Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ²

Abstract

We analyze the two-dimensional parabolic-elliptic Patlak-Keller-Segel model in the whole Euclidean space R 2 . Under the hypotheses of integrable initial data with finite second moment and entropy, we first show local in time existence for any mass of "free-energy solutions", namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlled from above. The main result of the paper is to show the global existence of free-energy solutions with initial data as before for the critical mass 8 π/χ. Actually, we prove that solutions blow-up as a delta Dirac at the center of mass when t → ∞ keeping constant their second moment at any time. Furthermore, all moments larger than 2 blow-up as t → ∞ if initially bounded.

The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels

“…(1.2) This model shares some features with the classical Patlak-Keller-Segel model for chemotaxis [33,46] without diffusion, see [16,12,13,26] for the state of the art in this problem. Here, the main similarity is the possible formation of a finite time point concentration and the main difference the strong singularity of the potential in the Patlak-Keller-Segel system.…”

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

“…In fact, the classical Patlak-Keller-Segel [35,24] system, see [12,10,11], corresponds to the choice of the Newtonian potential in R 2 as interaction, W = 1 2π log |x| with linear diffusion. In the case without diffusion, a notion of weak measure solutions was introduced in [36] for which the author proved global-in-time existence, although uniqueness is lacking.…”

Finite-Time Blowup in a Supercritical Quasilinear Parabolic-Parabolic Keller-Segel System in Dimension 2

“…Thanks to more careful studies, more refined conditions on initial data yielding blowup have been identified (see [2]) and the detailed quantitative way of the occurrence of blowup has been shown, see for instance [11,16,17]. Next the situation when the initial data have critical mass has been studied ( [4][5][6]). Finally, in the case of a quasilinear system, for any space dimension n critical nonlinearities have been identified such that if φ and ψ satisfy the subcritical relation, then solutions to (1.1) stay bounded for any time, while for those satisfying the supercritical relation solutions blow up in finite time independently of the magnitude of initial mass provided the data are concentrated enough, see [10].…”