Adrien Blanchet

Eric A Carlen

José A Carrillo

Functional inequalities, thick tails and asymptotics for the critical mass

Keywords:

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 ̺ λ , λ > 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 H λ 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 H λ . While the entropy dissipation for H λ 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 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 ̺ λ . 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.

1 Introduction

The PKS system and its critical mass

The Patlak-Keller-Segel system [START_REF] Patlak | Random walk with persistence and external bias[END_REF][START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF] is one of the simplest models of chemotaxis, describing the evolution of the population density of a cell colony which is diffusing across a two dimensional surface. In addition to the diffusion, as the cells move across the surface, they continually emit a chemical attractant, which itself diffuses across the surface. The cells tend to move towards higher concentrations of the attractant, and this induces a drift term tending to concentrate the population, and countering the spreading effects of the diffusion. A model organism for this type of behavior is the Dictyostelium Discoideum which segregates cyclic adenosine monophosphate, another important example of chemotactic movement are endothelial cells who react to VEGF to form blood vessels. See [START_REF] Hillen | A user's guide to PDE models for chemotaxis[END_REF][START_REF] Perthame | Transport equations in biology[END_REF] for recent reviews on chemotaxis models.

The model is mathematically interesting on account of this competition between the concentrating effects of the drift induced by the chemical attractant and the spreading effects of the diffusion, and there is a critical value to the total mass of the initial data, so that for masses above this value, the concentration wins, and the density collapses in a finite time. However, for masses below this critical mass, diffusion dominates, and the colony smoothly diffuses off to infinity. At the critical mass, there is a continuous family of stationary solutions, and this paper is concerned with determining their stability properties, and since they all turn out to be stable, basins of attraction for each of them. We begin by introducing the model and the critical mass associated with it. If ρ denotes the population density, and c the concentration of the chemical attractant, the system of equations is

             ∂ρ ∂t (t, x) = div [∇ρ(t, x) -ρ(t, x)∇c(t, x)] t > 0 , x ∈ R 2 , c(t, x) = - 1 2 π R 2 log |x -y|ρ(t, y) dy , t > 0 , x ∈ R 2 , ρ(0, x) = ρ 0 (x) ≥ 0 x ∈ R 2 , (1.1)
with an appropriate choices of units, so that all dimensional constants are unity.

In most of this paper, we consider initial data ρ 0 that belongs to L 1 (R 2 , log(e + |x| 2 ) dx), and such that ρ 0 log ρ 0 is integrable. The relevance of these conditions shall be explained shortly, but at the very least, they insure that c(0, x) is well defined. It will sometimes be convenient to write the second equation in (1.1) in the compact form c(t, x) = G * ρ(t, x) where G(x) = -1/(2 π) log |x| is the Green's function for -∆ in R 2 . That is, -∆c = ρ.

Also throughout the paper, the term density shall always refer to a non-negative integrable function on R 2 , and we shall use the term mass to refer to the total integral of a density ρ. Because of the divergence form structure of the system, solutions formally satisfy the conservation of mass

R 2 ρ(t, x) dx = R 2
ρ 0 (x) dx := M for all t ≥ 0; i.e., the mass M is conserved in time.

The PKS system can be rewritten advantageously as follows: Introduce the free energy functional F PKS

F PKS [ρ] = R 2 ρ(x) log ρ(x) dx + 1 4π R 2 ×R 2
ρ(x) log |x -y|ρ(y) dx dy .

The first integral is well defined if ρ log ρ is integrable, and the positive part of ρ(x) log |x -y|ρ(y) is integrable when ρ belongs to L 1 (R 2 , log(e + |x| 2 ) dx), so that the second integral is at least well-defined under this condition. Now suppose that the density ρ belongs to L 1 (R 2 , log(e + |x| 2 ) dx), and moreover, ρ log ρ is integrable. Then a simple formal calculation shows that for all u ∈ C ∞ c (R 2 ) with zero mean,

lim ǫ→0 1 ǫ (F PKS [ρ + ǫu] -F PKS [ρ]) = R 2 δF PKS [ρ] δρ (x) u(x) dx where δF PKS [ρ] δρ (x) := log ρ(x) + 1 2π R 2 log |x -y|ρ(y) dy = log ρ(x) -G * ρ(x) .
It is then easy to see that the evolution equation in (1.1) can be rewritten as

