In this paper, we propose a new nonlinear model describing the dynamical interaction of two species within a viscous flow. The proposed model is a cross-diffusion system coupled with the Brinkman problem written in terms of velocity fluid, vorticity, and pressure and describing the flow patterns driven by an external source depending on the distribution of species. In the first part, we derive macroscopic models from the kinetic-fluid equations by using the micro-macro decomposition method. On the basis of the Schauder fixed-point theory, we prove the existence of weak solutions for the derived model in the second part. The last part is devoted to developing a one-dimensional finite volume approximation for the kinetic-fluid model, which is uniformly stable along the transition from kinetic to macroscopic regimes. Our computation method is validated with various numerical tests.
KEYWORDSBrinkman flows, cross-diffusion, finite volume method, kinetic theory, Schauder fixed-point theory 6288
This paper is concerned with the modeling and mathematical analysis of vehicular traffic phenomena. We adopt a kinetic theory point of view, under which the microscopic state of each vehicle is described by: (i) position, (ii) velocity and also (iii) activity, an additional varible that we use to describe the quality of the driver-vehicle micro-system. We use methods coming from game theory to describe interactions at the microscopic scale, thus constructing new models within the framework of the Kinetic Theory of Active Particles; the resulting models incorporate some of the symmetries that are commonly found in the mathematical models of the kinetic theory of gases. Short-range interactions and mean field interactions are introduced and modeled to depict velocity changes related to passing phenomena. Our main goal is twofold: (i) to use continuum-velocity variables and (ii) to introduce a non-local acceleration term modeling mean field interactions, related to, for example, the presence of tollgates or traffic highlights.
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