A mixed Darcy–power-law mobility model for pressure diffusion in complex fluids: Constitutive structure, global well-posedness, regularization, and asymptotics

José Luis Díaz Palencia; José Luis Díaz Palencia

doi:10.3934/era.2026395

Electronic Research Archive

2026, Volume 34, Issue 12: 9406-9437. doi: 10.3934/era.2026395

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A mixed Darcy–power-law mobility model for pressure diffusion in complex fluids: Constitutive structure, global well-posedness, regularization, and asymptotics

José Luis Díaz Palencia ^,

Department of Mathematics and Education, Universidad a Distancia de Madrid (UDIMA), Madrid 28400, Spain

Received: 12 July 2026 Revised: 04 September 2026 Accepted: 10 September 2026 Published: 11 November 2026

A scalar parabolic model was proposed for pressure equalization in a fluid-saturated complex medium in which two dissipative mobility channels act in parallel. The pressure flux contains a persistent Darcy contribution and a gradient-activated power-law contribution. In dimensional variables, the closure is
$ Q = -K_1\nabla P-K_p\left|{\nabla P}\right|^{p-2}\nabla P, \qquad K_1, K_p>0, \quad p>2, $
and the corresponding nondimensional equation is
$ \partial_tu- \operatorname{div}\!\left(\nu\nabla u+\kappa\left|{\nabla u}\right|^{p-2}\nabla u\right) = f. $
This is a standard monotone operator with $ (2, p) $-growth. No novelty was claimed for the algebraic sum of the Laplacian and the $ p $-Laplacian. The model-level contribution was the constitutive interpretation and its measurable crossover observables. On bounded connected Lipschitz domains, including homogeneous mixed boundaries, an implicit-Euler/Browder–Minty construction proved global well-posedness, energy balance, comparison, positivity, continuous dependence, and pure-Neumann mass conservation. The convex gradient-flow structure gave positive-time energy regularization and sign-preserving $ L^2 $-to-$ L^\infty $ smoothing. A separate whole-space theorem proved conditional Gaussian asymptotics with a first-moment correction. For $ p\ge3 $ and $ L^1\cap L^2 $ initial data, a new lemma verified the basic flux hypotheses for $ 1\le r\le p' = p/(p-1) $. The high-gradient similarity and crossover-time predictions were explicitly heuristic, not transition-time bounds.
- nonlinear parabolic equation,
- Darcy law,
- power-law fluid,
- monotone operator,
- gradient flow,
- heat kernel,
- asymptotic expansion
Citation: José Luis Díaz Palencia. A mixed Darcy–power-law mobility model for pressure diffusion in complex fluids: Constitutive structure, global well-posedness, regularization, and asymptotics[J]. Electronic Research Archive, 2026, 34(12): 9406-9437. doi: 10.3934/era.2026395

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Abstract

A scalar parabolic model was proposed for pressure equalization in a fluid-saturated complex medium in which two dissipative mobility channels act in parallel. The pressure flux contains a persistent Darcy contribution and a gradient-activated power-law contribution. In dimensional variables, the closure is

$ Q = -K_1\nabla P-K_p\left|{\nabla P}\right|^{p-2}\nabla P, \qquad K_1, K_p>0, \quad p>2, $

and the corresponding nondimensional equation is

$ \partial_tu- \operatorname{div}\!\left(\nu\nabla u+\kappa\left|{\nabla u}\right|^{p-2}\nabla u\right) = f. $

This is a standard monotone operator with $ (2, p) $-growth. No novelty was claimed for the algebraic sum of the Laplacian and the $ p $-Laplacian. The model-level contribution was the constitutive interpretation and its measurable crossover observables. On bounded connected Lipschitz domains, including homogeneous mixed boundaries, an implicit-Euler/Browder–Minty construction proved global well-posedness, energy balance, comparison, positivity, continuous dependence, and pure-Neumann mass conservation. The convex gradient-flow structure gave positive-time energy regularization and sign-preserving $ L^2 $-to-$ L^\infty $ smoothing. A separate whole-space theorem proved conditional Gaussian asymptotics with a first-moment correction. For $ p\ge3 $ and $ L^1\cap L^2 $ initial data, a new lemma verified the basic flux hypotheses for $ 1\le r\le p' = p/(p-1) $. The high-gradient similarity and crossover-time predictions were explicitly heuristic, not transition-time bounds.

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