Global Existence and Blow-Up for a Weakly Dissipative Modified Two-Component Camassa-Holm System

On the Blow-Up of Solutions of a Weakly Dissipative Modified Two-Component Periodic Camassa-Holm System

Wave-breaking for the Weakly Dissipative Modified Camassa-Holm Equation

The equation (1) arises from an intrinsic (arc-length preserving) invariant planar curve flow in Euclidean geometry and it can be regarded as a Euclidean-invariant version of the Camassa-Holm equation in [1]. It has the form of a modified Camassa-Holm equation with cubic nonlinearity. By Fuchssteiner [2] and Olver and Rosenau[3], it can be derived as a new integrable system by applying the gene...

Local Well-posedness and Blow-up Phenomenon for a Modified Two-component Camassa-Holm System in Besov Spaces

ρt + (uρ)x = 0, t > 0, x ∈ R, (1.1b) m(0, x) = m0(x), x ∈ R, (1.1c) ρ(0, x) = ρ0(x), x ∈ R, (1.1d) where (u0, ρ0) is given modified profile,m = u−uxx and ρ = (1−∂ x)(ρ−ρ0) ,g is a positive constant. For convenience, we let g = 1 in this paper. The modified two-component Camassa-Holm equation is written in terms of velocityu and locally averaged density ρ . With m = u− uxx ,ρ = γ − γxx and γ = ρ...

Wave-Breaking and Global Existence for a Generalized Two-Component Camassa-Holm System

In this paper we study a generalized two-component Camassa-Holm system which can be derived from the theory of shallow water waves moving over a linear shear flow. This new system also generalizes a class of dispersive waves in cylindrical compressible hyperelastic rods. We show that this new system can still exhibit the wave-breaking phenomenon. We also determine the exact blow-up rate of such...

Global Dissipative Solutions for the Two-component Camassa-holm Shallow Water System

This article presents a continuous semigroup of globally defined weak dissipative solutions for the two-component Camassa-Holm system. Such solutions are established by using a new approach based on characteristics a set of new variables overcoming the difficulties inherent in multi-component systems.