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For the famous Camassa-Holm equation, the well-posedness in$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p\leq2$ and the ill-posednessin $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $1\leq p\leq+\infty,\1<r\leq+\infty$ had been studied in \cite{d1,d2,glmy}. That is to say, it leftan open problem in the critical case $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with$2<p\leq+\infty$ proposed by Danchin in \cite{d1,d2}. In this paper, we solvethis problem. The main difficulty is to prove the uniqueness, which usuallyneeds to use the Moser-type inequality, resulting in the index $p$ belongs to$[1,2]$. To overcome the difficulty, inspired by Linares, Ponce and Thomas\cite{lps}, we combine the Lagrange coordinate transformation and small timeconditions to avoid using the Moser-type inequality. As a result, we obtain thelocal well-posedness for the Camassa-Holm equation in critical Besov spaces$B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p<+\infty$. It is worthmentioning that our method is suitable for many Camassa-Holm type equationssuch as the Novikov equation and the two-component Camassa-Holm system, whichcan also improve their index on the local well-posedness.

A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces $B^{1+\frac{1}{p}}_{p,1},1\leq p<+\infty$