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

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.