On the ill-posedness result for the BBM equation

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.

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