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February  2005, 13(2): 277-290. doi: 10.3934/dcds.2005.13.277

One dimensional Dirac equation with quadratic nonlinearities

Shuji Machihara 1,

1. 

Department of Mathematics, Shimane University, Matsue 690-8504, Japan

Received  August 2004 Revised  April 2005 Published  April 2005

The local well-posedness for the nonlinear Dirac equation with special forms of quadratic nonlinearities in one space dimension is obtained by two approaches. One is to apply the Fourier restriction norm method of Bourgain [2, 3] by showing the bilinear estimates for the nonlinearities. Another is to study the explicit solutions for wave equations and derive another bilinear estimates similar with Bournaveas [4].
Keywords: Fourier restriction norm, Nonlinear Dirac equation, bilinear estimates..
Mathematics Subject Classification: 35L05, 35L6.
Citation: Shuji Machihara. One dimensional Dirac equation with quadratic nonlinearities. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 277-290. doi: 10.3934/dcds.2005.13.277
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