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The Hölder continuity of the solution map to the \(b\)-family equation in weak topology

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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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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Robin Ming Chen

  2. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019-0408, USA

    Yue Liu

  3. Department of Mathematics, University of Jiangsu, Jiangsu, 212013, Peoples Republic of China

    Pingzheng Zhang

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  1. Robin Ming Chen
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  2. Yue Liu
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  3. Pingzheng Zhang
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Correspondence to Yue Liu.

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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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