Communications on Pure and Applied Analysis (CPAA)

Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge

Pages: 1669 - 1683, Volume 13, Issue 4, July 2014      doi:10.3934/cpaa.2014.13.1669

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Magdalena Czubak - Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, United States (email)
Nina Pikula - Department of Mathematics, University of California, San Diego (UCSD), La Jolla, CA 92093-0112, United States (email)

Abstract: We consider the Maxwell-Klein-Gordon equation in 2D in the Coulomb gauge. We establish local well-posedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution $\phi$ of the gauged Klein-Gordon equation. The main tool for handling the wave equations is the product estimate established by D'Ancona, Foschi, and Selberg. Due to low regularity, we are unable to use the conventional approaches to handle the elliptic variable $A_0$, so we provide a new approach.

Keywords:  Maxwell-Klein-Gordon, null forms, Coulomb gauge, local well-posedness, low regularity.
Mathematics Subject Classification:  Primary: 35L70; Secondary: 35J15.

Received: December 2013;      Revised: January 2014;      Available Online: February 2014.