Smoothing for the Zakharov & Klein-Gordon-Schr\"{o}dinger Systems on Euclidean Spaces

TL;DR

This paper demonstrates that solutions to the Zakharov and Klein-Gordon-Schrödinger systems become smoother over time at low regularity, using new bilinear estimates, with implications for long-term dynamics and global well-posedness.

Contribution

It introduces a novel bilinear $X^{s,b}$ estimate and applies smoothing techniques to establish global attractors and well-posedness results for these systems.

Findings

Proves solutions gain regularity over time at low initial regularity.

Establishes existence of global attractors in energy space for 2D and 3D.

Shows global well-posedness in 4D below energy space for small data.

Abstract

This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schr\"{o}dinger 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-Schr\"{o}dinger flow in the energy space for dimensions $d = 2,3$. Secondly, we use smoothing in conjunction with a high-low decomposition to show global well-posedness of the Klein-Gordon-Schr\"{o}dinger evolution on $\mathbb{R}^4$ below the energy space for sufficiently small initial data.