12:00
On this talk we will focus on the family of aggregation-diffusion equations
∂ρ∂t=div(m(ρ)∇(U′(ρ)+V)).
Here, m(s) represents a continuous and compactly supported nonlinear mobility (saturation) not necessarily concave. U corresponds to the diffusive potential and includes all the porous medium cases, i.e. U(s)=1m−1sm for m>0 or U(s)=slog(s) if m=1. V corresponds to the attractive potential and it is such that V≥0, V∈W2,∞.
Taking advantage of a family of approximating problems, we show the existence of C0-semigroups of L1 contractions. We study the ω-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. Furthermore, since this problem has a formal gradient-flow structure, we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the L∞-constrained gradient flow of probability densities. Finally, we explore the properties of a corresponding implicit finite volume scheme introduced by Bailo, Carrillo and Hu.
The talk presents joint work with Prof. J.A. Carrillo and Prof. D. Gómez-Castro.