Continuity and analyticity for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

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• 作者：Shouming Zhou
• 关键词：Non ; uniform dependence ; Hölder continuity ; Analyticity ; Gevrey regularity ; Cross ; coupled Camassa ; Holm
• 刊名：Monatshefte für Mathematik
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：182
• 期：1
• 页码：195-238
• 全文大小：
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081
• 卷排序：182

文摘

In this paper, we mainly study the well-posedness in the sense of Hadamard, non-uniform dependence, Hölder continuity and analyticity of the data-to-solution map for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs on both the periodic and the nonperiodic case. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces \(H^{s} \times H^{s},s>5/2\) in the sense of Hadamard, that is, the data-to-solution mapis continuous. In conjunction with the well-posedness estimate, it is also proved that this dependence is sharp by showing that the solution map is not uniformly continuous. Furthermore, the Hölder continuous in the \(H^r \times H^r\) topology when \(0\le r< s\) with Hölder exponent \(\alpha \) depending on both s and r are shown. Finally, applying generalized Ovsyannikov type theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of the CCCH system. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map