Computing the potential from the continuity equation

[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?

If you have any knowledge about this problem, please help.

Thanks.

MF

 

[jvicens]

Dear MF,

Thank you for reaching out for help with your issue. The problem you have described is known as the "inverse potential problem" and it has been studied extensively in the field of fluid mechanics. The goal of this problem is to find the potential given the density at all times, as you have stated.

The PDE you have written is correct, and it is known as the Poisson equation for the potential. This equation can be solved numerically using various methods, such as finite difference or finite element methods. However, the existence and regularity of the solution depends on the boundary conditions and the smoothness of the density function.

If you impose homogeneous Neumann boundary conditions, which means that the normal derivative of the potential is zero at the boundaries, then the solution may not exist or may not be unique. This is because the Neumann boundary conditions do not fully constrain the solution.

In order to guarantee the existence and uniqueness of the solution, you may need to impose additional boundary conditions, such as Dirichlet boundary conditions which specify the value of the potential at the boundaries. This would make the problem well-posed and the solution can be found using numerical methods.

Regarding the regularity of the solution, it depends on the smoothness of the density function. If the density function is sufficiently smooth, then the potential solution will also be smooth. However, if the density function has discontinuities or singularities, then the potential solution may also have similar irregularities.

I hope this helps you with your problem. If you have any further questions, please do not hesitate to ask.