Refined Asymptotics for the subcritical KellerSegel system and Related Functional Inequalities
Abstract
We analyze the rate of convergence towards selfsimilarity for the subcritical KellerSegel system in the radially symmetric twodimensional case and in the corresponding onedimensional case for logarithmic interaction. We measure convergence in Wasserstein distance. The rate of convergence towards selfsimilarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the logarithmic HardyLittlewoodSobolev inequality in the one dimensional and radially symmetric two dimensional case based on optimal transport arguments. In addition we prove that the onedimensional equation is a contraction with respect to Fourier distance in the subcritical case.
1 Introduction
We will concentrate on seeking decay rates towards equilibria or selfsimilarity profiles for aggregation equations with linear diffusion in the fair competition regime. These models describe the evolution of a population of individuals which are diffusing by standard Brownian motion and attracting each other by a pairwise symmetric potential . We focus on a logarithmic interaction potential , with . The FokkerPlanck equation governing the evolution of the probability density function associated to this particle system reads as
(1.1) 
Due to translational invariance and mass conservation, in the rest of this work we restrict to zero center of mass probability densities,
By fair competition, we mean that the dynamics of (1.1) are driven by a simple dichotomy as in the classical KellerSegel system in two dimensions [24, 30, 20, 12], the modified KellerSegel system in one dimension [15, 7] or the KellerSegel model with suitable nonlinear diffusion in larger dimensions [9]. In all these examples there is a critical parameter which makes the distinction between global existence of solutions and finitetime blowup. More precisely, we will discuss the modified onedimensional KellerSegel equation [15, 7]:
(1.2) 
and the radially symmetric twodimensional classical KellerSegel equation:
(1.3) 
where denotes the cumulated mass of inside balls,
Both equations (1.2) and (1.3) exhibit a transition depending on the sensitivity coefficient :

Critical Case. For solutions exist globallyintime. There are infinitely many stationary solutions with infinite second moment. Solutions having finite initial second moment concentrate in infinite time towards the Dirac mass [10, 5, 23]. Solutions of infinite initial second moment close enough to a stationary solution converge to it as [8].
The critical parameter can be obtained from two formal computations at this stage. The evolution of the second moment satisfies in both cases the relation:
This implies that for solutions will necessarily blowup before the second moment touches zero. On the other hand, the KellerSegel equation (1.1) is equipped with a free energy (entropy minus potential energy),
(1.4) 
It is formally decreasing along the trajectories
(1.5) 
Moreover, it was shown in [20, 12] that for the free energy estimate from above implies an a priori bound in the entropy part of the functional which is at the basis of the construction of globalintime solutions. This was achieved by using the LogarithmicHLS inequality [4, 17] which relates the entropy and the interaction part of the functional.
Nontrivial equilibrium profiles or critical profiles, only exist for the critical parameter . They are solutions to the following EulerLagrange equations:
(1.6)  
(1.7) 
resp. in dimension and in dimension with radially symmetry. In fact, we have an explicit formulation of the stationary states,
(1.8) 
This coincides with the equality cases in the LogarithmicHLS inequality.
In the subcritical case , solutions are known to converge to unique selfsimilar profiles [12]. For studying convergence towards selfsimilarity, it is generally useful to rescale the space and time variables in the subcritical regime . The KellerSegel system rewrites as
(1.9) 
and the free energy is complemented with a quadratic confinement potential:
(1.10) 
Due to the change of variables, selfsimilar solutions correspond to equilibrium solutions of (1.9). The rate of convergence towards equilibrium for (1.9) in the subcritical case was recently studied in [11] where the same rate as for the heat equation was obtained for small mass.
Let us finally mention that both (1.1) and (1.9) are gradient flows of the free energy functionals (1.4) and (1.10) respectively, when the space of probability measures is endowed with the euclidean Wasserstein metric . We refer to the seminal papers [22, 29] and to [1] for a general theory. For instance, we can write (1.1) in short as
(1.11) 
This assertion was made rigorous in [7], where the variational minimizing movement scheme [1] was shown to converge for (1.1). This fact allows us to consider a way of measuring the distance towards equilibrium or selfsimilarity intimately related to the evolution due to (1.11). In fact, we will show that optimal transport tools are key techniques to describe this behavior at least in the one dimensional case (1.2) and in the radial case in two dimensions (1.3).
