On the weak solutions for the rotation-two-component Camassa–Holm equation

2020 ◽  
Vol 61 (6) ◽  
pp. 061514
Author(s):  
Li Yang ◽  
Chunlai Mu ◽  
Shouming Zhou ◽  
Xinyu Tu
Keyword(s):  
Weak Solutions ◽  
Holm Equation ◽  
Two Component

scholarly journals Geometric Flows of Curves, Two-Component Camassa-Holm Equation and Generalized Heisenberg Ferromagnet Equation

2021 ◽  
Vol 2090 (1) ◽  
pp. 012068
Author(s):  
Gulgassyl Nugmanova ◽  
Aigul Taishiyeva ◽  
Ratbay Myrzakulov ◽  
Tolkynai Myrzakul
Keyword(s):  
Heisenberg Ferromagnet ◽  
Geometric Flows ◽  
Space Curves ◽  
Holm Equation ◽  
Two Component

Abstract In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established.

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.

Global weak solutions for a generalized Camassa-Holm equation

2018 ◽  
Vol 291 (16) ◽  
pp. 2457-2475
Author(s):  
Xi Tu ◽  
Zhaoyang Yin
Keyword(s):  
Weak Solutions ◽  
Global Weak Solutions ◽  
Holm Equation

Wave breaking phenomenon for a modified two-component Dullin-Gottwald-Holm equation

2014 ◽  
Vol 55 (9) ◽  
pp. 093101 ◽  
Author(s):  
Panpan Zhai ◽  
Zhengguang Guo ◽  
Weiming Wang
Keyword(s):  
Wave Breaking ◽  
Holm Equation ◽  
Two Component

scholarly journals On measures of accretion and dissipation for solutions of the Camassa–Holm equation

2017 ◽  
Vol 14 (04) ◽  
pp. 721-754
Author(s):  
Grzegorz Jamróz
Keyword(s):  
Lebesgue Measure ◽  
Weak Solutions ◽  
Representation Formula ◽  
Structural Features ◽  
Dissipative Solutions ◽  
Holm Equation

We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa–Holm equation. Demonstrating certain novel properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such as the fact that they are singular with respect to the Lebesgue measure. We apply these results to gain new insights into the structure of weak solutions, proving in particular that measures of accretion vanish for dissipative solutions of the Camassa–Holm equation.

On Solutions to a Two-Component Generalized Camassa-Holm Equation

2010 ◽  
Vol 124 (3) ◽  
pp. 307-322 ◽  
Author(s):  
Zhengguang Guo ◽  
Yong Zhou
Keyword(s):  
Holm Equation ◽  
Two Component

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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