Ill-posedness for the two component Degasperis-Procesi equation in critical Besov space

TL;DR

This paper demonstrates ill-posedness for the two component Degasperis-Procesi equation in a critical Besov space by constructing initial data that causes solution norm inflation, highlighting the equation's sensitivity to initial conditions.

Contribution

It introduces a new initial data construction to prove ill-posedness in critical Besov space for the coupled Degasperis-Procesi system, contrasting with known well-posedness results.

Findings

Norm inflation in critical Besov space $B^1_{ty,1}$

Ill-posedness due to coupled structure of the equation

Difference from local well-posedness results

Abstract

In this paper, we study the Cauchy problem for the two component Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$. By presenting a new construction of initial data, we proved the norm inflation of the corresponding solutions in $B^1_{\infty,1}(\mathbb R)$ and hence ill-posedness. This is quite different from the local well-posedness result for the Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$ due to the coupled structure of density function.