Simone Fagioli: Solutions to aggregation

The lecture was held within the framework of the Hausdorff Trimester Program: Kinetic Theory

Abstract:
Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation Abstract: We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility coupled with nonlinear diffusion, arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the Followthe-Leader scheme. In the purely aggregative case, we rigorously prove that a suitable discrete piecewise density reconstructed from the particle scheme converges strongly towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. In presence of diffusion, we prove convergence of
the aforementioned scheme to weak solutions of the aggregation–diffusion PDEs. The main novelties concern the presence of a nonlinear mobility term and the non-strict monotonicity of the diffusion function. As a consequence, our result applies also to strongly degenerate diffusion equations. This is a joint work with M.Di Francesco and E. Radici (University of L’Aquila)