Degenerate diffusion in porous media with hysteresis-dependent permeability

TL;DR

This work addresses degenerate diffusion in unsaturated porous media with a hysteresis-dependent permeability by extending the convexification approach to a model where $s_t=\mathrm{div}(\kappa(x,s)\nabla u)$ and $s=G[u]$ with $G$ a convexifiable Preisach operator. The authors develop a time-discretization scheme and establish uniform bounds and a convexity estimate that remain valid when the permeability depends on the hysteresis saturation. They introduce anisotropic Hilbert-space embeddings and Orlicz-space tools, notably using $\Phi_{log}(v)=v\log(1+v)$, to obtain compactness and pass to the limit, proving existence of a solution with $u\in L^{\infty}$, $\nabla u\in L^2$, and $u_t,s_t\in L^{\Phi_{log}}$. The main novelty lies in handling saturation-dependent permeability within the convexification framework, while the question of uniqueness remains open and warrants further investigation.

Abstract

Hysteresis in the relation between the capillary pressure and the moisture content in unsaturated porous media, which is due to surface tension at the liquid-gas interface, exhibits strong degeneracy in the resulting mass balance equation. Solutions to such degenerate equations have been recently constructed by the method of convexification. We show here that the convexification argument works even if the permeability coefficient depends on the hysteretic saturation. The problem of uniqueness remains open in this case.

Theorems & Definitions (11)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Theorem 2.6
• Proposition 3.1
• Lemma 4.1
• Lemma 5.1
• Lemma 5.2
• Lemma 5.3
• Proposition 6.1