Homogenized Model of Non-Stationary Diffusion in Porous Media with the Drift

Abstract

We consider an initial boundary-value problem for a parabolic equation describing non-stationary diffusion in porous media with non-linear absorption on the boundary and the transfer of the diffusing substance by fluid. We prove the existence of the unique solution for this problem. We study the asymptotic behavior of a sequence of solutions when the scale of microstructure tends to zero and obtain the homogenized model of the diffusion process.

Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

“…The existence and uniqueness of solutions of is quite classical. It can be obtained using an adaptation of the Galerkin method to nonlinear problems: See Ladyzhenskaya et al and Lions in connection with the method and different nonlinear problems; see, eg, Goncharenko and Khilkova, Goncharenko and Jagër et al for different methods in volume perforated media. Note that under the conditions ( n ≤ 4, respectively) the integral ( , respectively) arising in the weak formulation is well defined, cf Brillard et al and .…”

“…Indeed, taking into account that the Sobolev embedding from H 1 (Ω ϵ , ∂ Ω) into L q (Ω ϵ ) is continuous when , we can write The same restriction holds for the weak formulations of the homogenized problems due to the term . Let us refer to Brillard et al for more details about the proof of and to Goncharenko and Khilkova for different restrictions on that could avoid the restriction on n but that might not work when justifying the weak formulation of the homogenized problems, cf, eg, and . Note that the estimate proves to be essential in order to obtain uniform bounds for the solutions, cf , and show convergence.…”

“…The convergence in the general case mentioned above is currently the subject of research. For stationary models without an advection term, the homogenized problems read .Finally, we refer to Goncharenko and Khilkova for the convergence in a model with an advection term, as in , but with the obstacles distributed over the whole volume and with very different assumptions on σ , on the parameters of the problem, and on the velocity field.…”

“…We refer to Gómez et al for a precise description of the physical model under consideration in the stationary case. See also Goncharenko and Khilkova, and references therein, in connection with these kinds of models in perforated domains over the whole volume.…”

“…See Cioranescu et al and Conca for some critical relations of parameters on the boundary conditions of fluid models in porous media. For recent works on non‐stationary diffusion models in porous media, cf, eg, Goncharenko and Khilkova, Allaire and Hutridurga, Cabarrubias and Donato and references therein.…”