# Aggregation

## Summary

Consider the active scalar problem of motion of a function $\rho(x,t)$ by a velocity field $$\label{eq:aggregation} v=-\nabla N*\rho$$ where $N$ is the Newtonian potential. Non positive solutions that are compactly supported converge in the long time to a radially symmetric self similar solution.

Equation \eqref{eq:aggregation} has solutions of the form $$\rho(x,t)=h(t)\textbf{1}_{\Omega(t)}(x)$$ which we call patch solutions. $\Omega(t)$ is a subset of $\mathbf{R}^N$. Non negative patch solutions blow up in finite time. Numerically the patches collapse on skeletons of codimension $1$ (see movies, pictures).

## Movies!

Below are movies of the time evolution of the boundary of a patch $\Omega(t)$, both in 3D and 2D, for non-negative solutions (attraction) as well as (rescaled) non-positive solutions (repulsion).

For the 2D movies, the red curve is the boundary $\partial\Omega(0)$ of the patch at $t=0$, and the evolving green curve is the boundary $\partial\Omega(t)$ of the patch.

Cube (attractive, 3D)
Square torus (attractive, 3D)
Smooth square (attraction, 2D)
Elliptical annulus (attraction, 2D)
Non connected shape: two disks (attraction, 2D)
Smooth star (repulsion, 2D)

## Pictures

Two examples of contour dynamics for the rescaled spreading problem in two dimensions.

Fig. (a)-(b) show snapshots of the boundary evolution of a patch. At time s = 10 the patch has reached steady state. The initial boundary is shown as a red dashed line and the long time limit (s = 10) is shown as a green solid line.

Fig. (b) exhibits the "pinching phenomena" of the boundary.

(a) Smooth star, no pinching
(b) Smooth star with pinching

## References

A. Bertozzi, T. Laurent, and F. Léger. Aggregation via the Newtonian potential and aggregation patches. M3AS, vol. 22, Supp. 1, 2012.