Upscaling of a system of semilinear parabolic partial differential equations coupled with a system of nonlinear ordinary differential equations originating in the context of crystal dissolution and precipitation inside a porous medium: existence theory and periodic homogenization - Mahato, Hari Shankar; Kräutle, Serge; Böhm, Michael; Knabner, Peter (Advances in Mathematical Sciences and Applications, 2017)
In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible, i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is two- fold: first we show the existence of a unique positive global weak solution for the coupled systems and then we upscale (homogenize) the model from the micro scale to the macro scale. For the existence of solution, some regularization techniques, Schaefer’s fixed point theorem and Lyapunov type arguments have been used whereas the concepts of two-scale convergence and periodic unfolding are used for the homogenization.<br
Keywords: semilinear parabolic partial differential equations; nonlinear ordinary differential equations; reversible reactions; fixed point theorems; Lyapunov functionals; maximal regularity; periodic homogenization; two-scale convergence; periodic unfolding