The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H 1 ( R )

Dissipative Solutions for the Camassa–holm Equation

has been extensively studied since the first systematic analysis in [5, 6]. Part of the attraction is the surprising complexity of the equation and its deep and nontrivial properties. To list a few of its peculiarities: The Camassa–Holm equation has a bi-Hamiltonian structure [16], it is completely integrable [5], and it has infinitely many conserved quantities [5]. Here we study the equation w...

Global Dissipative Solutions of the Camassa-Holm Equation

This paper is concerned with the global existence of dissipative solutions to the Camassa-Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as an O.D.E. in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation along a suitable cone of directions. For a give...

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...

Time-Periodic Solution of the Weakly Dissipative Camassa-Holm Equation

The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

and Applied Analysis 3 2. Main Results Firstly, we give some notation. The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ ×R is denoted byC∞ 0 . L L R 1 ≤ p < ∞ is the space of all measurable functions h such that ‖h‖pLp ∫ R |h t, x |pdx < ∞. We define L∞ L∞ R with the standard norm ‖h‖L∞ infm e 0supx∈R\e|h t, x |. For any real number s, H H R denotes the S...