Search

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., 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.

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.

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer. This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts., Homogenization and dimension reduction of thin heterogeneous layers

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.