Revista Matemática Iberoamericana

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

Shuji Machihara, Kenji Nakanishi, and Tohru Ozawa

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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.

Article information

Rev. Mat. Iberoamericana, Volume 19, Number 1 (2003), 179-194.

First available in Project Euclid: 31 March 2003

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Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations

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), no. 1, 179--194.

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