Open Access Research

The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H 1 ( R )

Shaoyong Lai1*, Nan Li1 and Yonghong Wu2

Author Affiliations

1 Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 610074, China

2 Department of Mathematics and Statistics, Curtin University, Perth, WA, 6845, Australia

For all author emails, please log on.

Boundary Value Problems 2013, 2013:26  doi:10.1186/1687-2770-2013-26

Published: 12 February 2013

Abstract

The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M2">View MathML</a> under the assumption that the initial value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M3">View MathML</a> only belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/26/mathml/M1">View MathML</a>. The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution.

MSC: 35G25, 35L05.

Keywords:
global weak solution; Camassa-Holm type equation; existence