In order to investigate further the bounds of the free energy functional leading to the dichotomy discussed above and the characterization of the critical profiles, the LogarithmicHLS inequality proved in [4, 17] is essential. In Section 2, we show an alternative proof based on optimal transport tools in the one dimensional case and in the radial case in two dimensions. There is another recent proof of this inequality with sharp constants in the two dimensional case by fast diffusion flows [16]. The LogarithmicHLS inequality can be restated with our notations as:
Theorem 1.1 (Logarithmic HLS inequality).
Assume . The functional is bounded from below. The extremal functions are uniquely given by (1.8) up to dilations in the set of probability densities with zero center of mass.
In short, we demonstrate that any critical point of the free energy is in fact a global minimizer. This is a property which holds true for convex functionals, although the functional is not displacement convex in the sense of McCann [27].
The ideas behind the proof of the sharp LogarithmicHLS inequality allow us to tackle the rate of convergence in by similar methods for the rescaled version (1.9) in one dimension and for radial densities in the two dimensional case provided . We prove in Section 3 the following result.
Theorem 1.2 (Longtime asymptotics).
Assume that being the initial data radially symmetric if . In the subcritical case , solutions of (1.9) in the rescaled variables converge exponentially fast towards the unique equilibrium configuration . More precisely, the following estimate holds true
Surprisingly enough, the rate of convergence that we obtain does not depend on the parameter . Our estimate is uniform as long as remains subcritical and is equal to the rate of convergence towards selfsimilarity for the heat equation. This is due to the fact that entropy and interaction contributions cancel each other, and only the confinement contribution remains yielding a uniform estimate. Although convergence is likely to be uniform, notice that the asymptotic profile becomes more and more singular as , as shown by the simple second moment identity
(1.12) 
Finally, we devote Section 4 to propose an alternative method of measuring the distance towards selfsimilarity in the one dimensional case. We make a connection between the one dimensional modified KellerSegel model (1.2) and certain Boltzmannlike equations used in granular gases and wealthdistribution models, see [18, 21] and the references therein. This connection is due to the fact that (1.2) can be written in Fourier variables like the referred Boltzmann equations. Following the ideas of [18] we prove that equation (1.2) is indeed a contraction for the socalled Fourier distances defined in Section 4.
Theorem 1.3.
Assume and the initial data have finite second moments. The onedimensional KellerSegel system (1.2) is a contraction for the distance . It is a uniformly strict contraction in the rescaled frame, with a contraction factor which does not depend on .
2 An alternative proof of the logarithmic HLS inequality
2.1 Preliminaries on Optimal Transport Tools
Let and be two density probabilities. According to [13, 26] there exists a convex function whose gradient pushes forward the measure onto : . This convex function satisfies the MongeAmpre equation in the weak sense,
Regularity of the transport map is a big issue in general. Here we will use the fact that the Hessian measure can be decomposed in an absolute continuous part and a positive singular measure [31, Chapter 4]. In particular we have . The formula for the change of variables will be important when dealing with the entropy contribution. For any measurable function , bounded below such that we have [27, 31]
(2.1) 
In fact this paper will only be concerned with the onedimensional case, and the twodimensional radial case. The complexity of Brenier’s transport problem dramatically reduces in both cases. In dimension one, the transport map is explicitely given by: where and denote respectively the pseudoinverse cumulative distribution function of the densities and . The singular part of the positive measure corresponds to having holes in the support of the density .
In the twodimensional radial case, the Brenier’s map can be expressed as the onedimensional transport between the densities and . The determinant of the Hessian is given by
where the derivative of has to be understood in the distributional sense.
The following Lemma will be used to estimate the interaction contribution in the free energy, and in the evolution of the Wasserstein distance. For notational convenience we denote the convex combination of and by .
Lemma 2.1.
Let be an increasing and concave function such that . Then
(2.2) 
Equality is achieved in (2.2) if and only if the distributional derivative of the transport map is a constant function.
Analogously in the twodimensional radially symmetric case we deduce
(2.3) 
Equality is achieved in (2.3) if and only if is a multiple of the identity.
Proof.
