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

Machedon, Matei; Sterbenz, Jacob

doi:10.1090/S0894-0347-03-00445-4

Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations
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by Matei Machedon and Jacob Sterbenz

J. Amer. Math. Soc. 17 (2004), 297-359

DOI: https://doi.org/10.1090/S0894-0347-03-00445-4

Published electronically: November 13, 2003

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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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