Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes
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Journal of Evolution Equations
Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes Adam Bobrowski
Abstract. Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.
1. Introduction 1.1. Main results The paper is devoted to a semigroup-theoretic approach to the problem of approximating solutions to an equation modeling diffusion in two thin 3D layers separated by a semi-permeable membrane: particles diffusing in the upper layer may, via a stochastic mechanism, filter through the membrane to the lower layer and continue their chaotic movement there, and vice versa. To this end, we study the reaction–diffusion equation ∂t u = 3D u + F(u)
(1.1)
where 3D is a 3D Laplace operator, and F is a Lipschitz continuous forcing (reaction) term, considered in two layers of thickness ε, equipped with boundary and transmission conditions (see (2.3) and (3.1), further down) that describe in particular the way the membrane works. As a starting point, we prove appropriate generation theorems in the Mathematics Subject Classification: 35K57, 47D06, 35B25, 35K58 Keywords: Semigroups of operators, Semilinear equations, Reaction–diffusion equations, Irregular convergence, Singular perturbations, Boundary and Transmission conditions, Thin layers. This research is supported by National Science Center (Poland) Grant 2017/25/B/ST1/01804.
A. Bobrowski
J. Evol. Equ.
spaces of square integrable and continuous functions, respectively (thus establishing existence and uniqueness of mild solutions of the equation). Next, we show that, as ε → 0, the approximating processes resemble more and more 2D Brownian motions on the upper and lower sides of the membrane. Remarkably, the limit process allows also jumps from one side to the other: this possibility is the limit equivalent of the mechanism of filtering through the membrane in the approximating process. More specifically, as ε → 0 and as looked upon through a magnifying glass (see below), solutions of (1.1) become more and more flat in the ve
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