Blow-up and global solutions to a new integrable model with two components

Abstract

We will discuss a new integrable model which describes the motion of fluid. The present work is mainly concerned with global existence and blow-up phenomena which are largely due to the application of conservation laws for this integrable equations. Moreover, a new blow-up criterion for nonperiodic case is also established via the associated potential. Some interesting examples are also given to illustrate the application of our results. The precise blow-up rate is also investigated. Finally, we will emphasize the relations of classical Camassa-Holm equation and our model by analyzing the existence of global solutions.

Global Existence for the Generalized Two-Component Hunter–Saxton System

“…[47]). We observe that the rate of break-down we obtained is in accordance the one computed for the Camassa-Holm system [18].…”

“…This system, reading as (1.1) with m replaced by (1 − ∂ 2 xx )u, has recently been the object of intensive study (see [10,[16][17][18][19]28,33,34,51]). Constantin & Ivanov [10] derived the Camassa-Holm system from the Green-Naghdi equations, which themselves originate in the governing equations for water waves [23].…”

Blow‐up for the b‐family of equations

“…The local well‐posedness in H s , with for the Cauchy non periodic problem was elaborated in , and for the Cauchy periodic problem. With respect to blow‐up criteria on the line we refer to and, for the unit torus, to . For the existence globally of the solution, see .…”

“…In this paper, we rather focus on blow‐up criteria as well in estimates about the lifespan of the solutions. The blowup problem for the b ‐family of equations has been already treated, for example, in : in these references the condition on the initial datum u 0 leading to the blowup typically involves the computation of some global quantities (the Sobolev norm , or some other integral expressions of u 0 ). Motivated by the recent paper (where earlier blowup results for the Camassa—Holm equations were unified in a single theorem), we address the more subtle problem of finding a local‐in‐space blowup criterion for the b ‐family of equations, that is, a blowup condition involving only the properties of u 0 in a neighborhood of a single point .…”

Wave Breaking and Measure of Momentum Support for an Integrable Camassa‐Holm System with Two Components

“…The extended supersymmetric Camassa‐Holm equation was also presented by Z. Popowicz in . Stability of traveling wave solutions and wave breaking phenomenon are the subjects of . Very recently, smooth traveling wave solutions with was investigated by Mustafa in, the persistence properties of strong solutions were discussed in and the investigation of other related two‐component models can be found in .…”