  We study the global wellposedness of pressure-less Eulerian dynamics in
multi-dimensions, with radially symmetric data. Compared with the 1D system, a
major difference in multi-dimensional Eulerian dynamics is the presence of the
spectral gap, which is difficult to control in general. We propose a new pair
of scalar quantities that provides a significant better control of the spectral
gap. Two applications are presented. (i) the Euler-Poisson equations: we show a
sharp threshold condition on initial data that distinguish global regularity
and finite time blowup; (ii) the Euler-alignment equations: we show a large
subcritical region of initial data that leads to global smooth solutions.

Analysis of PDEs