"Regularization" of the stationnary solution for diffusion in spherical coordinates

Question

In spherical coordinates $(r,\theta,\phi)$, the stationnary diffusion equation is a Laplace equation $\Delta f =0$. The solution in a radial symmetry : $f=f_{\infty}-r_0/r$ where $f_{\infty}$ and $r_0$ are constant has several issues :

1. First, $f$ is negative under a certain radius which is annoying if $f$ is for example a concentration

2. $f'(r)$ is not null for $r=0$, which is not compatible with the radial symmetry, that would impose $f'(0)=0$.

3. If $f$ is a concentration, the velocity is proportional to the derivative $v\sim\partial_r f$ and the flux is not zero $F\sim r^2 v \neq 0 $.

How can one make sense of that solution ? And what other phenomena help to regularize the solution by changing the equation ? Does that mean that there is no stationnary solution ?

Answer 1

If there is no sink/source at $r=0$ (which is implied by your equation), then putting $r_0=0$ solves your problem, doesn't it?

You may have some stationary diffusion due to difference in the boundary conditions. Then $f(\vec{x})\ne\text{const}$, but it has no singularity at $r=0$.