# Computing the potential from the continuity equation

1. Mar 20, 2009

### mathf

Dear All,
I need help on the following issue. Assuming the flow to be potential, I want to compute the potential given the density at all times, that is :

From the continuity equation:

$$\partial _t \rho + \nabla \cdot \left( {\rho \nabla \phi } \right) = 0$$

One can write down an elliptic PDE for finding the potential at a
fixed time t (this is a PDE in space only):

$$- \rho \Delta \phi - \nabla \rho \cdot \nabla \phi = \partial _t \rho$$

If one wants to impose the homogeneous Neumann boundary conditions,
what are the results about the existence and the regularity of the
solution?