Advanced Studies in Pure Mathematics

Well-posedness of the Cauchy problem for the Maxwell–Dirac system in one space dimension

Mamoru Okamoto

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Abstract

We determine the range of Sobolev regularity for the Maxwell–Dirac system in $1+1$ space time dimensions to be well-posed locally. The well-posedness follows from the null form estimates. Outside the range for the well-posedness, we show either the flow map is not continuous or not twice differentiable at zero.

Article information

Source
Nonlinear Dynamics in Partial Differential Equations, S. Ei, S. Kawashima, M. Kimura and T. Mizumachi, eds. (Tokyo: Mathematical Society of Japan, 2015), 497-505

Dates
Received: 16 December 2011
Revised: 7 February 2012
First available in Project Euclid: 30 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1540934248

Digital Object Identifier
doi:10.2969/aspm/06410497

Mathematical Reviews number (MathSciNet)
MR3381317

Zentralblatt MATH identifier
1335.35212

Subjects
Primary: 35Q40: PDEs in connection with quantum mechanics 35L70: Nonlinear second-order hyperbolic equations

Keywords
Maxwell–Dirac system null structure local well-poseness

Citation

Okamoto, Mamoru. Well-posedness of the Cauchy problem for the Maxwell–Dirac system in one space dimension. Nonlinear Dynamics in Partial Differential Equations, 497--505, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06410497. https://projecteuclid.org/euclid.aspm/1540934248


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