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

Abstract

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.

On the mean field limit for Brownian particles with Coulomb interaction in 3D

“…Recall that ρ dx ∈ C([0, T ]; C b (R 3 ) ′ ), we have ρ ε ∈ C([0, T ]; L 1 (R 3 )). Using basically the same argument as in Step 1 of the proof of Lemma 2.5 in [43], we obtain that ρ ε is a Cauchy sequence in L 1 (δ, T ; L 1 loc ) as ε → 0. In fact, for convex function β ∈ C 1 (R 3 ), we have the following chain rule: ∂ t β(ρ ε ) = ∇β(ρ ε ) · ∇h + β ′ (ρ ε )r ε + ∆β(ρ ε ) − β ′′ (ρ ε )|∇ρ ε | 2 .…”

“…Next, following the proof of Lemma 2.5 in [43], taking some non-negative test function χ ∈ C ∞ c (R 3 ) and integrate (B.6) over [δ 1 , t 1 ] where δ ≤ δ 1 ≤ t 1 , we find…”

Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos

“…The uniqueness of weak solutions to the KS model has been concerned by many scholars. The optimal transport method [16] and the renormalizing argument [18] have been used to prove the uniqueness of weak solutions to the classical KS model with normal Laplacian term. Here we will follow the method in [31] to prove the uniqueness for the generalized KS model (1).…”

Fractional Keller–Segel equation: Global well-posedness and finite time blow-up

“…This assumption is sufficient to enjoy the Log-Lipschitz regularity of the nonlinear drift K * ρ, as in this case K is the Newtonian kernel (see for instance [29]). It is possible to relax this assumption to L ln L initial data [16] or even measure initial data [1]. Large time behavior is also studied in [8,10,16].…”

“…It is possible to relax this assumption to L ln L initial data [16] or even measure initial data [1]. Large time behavior is also studied in [8,10,16]. In higher dimension, the variant case α = 2, a = d = 3 is studied in [15], where a finite time blow-up is obtained under a concentration of initial mass condition.…”