Discrete and continuum links to a nonlinear coupled transport problem of interacting populations
This work advances theoretical understanding of coupled transport in interacting populations, but it is incremental as it extends known variational structures and hydrodynamic limits to a specific problem.
The paper develops microscopic models for coupled transport flux in interacting populations, inspired by pedestrian dynamics, and establishes links to cross-diffusion and thermo-diffusion problems. It provides closed-form solutions, Lyapunov functionals, and numerical approximations for both the continuum limit and particle systems.
We are interested in exploring interacting particle systems that can be seen as microscopic models for a particular structure of coupled transport flux arising when different populations are jointly evolving. The scenarios we have in mind are inspired by the dynamics of pedestrian flows in open spaces and are intimately connected to cross-diffusion and thermo-diffusion problems holding a variational structure. The tools we use include a suitable structure of the relative entropy controlling TV-norms, the construction of Lyapunov functionals and particular closed-form solutions to nonlinear transport equations, a hydrodynamics limiting procedure due to Philipowski, as well as the construction of numerical approximates to both the continuum limit problem in 2D and to the original interacting particle systems.