### 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_{t}
^{2} 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 |

Publication status | Published - 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Revista Matematica Iberoamericana*,

*19*(1), 179-194.

**Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation.** / Machihara, Shuji; Nakanishi, Kenji; Ozawa, Tohru.

Research output: Contribution to journal › Article

*Revista Matematica Iberoamericana*, vol. 19, no. 1, pp. 179-194.

}

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 Lt 2 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 Lt 2 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

M3 - Article

VL - 19

SP - 179

EP - 194

JO - Revista Matematica Iberoamericana

JF - Revista Matematica Iberoamericana

SN - 0213-2230

IS - 1

ER -