Local well-posedness for the (n+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

TL;DR

This paper proves local well-posedness for the Maxwell-Klein-Gordon equations in (n+1) dimensions with low regularity data, extending previous results and utilizing null structures and bilinear estimates.

Contribution

It extends well-posedness results to higher dimensions for the Maxwell-Klein-Gordon equations, including below energy regularity in 3+1 dimensions, using advanced harmonic analysis techniques.

Findings

Established local well-posedness for n+1 dimensions with low regularity data.

Improved regularity results in 3+1 dimensions below energy level.

Utilized null structure and bilinear estimates in wave-Sobolev spaces.

Abstract

This is an extension of the paper [14] by the author for the 2+1 dimensional Maxwell-Klein-Gordon equations in temporal gauge to the n+1 dimensional situation for $n \ge 3$. They are shown to be locally well-posed for low regularity data, in 3+1 dimensions even below energy level improving a result by Yuan. Fundamental for the proof is a partial null structure of the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces, in 3+1 dimensions proven by d'Ancona, Foschi and Selberg, on an $(L^{\frac{2(n+1)}{n-1}}_x L^2_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.