Consider the active scalar problem of motion of a function $\rho(x,t)$ by a velocity field
\begin{equation}
\label{eq:aggregation}
v=-\nabla N*\rho
\end{equation}
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.