;31:1–33.
   Google Scholar
• Constantin A. Finite propagation speed for the Camassa--Holm equation. J. Math. Phys. 2005;46:023506, 4 pp.
   Web of Science ®Google Scholar
• Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equation. Acta Math. 1998;181:229–243.
   Web of Science ®Google Scholar
• Constantin A, Escher J. On the blow-up rate and the blow--up set of breaking waves for a shallow water equation. Math. Z. 2000;233:75–91.
   Web of Science ®Google Scholar
• Degasperis A, Procesi M. Asymptotic integrability. In: Degasperis A, Gaeta G, editors. Symmetry and perturbation theory. New Jersey (NJ): World Scientific; 1999. p. 23–37.
   Google Scholar
• Holm DD, Staley MF. Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2003;2:323–380.
   Web of Science ®Google Scholar
• Degasperis A, Holm DD, Hone ANW. A new integrable equation with peakon solution. Theoret. Math. Phys. 2002;133:1463–1474.
   Web of Science ®Google Scholar
• Yin Z. Global weak solutions to a new periodic integrable equation with peakon solutions. J. Funct. Anal. 2004;212:182–194.
   Web of Science ®Google Scholar
• Coclite GM, Karlsen KH, Risebro NH. Numberical schemes for computing discontinuous solutions of the Degasperis--Procesi equation. IMA J. Numer. Anal. 2008;28:80–105.
   Web of Science ®Google Scholar
• Lundmark H. Formation and dynamics of shock waves in the Degasperis--Procesi equation. J. Nonlinear Sci. 2007;17:169–198.
   Web of Science ®Google Scholar
• Escher J, Liu Y, Yin Z. Shock waves and blow-up phenomena for the periodic Degasperis--Procesi equation. Indiana Univ. Math. J. 2007;56:87–117.
   Web of Science ®Google Scholar
• Hone ANW, Lundmark H, Szmigielski J. Explicit multipeakon solutions of Novikovs cubically nonlinear integrable Camassa-Holm type equation. Dynam. Part. Differ. Equat. 2009;6:253–289.
   Web of Science ®Google Scholar
• Wu X, Yin Z. Well-posedness and global existence for the Novikov equation. Annali Sc. Norm. Sup. Pisa: XI;2012:707–727.
   Google Scholar
• Tiglay F. The periodic Cauchy problem for Novikov’s equation. Int. Math. Res. Not. 2011;20:4633–4648.
   Google Scholar
• Wu X, Yin Z. Global weak solutions for the Novikov equation. J. Phys. A: Math. Theor. 2011;44:055202, 17 pp.
   Web of Science ®Google Scholar
• Constantin A, Molinet L. Global weak solutions for a shallow water equation. Comm. Math. Phys. 2000;211:45–61.
   Web of Science ®Google Scholar
• Kato T. Quasi-linear equation of evolution, with applications to partical differential equations. In: Spectral theory and differential equation. Vol. 488, Lecture notes in mathematics. Spring-Verlag: Berlin; 1975. p. 25–70.
   Google Scholar
• Kato T, Ponce G. Commutator estimation and the Euler and Navier--Stokes equation. Commun. Pure Appl. Math. 1998;41:891–907.
   Web of Science ®Google Scholar
• Constantin A, Molinet L. The initial value problem for a generalized Boussinesq equation. Differ. Int. Equat. 2002;15:1061–1072.
   Google Scholar
• Constantin A, Escher J. Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 1998;51:475–504.
   Web of Science ®Google Scholar
• Amann H. Ordinary differential equation. Berlin: W. de Gruyter; 1990.
   Google Scholar
• Yin Z. On the Cauchy problem for an integrable equation with peakon solutins. Illinois J. Math. 2003;47:649–666.
   Web of Science ®Google Scholar
• Malek J, Necas J, Rokyta M, Ruzicka M. Weak and measure-valued solutions to evolutionary PDEs. London: Chapman & Hall; 1996.
   Google Scholar
• Natanson IP. Theory of functions of a real variable. New York (NY): F. Ungar Publ. Co.; 1998.
   Google Scholar