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

Abstract

We study a reaction-diffusion-convection problem with nonlinear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, 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 an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.