# adjoint method for degenerate problems

For the sake of the rest of the question, I'm interested in the porous medium equation

$$S\frac{\partial\phi}{\partial t} = \nabla\cdot K\phi\,\nabla\phi$$

where $$S$$ and $$K$$ are spatially-variable coefficients and $$\psi$$ (the hydraulic head) is the field to be solved for. This is a parabolic PDE, but it's degenerate since the "diffusivity" $$K\phi$$ of the medium can go to zero whenever $$\phi \to 0$$. As a consequence, the $$\phi = 0$$ contour propagates with finite speed in contrast to the usual infinite propagation speed of disturbances for non-degenerate parabolic PDE.

There are well-posedness results for the porous medium equation and convergent discretization schemes that don't regularize away the degeneracy. So in practice the weirdness of a diffusion coefficient going to zero can be overcome.

Now suppose we want to estimate the hydraulic conductivity $$K$$ from observational data. In order to do that, we'll probably use the adjoint method. The adjoint of the linearization of the PDE is

$$S\frac{\partial\psi}{\partial t} = -\nabla\cdot K\phi\,\nabla\psi + K\nabla\phi\cdot\nabla\psi + \ldots$$

where the ellipses denote some source terms that don't particularly matter. This is solved for the adjoint state $$\psi$$ backwards on some interval $$[0, T]$$ with a final condition imposed on $$\psi|_{t = T}$$. The adjoint linearization is an advection-diffusion equation, again with a diffusion coefficient ($$K\phi$$) that can and probably does go to zero somewhere.

Does the degeneracy of the adjoint equation pose a problem in practice? I've seen papers on overcoming the degeneracy of the forward problem, but I haven't seen anyone discuss the impact (if any) on the inverse problem. I'd add that degenerate problems show up in more fields than just hydrology -- you get similar equations for ice flow and wildfire spread.