Communicated by N. Bellomo and F. Brezzi 24 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. 26We prove well-posedness of compactly supported L ∞ ∩ L 1 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. 29Whereas 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.

We present a primal-dual method to solve L 1 -type non-smooth optimization problems independently of the grid size. We apply these results to two important problems : the Rudin-Osher-Fatemi image denoising model and the L 1 earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 × 4096 in a few minutes without parallelization.

We present a method to efficiently solve the optimal transportation problem for a general class of strictly convex costs. Given two probability measures supported on a discrete grid with n points we compute the optimal transport map between them in O(n log(n)) operations and O(n) storage space. Our approach allows us to solve optimal transportation problems on spatial grids as large as 4096 × 4096 and 384 × 384 × 384 in a matter of minutes.

We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L 1 norm of the velocity field. Existing results in the literature use an L p norm with p > 1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.