Speaker
Description
Nonlocal aggregation equations with nonlinear mobility arise naturally
in the study of interacting particle systems subject to congestion and density
constraints. Their evolution can be formally interpreted as a gradient
flow of the interaction energy with respect to a generalized transport geometry
induced by the mobility.
In this talk, I will investigate the long-time behaviour of entropy solutions
to a class of aggregation equations with attractive interaction potentials
and saturating Lipschitz mobility. After establishing global wellposedness
and confinement of mass, I will focus on the case of quadratic
interactions, where the dynamics exhibit a particularly rich interplay between
energy dissipation and geometric constraints.
A key result is the strong L^1 relative compactness of trajectories, obtained
without uniform-in-time BV estimates. Combined with the dissipation
of the interaction energy, this allows us to prove that every asymptotic
accumulation point is a fully saturated stationary state, taking only
the values zero and the maximal admissible density.
Finally, I will discuss the geometry of these limiting configurations,
showing that stationary states are characterized by a measure-theoretic
star-shapedness condition. In particular, despite the attractive nature
of the interactions, the dynamics do not necessarily select the energyminimizing
disk, revealing a nontrivial distinction between stationarity
and energy minimization in aggregation models with degenerate mobility.
This is a joint work with M. Perugini.