Some new well-posedness results for the Klein-Gordon-Schr\"odinger system

We consider the Cauchy problem for the 2D and 3D Klein-Gordon-Schr"odinger system. In 2D we show local well-posedness for Schr"odinger data in H^s and wave data in H^{\sigma} x 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- 1/2 \le \sigma < s+ 3/2. In 3D we show global well-posedness for s \ge 0, s - 1/2 < \sigma \le s+1. Fundamental for our results are the studies by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr for the Zakharov system, and also the global well-posedness results for the Zakharov and Klein-Gordon-Schr"odinger system by Colliander, Holmer and Tzirakis.