We have on the one hand . We next use the concavity of to conclude. Equality occurs if is absolutely continuous and if is constant. In the twodimensional case we use . ∎
Optimal transport is a powerful tool for reducing functional inequalities onto pointwise inequalities (e.g. matrix inequalities). We highlight for example the seminal paper by McCann [27] where the displacement convexity issue for some energy functional is reduced to the concavity of . We also refer to the works of Barthe [2, 3] and CorderoErausquin et al. [19]. We require simple pointwise inequalities which are extensions of the classical Jensen’s inequality.
Lemma 2.2.
We have the following convexlike inequality for some exponent and any positive ,
(2.4) 
Equality occurs if and only if . The continuous version reads as follows. For any measurable function :
(2.5) 
Proof.
We only prove (2.4). The continuous version (2.5) is obtained by an approximation procedure. We introduce the auxiliary function defined as follows.
Clearly, diverges towards as or , and as or , and is bounded below. Then there exists at least one critical point. Any critical point satisfies
Hence and
We conclude that . Therefore the unique critical point of is . ∎
2.2 The onedimensional case
The novelty here is contained in the proof of the logarithmic HLS inequality. This brings no information by itself since the uniqueness of the extremal functions is already known [17]. We show below that the logarithmic HLS inequality is a simple consequence of the Jensen’s inequality. However our proof relies on the existence of a critical point of the free energy . In short, we demonstrate that any critical point of the free energy is in fact a global minimizer. This is a property which holds true for convex functionals. However the functional here is not convex.
Our first Lemma is a reformulation of the EulerLagrange equation for the extremal function (1.6).
Lemma 2.3 (Characterization of extremal functions).
The critical profiles satisfy the following identity,
(2.6) 
In the subcritical regime , the equilibrium in the rescaled frame satisfies the following identity,
(2.7) 
Proof.
The formulation (2.6) is equivalent to integrating once the equation for the critical profile. We integrate equation (1.6) against some test function .
where we have finally used the change of variables: . This holds true for any derivative , so we obtain identity (2.6) up to a constant. Since both sides of (2.6) have mass one, the constant is zero. The identity (2.7) is obtained in a similar way. ∎
Proof of Theorem 1.1..
Applying the change of variables formula (2.1) for , the functional rewrites as follows,
Using Lemma 2.1 for which is increasing and concave, we deduce
Equality arises if and only if the transport map is a constant function. Such a map corresponds exactly to the dilations of the critical profile . ∎
Theorem 2.4 (Logarithmic HLS inequality with a quadratic confinement).
Assume and . The functional is bounded from below. The extremal functions are unique in the set of probability densities with zero center of mass.
We give below the main lines of the proof following a direct argument analogous to the proof of Theorem 1.1. Note that the uniqueness of the extremal functions in dimension or in dimension in the class of radially symmetric densities is a consequence of Theorem 1.2.
Sketch of proof of Theorem 2.4.
The key point consists in replacing the Jensen’s inequality with the following convexlike inequality. For any positive the following inequality holds true.
(2.8) 
Equality occurs if and only if . It reduces to the usual Jensen’s inequality when . The proof of (2.8) is analogous to Lemma 2.2. The proof of uniqueness for the extremal functions of is a mixture between the proofs of Theorem 1.1 and Theorem 1.2. ∎
2.3 The twodimensional case
We restrict to radially symmetric functions in the twodimensional case due to decreasing rearrangement [25, 4, 17]. We recall the Newton’s theorem for Poisson potential: the field induced by a radially symmetric distribution of masses outside a given ball is equivalent to the field induced by a point at the center of the ball [25]. Equivalently it reads
(2.9) 
As a consequence we can rewrite the functional simpler under radial symmetry:
Lemma 2.5 (Characterization of extremal functions under radial symmetry).
The critical profiles satisfy the following identity
(2.10) 
In the subcritical regime , the radiallysymmetric equilibrium satisfies the following identity
(2.11) 
We are now ready to examinate the logarithmic HardyLittlewoodSobolev inequality in the twodimensional radial setting.
Proof of Theorem 1.1..
We apply the change of variables formula (2.1) for to get:
where we have used . We have consequently,
(2.12) 
The last contribution of (2.12) can be evaluated using Lemma 2.1
We obtain from the characterization (2.10) . Again equality occurs if and only if the transport map is a multiple of the identity. ∎
2.4 Obstruction in dimension higher than three
We explain in this Section why the above strategy fails to work in dimension higher than three, even in the radiallysymmetric setting. A first remark is that Newton’s Theorem is not valid, since the logarithm kernel is not the fundamental solution of the Poisson equation, although this is not essential as shown in dimension one. It turns out that our strategy works fine for any interaction kernel , for . The case corresponds to . The case is critical for integrability reasons. The case is exactly the harmonic case for which the Newton’s Theorem holds true. We refer to [14] for details in the case . Hence the case is out of range when . We sketch below where some obstruction appears when .
