Diffusion in Fissured Media

Source abstract

The nonlinear initial-boundary value problem \[\begin{gathered} \frac{{\partial u}}{{\partial t}} + \frac{1}{\varepsilon }(\alpha (u) - v) = f_1 ,\quad - {\operatorname{div}}\,(k{\operatorname{grad}}\,v) + \frac{1}{\varepsilon }(v - \alpha (u)) = f_2 \quad {\text{in }}G \times (0,T), \hfill \\ u(x,0) = u_0 (x)\quad {\text{in }}G,\quad v(s,t) = 0\quad {\text{on }}\partial G \times (0,T) \hfill \\ \end{gathered} \] is a well-posed model of diffusion in a fissured porous medium. Special features of the solution include the perseverance of local spatial continuity or singularities in the concentration u, the instantaneous propagation of the partially-saturated region throughout G, the delayed and limited advance of the fully-saturated region into G, and the concentration discontinuity on the boundary of the fully-saturated region. Weak maximum and order-comparison principles are obtained as $L^\infty$ and $L^1$ estimates on a solution and a difference of solutions, respectively.