Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space

TL;DR

This paper proves that the solution map for the Camassa-Holm equation in the critical Besov space is not uniformly continuous, addressing an open question and advancing understanding of the equation's sensitivity to initial data.

Contribution

It provides a positive answer to the open problem of uniform continuity of the data-to-solution map in the critical Besov space for the Camassa-Holm equation.

Findings

The data-to-solution map is not uniformly continuous in the critical Besov space.

Addresses an open problem left by previous research.

Enhances understanding of the equation's dependence on initial data.

Abstract

Whether or not the data-to-solution map of the Cauchy problem for the Camassa-Holm equation and Novikov equation in the critical Besov space $B_{2,1}^{3/2}(\R)$ is not uniformly continuous remains open. In the paper, we aim at solving the open question left the previous works in \cite{Li3,Li4} and give a positive answer to this problem.