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

TL;DR

This paper establishes the local well-posedness of the Camassa-Holm equation in critical Besov spaces for a broader range of p, solving an open problem and introducing a novel method avoiding Moser-type inequalities.

Contribution

It proves local well-posedness in critical Besov spaces $B^{1+rac{1}{p}}_{p,1}$ for all $1 extless p extless \infty$, extending previous results and applying a new technique.

Findings

Proves well-posedness for $1 extless p extless \infty$ in critical Besov spaces.

Introduces a method combining Lagrange coordinates and small time conditions.

Applicable to other Camassa-Holm type equations, improving their well-posedness indices.

Abstract

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-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $1\leq p\leq+\infty,\ 1<r\leq+\infty$ had been studied in \cite{d1,d2,glmy}. That is to say, it left an 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 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 $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p<+\infty$. 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.