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

2006/03/27 by Jim Colliander, Colliander, Jim, Justin Holmer +3 · 3 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.math/0603595

openalex publication_date 2006/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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:ℝ_x^d× ℝ_t → ℂ and n:ℝ^d_x× ℝ_t→ ℝ. The Zakharov system is known to be locally well-posed in (u,n)∈ L²× H^-1/2 and the Klein-Gordon-Schrödinger system is known to be locally well-posed in (u,n)∈ L²× L². 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 L² norm of u and controlling the growth of n via the estimates in the local theory.

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