Abstract
We prove that the solution map of the \(b\)-family equation is Hölder continuous as a map from a bounded set of \(H^s(\mathbb{R }), s>\frac{3}{2}\) with \(H^r(\mathbb{R })\) (\(0\le r<s\)) topology, to \(C([0, T], H^r(\mathbb{R }))\) for some \(T>0\). Moreover, we show that the obtained exponent of the Hölder continuity is optimal when \(s-1<r<s\).
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Acknowledgments
This research is partially supported by the AMS Fan Fund Travel Grant 2010. The work of Chen is partially supported by the NSF Grant DMS-0908663. The work of Liu is partially supported by the NSF Grants DMS-0906099 and DMS-1207840, the NHARP Grant-003599-0001-2009, and the NSF-China Grant-11271192. The work of Zhang is supported in part by the NSF-China grants No. 11171135 and 11271164.
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Chen, R.M., Liu, Y. & Zhang, P. The Hölder continuity of the solution map to the \(b\)-family equation in weak topology. Math. Ann. 357, 1245–1289 (2013). https://doi.org/10.1007/s00208-013-0939-9
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DOI: https://doi.org/10.1007/s00208-013-0939-9