Venue
Sala Seminari (Dip. Matematica).
Abstract
We describe the one dimensional dynamics of a biological population influenced by the presence of a nonlocal attractive potential and a diffusive term, under the constraint that no overcrowding can occur. This setting can be expressed by a class of aggregation-diffusion PDEs with nonlinear mobility. We investigate the existence of weak type solutions obtained as many particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to non negative initial data with finite total variation, away from vacuum and supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of the nonlinear mobility term and the non strict monotonicity of the diffusion function, thus our result applies also to strongly degenerate diffusion equations. We also address the pure attractive regime, where we are able to achieve the stronger notion of entropy solution. Finally, we apply this deterministic method to a slightly different class of aggregation-diffusion equations, where the mobility is still linear in the density but not linear in the space variable and the nonlocal attractive term is a standard choice in the theory of opinion dynamics. This is based on joint works with M. Di Francesco and S. Fagioli.