Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift

Raveendran, Vishnu; Cirillo, Emilio N. M.; Muntean, Adrian

doi:10.1090/qam/1622

Cited by 3 publications

(2 citation statements)

References 24 publications

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“…Indeed the importance of detecting the overall behaviour of a material which might include periodically distributed heterogeneties is very important in many applications, from nonlinear elasticity, mean-field games, to micro and ferromagnetic, conductivity, evolutions problems, polycrystals, discrete models, etc. Indeed we refer to [4,7,10,22,25,26,27,45] among a much wider bibliography.…”

Section: Introductionmentioning

confidence: 99%

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Tachago¹,

Gargiulo²,

Nnang³

et al. 2021

Evolution Equations &Amp; Control Theory

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Section: Introductionmentioning

confidence: 99%

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Tachago¹,

Gargiulo²,

Nnang³

et al. 2021

Evolution Equations &Amp; Control Theory

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“…In this section, we collect the main results from Paper I ( [38]), Paper II ( [37]), Paper III ( [36]), and Paper IV ([39]).…”

Section: Resultsmentioning

confidence: 99%

Scaling effects and homogenization of reaction-diffusion problems with nonlinear drift

Raveendran

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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.

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Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

Raveendran,

de Bonis,

Cirillo

et al. 2024

Quart. Appl. Math.

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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.

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Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift

Cited by 3 publications

References 24 publications

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Scaling effects and homogenization of reaction-diffusion problems with nonlinear drift

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

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