∂ρ ∂t (t, x) = div ρ(t, x)∇ δF PKS [ρ(t)] δρ (x) . (1.2)
It follows that at least along well-behaved classical solutions (for which we may integrate by parts),

d dt F PKS [ρ(t)] = - R 2 ρ(t, x) ∇ δF PKS [ρ(t)] δρ (x) 2 dx . (1.3)
In particular, along such solutions, t → F PKS [ρ(t)] is monotone non-increasing. The key to exploiting this monotonicity, as discovered in [START_REF] Dolbeault | Optimal critical mass in the two-dimensional Keller-Segel model in R 2[END_REF], is the sharp logarithmic Hardy-Littlewood-Sobolev (Log HLS) inequality [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF][START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF]:

1.1 LEMMA (Logarithmic Hardy-Littlewood-Sobolev inequality). Let f be a non-negative measurable function in R 2 such that f log f and f log(e + |x| 2 ) belong to L 1 (R 2 ). Let M := R 2 f dx. Then

R 2 f (x) log f (x) dx + 2 M R 2 ×R 2 f (x) log |x -y|f (y) dx dy ≥ -C(M ) , (1.4)
with C(M ) := M (1 + log πlog(M )). There is equality if and only if f (x) = ̺ λ (xx 0 ) for some λ > 0 and some x 0 ∈ R 2 , where

̺ λ (x) := M π λ (λ + |x| 2 ) 2 .
Following [START_REF] Dolbeault | Optimal critical mass in the two-dimensional Keller-Segel model in R 2[END_REF], one may apply sharp log HLS inequality (1.4) to deduce that

F PKS [ρ] = M 8π R 2 ρ(x) log ρ(x) dx + 2 M R 2 ×R 2 ρ(x) log |x -y|ρ(y) dx dy + 1 - M 8π R 2 ρ(x) log ρ(x) dx ≥ - M 8π C(M ) + 1 - M 8π R 2 ρ(x) log ρ(x) dx . (1.5)
It follows from this and the monotonicity of F PKS [ρ(t)] that for solutions ρ of the PKS system for which M < 8π,

E[ρ(t)] := R 2 ρ(t, x) log ρ(t, x) dx ≤ 8πF [ρ 0 ] -M C(M ) 8π -M .
Therefore, for M < 8π, the entropy E[ρ(t)] stays bounded from above, uniformly in time. This precludes the collapse of mass into a point mass for such initial data. In [START_REF] Dolbeault | Optimal critical mass in the two-dimensional Keller-Segel model in R 2[END_REF][START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF], this formal analysis is made rigorous, and the global existence of solutions below the critical value 8π and a number of their properties as well are established. Previous work in this direction, by Jäger and Luckhaus [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF], had shown that for initial data of sufficiently small mass, the entropy E[ρ(t)] stayed bounded uniformly in t. Their analysis used the Gagliardo-Nirenberg-Sobolev inequality for functions f in R 2 that bounds f 4 in terms of ∇f 2 and f 2 , and not the Log HLS inequality, but their global existence result requires the mass to lie below a threshold that is strictly less than 8π.

That 8π is the actual critical value at which diffusive and concentrating effects are balanced, and not only a better lower bound, can be seen by computing moments: When the initial data has a finite second moment, and M > 8π such collapse, or "blow-up" does indeed occur in a finite time. To see this, we first note a weak formulation of our the PKS evolution equation that will useful to us later on. Let ψ be any test function. Then

d dt R 2 ψ(x)ρ(t, x) dx = R 2 ∆ψ(x)ρ(t, x) dx - 1 4π R 2 ×R 2 ρ(t, x) (∇ψ(x) -∇ψ(y)) • (x -y)
|x -y| 2 ρ(t, y) dx dy .

(1.6)

In addition to the usual integration by parts, we have symmetrized the second term on the right in x and y.

Fixing any a ∈ R 2 and taking ψ(x) = a • x, we see from (1.6) that d dt R 2 xρ(t, x) dx = 0; i.e., the center of mass is conserved. Due to the translational invariance, we henceforth assume zero center of mass. More interestingly, taking ψ(x) = |x| 2 , so that (∇ψ(x) -∇ψ(y)) • (xy) = 2, we find

d dt R 2 |x| 2 ρ(t, x) dx = 4M - 1 2π M 2 = 4M 1 - M 8π .
Thus, if M > 8π, the right hand side is strictly negative, and this shows that the second moment of ρ(t) reaches zero in a finite time if initially bounded, or else some sort of singularity develops that would invalidate the formal calculation we have just made. Thus, the mass value M = 8π is the critical mass for the PKS system: For M < 8π, one has global solutions for which diffusion dominates so that all of the mass tends to infinity as the time tends to infinity, see [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF], while for M > 8π, solutions develop singularities, see [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF].

