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Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0001-5168-0841
Sapienza Università di Roma, Italy.
Sapienza Università di Roma, Italy.ORCID iD: 0000-0003-3673-2054
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0002-1160-0007
2024 (English)In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485Article in journal (Refereed) Epub ahead of print
Abstract [en]

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2024.
Keywords [en]
Homogenization, reaction-diffusion equations with large nonlinear drift, two-scale convergence with drift, strong convergence in moving coordinates, effective dispersion tensors for reactive flow in porous media
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-97991DOI: 10.1090/qam/1687ISI: 001162782500001OAI: oai:DiVA.org:kau-97991DiVA, id: diva2:1827526
Funder
Swedish Research Council, 2018-03648Available from: 2024-01-14 Created: 2024-01-14 Last updated: 2024-03-08Bibliographically approved
In thesis
1. Scaling effects and homogenization of reaction-diffusion problems with nonlinear drift
Open this publication in new window or tab >>Scaling effects and homogenization of reaction-diffusion problems with nonlinear drift
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We study the periodic homogenization of reaction-diffusion problems with nonlinear drift describing the transport of interacting particles in composite materials. The microscopic model is derived as the hydrodynamic limit of a totally asymmetric simple exclusion process for a population of interacting particles crossing a domain with obstacles. We are particularly interested in exploring how the scalings of the drift affect the structure of the upscaled model.

We first look into a situation when the interacting particles cross a thin layer that has a periodic microstructure. To understand the effective transmission condition, we perform homogenization together with the dimension reduction of the aforementioned reaction-diffusion-drift problem with variable scalings.

One particular physically interesting scaling that we look at separately is when the drift is very large compared to both the diffusion and reaction rate. In this case, we consider the overall process taking place in an unbounded porous media. Since we have the presence of a large nonlinear drift in the microscopic problem, we first upscale the model using the formal asymptotic expansions with drift. Then, with the help of two-scale convergence with drift, we rigorously derive the homogenization limit for a similar microscopic problem with a nonlinear Robin-type boundary condition. Additionally, we show the strong convergence of the corrector function. 

In the large drift case, the resulting upscaled equation is a nonlinear reaction-dispersion equation that is strongly coupled with a system of nonlinear elliptic cell problems. We study the solvability of a similar strongly coupled two-scale system with nonlinear dispersion by constructing an iterative scheme. Finally, we illustrate the behavior of the solution using the iterative scheme.

Abstract [en]

We study the homogenization of reaction-diffusion problems with nonlinear drift. The microscopic model is derived as the hydrodynamic limit of a totally asymmetric simple exclusion process of interacting particles. We first look into a situation when the interacting particles cross a thin composite layer. To understand the effective transmission condition, we perform the homogenization and dimension reduction of the model with variable scalings. One physically interesting scaling that we look at separately is when the drift is large. In this case, we consider the overall process taking place in an unbounded porous media. We first upscale the model using the asymptotic expansions with drift. Then, using two-scale convergence with drift, we rigorously derive the homogenization limit for a similar microscopic problem with a nonlinear boundary condition. Additionally, we show the strong convergence of the corrector function. In the large drift case, the resulting upscaled model is a nonlinear reaction-dispersion equation strongly coupled with a system of nonlinear elliptic cell problems. We study the solvability of a similar strongly coupled two-scale system with nonlinear dispersion by constructing an iterative scheme. Finally, we illustrate the behavior of the solution using the iterative scheme.

Place, publisher, year, edition, pages
Karlstads universitet, 2024. p. 24
Series
Karlstad University Studies, ISSN 1403-8099 ; 2024:7
Keywords
homogenization, asymptotic expansion with drift, two-scale convergence with drift, effective transmission condition, dimension reduction, two-scale system, nonlinear dispersion
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-98720 (URN)10.59217/fjww2863 (DOI)978-91-7867-440-4 (ISBN)978-91-7867-441-1 (ISBN)
Public defence
2024-04-18, Sjöström lecture hall, 1B309, Karlstads universitet, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2024-03-28 Created: 2024-02-29 Last updated: 2024-03-28Bibliographically approved

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Raveendran, VishnuMuntean, Adrian

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