This paper studies the transport of a mass µ in R d, d ≥ 2, by a flow field v = −∇K ∗µ. We focus on kernels K = |x | α /α for 2 − d ≤ α < 2. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius. The monotonicity is preserved for all time, in contrast to the case α> 2 where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential (α = 2 − d) we show that under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. It follows that there exists a unique classical solution for all time in the case of monotone data, and a solution defined by a choice of a jump condition in the case of general radially symmetric data. In the case 2 − d < α < 2 and at the critical exponent p we exhibit initial data in L p for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.

In this paper we derive evolution equations for the 2D active scalar problem when the solution is supported on 1D curve(s). These equations are a generalization of the Birkhoff-Rott equation when vorticity is the active scalar. The formulation is Lagrangian and they are valid for nonlocal kernels K that may include both a gradient and an incompressible term. We develop a numerical method for implementing the model which achieves second order convergence in space and fourth order in time. We verify the model by simulating classic active scalar problems such as the vortex sheet problem (in the case of inviscid, incompressible flow) and the collapse of delta ring solutions (in the case of pure aggregation), finding excellent agreement. We then study two examples with kernels of mixed type- i.e., kernels that contain both incompressible and gradient flows. The first example is a vortex density model which arises in superfluids. We analyze the effect of the added gradient component on the Kelvin-Helmholtz instability. In the second example, we examine a nonlocal biological swarming model and study the dynamics of density rings which exhibit complicated milling behavior.

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K ∗ ρ) in Rd, d ≥ 2, where K(r) = r γ /γ with γ> 2. It was previously observed 1 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ> 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ> 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two dimensional (in-)stability implies n dimensional (in-)stability.

dynamics for the aggregation equation with degenerate diffusion