Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D

TL;DR

This paper improves the understanding of well-posedness for the Maxwell-Klein-Gordon system in 2D by lowering regularity requirements for data in Sobolev and Fourier-Lebesgue spaces, approaching critical scaling.

Contribution

It provides new well-posedness results for the Maxwell-Klein-Gordon system in 2D with minimal regularity assumptions, reducing the gap to critical regularity in both Coulomb and Lorenz gauges.

Findings

Reduced regularity gap for Sobolev spaces in well-posedness

Extended results to Fourier-Lebesgue spaces near critical scaling

Achieved better understanding of minimal data regularity requirements

Abstract

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based Sobolev spaces $H^s$ and $H^l$ for the electromagnetic field $\phi$ and the potential $A$, respectively. The minimal regularity assumptions are $s > \frac{1}{2}$ and $l > \frac{1}{4}$ , which leaves a gap of $\frac{1}{2}$ and $\frac{1}{4}$ to the critical regularity with respect to scaling $s_c = l_c =0$ . This gap can be reduced for data in Fourier-Lebesgue spaces $\widehat{H}^{s,r}$ and $\widehat{H}^{l,r}$ to $s> \frac{21}{16}$ and $l > \frac{9}{8}$ for $r$ close to $1$ , whereas the critical exponents with respect to scaling fulfill $s_c \to 1$ , $ l_c \to 1 $ as $r \to 1$ . Here $\|f\|_{\widehat{H}^{s,r}} := \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 < r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, . $ Thus the gap is reduced for $\phi$ as well as $A$ in both gauges.