An approximate treatment of gravitational collapse

Abstract

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.