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

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.

Locations

• Journal of the American Mathematical Society - View - PDF