LOW REGULARITY WELL-POSEDNESS FOR THE 3D KLEIN - GORDON - SCHRODINGER SYSTEM

The Klein-Gordon-Schrodinger system in 3D is shown to be locally well-posed for Schrodinger data in H-s and wave data in H-sigma x H sigma-1, if s > -1/4 sigma > -1/2, sigma - 2s > 3/2 and sigma - 2 < s < sigma + 1. This result is optimal up to the endpoints in the sense that the local flow map is not C-2 otherwise. It is also shown that (unconditional) uniqueness holds for s = sigma = 0 in the natural solution space C-0 ([0, T], L-2) x C-0 ([0, T], L-2) x C-0 ([0, T], H-1/2). This solution exists even globally by Colliander, Holmer and Tzirakis [6]. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru [3], and Bejenaru and Herr [4].