The aim of this survey is to serve as an introduction to the different techniques available in
the broad field of aggregation-diffusion equations. We aim to provide historical context, key …

We solve explicitly a certain minimization problem for probability measures in one
dimension involving an interaction energy that arises in the modeling of aggregation …

Densities of particles on R n which interact pairwise through an attractive-repulsive power-
law potential W α, β (x)=| x| α/α-| x| β/β have often been used to explain patterns produced by …

This paper is devoted to a new family of reverse Hardy–Littlewood–Sobolev inequalities
which involve a power law kernel with positive exponent. We investigate the range of the …

Particles interacting through long-range attraction and short-range repulsion given by power-
laws have been widely used to model physical and biological systems and to predict or …

We focus on a family of nonlinear continuity equations for the evolution of a non-negative
density $\rho $ with a continuous and compactly supported nonlinear mobility $\mathrm …

In this paper we derive a variety of functional inequalities for general homogeneous
invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities …

We study well-posedness and long-time behaviour of aggregation–diffusion equations of the
form∂ ρ∂ t= Δ ρ m+∇⋅(ρ (∇ V+∇ W⁎ ρ)) in the fast-diffusion range, 0< m< 1, and V and W …

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure
space with two negative exponents. For applications we show the reverse Hardy–Littlewood …

We solve explicitly a certain minimization problem for probability measures involving an
interaction energy that is repulsive at short distances and attractive at large distances. We …