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Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations


Authors: Matei Machedon and Jacob Sterbenz
Journal: J. Amer. Math. Soc. 17 (2004), 297-359
MSC (2000): Primary 35Q60, 35L70
DOI: https://doi.org/10.1090/S0894-0347-03-00445-4
Published electronically: November 13, 2003
MathSciNet review: 2051613
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Abstract: We prove that the evolution problem for the Maxwell-Klein-Gordon system is locally well posed when the initial data belong to the Sobolev space $H^{\frac{1}{2} + \epsilon}$ for any $\epsilon > 0$. This is in spite of a complete failure of the standard model equations in the range $\frac{1}{2} < s < \frac{3}{4}$. The device that enables us to obtain inductive estimates is a new null structure which involves cancellations between the elliptic and hyperbolic terms in the full equations.


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Additional Information

Matei Machedon
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: mxm@math.umd.edu

Jacob Sterbenz
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: jks@math.umd.edu, sterbenz@math.princeton.edu

DOI: https://doi.org/10.1090/S0894-0347-03-00445-4
Received by editor(s): October 16, 2002
Published electronically: November 13, 2003
Additional Notes: Both authors were supported by NSF grant DMS-0100406.
Article copyright: © Copyright 2003 American Mathematical Society