WELL-POSEDNESS AND ANALYTICITY FOR AN INTEGRABLE TWO-COMPONENT SYSTEM WITH CUBIC NONLINEARITY

2013 ◽  
Vol 10 (04) ◽  
pp. 703-723 ◽  
Author(s):  
YONGSHENG MI ◽  
CHUNLAI MU

We study the Cauchy problem associated with a new integrable two-component system with cubic nonlinearity, which was recently proposed by Song, Qu and Qiao. We establish the local well-posedness in a range of the Besov spaces. Moreover, for analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend a result by Danchin, and Himonas et al. to more complex equations.

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao

We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin's cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.


2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Jiangbo Zhou ◽  
Lu Yao ◽  
Lixin Tian ◽  
Wenbin Zhang

We consider the Cauchy problem for an integrable modified two-component Camassa-Holm system with cubic nonlinearity. By using the Littlewood-Paley decomposition, nonhomogeneous Besov spaces, and a priori estimates for linear transport equation, we prove that the Cauchy problem is locally well-posed in Besov spaces Bp, rs with 1≤p, r≤+∞ and s>max{2+(1/p),5/2}.


2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao

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.


2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Elena Cordero ◽  
Kasso A. Okoudjou

We give a sharp estimate on the norm of the scaling operatorUλf(x)=f(λx)acting on the weighted modulation spacesMs,tp,q(ℝd). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.


2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Han Yang ◽  
Xiaoming Fan ◽  
Shihui Zhu

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is inHswiths> 2/5, then there exists a global solution in time, and theHsnorm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which generates a new space-timeLt,x4estimate for nonlinear equation with the relatively general defocusing power nonlinearity.


2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Ls Yong

The dissipative periodic 2-component Degasperis-Procesi system is investigated. A local well-posedness for the system in Besov space is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking criterions for strong solutions to the system with certain initial data are derived.


2008 ◽  
Vol 05 (03) ◽  
pp. 547-568 ◽  
Author(s):  
RINALDO M. COLOMBO ◽  
CRISTINA MAURI

Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler system describes the evolution of the fluid. At the junction, suitable physical conditions couple the n Euler systems. In this paper we prove the well posedness of the Cauchy problem for the model so obtained, provided the total varaiatio of the initial data is sufficiently small.


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