scholarly journals On the Study of Local Solutions for a Generalized Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Meng Wu
Keyword(s):  
Sobolev Space ◽  
Strong Solutions ◽  
Well Posedness ◽  
Regularization Technique ◽  
Holm Equation ◽  
Dispersive Model ◽  
Local Solutions

The pseudoparabolic regularization technique is employed to study the local well-posedness of strong solutions for a nonlinear dispersive model, which includes the famous Camassa-Holm equation. The local well-posedness is established in the Sobolev spaceHs(R)withs>3/2via a limiting procedure.

scholarly journals The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Nan Li ◽  
Shaoyong Lai ◽  
Shuang Li ◽  
Meng Wu
Keyword(s):  
Sobolev Space ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Unique Global Solution ◽  
Initial Value ◽  
Regularization Technique ◽  
Global Existence Of Solutions ◽  
Holm Equation ◽  
Sign Condition ◽  
Nonlinear Generalization

A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of the pseudoparabolic regularization technique, its local well posedness in Sobolev spaceHS(R)withs>3/2is established via a limiting procedure. Provided that the initial valueu0satisfies the sign condition andu0∈Hs(R)  (s>3/2), it is shown that there exists a unique global solution for the equation in spaceC([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)).

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.

WELL-POSEDNESS AND BLOWUP PHENOMENA FOR A THREE-COMPONENT CAMASSA–HOLM SYSTEM WITH PEAKONS

2012 ◽  
Vol 09 (03) ◽  
pp. 451-467 ◽  
Author(s):  
QIAOYI HU ◽  
LIYUN LIN ◽  
JI JIN
Keyword(s):  
Initial Value Problem ◽  
Strong Solutions ◽  
Well Posedness ◽  
Initial Value ◽  
Blowup Rate ◽  
Holm Equation ◽  
Two Component

First, we establish the local well-posedness of the initial value problem for a new three-component Camassa–Holm system with peakons. We then present a precise blowup scenario and several blowup results for strong solutions to that system. Finally, we determine the blowup rate of strong solutions to the system when a blowup occurs. Our results include all earlier results on the Camassa–Holm equation and on a two-component Camassa–Holm system with peakons.

scholarly journals Well-Posedness, Blow-Up Phenomena, and Asymptotic Profile for a Weakly Dissipative Modified Two-Component Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu
Keyword(s):  
Cauchy Problem ◽  
Blow Up ◽  
Strong Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Profile ◽  
Holm Equation ◽  
Two Component ◽  
The Cauchy Problem ◽  
Blow Up Rate

We study the Cauchy problem of a weakly dissipative modified two-component Camassa-Holm equation. We firstly establish the local well-posedness result. Then we present a precise blow-up scenario. Moreover, we obtain several blow-up results and the blow-up rate of strong solutions. Finally, we consider the asymptotic behavior of solutions.

scholarly journals The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Shaoyong Lai
Keyword(s):  
Sobolev Space ◽  
Differential Equations ◽  
Weak Solutions ◽  
Lower Order ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Existence Of Weak Solutions ◽  
Abstract Differential Equations ◽  
Holm Equation

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in spaceC([0,T),Hs(R))∩C1([0,T),Hs-1(R))withs>3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev spaceHs(R)with1≤s≤3/2is developed.

scholarly journals The Uniqueness of Strong Solutions for the Camassa-Holm Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Meng Wu ◽  
Chong Lai
Keyword(s):  
Sobolev Space ◽  
Strong Solution ◽  
Strong Solutions ◽  
Initial Value ◽  
Holm Equation

Assume that there exists a strong solution of the Camassa-Holm equation and the initial value of the solution belongs to the Sobolev spaceH1(R). We provide a new proof of the uniqueness of the strong solution for the equation.

Well-posedness and blow-up phenomena for a periodic two-component Camassa–Holm equation

2011 ◽  
Vol 141 (1) ◽  
pp. 93-107 ◽  
Author(s):  
Qiaoyi Hu ◽  
Zhaoyang Yin
Keyword(s):  
Blow Up ◽  
Strong Solutions ◽  
Well Posedness ◽  
Holm Equation ◽  
Two Component ◽  
Blow Up Rate

We establish the local well-posedness for a periodic two-component Camassa–Holm equation. We then present precise blow-up scenarios. Finally, we obtain several blow-up results and the blow-up rate of strong solutions to the equation.

Local well-posedness for the quantum Zakharov system in one spatial dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 157-192 ◽  
Author(s):  
Yung-Fu Fang ◽  
Hsi-Wei Shih ◽  
Kuan-Hsiang Wang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Electric Field ◽  
Classical Result ◽  
Spatial Dimension ◽  
Well Posedness ◽  
Ion Density ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Quantum Zakharov System

We consider the quantum Zakharov system in one spatial dimension and establish a local well-posedness theory when the initial data of the electric field and the deviation of the ion density lie in a Sobolev space with suitable regularity. As the quantum parameter approaches zero, we formally recover a classical result by Ginibre, Tsutsumi, and Velo. We also improve their result concerning the Zakharov system and a result by Jiang, Lin, and Shao concerning the quantum Zakharov system.

Sign in / Sign up

Export Citation Format

Share Document