Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation

Abstract

After some remarks on a possible zero-curvature formulation we first establish local well-posedness for the 2-component Camassa-Holm equation. Then precise blow-up scenarios for strong solutions to the system are derived. Finally we present two blow-up results for strong solutions to the system.

On the Cauchy problem for the two‐component b‐family system

“…The two‐component integrable CH system was first derived in and can be viewed as a model in the context of shallow water theory . Many recent works are devoted in studying this system (see, for instance, and references therein). With the presence of a linear shear flow and nonzero vorticity, the two‐component b ‐family system with any b ≠ − 1 followed by Ivanov's approach is recently derived in .…”

Blow-up Phenomena and Persistence Properties of Solutions to the Two-Component DGH Equation

“…The variable ( , ) describes the horizontal velocity of the fluid, and ( , ) denotes the horizontal deviation of the surface from equilibrium, all measured in dimensionless units. This model can be derived by Constantin and Ivanov's approach [25] from shallow water theory, which includes the two-component Camassa-Holm system [26][27][28][29][30][31][32] as its special case with = 1 and = 0 in (7). From a geometric point of view, (7) is the model for geodesic motion on the semidirect product Lie group of diffeomorphisms acting on densities, with respect to the 1 -norm of velocity and the 2 -norm on the density.…”

Initial Boundary Value Problem of the General Three-Component Camassa-Holm Shallow Water System on an Interval

“…They investigated the local wellposedness and blow-up solutions of system (1) by means of Kato's semigroup approach to nonlinear hyperbolic evolution equation and obtained a criterion and condition on the initial data guaranteeing the development of singularities in finite time for strong solutions of system (1) by energy estimates. Recently the initial boundary value problem for the system (1) has been established in [17]; moreover, the local well-posedness and blow-up phenomena for the coupled Camassa-Holm equation were also established in [16,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. In [33], Tian and Xu obtained the compact and bounded absorbing set and the existence of the global attractor for viscous system (1) with the periodic boundary condition in by uniform prior estimate.…”