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Low regularity well-posedness for some nonlinear Dirac equations in one space dimension. (English) Zbl 1240.35362
Summary: 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 ${H}^{\epsilon }$, the critical space being ${L}^{2}$, and globally well posed for initial data in ${H}^{1/2+\epsilon }$, for any $\epsilon >0·$ 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 ${H}^{\epsilon }·$

##### MSC:
 35L72 Quasilinear second-order hyperbolic equations 35Q40 PDEs in connection with quantum mechanics 35B30 Dependence of solutions of PDE on initial and boundary data, parameters
##### Keywords:
well-posedness; nonlinear Dirac equation; Cauchy problem