TITLE:
A Formulation of the Porous Medium Equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results
AUTHORS:
Koffi B. Fadimba
KEYWORDS:
Porous Medium Equation, Porosity, Saturation Equation, A Priori Estimates, Regularity Results
JOURNAL NAME:
Applied Mathematics,
Vol.15 No.10,
October
30,
2024
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
)
:
∂(
ϕS
)
∂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=ϕS
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
.