Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

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.