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arXiv · 2206.06523

Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa-Holm equation in time weighted $H^1(\mathbb{R})$ space

Zhiying Meng·Zhaoyang Yin
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Abstract

In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted $H^1$ space. Second, we show that the peakon solutions are conservative weak solutions in $H^1.$ Finally, given a conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables.

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Zhiying Meng, Zhaoyang Yin. 2022-06-13. Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa-Holm equation in time weighted $H^1(\mathbb{R})$ space. https://arxiv.org/abs/2206.06523

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