Stability of blow-up solution for the two component Camassa–Holm equations

2020 ◽  
Vol 120 (3-4) ◽  
pp. 319-336
Author(s):  
Xintao Li ◽  
Shoujun Huang ◽  
Weiping Yan

This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa–Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler’s equation in the shallow water regime.

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yunxi Guo ◽  
Tingjian Xiong

The two-component μ-Hunter-Saxton system is considered in the spatially periodic setting. Firstly, a wave-breaking criterion is derived by employing the localization analysis of the transport equation theory. Secondly, several sufficient conditions of the blow-up solutions are established by using the classic method. The results obtained in this paper are new and different from those in previous works.


Author(s):  
Joachim Escher

This paper is devoted to the study of a recently derived periodic shallow water equation. We discuss in detail the blow-up scenario of strong solutions and present several conditions on the initial profile, which ensure the occurrence of wave breaking. We also present a family of global weak solutions, which may be viewed as global periodic shock waves to the equation under discussion.


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