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

Magdalena Czubak; Nina Pikula

doi:10.3934/cpaa.2014.13.1669

Article Contents

2014, Volume 13, Issue 4: 1669-1683. Doi: 10.3934/cpaa.2014.13.1669

This issue Previous Article Existence and uniqueness of a positive connection for the scalar viscous shallow water system in a bounded interval Next Article

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

Magdalena Czubak^1, and
Nina Pikula^2,

1.
Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000
2.
Department of Mathematics, University of California, San Diego (UCSD), La Jolla, CA 92093-0112

Received: November 30, 2013

Revised: December 31, 2013

Published: June 2015

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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.

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Mathematics Subject Classification: Primary: 35L70; Secondary: 35J15.

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References

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