The Cauchy problem for generalized fractional Camassa–Holm equation in Besov space

2021 ◽  
Author(s):  
Lei Mao ◽  
Hongjun Gao
Keyword(s):  
Cauchy Problem ◽  
Besov Space ◽  
Holm Equation ◽  
The Cauchy Problem

scholarly journals The Cauchy problem for the generalized Camassa–Holm equation in Besov space

2014 ◽  
Vol 256 (8) ◽  
pp. 2876-2901 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Yimin Zhang
Keyword(s):  
Cauchy Problem ◽  
Besov Space ◽  
Holm Equation ◽  
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.

scholarly journals A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION

2020 ◽  
Vol 8 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
KATRIN GRUNERT ◽  
HELGE HOLDEN
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wasserstein Metric ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Lipschitz Metric

We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

On the Cauchy problem for a modified Camassa–Holm equation

2020 ◽  
Vol 193 (4) ◽  
pp. 857-877
Author(s):  
Zhaonan Luo ◽  
Zhijun Qiao ◽  
Zhaoyang Yin
Keyword(s):  
Cauchy Problem ◽  
Holm Equation ◽  
The Cauchy Problem

On the Cauchy problem for the generalized Camassa–Holm equation

2014 ◽  
Vol 176 (3) ◽  
pp. 423-457 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Holm Equation ◽  
The Cauchy Problem

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.

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