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Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems
Authors:James Colliander  Justin Holmer  Nikolaos Tzirakis
Institution:Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4

Justin Holmer ; Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Nikolaos Tzirakis ; Department of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, Canada M5S 2E4

Abstract:We prove low regularity global well-posedness for the 1d Zakharov system and the 3d Klein-Gordon-Schrödinger system, which are systems in two variables $ u:\mathbb{R}_x^d\times \mathbb{R}_t \to \mathbb{C}$ and $ n:\mathbb{R}^d_x\times \mathbb{R}_t\to \mathbb{R}$. The Zakharov system is known to be locally well-posed in $ (u,n)\in L^2\times H^{-1/2}$ and the Klein-Gordon-Schrödinger system is known to be locally well-posed in $ (u,n)\in L^2\times L^2$. Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the $ L^2$ norm of $ u$ and controlling the growth of $ n$ via the estimates in the local theory.

Keywords:Zakharov system  Klein-Gordon-Schr\"odinger system  global well-posedness
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