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

For the famous Camassa-Holm equation, the well-posedness in $B_{p,1}^{1+\frac{1}{p}}(\mathbb{R})$ with $1\le p\le 2$ and the ill-posedness in $B_{p,r}^{1+\frac{1}{p}}(\mathbb{R})$ with $1\leq p\leq+\infty,
1<r\leq+\infty$ had been studied in \cite{d1,d2,glmy}. That is to say, it left an open problem in the critical case $B_{p,1}^{1+\frac{1}{p}}(\mathbb{R})$ with $2<p\le +\mathrm{\infty }$ proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem. The main difficulty is to prove the uniqueness, which usually needs 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 time conditions to avoid using the Moser-type inequality. As a result, we obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces $B_{p,1}^{1+\frac{1}{p}}(\mathbb{R})$ with $1\le p<+\mathrm{\infty }$. It is worth mentioning that our method is suitable for many Camassa-Holm type equations such as the Novikov equation and the two-component Camassa-Holm system, which can also improve their index on the local well-posedness.