The Hölder continuity of the solution map to the $$b$$ -family equation in weak topology

• DOI: 10.1007/s00208-013-0939-9
• Publication Date: 2013-04-20T06:59:53Z
• Authors: Robin Ming Chen; Yue Liu; Pingzheng Zhang
• Keywords: Hölder condition; Exponent; Continuous map

We prove that the solution map of the $$b$$ -family equation is Holder continuous as a map from a bounded set of $$H^s(\mathbb{R }), s>\frac{3}{2}$$ with $$H^r(\mathbb{R })$$ ( $$0\le r0$$ . Moreover, we show that the obtained exponent of the Holder continuity is optimal when $$s-1<r<s$$ .