Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Blow up phenomena and global existence for the nonlocal periodic Rotation-Camassa-Holm system

Pages: 1315 – 1335



Min Zhu (Department of Mathematics, Nanjing Forestry University, Nanjing, China)

Ying Wang (Department of Mathematics, University of Electronic Science and Technology of China, Chengdu, China)


Under consideration in the present paper is a mathematical model proposed as an equation of long-crested shallow-water waves propagating in one direction with the effect of Earth’s rotation. The system is called Rotation-Camassa-Holm system (RCH2). The local well-posedness of the periodic Cauchy problem is then established by the linear transport theory. Then, wave-breaking phenomena is investigated based on the method of characteristics and the Riccati-type differential inequality with two different kinds of methods. Finally, the wave-breaking data are illustrated and the existence of global solutions is obtained in detail for the periodic RCH2 system.


Rotation-Camassa-Holm system, blow up, wave breaking, persistence

2010 Mathematics Subject Classification

35B44, 35G25

The work of Zhu was partially supported by the NSF of China under the grant 11401309.

The work of Wang was partially supported by ZYGX2015J096 and NSF of China under the grant 11571063.

Received 18 September 2019

Accepted 13 February 2020

Published 23 September 2020