The Gelfand–Shilov smoothing effect for the radially symmetric homogeneous Landau equation with Shubin initial datum

Abstract

In this paper, we study the Cauchy problem associated with the radially symmetric spatially homogeneous non-cutoff Landau equation with Maxwellian molecules, while the initial datum belongs to negative-index Shubin space, which can be characterized by spectral decomposition of the harmonic oscillators. Based on this spectral decomposition, we construct the weak solution with Shubin's class initial datum, and then we prove the uniqueness and the Gelfand–Shilov smoothing effect of the solution to this Cauchy problem.