The periodic Cauchy problem for a combined CH–mCH integrable equation. (September 2016)

• Record Type: Journal Article
• Authors: Liu, Xingxing
• Is Part Of: Nonlinear analysis. Volume 143(2016)
• Page Start: 138
• Page End: 154
• Publication Date: 2016-09
• Subjects: 35G25 -- 35L05; Combined CH–mCH equation -- Non-uniform dependence -- Hölder continuity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.05.013
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper is concerned with the periodic Cauchy problem for a generalized Camassa–Holm integrable equation, which can be viewed as a generalization to both the Camassa–Holm (CH) and modified Camassa–Holm (mCH) equations. We mainly make a detailed presentation on the effects of varying the CH and mCH nonlocal nonlinearities on the non-uniform dependence and Hölder continuity of the solution map. Using a Galerkin-type approximation method, we first establish the local well-posedness result in Sobolev spaces H s, s > 5 2, with continuous dependence on the initial data. Then we prove that this dependence is sharp by showing that the data-to-solution map is not uniformly continuous, which is based on well-posedness estimates and the method of approximate solutions. Furthermore, we demonstrate that the solution map is Hölder continuous in the H σ topology with 0 ≤ σ < s .