Multi-species PDE systems with nonlocal interactions and small inertia

The present PhD thesis deals with the study of aggregation PDE systems with many species coupled through nonlocal interaction and considering inertial effects. We first study a multi-dimensional system, considering both smooth and singular self-interaction potentials and requiring smooth assumptions on cross-interaction potentials. We provide existence and uniqueness results of measure solutions considering initial data in a Wasserstein space of probability measures. Then, we investigate the small inertia limits for both the smooth and singular case, proving convergence results towards the corresponding macroscopic first order systems. These results extend to the many species case previous results by Fetecau-Sun and Choi-Jeong. We construct an upwind finite volume scheme for a kinetic system with two species. Here, the inertia term is not considered, and we require smooth assumptions on interaction potentials. A convergence result for the scheme is provided, without any restriction on the mesh size. This result is inspired by previous result by Filbet with minor modifications and a slight improvement of the rate of convergence. Furthermore, we study a one-dimensional macroscopic system for two species coupled through nonlocal interactions, with an additional damping parameter. This system describes the dynamics of interacting particles; in case of collisions a sticky particles condition is adopted. We prove existence and uniqueness of measure solutions by using optimal transportation theory and taking initial data in a space of probability measures with finite second moments. A large-time large-damping result is obtained, proving the convergence towards the corresponding first order system. Finally, we investigate the case with Newtonian potentials for the self-interaction terms, with additional confining external potentials. For the latter case, we prove existence of solutions and a large time collapse result, showing the convergence towards Dirac delta solutions. The results are complemented with numerical simulations. Previous results on this problem only dealt with the one species case, see Brenier et al. for the existence of sticky particles and Carrillo, Choi and Tse for the large damping limit. We stress that the technique we use in the large damping limit is totally new.