# Revista Matemática Iberoamericana

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**Volume 19, Issue 1, 2003, pp. 179–194**

**DOI: 10.4171/RMI/342**

Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation

Shuji Machihara^{[1]}, Kenji Nakanishi

^{[2]}and Tohru Ozawa

^{[3]}(1) Department of Mathematics, Shimane University, 690-8504, MATSUE, JAPAN

(2) Department of Mathematics, Kyoto University, 606-8502, KYOTO, JAPAN

(3) Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-ku, 060-0810, SAPPORO, JAPAN

In this paper we study the Cauchy problem for the nonlinear Dirac equation in the Sobolev space $H^s$. We prove the existence and uniqueness of global solutions for small data in $H^s$ with $s>1$. The method of proof is based on the Strichartz estimate of $L^2_t$ 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 Schröodinger equation as the speed of light tends to infinity.

*Keywords: *Nonlinear Dirac equation, Strichartz’s estimate, nonrelativistic limit, nonlinear Schrödinger equation

Machihara Shuji, Nakanishi Kenji, Ozawa Tohru: Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. *Rev. Mat. Iberoamericana* 19 (2003), 179-194. doi: 10.4171/RMI/342