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

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

2011-06-10

arxiv.org/abs/1106.2116v3

Mathematics

Analysis of PDEs

19 pages. Some typos corrected. Final version to be published in Differential and Integral Equations

Scientific paper

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.

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