Continuity for a generalized cross-coupled Camassa–Holm system with waltzing peakons and higher-order nonlinearities. (February 2020)

• Record Type: Journal Article
• Authors: Zhou, Shouming; Qiao, Zhijun; Mu, Chunlai
• Is Part Of: Nonlinear analysis. Volume 51(2020)
• Publication Date: 2020-02
• Subjects: Cross-coupled Camassa–Holm -- Well-posedness -- Non-uniform dependence -- Hölder continuity; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2019.102970
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper, we consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and Hölder continuity of the data-to-solution map for a generalized cross-coupled Camassa–Holm system with waltzing peakons on both the periodic and the non-periodic case. In light of a Galerkin-type approximation scheme, the system is shown well-posed in the Sobolev spaces H s × H s, s > 5 ∕ 2 in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous. Furthermore, we prove the Hölder continuity in the H r × H r topology when 0 ≤ r < s with Hölder exponent α depending on both s and r .