Asymptotic Properties of the Spinor Field and the Application to Nonlinear Dirac Models

In this paper we study the asymptotic behavior of the solutions of massless Dirac equations in $\mathbb{R}^{1+3}$ via a vector field method. It is proved that the solutions decay in a sharp rate and enjoy the so-called peeling properties. Based on this decay mechanism, we give a new insight to investigate the spinor null structure. Then a small-data global existence result of a nonlinear Dirac model follows. The application of our nonlinear Dirac model also anwers an open question raised by N. Tzvetkov concerning the Lorentz-invariant quadratic Dirac equations.