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

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 $\varphi$ 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 ${\hat{H}}^{s,r}$ and ${\hat{H}}^{l,r}$ 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 $\Vert f{\Vert }_{{\hat{H}}^{s,r}}:=\Vert ⟨\xi {⟩}^s\tilde{f}{\Vert }_{L_{\tau \xi }^{r^{\prime }}},1<r\le 2,\frac{1}{r}+\frac{1}{r^{\prime }}=1.$ Thus the gap is reduced for $\varphi$ as well as $A$ in both gauges.

24 pages. arXiv admin note: text overlap with arXiv:2010.06170