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Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension. (English) Zbl 1144.35306
Summary: 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 $\widehat{H^{s,p}(\bbfR)}$ whose scaling dimension is arbitrarily close to the critical scaling dimension.

35A07Local existence and uniqueness theorems (PDE) (MSC2000)
35L70Nonlinear second-order hyperbolic equations
35Q40PDEs in connection with quantum mechanics
35B33Critical exponents (PDE)
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