Low regularity well-posedness for the 3D Klein-Gordon-Schr\"odinger system

TL;DR

This paper establishes low regularity local well-posedness for the 3D Klein-Gordon-Schrödinger system, identifying optimal conditions for initial data regularity and proving unconditional uniqueness in natural solution spaces.

Contribution

It provides the first low regularity well-posedness results for the 3D Klein-Gordon-Schrödinger system, extending previous work by leveraging new results for the Zakharov system.

Findings

Well-posedness for s > -1/4, σ > -1/2 under certain conditions

Optimal regularity thresholds up to endpoints

Unconditional uniqueness for s=σ=0 in L^2-based spaces

Abstract

The Klein-Gordon-Schr\"odinger system in 3D is shown to be locally well-posed for Schr\"odinger data in H^s and wave data in H^{\sigma} \times 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) \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.