Parabolic Elliptic Keller Segel System Research Articles

The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in ℝ d , d ≥ 3

It is known that, for the parabolic-elliptic Keller-Segel system with critical porous-medium diffusion in dimension ℝ d , d ≥ 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 ℝ d , d ≥ 3, when the chemotactic sensitivity is below the same critical value. The solution is constructed by using a minimizing scheme involving the Monge-Kantorovich metric for the first component and the L2-norm for the second component. The cornerstone of the proof is the derivation of additional estimates based on a method due to [22].

Blow-up dynamics of self-attracting diffusive particles driven by competing convexities

In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is an approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel system in high dimensions.

Sharp condition for blow-up and global existence in a two species chemotactic Keller–Segel system in 2

For the parabolic–elliptic Keller–Segel system in 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.

Existence and uniqueness theorem on weak solutions to the parabolic–elliptic Keller–Segel system

In Rn (n⩾3), we first define a notion of weak solutions to the Keller–Segel system of parabolic–elliptic type in the scaling invariant class Ls(0,T;Lr(Rn)) for 2/s+n/r=2 with n/2<r<n. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for every initial data in Ln/2(Rn). We prove also their uniqueness. As for the marginal case when r=n/2, we show that if n⩾4, then the class C([0,T);Ln/2(Rn)) enables us to obtain the only weak solution.

Convergence of a Stochastic Particle Approximation for Measure Solutions of the 2D Keller-Segel System

The analysis of a stochastic interacting particle scheme for the approximation of measure solutions of the parabolic-elliptic Keller–Segel system in 2D is continued. In previous work it has been shown that solutions of a regularized scheme converge to solutions of the regularized Keller–Segel system, when the number of particles tends to infinity. In the present work, the regularization is eliminated in the particle model, which requires an application of the framework of time dependent measures with diagonal defects, developed by Poupaud. The subsequent many particle limit of the BBGKY hierarchy can be solved using measure solutions of the Keller–Segel system and the molecular chaos assumption. However, a uniqueness result for the limiting hierarchy and therefore a proof of propagation of chaos is missing. Finally, the dynamics of strong measure solutions, i.e., sums of smooth distributions and Delta measures, of the particle model is discussed formally for the cases of 2 and 3 particles. The blow-up behavior for more than 2 particles is not completely understood.

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

In two space dimensions, the parabolic-parabolic Keller-Segel system shares many properties with the parabolic-elliptic Keller-Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold M(c). However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above M(c) always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller-Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above M(c), which is forbidden in the parabolic-elliptic case.