A vector field method for some nonlinear Dirac models in Minkowski spacetime

Abstract

Firstly we use the Lie derivatives of spinors with respect to Killing vector fields and conformal Killing vector fields to obtain the generalized commuting vector field set adapted to the Dirac operator. By the conservation law of the charge current and total angular momentum, as well as the weighted Sobolev estimates, it gives the optimal decay for the linear massless spinor fields and more decay for certain decomposition parts in any dimensions. Moreover, by exploiting the Dirac null structure, we also show the small data global existence and asymptotic decay properties for some well-known nonlinear models with cubic nonlinearities in space dimension two and quadratic nonlinearities in space dimension three.