Low regularity well-posedness for the 3D Klein-Gordon-Schrödinger system

Abstract

The Klein-Gordon-Schrödinger system in 3D is shown to be locally well-posed for Schrödinger data in H^s and wave data in H^σ \times H^{σ-1}, if s > - 1/4, σ> - 1/2, σ-2s > 3/2 and σ-2 < s < σ+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=σ=0 in the natural solution space C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-1/2}) . This solution exists even globally by Colliander, Holmer and Tzirakis. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru, and Bejenaru and Herr.

14 pages. Final version to appear in "Communications on Pure and Applied Analysis"