Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension

Pages: 605 - 616, Volume 20, Issue 3, March 2008      doi:10.3934/dcds.2008.20.605

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Nikolaos Bournaveas - University of Edinburgh, School of Mathematics, Edinburgh EH9 3JZ, United Kingdom (email)

Abstract: We study a nonlinear Dirac system in one space dimension with a quadratic nonlinearity which exhibits null structure in the sense of Klainerman. Using an $L^{p}$ variant of the $L^2$ restriction method of Bourgain and Klainerman-Machedon, we prove local well-posedness for initial data in a Sobolev-like space $\hat{H^{s,p}}(\R)$ whose scaling dimension is arbitrarily close to the critical scaling dimension.

Keywords:  Klainerman null forms, restriction method.
Mathematics Subject Classification:  Primary: 35L70; Secondary: 35Q40.

Received: October 2006;      Revised: October 2007;      Available Online: December 2007.