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On the global weak solutions for a modified two‐component Camassa‐Holm equation
Author(s) -
Guan Chunxia,
Yin Zhaoyang
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200193
Subject(s) - mathematics , camassa–holm equation , a priori and a posteriori , initial value problem , component (thermodynamics) , limit (mathematics) , order (exchange) , weak solution , cauchy problem , mathematical analysis , mathematical physics , combinatorics , physics , quantum mechanics , integrable system , philosophy , epistemology , finance , economics
In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim x → ±∞ u 0 ( x ) = u ± . By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u − ≤ u + . The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1(\mathbb {R})\times H^1(\mathbb {R})$\end{document} and some a priori estimates on the first‐order derivatives of approximation solutions.