An approximate treatment of gravitational collapse
  This work studies a simplified model of the gravitational instability of an
initially homogeneous infinite medium, represented by $\TT^d$, based on the
approximation that the mean fluid velocity is always proportional to the local
acceleration. It is shown that, mathematically, this assumption leads to the
restricted Patlak-Keller-Segel model considered by J\"ager and Luckhaus or,
equivalently, the Smoluchowski equation describing the motion of
self-gravitating Brownian particles, coupled to the modified Newtonian
potential that is appropriate for an infinite mass distribution. We discuss
some of the fundamental properties of a non-local generalization of this model
where the effective pressure force is given by a fractional Laplacian with
$0<\alpha<2$, and illustrate them by means of numerical simulations. Local
well-posedness in Sobolev spaces is proven, and we show the smoothing effect of
our equation, as well as a \emph{Beale-Kato-Majda}-type criterion in terms of
$\rhomax$. It is also shown that the problem is ill-posed in Sobolev spaces
when it is considered backward in time. Finally, we prove that, in the critical
case (one conservative and one dissipative derivative), $\rhomax(t)$ is
uniformly bounded in terms of the initial data for sufficiently large pressure
forces.
