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

Shuji Machihara; Kenji Nakanishi; Tohru Ozawa

doi:10.4171/RMI/342

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

Shuji Machihara^*, Kenji Nakanishi, Tohru Ozawa

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

44 Citations (Scopus)

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_t² 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.

Original language	English
Pages (from-to)	179-194
Number of pages	16
Journal	Revista Matematica Iberoamericana
Volume	19
Issue number	1
DOIs	https://doi.org/10.4171/RMI/342
Publication status	Published - 2003
Externally published	Yes

Keywords

Nonlinear Dirac equation
Nonlinear Schrödinger equation
Nonrelativistic limit
Strichartz's estimate

ASJC Scopus subject areas

Mathematics(all)

Access to Document

10.4171/RMI/342

Cite this

@article{c87bbf4130b64ffe9cb8cc9cacd6894c,

title = "Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation",

abstract = "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 Schr{\"o}dinger equation as the speed of light tends to infinity.",

keywords = "Nonlinear Dirac equation, Nonlinear Schr{\"o}dinger equation, Nonrelativistic limit, Strichartz's estimate",

author = "Shuji Machihara and Kenji Nakanishi and Tohru Ozawa",

year = "2003",

doi = "10.4171/RMI/342",

language = "English",

volume = "19",

pages = "179--194",

journal = "Revista Matematica Iberoamericana",

issn = "0213-2230",

publisher = "Universidad Autonoma de Madrid",

number = "1",

}

TY - JOUR

T1 - Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation

AU - Machihara, Shuji

AU - Nakanishi, Kenji

AU - Ozawa, Tohru

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

KW - Nonlinear Dirac equation

KW - Nonlinear Schrödinger equation

KW - Nonrelativistic limit

KW - Strichartz's estimate

UR - http://www.scopus.com/inward/record.url?scp=0037570590&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037570590&partnerID=8YFLogxK

U2 - 10.4171/RMI/342

DO - 10.4171/RMI/342

M3 - Article

AN - SCOPUS:0037570590

VL - 19

SP - 179

EP - 194

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 1

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this