The identity which generalizes (2.9) reads as follows. If we have
where is defined as follows for ,
To continue our strategy, it is required to decouple the variables and , and more precisely to make the quantity appearing. As a matter of fact this is homogeneous to the determinant (under radial symmetry), which is the key quantity to look at in dimension higher than two. Therefore we seek a convexlike inequality
where and are suitable constants determined by zero and first order conditions. If we denote , this is equivalent to say that is convex. However simple computations show that it is indeed a concave function. In the case of an interaction kernel having homogeneity we show in [14] that the corresponding function is convex.
3 Exponential convergence towards the selfsimilar profile
3.1 The onedimensional case
To illustrate the strategy of proof of Theorem 1.2, we show a formal computation in the critical case . Up to our knowledge, the regularity of solutions under very weak assumptions is still an open problem In particular it is not known whether the solutions satisfy the identity (1.5) or not. So the following computation is questionable because the velocity field is not clearly defined in .
We compute formally the evolution of the Wasserstein distance to one of the equilibria (1.8) in the critical case . Notice that equilibria are infinitely far from each other with respect to the Wasserstein distance [8]. Using the gradient flow structure with respect to , one obtains the following formula for the derivative of , see [31, Chapter 8] and [1].
We recognize the characterization (2.6). Hence, we have at least formally . Observe that the Lemma 2.1 has been used with .
The same strategy is valid in the subcritical case for which we know that solutions are regular enough to ensure the validity of the computations. As a matter of fact, the density is everywhere positive and thus is absolutely continuous. On the other hand the dissipation of energy is welldefined and the dissipation estimate (1.5) holds true [12].
Proof of Theorem 1.2.
We compute the evolution of :
where we have used the fact that the center of mass is zero to double the variables. We rewrite each contribution using the reverse transport map :
where we have used that the second moment of the stationary state is explicitly given by (1.12). Applying now (2.5) for , and , we deduce
giving the desired inequality. ∎
3.2 The twodimensional radiallysymmetric case
Proving convergence towards a selfsimilar profile in the rescaled logarithmic case under radial symmetry goes as previously.
Proof of Theorem 1.2..
The virial computation reads equivalently
We compute again the evolution of the Wasserstein distance .
The last step in the inequality is a consequence of the arithmetic and geometric means inequality: . Next we use Lemma 2.2 to handle the interaction contribution. More precisely, we choose , and . One gets finally:
We conlude using characterization (2.11). ∎
4 Contraction in the onedimensional case
The aim of this Section is to point out the peculiar structure of the modified onedimensional KellerSegel system (1.2).
Lemma 4.1.
Equation (1.2) rewrites in Fourier variables as:
(4.1) 
Proof.
According to (4.1) the information propagates from lower to higher frequencies. The evolution of requires the knowledge of lower frequencies due to the integral contribution. This is of particular importance for designing a numerical scheme. Indeed there is no loss of information after truncation of the frequency box.
Remark 4.2 (Analogy with 1D Boltzmann).
Remark 4.3 (Evidence for blowup in the supercritical case).
We can directly notice the occurence of blowup when from (4.1). Observe that for , the righthandside is equivalent to:
(4.2) 
This argues in favor of blowup at low modes although misleadingly. We have plotted in Figure 1 numerical simulation of (4.1) in the supercritical case. Observe that blowup arises for , on the contrary to the misleading heuristics (4.2). The integrodifferential equation (4.1) makes perfect sense even in the supercritical regime after the first blowup event. However the outcoming fonction is no longer the Fourier transform of a probability measure. In fact the blowup time coincides with the formation of the first dirac mass, namely when the frequency distribution is flat at infinity. This contradictory intuition is similar to the proof of blowup based on the virial identity: the second momentum provides information at infinity but is used to prove blowup which is a local behaviour.
Recall the definition of Fourier distances [18] as they have been introduced for the analysis of the Boltzmann equation.
Definition 4.4 (Fourier distances).
Let , being two probability measures having the same center of mass. The distance is defined as follows: