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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 \(\widehat{H^{s,p}(\mathbb{R})}\) whose scaling dimension is arbitrarily close to the critical scaling dimension.

MSC:

35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
35Q40 PDEs in connection with quantum mechanics
35B33 Critical exponents in context of PDEs
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