Our focus in this paper is on the case M = 8π. Notice that for M = 8π, F PKS is exactly the functional that is on the left hand side in (1.4). Since the densities ̺ λ are minimizers of F PKS for M = 8π, it follows that δF PKS [̺ λ ] δρ (x) = 0 , and then from (1.2) that each ̺ λ -and each of their translates -is a stationary solution of (1.2); i.e., of (1.1).

Of course, this can also be checked directly. Our main goal in this paper is to determine the stability of these solutions, and to determine basins of attraction for them. In achieving this goal, we develop several novel functional inequalities, and a strategy of concentration control that may be useful elsewhere, and may be the main contribution of the paper.

Each of the ̺ λ has an infinite second moment, and so shall all of the functions in the basins of attraction that we find for them. This must be the case according to previous work [START_REF] Blanchet | Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2[END_REF] on the case M = 8 π for initial data with a finite second moment. The paper [START_REF] Blanchet | Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2[END_REF] proves the global existence of weak solutions with finite second moment that satisfy the free energy dissipation inequality

F PKS [ρ(T )] + T 0 R 2 ρ(t, x) ∇ δF PKS [ρ(t)] δρ (x) 2 dx dt ≤ F PKS [ρ(0)] ,
which is what one would guess should hold from (1.3). Moreover, [START_REF] Blanchet | Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2[END_REF] proves that every such solution blows up at infinite time. That is, the t → ∞ limit of any such solution is a Dirac mass 8 π at the center of mass of the initial data. Furthermore, a point mass of mass M is a stationary measure-valued solution in the sense introduced in [START_REF] Dolbeault | The two-dimensional Keller-Segel model after blow-up[END_REF] if and only if M ≥ 8π. Let us finally mention that an analysis of basins of attraction of ̺ λ in the radial case was done in [START_REF] Biler | The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane[END_REF] in terms of certain relative moment conditions. From this point of view, the solutions in the critical mass case with finite initial second moment are choosing in their large time asymptotics the only possible stationary state with a finite second moment.

Let us finally comment that the family of stationary solutions ̺ λ play a role too in the conjectured profile of blow-up for any point singularity of the solutions for masses M > 8π. Velazquez has proved [START_REF] Velázquez | Stability of some mechanisms of chemotactic aggregation[END_REF][START_REF]Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions[END_REF] that the inner part of the matched-asymptotics expansion for the blow-up profile is given by these stationary solutions for the critical mass value.

The second Lyapunov functional

The essential tool in our construction and analysis of solutions of the critical mass PKS system is an interesting and somewhat surprising interplay between the PKS system and another evolution equation which also has the ̺ λ as stationary solutions -the Fokker-Planck version of the fast diffusion equation in R 2 with exponent 1/2:

     ∂u ∂t (t, x) = ∆ u(t, x) + 2 π λM div(x u(t, x)) t > 0 , x ∈ R 2 , u(0, x) = u 0 (x) ≥ 0 x ∈ R 2 , (1.7)
corresponding to the fast diffusion equation ∂u ∂t = ∆ √ u by a self-similar change of variable, see [START_REF] Vázquez | The Porous Medium Equation. Mathematical theory[END_REF]. In the interest of brevity we refer to (1.7) as the fast-diffusion equation.

This equation can also be written in a form analogous to (1.2): for λ > 0, define the functional H λ on the non-negative functions in L 1 (R 2 ) by

H λ [u] := R 2 u(x) -̺ λ (x) 2 ̺ λ -1/2 (x) dx .
This functional is the relative entropy of the fast diffusion equation with respect to the stationary solution ̺ λ . The unique minimizer of H λ is ̺ λ , and a simple formal computation yields

δH λ [u] δu = 1 √ ̺ λ - 1 √ u , (1.8)
from which one sees that (1.7) can be rewritten as

∂u ∂t (t, x) = div u(t, x)∇ δH λ [u(t)] δu (x) , (1.9)
It follows that for classical solutions u of (1.7) for which one can integrate by parts,

