Khilkova L. Homogenized models of diffusion in a porous medium with nonlinear adsorption at the boundary

The thesis is devoted to the question of homogenization of boundary value problems for equations of diffusion in strongly perforated domains Ω_ε = Ω \ F_ε with a non-linear Robin's boundary condition. The domain of diffusion Ω_ε depends on the small parameter such that the perforating set F_ε becomes more and more loosened and distributes more densely in the fixed domain Ω as ε→0. Diffusion processes are considered in the perforated domains Ω_ε of two types: in strongly connected domains and in domains with fine-grained boundary. For both types of perforated structures, the theory of homogenization of the third boundary value problem with a non-linear boundary condition is of great interest and has not yet been constructed. The first part of the thesis is devoted to the homogenizing of diffusion processes in strongly connected domains and contains sections 2-4. Section 2 concerns a boundary value problem for a equation of stationary diffusion with a non-linear adsorption on the boundary of F_ε. For each fixed ε it is proved the existence of the unique solution u_ε(х). The asymptotic behavior of solutions u_ε(х) as ε→0 is studied, conditions of convergence are established and a homogenized equation is obtained. Section 3 deals with the domains of locally-periodic structure, for which explicit formulas of effective characteristics of the medium are obtained. Section 4 concerns an initial boundary value problem for an equation of non-stationary diffusion with non-linear absorption on the boundary and the transfer of the diffusing substance by fluid. For each fixed ε the existence of the unique solution u_ε(t,х) is proved. The asymptotic behavior of solutions u_ε(t,х) as ε→0 is studied, conditions of convergence are derived and a homogenized model of the diffusion process is obtained. Section 5 is devoted to diffusion processes in the domains with a fine-grained boundary. The corresponding boundary value problem is studied for the stationary diffusion equation in the domain, which is additional to a large number of fine (of radius O(ε^α), α >1) adsorbing grain-balls . The absorption function on surfaces of the grains is O(ε^β), β=β(α). The influence of the parameters α, β on the asymptotic of the solutions u_ε(х) is analysed, the conditions of convergence are determined and a homogenized equation is obtained.