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 3d Klein-Gordon-Schrödinger system, which are systems in two variables $u:{\mathbb{R}}_x^d\times {\mathbb{R}}_t\to \mathbb{C}$ and $n:{\mathbb{R}}_x^d\times {\mathbb{R}}_t\to \mathbb{R}$. 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.

References in corpus (1)

• Local ill-posedness of the 1D Zakharov system

Cited by in corpus (2)

• Low Regularity Global Well-Posedness for the Klein-Gordon-Schrödinger System with the Higher Order Yukawa Coupling
• Global Well-Posedness and Non-linear Stability of Periodic Traveling Waves for a Schrodinger-Benjamin-Ono System