  Motivated by the open problem of large-data global existence for the
non-cutoff Boltzmann equation, we introduce a model equation that in some sense
disregards the anisotropy of the Boltzmann collision kernel. We refer to this
model equation as isotropic Boltzmann, by analogy with the isotropic Landau
equation introduced by Krieger and Strain [Comm. Partial Differential Equations
37(4), 2012, 647--689]. The collision operator of our isotropic Boltzmann model
converges to the isotropic Landau collision operator under a scaling limit that
is analogous to the grazing collisions limit connecting (true) Boltzmann with
(true) Landau.
  Our main result is global existence for the isotropic Boltzmann equation in
the space homogeneous case, for certain parts of the "very soft potentials"
regime in which global existence is unknown for the space homogeneous Boltzmann
equation. The proof strategy is inspired by the work of Gualdani-Guillen [J.
Funct. Anal. 283(6), 2022, Paper No. 109559] on isotropic Landau, and makes use
of recent progress on weighted fractional Hardy inequalities.
