Abstract
In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in \(L^\infty (0,T;B^s_{2,r}(\mathbb {R}))\) to the inviscid Burgers equation as the filter parameter \(\alpha \) tends to zero with the given initial data \(u_0\in B^s_{2,r}(\mathbb {R})\). Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in \(B^s_{2,r}(\mathbb {R})\).
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Acknowledgements
M. Li is supported by the Jiangxi Provincial Natural Science Foundation (20232BAB2 01013) and China Postdoctoral Science Foundation (No. 2023M731431). J. Li is supported by the National Natural Science Foundation of China (12161004), Training Program for Academic and Technical Leaders of Major Disciplines in Ganpo Juncai Support Program (20232BCJ23009) and Jiangxi Provincial Natural Science Foundation (20224BAB201008).
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Cheng, Y., Lu, J., Li, M. et al. Zero-filter limit issue for the Camassa–Holm equation in Besov spaces. Monatsh Math (2024). https://doi.org/10.1007/s00605-024-01944-4
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DOI: https://doi.org/10.1007/s00605-024-01944-4