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

By Adrien Blanchet, Eric A. Carlen, José Antonio Carrillo. 2010

181 citations recorded by the source corpus.

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 $\rhohls$, $\lambda>0$, 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 $\Fcfd$ 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 $\Fcfd$. While the entropy dissipation for $\Fcfd$ 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 {\em 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 $\rhohls$. 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.

Subfields: Applied Mathematics, Modeling and Simulation, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d81f2822dfba0f059d31ce0ed96afded399a26b9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Modeling and Simulation

In Statistics and Probability

Citations in the corpus

Refined asymptotics for the subcritical Keller-Segel system and related functional inequalities (2010, background): “y the evolution of the Wasserstein distance to one of the equilibria (1.8) in the critical case χ= 1. Notice that equilibria are inﬁnitely far from each other with respect to the Wasserstein distance [8]. Using the gradient ﬂow structure with respect to W 2, one obtains the following formula for the”
Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, background): “For this reason, the rigorous results of the gradient flow theory are not directly applicable, however, certain aspects may be recovered, such as the use of steepest descent schemes [11, 12].”
Equilibria of homogeneous functionals in the fair-competition regime (2016, background): “The critical case χ = 1 was analyzed further in [7, 5, 15] in terms of stability of stationary states.”
Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, background): “1) possesses a critical mass: if ‖u0‖1 8π then classical solutions exist for all time (see for example [8,9,11,13,63]) and if ‖u0‖1 > 8π then all classical solutions with finite second moment blow up in finite time [13,46,54] and are known to concentrate at least 8π mass into a single point at blow-”
Asymptotic Estimates for the Parabolic-Elliptic Keller-Segel Model in the Plane (2012, background): “simple computation of the second moment shows that smooth solutions with mass larger than 8πblow-up in ﬁnite time; see for instance [27]. The case M= 8πhas been extensively studied. We shall refer to [5, 6, 7] for some recent papers on this topic. The asymptotic regime is of a very diﬀerent nature i”
Sparse stabilization and optimal control of the Cucker-Smale model (2013, methodology): “Models in microbiology, such as the Patlak-Keller-Segel model [36, 46], describing the chemotactical aggregation of cells and multicellular micro-organisms, inspired a very rich mathematical literature [32, 33, 48], see also the very recent work [3] and references therein.”
Uniqueness for Keller-Segel-type chemotaxis models (2012, methodology): “s a trivial case since the Newtonian potential is given by B0,1(x) = |x|. For α = ε = 0 and d = 2 since log(e + |x|2)ρ ∈ L1(Rd) then vρ ∈ L1(Rd) using the logarithmic HLS inequality, see for instance [10]. Notation for the ambient metric space. We let Xε := M2(Rd;m) × L2(Rd) endowed with the distanc”
On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, background): “ 4 p Mπ/λ(1−M/8π)t. So that in the case M>8π, if Hλ[ρ0] is initially bounded, and thus of inﬁnite 2-moment, then there cannot be global-in-time solutions. Based on this remark, the main results of [9] in the critical case can be summarised in the following: Theorem 10(Existence of global solution”
Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime (2020, background): “ solutions are expected, whereas when the self-attraction dominates the diﬀusion (M > Mc), blow-up solutions at ﬁnite time may be detected. (As to the corresponding asymptotic analysis we refer to [6, 10, 13].) 2010 Mathematics Subject Classiﬁcation. Primary: 35K55, 35A01, 35Q92. Secondary: 92C17”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.”