Some new well-posedness results for the Klein-Gordon-Schrödinger system

Abstract

We consider the Cauchy problem for the 2D and 3D Klein-Gordon Schrödinger system. In 2D we show local well posedness for Schrödinger data in $H^s$ and wave data in $H^{\sigma} \times H^{\sigma -1}$ for $s=-1/4 \, +$ and $\sigma = -1/2$, whereas ill posedness holds for $s < - 1/4$ or $\sigma < -1/2$, and global well-posedness for $s\ge 0$ and $s-\frac{1}{2} \le \sigma < s+ \frac{3}{2}$. In 3D we show global well posedness for $s \ge 0$, $ s - \frac{1}{2} < \sigma \le s+1$. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru [2], and Bejenaru and Herr [3] for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schrödinger system by Colliander, Holmer and Tzirakis [5].