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Raveendran, Vishnu
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Department of Mathematics and Computer Science (from 2013)
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MathematicsMathematical Analysis
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Homogenization of reaction-diffusion problems with nonlinear drift in thin structuresPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2022 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Abstract [en]

##### Place, publisher, year, edition, pages

Karlstad: Karlstads universitet, 2022. , p. 112
##### Series

Karlstad University Studies, ISSN 1403-8099 ; 2022:14
##### Keywords [en]

Thin layer, homogenization, dimension reduction, reaction-diffusion-convection problem, two-scale convergence, effective transmission condition, fast drift, weak solvability of quasi-linear parabolic systems in unbounded domains.
##### National Category

Mathematics Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:kau:diva-89580ISBN: 978-91-7867-277-6 (print)ISBN: 978-91-7867-288-2 (electronic)OAI: oai:DiVA.org:kau-89580DiVA, id: diva2:1653018
##### Presentation

2022-05-31, 1B364, Karlstad University, Karlstad, 10:00 (English)
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##### Funder

Swedish Research Council, VR 2018-03648Available from: 2022-05-11 Created: 2022-04-20 Last updated: 2022-05-11Bibliographically approved
##### List of papers

We study the question of periodic homogenization of a variably scaled reaction-diffusion equation with non-linear drift of polynomial type. The non-linear drift was derived as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle. We consider three different geometries: (i) Bounded domain crossed by a finitely thin flat composite layer; (ii) Bounded domain crossed by an infinitely thin flat composite layer; (iii) Unbounded composite domain.\end{itemize} For the thin layer cases, we consider our reaction-diffusion problem endowed with slow or moderate drift. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive homogenized evolution equations and the corresponding effective model parameters. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interfaces. As a special scaling, the problem with large drift is treated separately for an unbounded composite domain. Because of the imposed large drift, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. To deal with this special case, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder's fixed point Theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in the unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials which are resistant to slow, moderate, and high velocity impacts.

We study the question of periodic homogenization of a variably scaled reaction-diffusion equation with non-linear drift of polynomial type. The non-linear drift was derived as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle. We consider three different geometries: (i) Bounded domain crossed by a finitely thin composite layer; (ii) Bounded domain crossed by an infinitely thin composite layer; (iii) Unbounded composite domain. For the thin layer cases, we consider our reaction-diffusion problem endowed with slow or moderate drift. Using energy-type estimates, concepts like thin-layer convergence and two-scale convergence, we derive homogenized equations. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interfaces. The problem with large drift is treated separately for an unbounded composite domain. Because of the imposed large drift, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. This study wants to contribute with theoretical understanding needed when designing thin composite materials which are resistant to slow, moderate, and high velocity impacts.

1. Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer$(function(){PrimeFaces.cw("OverlayPanel","overlay1607228",{id:"formSmash:j_idt799:0:j_idt804",widgetVar:"overlay1607228",target:"formSmash:j_idt799:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift$(function(){PrimeFaces.cw("OverlayPanel","overlay1650978",{id:"formSmash:j_idt799:1:j_idt804",widgetVar:"overlay1650978",target:"formSmash:j_idt799:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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