scholarly journals A CONTINUOUS INTERPOLATION BETWEEN CONSERVATIVE AND DISSIPATIVE SOLUTIONS FOR THE TWO-COMPONENT CAMASSA–HOLM SYSTEM

2015 ◽  
Vol 3 ◽  
Author(s):  
KATRIN GRUNERT ◽  
HELGE HOLDEN ◽  
XAVIER RAYNAUD
Keyword(s):  
Cauchy Problem ◽  
Wave Breaking ◽  
Solution Concept ◽  
Global Weak Solutions ◽  
Dissipative Solutions ◽  
Lagrangian Variables ◽  
Two Component ◽  
The Cauchy Problem ◽  
Eulerian Coordinates ◽  
Continuous Interpolation

We introduce a novel solution concept, denoted ${\it\alpha}$-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the two-component Camassa–Holm system on the line with vanishing asymptotics. All the ${\it\alpha}$-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

scholarly journals On the Cauchy problem for the two-component Euler–Poincaré equations

2014 ◽  
Vol 267 (8) ◽  
pp. 2698-2730 ◽  
Author(s):  
Renjun Duan ◽  
Zhaoyin Xiang
Keyword(s):  
Cauchy Problem ◽  
Two Component ◽  
The Cauchy Problem

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

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component periodic Camassa-Holm equation. We first establish the local well-posedness result. Then we derive the precise blow-up scenario and the blow-up rate for strong solutions to the system. Finally, we present two blow-up results for strong solutions to the system.

The Cauchy problem for an integrable two-component model with peakon solutions

2013 ◽  
Vol 93 (4) ◽  
pp. 840-858 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Component Model ◽  
Two Component ◽  
The Cauchy Problem

scholarly journals On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Hatice Taskesen ◽  
Necat Polat ◽  
Abdulkadir Ertaş
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Type Equation ◽  
Global Solutions ◽  
Initial Energy ◽  
Potential Well ◽  
Global Weak Solutions ◽  
Power Type ◽  
Initial Value ◽  
The Cauchy Problem

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq-type equation with power-type nonlinearity and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq-type equation for the supercritical initial energy case.

scholarly journals TheH1(R)Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Zhaowei Sheng ◽  
Shaoyong Lai ◽  
Yuan Ma ◽  
Xuanjun Luo
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Space Variable ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Dissipative Term ◽  
First Order ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Derivatives Of

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞)×R)∩L∞([0,∞);H1(R))provided that its initial valueu0(x)belongs to the spaceH1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

scholarly journals GLOBAL DISSIPATIVE SOLUTIONS OF THE CAMASSA–HOLM EQUATION

2007 ◽  
Vol 05 (01) ◽  
pp. 1-27 ◽  
Author(s):  
ALBERTO BRESSAN ◽  
ADRIAN CONSTANTIN
Keyword(s):  
Initial Data ◽  
Wave Breaking ◽  
Integral Transformation ◽  
Local Source ◽  
Dissipative Solutions ◽  
Holm Equation ◽  
Directional Variation ◽  
The Cauchy Problem ◽  
Non Local ◽  
New Variables

This paper is devoted to the continuation of 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 a semilinear hyperbolic system in an L∞space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data [Formula: see text], and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

The global weak solutions to the Cauchy problem of the generalized Novikov equation

2014 ◽  
Vol 94 (7) ◽  
pp. 1334-1354 ◽  
Author(s):  
Yongye Zhao ◽  
Yongsheng Li ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Global Weak Solutions ◽  
The Cauchy Problem ◽  
Novikov Equation
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