Low regularity solutions for the (2+1) - dimensional Maxwell-Klein-Gordon equations in temporal gauge

TL;DR

This paper establishes local well-posedness for the (2+1)-dimensional Maxwell-Klein-Gordon equations at low regularity levels, leveraging null structures and advanced bilinear estimates, extending prior work from higher dimensions.

Contribution

It demonstrates low regularity solutions for the 2+1D Maxwell-Klein-Gordon equations using novel null structure analysis and bilinear estimates, building on techniques from related equations.

Findings

Proves local well-posedness below energy regularity.

Identifies a partial null structure in the nonlinearity.

Extends methods from (3+1)-dimensional cases to (2+1) dimensions.

Abstract

The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.