讲座题目：Well-posedness of Camassa-Holm equation in the endpoint space $H^{3/2}$

主讲人：霍朝辉 副研究员（中科院数学与系统科学研究院）

主持人：陈秀卿教授

时 间：2015年3月20日（周五）上午10:00-11:00

地 点：主楼1214

主要内容：

We mianly consider the well-posedness of the Cauchy problem for Camassa-Holm equation. The local well-posedness for Camassa-Holm equation was obtained applying Kato's semigroup approach [G.R.Blanco,2001] in $H^{s}$ with $s>3/2$. Moteover, by using the transport equation theory and the Besov spaces,Danchin [R.Danchin,2003] proved that the Cauchy problem for the Camassa-Holm equation is locally well-posed in $B^{3/2}_{2,1}$. Counterexamples to well-posedness in thecase $s<3/2 $ have been exhibited by Himonas and Misiolek [A. Himonas, G. Misiolek,2001] actually what they do prove there is that uniform continuity with respect to the data cannot hold in $H^s$ with $s<3/2 $). Therefore, in the Sobolev spaces framework, $s=3/2$ seems to be thecritical value for local well-posedness. We can show that if $u_0 \in H^{3/2}(\R)$ and $u_0-u_{0xx} $ does not change signand belongs to $L^1$; then theCauchy problem of Camassa-Holm equation  is locally well-posed in endpoint space $H^{3/2}$. The flow map$u_0\rightarrow u(t)$ is also continuous.

主讲人简介：

2007/03 – 至今，中科院，数学与系统科学研究院数学所，副研究员

2007/02 – 2009/02，香港城市大学，数学系，Research Fellow

2006/06 – 2007/03，中科院， 数学与系统科学研究院数学所，助理研究员

2004/06 – 2006/06，北京航空航天大学，数学系，博士后

2001/09 – 2004/06， 中国工程物理研究院北京研究生部

(主页http://sourcedb.amss.cas.cn/zw/zjrck/fyjy/200910/t20091016_2552552.html)