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

Nikolaos Bournaveas

doi:10.3934/dcds.2008.20.605

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2008, Volume 20, Issue 3: 605-616. Doi: 10.3934/dcds.2008.20.605

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

Nikolaos Bournaveas^1,

1.
University of Edinburgh, School of Mathematics, Edinburgh EH9 3JZ

Received: September 30, 2006

Revised: September 30, 2007

Published: August 2008

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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.

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