Modeling capillary hysteresis in unsatured porous media (Q2012560)

The authors model cyclic hysteresis phenomena for flows in unsaturated porous media. They model these hysteresis effects by the differential inclusion \[ 0 \in \left\{ \frac{\partial S_w}{\partial t} - \Delta F_c \left( S_w, \text{sign} \left( \frac{\partial S_w}{\partial t} \right) \right) \right\}, \] where the graph of the multivalued operator \(F_c\) represents the hysteresis loop. After replacing \(F_c\) by an appropriate function \(\Phi\), the above formal problem is transformed to the evolution problem \[ \frac{\partial v}{\partial t} - \Delta \Phi(v) =0 \qquad\text{in } (0,T) \times \Omega\tag{1} \] endowed with homogeneous Neumann boundary conditions and initial data \(v(0) = S_w (0)\), where \(\Omega \subset \mathbb{R}^d\) is a bounded domain with Lipschitz boundary. Using a dynamical regularization process of Sobolev type, namely the third order problems \[ \begin{cases} \frac{\partial v_\lambda}{\partial t} - \Delta \Phi(v_\lambda) - \lambda \Delta \frac{\partial v_\lambda}{\partial t} =0 & \qquad\text{in } (0,T) \times \Omega, \\ \frac{\partial}{\partial n} \left( \Phi(v_\lambda) + \lambda \frac{\partial v_\lambda}{\partial t} \right) =0 & \qquad\text{on } (0,T) \times \partial \Omega, \end{cases} \] for \(\lambda >0\), it is shown with the help of the Yosida regularization that in the limit \(\lambda \searrow 0\) a generalized solution to (1) is obtained. This generalized solution then describes the hysteresis effect. For the proofs the authors mainly refer to [\textit{L.C. Evans} and \textit{M. Portilheiro}, Math. Models Methods Appl. Sci. 14, No. 11, 1599--1620 (2004; Zbl 1064.35091)] and [\textit{L.C. Evans}, Bull. Am. Math. Soc., New Ser. 41, No. 4, 409--438 (2004; Zbl 1053.35004)].