A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results
- 1 Department of Computer Science, Engineering, and Mathematics, University of South Carolina Aiken, Aiken, USA
Abstract
We consider a generalized form of the porous medium equation where the porosity ϕ is a function of time t : ϕ = ϕ ( x , t ) : ∂ ( ϕ S ) ∂ t − ∇ ⋅ ( k ( S ) ∇ S ) = Q ( S ) . In many works, the porosity ϕ is either assumed to be independent of (or to depend very little of) the time variable t . In this work, we want to study the case where it does depend on t (and x as well). For this purpose, we make a change of unknown function V = ϕ S in order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕ is independent of the time t . These results are then extended to the full saturation equation with time-dependent porosity ϕ = ϕ ( x , t ) . In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕ by the introduced transport vector w , through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q ≡ 0 .
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