TITLE:
Comprehensive Variational Analysis and Finite Element Approximation of the p-Laplacian Equation Applied to Non-Newtonian Porous Media Flows
AUTHORS:
Serigne Diouf, Mamadou Diop, Mamadou Moustapha Diop
KEYWORDS:
Porous Media Flow, Non-Newtonian Fluids, -Laplacian, Variational Method, Monotone Operators, Finite Element Method, Newton-Raphson Method, Regularization
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.10,
October
29,
2026
ABSTRACT: In this paper, we present a comprehensive mathematical and numerical study of non-Newtonian fluid flow obeying a power-law model within a heterogeneous porous medium characterized by a spatial permeability field
K(
x
)
. By extending the classical Darcy’s law through the non-linear
p
-Laplacian differential operator, we account for local viscosity variations induced by steep pressure gradients. On the theoretical side, we establish the existence and uniqueness of a weak solution in the Sobolev space
W
1,p
(
Ω
)
restricted to parameter ranges where the lower-order term and energy functional are rigorously well-defined via appropriate continuous Sobolev embeddings (specifically
p≥max(
2,
2d
d+2
)
), using the Minty-Browder monotone operator theory and calculus of variations. Boundedness of the operator and full coercivity with respect to the full stated norm are explicitly established. To overcome the singularity or degeneracy of the operator, a parametric
ε
-regularization technique is developed, replacing the earlier convergence sketch with uniform estimates and a monotonicity argument to identify the weak limit and prove strong convergence of the full family. Explicit injection and production source functions with defined spatial supports and magnitudes are integrated. The power-law constitutive derivation is explicitly linked to the
p
-Laplacian form, reconciling the heterogeneous permeability field. Finally, the physical discussion is expanded to validate shear-thickening and shear-thinning profiles against Barenblatt-type similarity solutions and experimental benchmarks.