Global conservative solutions for a model equation for shallow water waves of moderate amplitude

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:256

Article

Zhou, Shouming^1,2  Mu, Chunlai¹ 

[1] Chongqing Univ, Coll Math & Stat, Chongqing 401331, Peoples R China

[2] Chongqing Normal Univ, Coll Math Sci, Chongqing 400047, Peoples R China

关键词: Shallow water waves of moderate amplitude;    Conservative solutions;    Wave breaking;   

DOI  :  10.1016/j.jde.2013.11.011

来源: Elsevier

【 摘 要 】

In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Crown Copyright (C) 2013 Published by Elsevier Inc. All rights reserved.

【 授权许可】

Free