Aggregation and Spreading via 5 the Newtonian Potential : 6 the Dynamics of Patch Solutions

This paper considers the multidimensional active scalar problem of motion of a function 25 ρ(x, t) by a velocity field obtained by v = −∇N ∗ρ, where N is the Newtonian potential. 26 We prove well-posedness of compactly supported L∞ ∩ L1 solutions of possibly mixed 27 sign. These solutions include an important class of solutions that are proportional to 28 characteristic functions on a time-evolving domain. We call these aggregation patches. 29 Whereas positive solutions collapse on themselves in finite time, negative solutions spread 30 and converge toward a self-similar spreading circular patch solution as t → ∞. We give 31 a convergence rate that we prove is sharp in 2D. In the case of positive collapsing 32 solutions, we investigate numerically the geometry of patch solutions in 2D and in 3D 33 (axisymmetric). We show that the time evolving domain on which the patch is supported 34 typically collapses on a complex skeleton of codimension one. 35