Jesús Rosado

TL;DR: A continuous analogue of the theorems of [F. Cucker and S. Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852–862] is shown to hold for the solutions on the kinetic model, which means that the solutions will concentrate exponentially fast in velocity to the mean velocity of the initial condition, while in space they will converge towards a translational flocking solution.

TL;DR: In this article, the authors give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others.

TL;DR: In this paper, the authors considered radially symmetric kernels where the singularity at the origin is of order |x|α, α > 2 − d, and proved local well-posedness of the aggregation equation ∂tu + div(uv) = 0, v = −▿K * u with initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb )$ in dimensions 2 and higher.

TL;DR: In this paper, the authors extend the method used in [22] to prove uniqueness of solutions to a family of several nonlocal equations containing aggregation terms and aggregation/diusion competition.

TL;DR: It is shown how the entropy method applies for quantifying explicitly the exponential decay towards Fermi–Dirac and Bose–Einstein distributions in the one-dimensional case.