Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions d ≥ 2 and in all of space for d ≥ 3, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow up is possible for initial data of arbitrary mass. 1

solutions via maximum principle

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.

The L1-critical parabolic-elliptic Patlak-Keller-Segel system is a classical model of chemotactic aggregation in micro-organisms well-known to have critical mass phenomena [10, 8]. In this paper we study this critical mass phenomenon in the context of Patlak-Keller-Segel models with spatially varying diffusivity and decay rate of the chemo-attractant. The primary tool for the proof of global existence below the critical mass is the use of pseudo-differential operators to precisely evaluate the leading order quadratic portion of the potential energy (interaction energy). Under the assumption of radial symmetry, blow-up is proved above critical mass using a maximum-principle type argument based on comparing the mass distribution of solutions to a barrier consisting of the unique stationary solutions of the scale-invariant PKS. Although effective where standard Virial methods do not apply, this method seems to be dependent on the assumption of radial symmetry. For technical reasons we work in dimensions three and higher where L1-critical variants of the PKS have porous media-type nonlinear diffusion on the organism density, resulting in finite speed of propagation and simplified functional inequalities. 1

We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to m> 1 of the McKean–Vlasov equation where here the “diffusive ” portion of the dynamics are governed by Porous medium self–interactions. We focus primarily on m ∈ (1, 2] with particular emphasis on m = 2. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For m = 2, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter. 1