Monatshefte für Mathematik

, Volume 187, Issue 4, pp 735–764 | Cite as

The local well-posedness in Besov spaces and non-uniform dependence on initial data for the interacting system of Camassa–Holm and Degasperis–Procesi equations

  • Shouming Zhou


This paper deals with the Cauchy problem for the interacting system of the Camassa–Holm and Degasperis–Procesi equations
$$\begin{aligned} m_t=-3m(2u_x+v_x)-m_x(2u+v), n_t=-2n(2u_x+v_x)-n_x(2u+v), \end{aligned}$$
where \(m=u-u_{xx}\) and \(n= v-v_{xx}\). By the transport equations theory and the classical Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomogeneous Besov spaces \(B^s_{p,r}\times B^s_{p,r}\) with \(1\le p,r \le +\infty \) and \(s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} \) is obtained, and the local well-posedness in critical Besov space \(B^{5/2}_{2,1}\times B^{5/2}_{2,1}\) is also established. Moreover, by the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space \(H^s({\mathbb {R}})\times H^s({\mathbb {R}})\) with \(s>5/2\), and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous.


Degasperis–Procesi equation Camassa–Holm equation Well-posedness Non-uniform dependence 

Mathematics Subject Classification

35G25 35L05 35Q50 



This work is partly supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJ1703043), Natural Science Foundation of Chongqing (Grant Nos. cstc2017jcyjAX0123 and cstc2016jcyjA0181) and National Natural Science Foundation of China (Grant Nos. 11771063 and 11601048).


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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.College of Mathematics ScienceChongqing Normal UniversityChongqingPeople’s Republic of China

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