We consider the motion of a density of particles rho(x,t) by a velocity field v(x,t) obtained by convolving the density of particles with the gradient of the Newtonian potential, that is v = - grad N* rho. An important class of solutions are the ones where the particles are uniformly distributed on a time evolving domain. We refer to these solutions as aggregation patches, by analogy to the vortex patch solutions of the 2D incompressible Euler equations. Numerical simulations as well as some exact solutions show that the time evolving domain on which the patch is supported typically collapses on a complex skeleton of codimension one. We also show that going backward in time, any bounded compactly supported solution converges as t goes to minus infinity toward a spreading circular patch. We provide a rate of convergence which is sharp in 2D. This is a joint work with Bertozzi and Leger.
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