Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $  \mathbb{R}^+ $

E. Compaan; N. Tzirakis

doi:10.3934/dcds.2019156

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2019, Volume 39, Issue 7: 3867-3895. Doi: 10.3934/dcds.2019156

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Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $ \mathbb{R}^+ $

E. Compaan^1, and
N. Tzirakis^2, ,

1.
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2.
University of Illinois, Urbana-Champaign, IL 61801, USA

^* Corresponding author: N. Tzirakis
^* Corresponding author: N. Tzirakis

Received: March 31, 2018

Revised: December 31, 2018

Published: April 2019

E.C. was supported by a National Physical Science Consortium fellowship and by NSF MSPRF #1704865. N.T. was supported by a grant from the Simons Foundation (#355523 Nikolaos Tzirakis) and by Illinois Research Board, RB18051

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Abstract

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schrödinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the $L^2$ conservation law of the Schrödinger part with a careful iteration of the rough wave part in lower order Sobolev norms in the spirit of the work in [5].

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Mathematics Subject Classification: Primary: 35Q70, 35M32.

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References

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