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

Abstract

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<r≤s≤r+1. Here we prove ill-posedness for almost all exponent pairs (s,r) outside of this well-posedness region. The proof based on the fact that the solution is explicitly written under the specific condition of initial data, or we also use the argument of Iwabuchi and Ogawa (2013) by which we estimate each step of iteration terms of the solutions for this problem. In the remaining exponent pairs, we show the flow map is not twice differentiable at zero. We give an example of the flow map which is not twice differentiable but generally continuous.