Local and global well-posedness for the Chern-Simons-Dirac system in one   dimension
  We consider the Cauchy problem for the Chern-Simons-Dirac system on $\mathbb{R}^{1+1}$ with initial data in $H^s$. Almost optimal local well-posedness is obtained. Moreover, we show that the solution is global in time, provided that initial data for the spinor component has finite charge, or $L^2$ norm. 
Nikolaos Bournaveas, Timothy Candy, and Shuji Machihara
math.AP