Kyoto Journal of Mathematics

Well-posedness for nonlinear Dirac equations in one dimension

Shuji Machihara, Kenji Nakanishi, and Kotaro Tsugawa

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Abstract

We completely determine the range of Sobolev regularity for the Dirac- Klein-Gordon system, the quadratic nonlinear Dirac equations, and the wave-map equation to be well posed locally in time on the real line. For the Dirac-Klein-Gordon system, we can continue those local solutions in nonnegative Sobolev spaces by the charge conservation. In particular, we obtain global well-posedness in the space where both the spinor and scalar fields are only in L2(R). Outside the range for well-posedness, we show either that some solutions exit the Sobolev space instantly or that the solution map is not twice differentiable at zero.

Article information

Source
Kyoto J. Math., Volume 50, Number 2 (2010), 403-451.

Dates
First available in Project Euclid: 7 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1273236821

Digital Object Identifier
doi:10.1215/0023608X-2009-018

Mathematical Reviews number (MathSciNet)
MR2666663

Zentralblatt MATH identifier
1248.35170

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35L71: Semilinear second-order hyperbolic equations 35B44: Blow-up 35B65: Smoothness and regularity of solutions 35Q41: Time-dependent Schrödinger equations, Dirac equations

Citation

Machihara, Shuji; Nakanishi, Kenji; Tsugawa, Kotaro. Well-posedness for nonlinear Dirac equations in one dimension. Kyoto J. Math. 50 (2010), no. 2, 403--451. doi:10.1215/0023608X-2009-018. https://projecteuclid.org/euclid.kjm/1273236821


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