On the ill-posedness result for the BBM equation

TL;DR

This paper demonstrates that the initial value problem for the BBM equation is ill-posed in Sobolev spaces with negative index, showing the flow map is discontinuous at zero for small times, which is a sharp result.

Contribution

It establishes the ill-posedness of the BBM equation in $H^s( )$ for $s<0$, providing a sharp threshold for well-posedness.

Findings

Flow map is discontinuous at zero in $H^s( )$ for $s<0$.

Ill-posedness holds for small times $t>0$.

Result is sharp, indicating the precise regularity threshold.

Abstract

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in $H^s(\R)$, $s<0$ in the sense that the flow-map $u_0\mapsto u(t)$ that associates to initial data $u_0$ the solution $u$ cannot be continuous at the origin from $H^s(\R)$ to even $\mathcal{D}'(\R)$ at any fixed $t>0$ small enough. This result is sharp.