Well-posedness and Continuity Properties of the Degasperis-Procesi equation in critical Besov space

TL;DR

This paper establishes local existence and uniqueness of solutions to the Degasperis-Procesi equation in a critical Besov space, and analyzes the continuity properties of the data-to-solution map.

Contribution

It proves local well-posedness in the critical Besov space and shows the solution map is continuous but not uniformly continuous.

Findings

Existence and uniqueness of solutions in $B^1_{ abla,1}$

Continuity of the data-to-solution map

Lack of uniform continuity of the solution map

Abstract

In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in $B^1_{\infty,1}(\R)$. Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly continuous in $B^1_{\infty,1}$.