Evolution of mass and velocity distributions under newtonian gravitation

Let $\rho(r,t)$ and $v(r,t)$ be mass and velocity distributions. Given $\rho(r,0)$ and $v(r,0)$ (initial conditions) what is the differential equation that describes the evolution of $\rho(r,t)$ and $v(r,t)$ in space and time, assuming Newtonian gravitation?

You may assume $r$ to be one dimensional, if it makes writing the equations convenient.

EDIT

I think $v(r,t)$ should be replaced with $p(r,t) = \rho(r,t)v(r,t)$ and name it as momentum distribution to avoid some technical faults.

EDIT 2

It seems that to completely characterize an inviscid fluid flow, Euler equations are needed.

The law of conservation of mass is already given in the answer. A component of pressure is missing in the equation related to conservation of momentum.(first equation). Lastly a third equation corresponding to law of conservation of energy should be introduced.(3rd one of the euler equations of inviscid fluid flow).For the third equation we assume that the internal energy per unit volume of the fluid is zero.

Answer

Newton's law of gravitation says that to get the acceleration at a given point, you take any distant bit of mass and make a vector pointing towards it with length proportional to the mass and inversely proportional to the square of the distance to it. Integrate that expression over the entire mass distribution.

$\frac{\partial{\vec{v}}(\vec{r},t)}{\partial t} + [\vec{v}(\vec{r},t)\cdot\nabla]\vec{v}(\vec{r},t) = G \int_{\textrm{space}}\frac{\rho(\vec{r'},t)}{|(\vec{r'}-\vec{r})|^3}(\vec{r'}-\vec{r}) \textrm{d}V(\vec{r'})$

$G$ is Newton's gravitational constant, and $\textrm{d}V(\vec{r'})$ is a volume element at the location $\vec{r'}$.

Alternatively, in terms of $\vec{p}$,

$\frac{\textrm{d}\vec{p}}{\textrm{d}t} = G\rho\int_{\textrm{space}}\frac{\rho'}{|\vec{r'}-\vec{r}|^3}(\vec{r'}-\vec{r}) \textrm{d}V(\vec{r}')$

Conservation of mass requires that the net flow of mass into a region of space result in an increase in the density there.

$\frac{\partial\rho(\vec{r},t)}{\partial t} = \nabla \cdot \vec{p}(\vec{r},t)$