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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 $\phi$ (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, surface water runoff, algae growth in ocean columns, and wildfire spread.

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  • $\begingroup$ I haven't worked with porous media equations, but I have worked with ones that are spatially varying where the diffusivity can be zero (or even infinity). If the diffusivity is zero this just means $\partial_t \phi = 0$, i.e. $\phi$ remains fixed at that location. In the problems that I solve this is equivalent to prescribing a Dirichlet condition (since it doesn't evolve). There's nothing too worrying about it, except maybe that in 2D and higher, if this occurs at a single point you get a logarithmic singularity. $\endgroup$
    – lightxbulb
    Dec 4, 2023 at 18:26

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