In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space Hs. We prove the existence and uniqueness of global solutions for small data in Hs with s > 1. The method of proof is based on the Strichartz estimate of Lt2 type for Dirac and Klein-Gordon equations. We also prove that the solutions of the nonlinear Dirac equation after modulation of phase converge to the corresponding solutions of the nonlinear Schrödinger equation as the speed of light tends to infinity.
- Nonlinear Dirac equation
- Nonlinear Schrödinger equation
- Nonrelativistic limit
- Strichartz's estimate
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