On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also stated.Comment: 26 page

From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of $N$ player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article we emphasize the role of optimal transport theory in: 1) the passage from Nash to Cournot-Nash equilibria as the number of players tends to infinity, 2) the analysis of Cournot-Nash equilibria

Topological interactions in a Boltzmann-type framework

We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However, it exhibits a spatial non-locality instead of the classical non-locality in velocity space. This result relies on the approximation properties of Bernstein polynomials

Remarks on existence and uniqueness of Cournot-Nash equilibria in the non-potential case

This article is devoted to various methods (optimal transport, fixed-point, ordinary differential equations) to obtain existence and/or uniqueness of Cournot-Nash equilibria for games with a continuum of players with both attractive and repulsive effects. We mainly address separable situations but for which the game does not have a potential. We also present several numerical simulations which illustrate the applicability of our approach to compute Cournot-Nash equilibria

The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in \RR^d, d≥3d \ge 3

It is known that, for the parabolic-elliptic Keller-Segel system with critical porous-medium diffusion in dimension \RR^d, $d \ge 3$ (also referred to as the quasilinear Smoluchowski-Poisson equation), there is a critical value of the chemotactic sensitivity (measuring in some sense the strength of the drift term) above which there are solutions blowing up in finite time and below which all solutions are global in time. This global existence result is shown to remain true for the parabolic-parabolic Keller-Segel system with critical porous-medium type diffusion in dimension \RR^d, $d \ge 3$, when the chemotactic sensitivity is below the same critical value. The solution is constructed by using a minimising scheme involving the Kantorovich-Wasserstein metric for the first component and the $L^2$-norm for the second component. The cornerstone of the proof is the derivation of additional estimates which relies on a generalisation to a non-monotone functional of a method due to Matthes, McCann, & Savar\'e (2009)

A gradient flow approach to the Keller-Segel systems

These notes are dedicated to recent global existence and regularity results on the parabolic-elliptic Keller-Segel model in dimension 2, and its generalisation with nonlinear diffusion in higher dimensions, obtained throught a gradient flow approach in the Wassertein metric. These models have a critical mass Mc such that the solutions exist globally in time if the mass is less than Mc and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out

A gradient flow approach to the Keller-Segel systems

These notes are dedicated to recent global existence and regularity results on the parabolic-elliptic Keller-Segel model in dimension 2, and its generalisation with nonlinear diffusion in higher dimensions, obtained throught a gradient flow approach in the Wassertein metric. These models have a critical mass Mc such that the solutions exist globally in time if the mass is less than Mc and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out

Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions $\rho_\lambda$, $\lambda>0$, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional ${\mathcal H}_\lambda$ coming from the critical fast diffusion equation in $\R^2$. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for ${\mathcal H}_\lambda$. While the entropy dissipation for ${\mathcal H}_\lambda$ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of "controlled concentration" to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards $\rho_\lambda$. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.Comment: This version of the paper improves on the previous version by removing the small size condition on the value of the second Lyapunov functional of the initial data. The improved methodology makes greater use of techniques from optimal mass transportation, and so the second and third sections have changed places, and the current third section completely rewritte

A family of functional inequalities

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows