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The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation. (English) Zbl 1187.35179

Summary: A generalization of the Camassa-Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space \(H^s(R)\) with \(s> \frac 3 2\) is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space \(H^s\) with \(1< s \leqslant \frac{3}{2}\) is developed.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q51 Soliton equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35D30 Weak solutions to PDEs
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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