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On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. (English) Zbl 1189.35254

Summary: Considered herein is a two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. A wave-breaking criterion for strong solutions is determined in the lowest Sobolev space \(H^s, s>\frac 3 2\) by using the localization analysis in the transport equation theory. Moreover, an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this Sobolev space \(H^s\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35R37 Moving boundary problems for PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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