Asymptotic behaviour for small mass in the two-dimensional parabolic–elliptic Keller–Segel model

Blanchet, Adrien; Dolbeault, Jean; Escobedo, Miguel; Fernandez, Javier Gonzalez

doi:10.1016/j.jmaa.2009.07.034

Cited by 36 publications

(52 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In Section 3.3, we prove that when the mass is sufficiently small, every solution to (1.1) with compactly supported initial data converges to the self-similar dissipating solution. We mention that for the PKS equation in 2D with linear diffusion, similar results for small-mass solutions are obtained by [11], and recently the small-mass assumption is removed by [15], where they prove that all solutions with subcritical mass converge to the self-similar solution with exponential rate after proper rescaling. Both of these results are based on a spectral gap method, hence cannot be generalized to (1.1) due to the nonlinear diffusion term.…”

Section: Introductionsupporting

confidence: 69%

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Yao¹

2014

Ann. Inst. H. Poincaré Anal. Non Linéaire

View full text Add to dashboard Cite

In this paper we study the long time asymptotic behavior for a class of diffusion-aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak-Keller-Segel problem with critical power m = 2 − 2/d, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. We also apply the mass comparison method to another aggregation model with repulsive-attractive interaction, and prove that radial solutions converge to the stationary solution exponentially fast.

show abstract

Section: Introductionsupporting

confidence: 69%

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Yao¹

2014

Ann. Inst. H. Poincaré Anal. Non Linéaire

View full text Add to dashboard Cite

show abstract

“…Theorem 1.5 drastically improves some anterior results which establish the same exponential rate of convergence for some particular class of initial data. On the one hand, for a radially symmetric initial datum with finite second moment it has been proved in [15 [7,16] for previous results in that direction) for any initial datum f 0 with mass M ∈ (0, 8π) and which satisfies (roughly speaking) the strong confinement condition f 0 ≤G for some self-similar profileG associated to some massM ∈ [M, 8π). In that last work [17], the uniform exponential stability (with optimal rate) of the linearized rescaled equation is established in L 2 (G −1/2 ) by the mean of the analysis of the associated linear operator in a well chosen (equivalent) Hilbert norm.…”

Section: 7)mentioning

confidence: 99%

Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case

Fernández

Mischler

2015

Arch Rational Mech Anal

View full text Add to dashboard Cite

Abstract. The present paper deals with the parabolic-elliptic Keller-Segel equation in the plane in the general framework of weak (or "free energy") solutions associated to initial datum with finite mass M , finite second moment and finite entropy. The aim of the paper is threefold:(1) We prove the uniqueness of the "free energy" solution on the maximal interval of existence [0, T * ) with T * = ∞ in the case when M ≤ 8π and T * < ∞ in the case when M > 8π. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the "optimal regularity" as well as an estimate of the difference of two possible solutions in the critical L 4/3 Lebesgue norm similarly as for the 2d vorticity Navier-Stokes equation.(2) We prove immediate smoothing effect and, in the case M < 8π, we prove Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables).(3) In the case M < 8π, we also prove weighted L 4/3 linearized stability of the self-similar profile and then universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.

show abstract

“…Proof. This result can be proved directly by applying the Method of Trap in [4]. So we omit the details.…”

Section: 21mentioning

confidence: 86%

The flux limited Keller–Segel system; properties and derivation from kinetic equations

2019

View full text Add to dashboard Cite

The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyperbolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently.We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.

show abstract

Asymptotic behaviour for small mass in the two-dimensional parabolic–elliptic Keller–Segel model

Cited by 36 publications

References 12 publications

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case

The flux limited Keller–Segel system; properties and derivation from kinetic equations

Contact Info

Product

Resources

About