The purpose of this thesis is the derivation of corrector estimates justifying the upscaling of systems of partial differential equations (PDEs) with coupled fluxes posed in media with microstructures (like porous media). Such models play an important role in the understanding of, for example, drug-delivery mechanisms, where the involved chemical species diffusing inside the domain are assumed to obey perhaps other transport mechanisms and certain non-dissipative nonlinear processes within the pore space and at the boundaries of the perforated media (e.g. interaction, chemical reaction, aggregation, deposition). In this thesis, our corrector estimates provide a quantitative analysis in terms of convergence rates in suitable norms, i.e. as the small homogenization parameter tends to zero, the differences between the micro- and macro-concentrations and between the corresponding micro- and macro-concentration gradients are controlled in terms of the small parameter. As preparation, we are first concerned with the weak solvability of the microscopic models as well as with the fundamental asymptotic homogenization procedures that are behind the derivation of the corresponding upscaled models. We report results on three connected mathematical problems: 1. Asymptotic analysis of microscopic semi-linear elliptic equations/systems. We explore the asymptotic analysis of a prototype model including the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems, and particularly, in justifying rigorously the obtained averaged descriptions. We prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations. Then we use the structure of the two-scale expansions to derive corrector estimates delimitating quantitatively the convergence rate of the asymptotic approximates to the macroscopic limit concentrations and their gradients. High-order corrector estimates are also obtained. The semi-linear auxiliary problems are tackled by a fixed-point homogenization argument. Our techniques include also Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as suitable energy estimates. 2. Corrector estimates for a Smoluchowski-Soret-Dufour model. We consider a thermodiffusion system, which is a coupled system of PDEs and ODEs that account for the heat-driven diffusion dynamics of hot colloids in periodic heterogeneous media. This model describes the joint evolution of temperature and colloidal concentrations in a saturated porous tissue where the Smoluchowski interactions for aggregation process and a linear deposition process take place. By a fixed-point argument, we prove the local existence and uniqueness results for the upscaled system. To obtain the corrector estimates, we exploit the concept of macroscopic reconstructions as well as suitable integral estimates to control boundary interactions. 3. Corrector estimates for a non-stationary Stokes-Nernst-Planck-Poisson system. We investigate a non-stationary Stokes-Nernst-Planck-Poisson system posed in a perforated domain as originally proposed by Knabner and his co-authors (see e.g. [98] and [99]). Starting off with the setting from [99], we complete the results by proving corrector estimates for the homogenization procedure. Main difficulties are connected to the choice of boundary conditions for the Poisson part of the system as well as with the scaling of the Stokes part of the system.

Corrector homogenization estimates for PDE Systems with coupled fluxes posed in media with periodic microstructures / Vo, ANK KHOA. - (2018 Jan 16).

Corrector homogenization estimates for PDE Systems with coupled fluxes posed in media with periodic microstructures

VO, ANK KHOA
2018-01-16

Abstract

The purpose of this thesis is the derivation of corrector estimates justifying the upscaling of systems of partial differential equations (PDEs) with coupled fluxes posed in media with microstructures (like porous media). Such models play an important role in the understanding of, for example, drug-delivery mechanisms, where the involved chemical species diffusing inside the domain are assumed to obey perhaps other transport mechanisms and certain non-dissipative nonlinear processes within the pore space and at the boundaries of the perforated media (e.g. interaction, chemical reaction, aggregation, deposition). In this thesis, our corrector estimates provide a quantitative analysis in terms of convergence rates in suitable norms, i.e. as the small homogenization parameter tends to zero, the differences between the micro- and macro-concentrations and between the corresponding micro- and macro-concentration gradients are controlled in terms of the small parameter. As preparation, we are first concerned with the weak solvability of the microscopic models as well as with the fundamental asymptotic homogenization procedures that are behind the derivation of the corresponding upscaled models. We report results on three connected mathematical problems: 1. Asymptotic analysis of microscopic semi-linear elliptic equations/systems. We explore the asymptotic analysis of a prototype model including the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems, and particularly, in justifying rigorously the obtained averaged descriptions. We prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations. Then we use the structure of the two-scale expansions to derive corrector estimates delimitating quantitatively the convergence rate of the asymptotic approximates to the macroscopic limit concentrations and their gradients. High-order corrector estimates are also obtained. The semi-linear auxiliary problems are tackled by a fixed-point homogenization argument. Our techniques include also Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as suitable energy estimates. 2. Corrector estimates for a Smoluchowski-Soret-Dufour model. We consider a thermodiffusion system, which is a coupled system of PDEs and ODEs that account for the heat-driven diffusion dynamics of hot colloids in periodic heterogeneous media. This model describes the joint evolution of temperature and colloidal concentrations in a saturated porous tissue where the Smoluchowski interactions for aggregation process and a linear deposition process take place. By a fixed-point argument, we prove the local existence and uniqueness results for the upscaled system. To obtain the corrector estimates, we exploit the concept of macroscopic reconstructions as well as suitable integral estimates to control boundary interactions. 3. Corrector estimates for a non-stationary Stokes-Nernst-Planck-Poisson system. We investigate a non-stationary Stokes-Nernst-Planck-Poisson system posed in a perforated domain as originally proposed by Knabner and his co-authors (see e.g. [98] and [99]). Starting off with the setting from [99], we complete the results by proving corrector estimates for the homogenization procedure. Main difficulties are connected to the choice of boundary conditions for the Poisson part of the system as well as with the scaling of the Stokes part of the system.
Corrector estimates, PDEs with coupled fluxes, Thermo-diffusion system, Driftdiffusion-reaction system, Weak solvability, Homogenization
Corrector homogenization estimates for PDE Systems with coupled fluxes posed in media with periodic microstructures / Vo, ANK KHOA. - (2018 Jan 16).
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/9693
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