Sharp well-posedness and ill-posedness of the Camassa-Holm equation in critical Triebel-Lizorkin spaces

Abstract

This paper is devoted to the sharp well-posedness and ill-posedness of the Cauchy problem for the Camassa-Holm (CH) equation in critical Triebel-Lizorkin spaces $F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})$ with $(p,q)\in[1,\infty)\times[1,\infty]$ or $p=q=\infty$. On the one hand, we establish the local well-posedness in the sense of Hadamard in $F^2_{1,q}(\mathbb{R})$ for $1\leq q<\infty$ via Lagrangian coordinate transformation. On the other hand, by means of smooth atomic decomposition, strong ill-posedness is then proved in $F^{1+\frac{1}{p}}_{p,q}(\mathbb{R})$ with $(p,q)\in(1,\infty)\times[1,\infty]$ or $p=q=\infty$ in the sense of norm inflation, which in particular yields the ill-posedness of CH in critical Sobolev spaces $W^{1+\frac{1}{p},p}(\mathbb{R})$ with $1<p<\infty$, and provides a new perspective on the ill-posedness of CH in $H^{\frac{3}{2}}(\mathbb{R})$.