Description
Determination of the long-time behaviour of solutions to the two-dimensional Keller-Segel system with critical mass
This presentation will describe the dynamics of solutions to the parabolic-elliptic Keller-Segel system in two dimensions. A first behaviour is that the solution is global and converges to a forward self-similar solution, which occurs if the mass of the solution is less than the explicit threshold of 8 pi [Blanchet-Dolbeault-Perthame and related works]. A second behaviour is that the solution blows up in finite time, which happens if the mass is more than 8 pi. Explicit examples are given by the collapse of a single stationary state [Herrero-Velazquez] or the collision of several collapsing stationary states [Collot-Ghoul-Masmoudi-Nguyen].
The presentation will focus on the case of the critical mass of 8 pi, for solutions with finite second moment. Such solutions are global and concentrate in infinite time [Carrillo-Masmoudi], and an explicit example of a solution concentrating a stationary state at a scale log(t)^(-1/2) was constructed in the radial class [Ghoul-Masmoudi] and in the non-radial class [Davilá-del Pino-Dolbeault-Musso-Wei]. The main result that will be presented shows that this is the universal dynamics at critical mass, i.e. that all general solutions will converge to a stationary state which concentrates at scale log(t)^(-1/2) around the center of mass of the solution. The proof combines soliton resolution techniques, with the control of the evolution of the solution in various critical spaces, and with the perturbative analysis around an approximate solution involving multiple scales. This is based on joint work with Federico Buseghin 1