Ill-posedness of the Cauchy problem for the Chern-Simons-Dirac system in one dimension

We consider the Cauchy problem for the Chern-Simons-Dirac system in one spatial dimension. For this problem, Bournaveas, Candy, and Machihara (2012) proved the local in time well-posedness in Hs(R)×Hr(R) with −1/2