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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Abstract

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

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

Dates
First available in Project Euclid: 31 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1049123084

Mathematical Reviews number (MathSciNet)
MR1993419

Zentralblatt MATH identifier
1041.35061

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations

Keywords
Nonlinear Dirac equation Strichartz's estimate nonrelativistic limit nonlinear Schrödinger equation

Citation

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. https://projecteuclid.org/euclid.rmi/1049123084


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