Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing

Pages: 5201 - 5221, Volume 36, Issue 9, September 2016      doi:10.3934/dcds.2016026

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Shihui Zhu - Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, China (email)

Abstract: In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well-posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the balance law and some new estimates, we prove the existence and uniqueness of global weak solutions for the C-H equation with a forcing in $H^1(\mathbb{R})$.

Keywords:  Camassa-Holm equation, forcing, weak solution, uniqueness, characteristic method.
Mathematics Subject Classification:  Primary: 35L05, 35D30; Secondary: 76B15.

Received: August 2015;      Revised: January 2016;      Available Online: May 2016.