Smoothing for the Zakharov and Klein--Gordon--Schrödinger Systems on Euclidean Spaces

Abstract

This paper studies the existence and regularity properties of solutions to the Zakharov and Klein--Gordon--Schrodinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear $X^{s,b}$ estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein--Gordon--Schrodinger flow in the energy space for dimensions d = 2,3. Second, we use smoothing in conjunction with a high-low decomposition to show global well-posedness of the Klein--Gordon--Schrodinger evolution on $\mathbb{R}^4$ below the energy space for sufficiently small initial data.