Continuity Equation for Momentum Momentum is a conserved quantity, which makes me wonder if we can write an equation for the local conservation of momentum in the form of a continuity equation.  If we're considering a system of discrete objects, momentum isn't really a vector field.  It is a quantity intrinsic to each object in the space.  So I'm not sure if a continuity equation makes sense in this case.
If there is such an equation, it should be the mathematical statement that the total momentum of a system of bodies plus any changes in momentum from forces (or potential) originating outside of the system is zero.  Can we write such an equation?
 A: Of course we can. First off, when discussing conversation laws, continuity typically refers to the conservation of mass specifically:
$$\frac{D \rho}{D t} = 0 $$
in a Lagrangian frame or:
$$\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_i}{\partial x_i} = 0$$
Now, for momentum. This equation holds for both fluids and solids and is an expression of Newton's Second Law. In a Lagrangian frame:
$$ \frac{D \rho u_i}{D t} + \frac{\partial \sigma_{ij}}{\partial x_j} = 0$$ 
where $\sigma_{ij}$ is the Cauchy stress tensor. And likewise, for an Eulerian frame, this is:
$$ \frac{\partial \rho u_i}{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} + \frac{\partial \sigma_{ij}}{\partial x_j} = 0 $$
The form of the Cauchy stress tensor depends on what is being studied of course. It is also possible to write relativistic forms of this equation (and this is probably a better link).
When this is specialized for a fluid in a gravitational field, we get:
$$ \frac{\partial \rho u_i}{\partial t} + \frac{\partial \rho u_i u_j}{\partial x_j} = - \frac{\partial p}{\partial x_i} + \mu \frac{\partial^2 u_i}{\partial x_i x_j} + \rho g_i$$
where the Cauchy stress tensor is split into the pressure and deviatoric stress components. A virtually identical equation set can be written for solids as well.
