Sharp regularity for the periodic Camassa--Holm equation in critical Triebel--Lizorkin spaces

Abstract

We establish a sharp well-posedness and norm inflation theory for the Camassa--Holm equation in critical Triebel--Lizorkin $F_{p,q}^{1+1/p}(\mathbb{T})$. At the endpoint $p=1$, we prove local Hadamard well-posedness for $1\le q<\infty$. In contrast, we prove norm inflation for $1<p<\infty$ and $1\le q\le \infty$. We also complement the local well-posedness in the critical Besov spaces and higher-regularity Triebel--Lizorkin spaces. The positive results rely on a Lipschitz stability theorem for the periodic Green operator under degree-one Lagrangian flows. The negative result is based on a nested smooth atomic construction on the torus, adapted from its real-line counterpart.