Abstract
We consider the homogenization of diffusion-convective problems with given divergence-free velocities in nonperiodic structures defined by sequences of characteristic functions the first sequence . These quence of concentration the second sequence is uniformly bounded in the space of square-summable functions with square-summable derivatives with respect to spatial variables. At the same time, the sequence of time-derivative of product of these concentrations on the characteristic functions, that define a nonperiodic structure, is bounded in the space of square-summable functions from time interval into the conjugated space of functions depending on spatial variables, withsquare-summable derivatives. We prove the strong compactness of the second sequences in the space of quadratically summable functions and use this result to homogenize the corresponding boundary value problems that depend on a small parameter.
Highlights
We establish an Aubin-type compactness lemma [3, 9] for nonperiodic structures and apply it to find the homogenization of diffusion-convection equations for such kind of structures
We consider a diffusion-convection of some admixture with concentration cε during the movement of the liquid in the pore space with given divergence-free velocity vε
In this paper, we prove the compactness lemma for the domains with nonperiodic structures
Summary
We establish an Aubin-type compactness lemma [3, 9] for nonperiodic structures and apply it to find the homogenization of diffusion-convection equations for such kind of structures. We denote the pore space as Qεf = Ωεf × (0, T ) , the solid skeleton as Qεs = Ωεs × (0, T ) , and through. Γε = Ωεf ∩ Ωεs we denote the boundary between the pore space and the solid skeleton. We consider a diffusion-convection of some admixture with concentration cε during the movement of the liquid in the pore space with given divergence-free velocity vε. As a step we will show that the sequence {χε(x, t0) c ε(x, t0)} weakly converges to the function m(x, t0) c(x, t0) for almost all t0 ∈ (0, T ). As a last step, we prove that the sequence {c εk } converges strongly in L2(ΩT ) to the function c(x, t). Throughout the text we use notations of [8, 9] for the functional spaces and norms there
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