Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system

Well-posedness and persistence properties for two-component higher order Camassa–Holm systems with fractional inertia operator

“…In [23], the authors shown that the solutions possess exponentially decaying profiles for large values of the…”

Continuity properties of the data-to-solution map for the two-component higher order Camassa–Holm system

“…In [11], Escher et al showed the local well-posedness of (1.2) under a geometrical framework, and studied the blow-up scenarios and global strong solutions of (1.2) on the circle. In [18], Guan et al considered the Cauchy problem of (1.2) in the Besov space and showed that the solutions have exponential decay if the initial data has exponential decay. When α = 0, b = 2 and κ = ±1, (1.2) becomes the two-component Camassa-Holm system, which admits Lax pair and bi-Hamiltonian structure, and thus is completely integrable [3].…”

The global Gevrey regularity and analyticity of a two-component shallow water system with higher-order inertia operators

“…where s > 1, a = 1 is a real parameter, α is a constant which represents the vorticity of underlying flow, and κ > 0 is an arbitrary real parameter. The system (1.1) is the generalization of the two component water wave system with s = 1, namely, m = (1 − ∂ 2 x )u = u − u xx (see [29], [40] and [52]). For α = 0, ρ ≡ 0, the system (1.1) becomes a family of one-component equations…”

“…In [29], the authors proved the local wellposedness of (1.1) with s = 1 by using a geometrical framework and they studied the blow-up scenarios and global strong solutions of (1.1) in the periodic case. In [40], Guan et al studied the local wellposedness of (1.1) with s = 1 on the line in supercritical Besov spaces, and several blow-up results and the persistence properties. In [52], He and Yin studied the local well-posedness of (1.1) with s = 1 in the critical Besov spaces on the line and the existence of analytic solutions of the system.…”