Low regularity well-posedness for some nonlinear Dirac equations in one space dimension

Abstract

We prove that the Cauchy problem for a nonlinear Dirac equation with vector self-interaction (Thirring model) and for a nonlinear system of two Dirac equations coupled through a vector-vector interaction (Federbusch model) are locally well posed, in one space dimension, for initial data in Sobolev spaces of almost critical dimension; i.e., in $H^\varepsilon$, the critical space being $L^2$, and globally well posed for initial data in $H^{1/2+\varepsilon}$, for any $\varepsilon>0$. We also consider a nonlinear Dirac equation with quadratic nonlinearity which was studied earlier by S.~Machihara and N.~Bournaveas. We prove that the Cauchy problem for this equation is locally well posed for initial data in $H^{\varepsilon}$.