Global well-posedness for the viscous Camassa–Holm equation. (English) Zbl 1160.35064
The authors consider the initial value problem of Camassa-Holm equation with viscosity. They established global solution for the IVP with \(u_0\in L^2\). Their result is sharper than those in [M. Stanislavova and A. Stefanov, Discrete Contin. Dyn. Syst. 18, No. 1, 159–186 (2007; Zbl 1387.35502)] regarding the viscous version of the immediate effect in \(H^1_{x}(\mathbb R)\).
Reviewer: Meng Wang (Hangzhou)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Citations:
Zbl 1387.35502References:
| [1] | Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 11, 1661-1664 (1993) · Zbl 0972.35521 |
| [2] | Camassa, R.; Holm, D.; Hyman, J. M., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1994) · Zbl 0808.76011 |
| [3] | Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Ann. Sc. Norm Sup. Pisa Cl. Sci. (4), 26, 2, 303-328 (1998) · Zbl 0918.35005 |
| [4] | Constantin, A.; Molinet, L., Global weak solutions for a shallow water equation, Comm. Math. Phys., 211, 1, 45-61 (2000) · Zbl 1002.35101 |
| [5] | Constantin, A.; Strauss, W., Stability of peakons, Comm. Pure Appl. Math., 53, 5, 603-610 (2000) · Zbl 1049.35149 |
| [6] | Himonas, A.; Misiolek, G., The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations, 14, 821-831 (2001) · Zbl 1009.35075 |
| [7] | Li, Y. A.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119 |
| [8] | M. Stanislavova, A. Stefanov, Global well-posedness for the Camassa-Holm equation in the energy norm, preprint; M. Stanislavova, A. Stefanov, Global well-posedness for the Camassa-Holm equation in the energy norm, preprint · Zbl 1175.35129 |
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