An approximate treatment of gravitational collapse

TL;DR

This paper analyzes a simplified gravitational collapse model, connecting it to known equations like Patlak-Keller-Segel, and explores properties such as well-posedness, smoothing effects, and bounds in the context of fractional Laplacian operators.

Contribution

It introduces a non-local generalization of gravitational collapse models with fractional Laplacian pressure, proving well-posedness and boundedness results, and providing numerical insights.

Findings

Proves local well-posedness in Sobolev spaces.

Demonstrates smoothing effects of the model.

Establishes bounds on maximum density for large pressure forces.

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.