Unconditional well-posedness below energy norm for the Maxwell-Klein-Gordon system

Abstract

The Maxwell-Klein-Gordon equation $ \partial^{\alpha} F_{\alpha \beta} = -Im(\Phi \overline{D_{\beta} \Phi}) $ , $ D^{\mu}D_{\mu} \Phi = m^2 \Phi $ , where $F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta} A_{\alpha}$, $D_{\mu} = \partial_{\mu} - iA_{\mu} $, in the (3+1)-dimensional case is known to be unconditionally well-posed in energy space, i.e. well-posed in the natural solution space. This was proven by Klainerman-Machedon and Masmoudi-Nakanishi in Coulomb gauge and by Selberg-Tesfahun in Lorenz gauge. The main purpose of the present paper is to establish that for both gauges this also holds true for data $\Phi(0)$ in Sobolev spaces $H^s$ with less regularity, i.e. $s < 1$, but $s$ sufficently close to $1$. This improves the (conditional) well-posedness results in both cases, i.e. uniqueness in smaller solution spaces of Bourgain-Klainerman-Machedon type, which were essentially known by Cuccagna, Selberg and the author for $s > \frac{3}{4}$ , and which in Coulomb gauge is also contained in the present paper. In fact, the proof consists in demonstrating that any solution in the natural solution space for some $s > s_0$ belongs to a Bourgain-Klainerman-Machedon space where uniqueness is known. Here $s_0 \approx 0.914$ in Coulomb gauge and $s_0 \approx 0.907$ in Lorenz gauge.