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

Machedon, Matei¹ ; Sterbenz, Jacob^1, 2

¹ Department of Mathematics, University of Maryland, College Park, Maryland 20742
² Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Journal of the American Mathematical Society, Tome 17 (2004) no. 2, pp. 297-359

Résumé

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.

Détail

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

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