Thomas Laurent

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005; Holm and Putkaradze, 2005; Mogilner and Edelstein-Keshet, 1999; Morale et al., 2005; Topaz and Bertozzi, 2004; Topaz et al., 2006).

Dimensionality of Local Minimizers of the Interaction Energy

In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated with a repulsive–attractive potential. We show how the dimensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin.

AGGREGATION AND SPREADING VIA THE NEWTONIAN POTENTIAL: THE DYNAMICS OF PATCH SOLUTIONS

This paper considers the multidimensional active scalar problem of motion of a function ρ(x, t) by a velocity field obtained by v = -∇N * ρ, where N is the Newtonian potential. We prove well-posedness of compactly supported L∞ ∩ L1 solutions of possibly mixed sign. These solutions include an important class of solutions that are proportional to characteristic functions on a time-evolving domain. We call these aggregation patches. Whereas positive solutions collapse on themselves in finite time, negative solutions spread and converge toward a self-similar spreading circular patch solution as t → ∞. We give a convergence rate that we prove is sharp in 2D. In the case of positive collapsing solutions, we investigate numerically the geometry of patch solutions in 2D and in 3D (axisymmetric). We show that the time evolving domain on which the patch is supported typically collapses on a complex skeleton of codimension one.

Characterization of radially symmetric finite time blowup in multidimensional aggregation equations

This paper studies the transport of a mass

Parabolic Behavior of a Hyperbolic Delay Equation

\mu

Spatiotemporal chemotactic model for ant foraging

in

The Regularity of the Boundary of a Multidimensional Aggregation Patch

\mathbb{R}^d, d \geq 2,

Enhanced lasso recovery on graph

by a flow field

The planar optics phase sensor: a study for the VLTI 2nd generation fringe tracker

v= -\nabla K*\mu

Finite-time blow-up of solutions of an aggregation equation in R n

. We focus on kernels