Ill-posedness for one-dimensional wave maps at the critical regularity

We show that the wave map equation in ${\mathbb{R}}^{1+1}$ is in general ill-posed in the critical space ${\dot{H}}^{1\mathrm{/}2}$, and the Besov space ${\dot{B}}_2^{1\mathrm{/}2,1}$. The problem is attributed to the bad behaviour of the one-dimensional bilinear expression $D^{-1}(fDg)$ in these spaces.