Chapters 5 and 6 are a joint work with J. Dolbeault. They are devoted to the study of the parabolic-elliptic Keller-Segel model
which describes the motion of unicellular amoebae, like dictyostelium discoideum. Here u denotes their spatial density and it makes sense to consider them in a two-dimensional setting. A straightforward computation (see [109, pages 122–124] and [122]) shows that solutions (with second moment initially finite) blow-up in finite time if the total mass is large enough (larger than 8⇡ with our conventions), while, for solutions with smaller masses, the diffusion dominates the large time asymptotics.
More precisely, it has been shown in [100, 72, 66, 88], that if n0 2L1+⇣ then there exists a solution u, in the sense of distributions, that is global in time and such that M = R
R3u(x, t) dx is preserved along the evolution. There is no non-trivial stationary solution to (2.3.1) and any solution converges to zero locally as time gets large. It is therefore convenient to study the asymptotic behavior of u in self-similar variables, where space and time scales are given respectively by R(t) :=p In this variables the system can be written as
8>
Existence of a stationary solution to (2.3.3) has been established in [76] by ODE techniques, and in [107]
by PDE methods. This stationary solution is unique according to [79]. Moreover it has been shown in [66]
that nand rc converge ast! 1, respectively in L1(R2) and L2(R2) to this unique stationary solution which involves smooth and radially symmetric functions.
A simple computation of the second moment shows that smooth solutions with mass larger than 8⇡ blow-up in finite time; see for instance [100]. The case M = 8⇡ has been extensively studied. We shall refer to [79, 80, 81] for some recent papers on this topic. The asymptotic regime is of a very different nature in such a critical case. In the chapters 5 and 6 we shall restrict our purpose to the sub-critical case M <8⇡.
The rate of convergence towards the stationary solution in self-similar variables gives, after undoing the change of variables, the rate of convergence towards the asymptotic profile for the solutions of (2.3.1).
In [65], it has been proved that ifM is less than some massM⇤ 2(0,8⇡), then convergence holds at an exponential rate, which is essentially governed by the linearization of System (2.3.3) around the stationary
solution. However, the estimate of the value ofM⇤ was found to be significantly smaller than 8⇡. In the radially symmetric setting, V. Calvez and J.A. Carrillo have found in [67] that the rate measured with respect to Wasserstein’s distance does not depend on the mass, in the whole range (0,8⇡). The goal of chapters 5 and 6 is to prove a similar estimate with no symmetry assumption.
In chapter 5 we define a proper functional framework and prove a functional inequality which is the crucial tool for obtaining the main result of chapter 6. Let us describe briefly this framework. InR2 the logarithmic Hardy-Littlewood-Sobolev has been established with optimal constants in [69] (also see [64]) and can be written as
Z
R3n dx. Equivalently, inequality (2.3.4) can be written as Z equivalent to the euclidian Onofri’s inequality.
To study the Keller-Segel system written in self-similar variables, equation (2.3.3), it turns out to be convenient to use an equivalent form of the logarithmic Hardy-Littlewood-Sobolev inequality, which, provided that M <8⇡, can be written as
Z where (nM, cM) denotes the unique stationary solution, of (2.3.3), given by
−∆cM =M e−12|x|2+cM R
R3e−12|x|2+c dx =:nM , x2R2 .
Following [69, 64, 68, 71], we show by duality that (2.3.5) corresponds to a new Onofri type inequality.
Theorem 4 For everyM 2(0,8⇡), and for all function φsmooth and compactly supported, one has log functional associated with the inequality around φ⌘1. By density, it is possible to attain an inequality in the functional space obtained when the set of smooth functions with compact support is completed with respect to the normkφk2 =R
R3|rφ|2 dx+ (R
R2φ dµM)2. This is the main result of chapter 5.
The inequality obtained by expanding aroundφ= 1 is a Poincar´e type inequality. It has a counterpart in the Hardy-Littlewood-Sobolev framework, which can be written as
Q1[f] :=
This suggests to write the linearized Keller-Segel system on the space of square integrable functions with respect todµM, which are orthogonal to the one-dimensional kernel of the linearized operatorL, endowed
CHAPTER 2. INTRODUCTION (ENGLISH VERSION) 23 with the scalar product associated to the quadratic form Q1. ThenL is self-adjoint and we prove that it has an explicit spectral gap:
Q1[f] =hf, fi hf, Lfi=:Q2[f].
Details onL and in particular on its spectrum are given later. With these preliminaries in hand, one can then study the large time asymptotics of the solutions to the Keller-Segel system and relate the spectral gap with rates of convergence.
In chapter 6 we provide refined asymptotics concerning the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables. It has been shown in [65] that there exists a positive massM? 8⇡ such that for any initial datan02L2(n−M1dx) of massM < M? satisfying (2.3.2), System (2.3.3) has a unique solution nsuch that
Z
R2|n(t, x)−nM(x)|2 dx
nM(x) C e−2δ t 8t≥0
for some positive constants C and δ. Moreover δ can be taken arbitrarily close to 1 as M ! 0. If M <8⇡, we may notice that the condition n0 2L2(n−M1dx) is stronger than (2.3.2). Our main result is that M? = 8⇡ and δ = 1, at least for a large subclass of solutions with initial datum n0 satisfying the following technical assumption
9"2(0,8⇡−M) such that Z s
0
n0,⇤(σ)dσ Z
B“ 0,p
s/⇡”nM+"(x)dx 8s≥0. (2.3.6)
Theorem 5 Assume thatn0 satisfies the technical assumption (2.3.6), n0 2L2+(n−M1dx) and M :=
Z
R3
n0dx <8⇡ . Then any solution of (2.3.3) with initial datum n0 is such that
Z
R2|n(t, x)−nM(x)|2 dx
nM(x) C e−2t 8t≥0
for some positive constant C, where nM is the unique stationary solution to (2.3)with mass M.
This result turns out to be consistent with the recent results of [67] for the two-dimensional radial model and its one-dimensional counterpart. The estimateδ= 1 is sharp. For completeness, let us mention that results of exponential convergence for problems with mean field have been obtained earlier in [89, 90], but only for interaction potentials involving much smoother kernels than G2.
To obtain this result, we establish uniform estimates on knkLp(R2) by applying symmetrization tech-niques as in [93, 94], and then prove the uniform convergence ofntonM using Duhamel’s formula. Our main tool are the spectral gap of the linearized operator L and the strict positivity of the linearized entropy in the appropriate functional space.