Behaviour of mass and momentum distributions under Newtonian Gravity

Question

In the context of this question should mass distribution $\rho(r,t)$ and momentum distribution $p(r,t)$ be well behaved ? By 'well behaved' it is meant that derivatives of all orders exist everywhere.

I don't see any reason why they should be well behaved. For the equations (given in the answer (by Mark Eichenlaub) to the above mentioned question) to be consistent the conditions required are

1. $\vec{p}$ should be continuously differentiable in order that $\nabla\cdot\vec{p}$ exists.

2. $\rho(r,t)$ should be partially differentiable wrt $t$ i.e., $\frac{\partial\rho(\vec{r},t)}{\partial t}$ exists.

   Please give strong reasons as to why it is required that $\rho(r,t)$ and $\vec{p}(r,t)$ to be always well behaved ?

Answer 1

Will $\rho$ always behave well? No, not at all.

This is physics, not mathematics -- if something is wrong with solutions, the explanation is simple: too bad for equations, probably approximations behind them lack some process or the mathematical description starts to be invalid.

In your case, it is obvious that a cloud with no angular momentum will be collapsing into one point, yet this is not a problem -- it is just "physical" that there will be some repulsion that would start to be significant in high density regions invalidating this simplified model.

Answer 2

They are not necessary well behaved. But if they're not you will use distributions instead of functions. For example, if $\rho$ is not continuous, $\partial\rho/\partial t$ will behave like a Dirac distribution. The maths are just more complex, rarely bring new insight about the physics, that's why physicists often assume the functions to be regular enough.