scholarly journals On the Cauchy problem for a higher-order μ-Camassa-Holm equation

2018 ◽  
Vol 38 (8) ◽  
pp. 4163-4187
Author(s):  
Feng Wang ◽  
◽  
Fengquan Li ◽  
Zhijun Qiao ◽  
◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Higher Order ◽  
Holm Equation ◽  
The Cauchy Problem

Well-posedness and limit behaviors for a stochastic higher order modified Camassa–Holm equation

2016 ◽  
Vol 16 (06) ◽  
pp. 1650019
Author(s):  
Lin Lin ◽  
Guangying Lv ◽  
Wei Yan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Higher Order ◽  
Well Posedness ◽  
Local Existence And Uniqueness ◽  
Holm Equation ◽  
The Cauchy Problem

This paper is devoted to the Cauchy problem for a stochastic higher order modified-Camassa–Holm equation [Formula: see text] The local existence and uniqueness with initial data [Formula: see text], [Formula: see text] and [Formula: see text], is established. The limit behaviors of the solution are examined as [Formula: see text].

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 The Cauchy Problem for Higher-Order KP Equations

1999 ◽  
Vol 153 (1) ◽  
pp. 196-222 ◽  
Author(s):  
J.C. Saut ◽  
N. Tzvetkov
Keyword(s):  
Cauchy Problem ◽  
Higher Order ◽  
The Cauchy Problem ◽  
Kp Equations

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 The Cauchy problem for higher-order modified Camassa–Holm equations on the circle

2019 ◽  
Vol 187 ◽  
pp. 397-433 ◽  
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Xiaoping Zhai ◽  
Yimin Zhang
Keyword(s):  
Cauchy Problem ◽  
Higher Order ◽  
The Cauchy Problem

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.

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