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 : R x d × R t → C u:\mathbb {R}_x^d\times \mathbb {R}_t \to \mathbb {C} and n : R x d × R t → R 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 ) ∈ L 2 × H − 1 / 2 (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 ) ∈ L 2 × L 2 (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 L^2 norm of u u and controlling the growth of n n via the estimates in the local theory.