Aggregation

Summary

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.

Repulsive smooth polygon, no pinching
(a) Smooth star, no pinching
Repulsive smooth polygon with 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.