d dt H λ [u(t)] = - R 2 u(t, x) ∇ δH λ [u(t)] δu (x) 2 dx = - R 2 1 2 ∇ log u(t, x) + 2 π λ M x u(t, x) 2 dx .
As one sees from (1.8) and (1.9), the densities ̺ λ are stationary solutions of the fast diffusion equation (1.7), as well as the PKS system (1.1). This is much more than a coincidence, and there are very close connections between the two evolution equations. Unlike the functional F PKS , the functional H λ is convex. In fact, it is not only convex in the usual sense, but is also displacement convex. This second type of convexity, arising in the theory of optimal mass transportation, will play a basic role in our analysis. We shall explain the relevant aspects of displacement convexity in Section 2 of this paper. For now, we return to the convexity of H λ in the usual sense. Making simple computations, one finds that Since G 1 [u] is affine on its domain of definition, and since G 2 [u] is convex on its domain of definition, one might formally conclude the convexity of H λ on its domain of definition. In fact, those who are familiar with displacement convexity will recognize that functionals G 1 [u] and G 2 [u] are displacement convex on their domains of definition. Unfortunately, separate consideration of G 1 [u] and G 2 [u] is not relevant in our context: Note that ̺ λ (x) is not integrable, and thus if H λ [u] is to be well defined, u(x) cannot be integrable either. Furthermore, since |x| 2 ̺ λ (x) is not integrable, it is clear that |x| 2 u(x) also will not be integrable on the whole domain of definition of H λ . Thus, cancelations are crucial to the definition of H λ , and the integral in (1.10) cannot be split into a sum of three integrals to be analyzed separately.

As far as the convexity (in the usual sense) of H λ is concerned, it is easy to give a rigorous proof: Indeed, H λ [u] can be written as

H λ [u] := R 2 [Φ(u(x)) -Φ(̺ λ (x)) -Φ ′ (̺ λ )(u(x) -̺ λ (x))] dx (1.11) with Φ(s) = -2 √
s, which is a convex function. However, displacement convexity is essential to our strategy, and even after we have properly introduced the notion of displacement convexity, we shall have to work much harder to prove that H λ is in fact strictly displacement convex on its domain of definition.

The convexity properties of H λ are relevant to the analysis of the PKS system due to the perhaps surprising fact that H λ is monotone decreasing also along solutions of the critical mass PKS system (1.1), and not only along solutions of the fast diffusion equation (1.7). This gives us a second Lyapunov function for the critical mass PKS system. To see why this should be so, we make a formal calculation that we shall revisit in full rigor later on: Let ρ be a sufficiently nice solution of the PKS system. Then

d dt H λ [ρ(t)] = R 2 δH λ [ρ] δρ div ρ(t, x)∇ δF PKS [ρ] δρ dx = - R 2 ρ∇ δH λ [ρ] δρ • ∇ δF PKS [ρ] δρ dx = - R 2 ρ∇ 1 √ ̺ λ - 1 √ ρ • ∇ [log ρ -G * ρ] dx = - R 2 2 π λ M x ρ + ∇ √ ρ • ∇ [log ρ -G * ρ] dx (1.12)
Integrating by parts once more on the term involving the Green's function,

R 2 ∇ √ ρ • ∇ [log ρ -G * ρ] dx = 1 2 R 2 |∇ρ| 2 ρ 3/2 + R 2 √ ρ ∆G * ρ = 1 2 R 2 |∇ρ| 2 ρ 3/2 - R 2 ρ 3/2 dx . Also, R 2 xρ • ∇ log ρ dx = -2M
and, making the same symmetrization that led to (1.6),

R 2 ρ(x) x • ∇G * ρ(x) dx = 1 4π R 2 ×R 2 ρ(t, x) (x -y) • x -y |x -y| 2 ρ(t, y) dx dy = M 2 4π
.

(1.13)

Using the last three calculations in (1.12), we find

d dt H λ [ρ(t)] = - 1 2 R 2 |∇ρ| 2 ρ 3/2 dx + R 2 ρ 3/2 dx + 4 M π λ 1 - M 8π .
Notice that the constant term vanishes in critical mass case M = 8π. Thus, in the critical mass case, formal calculation yields that for all T > 0,

H λ [ρ(T )] + T 0 1 2 R 2 |∇ρ| 2 ρ 3/2 (t, x) dx - R 2 ρ 3/2 (t, x) dx dt ≤ H λ [ρ 0 ] . (1.14)
In fact, the formal computation yields equality instead of merely inequality in (1.14), but it is this inequality that is useful to us, and this is what we shall actually prove for the solutions that we construct here.

The key to exploiting (1.14) is a particular case of the Gagliardo-Nirenberg-Sobolev (GNS) inequalities for which the sharp form was found by Del Pino and Dolbeault [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF].

1.2 LEMMA (Gagliardo-Nirenberg-Sobolev inequality). For all functions f in R 2 with a square integrable distri- butional gradient ∇f , π R 2 |f | 6 dx ≤ R 2 |∇f | 2 dx R 2 |f | 4 dx ,
and there is equality if and only if f is a multiple of a translate of ̺ λ 1/4 for some λ > 0.

To apply this, note that at least for strictly positive densities ρ,

R 2 |∇ρ 1/4 (x)| 2 dx = 1 16 R 2 |∇ρ| 2 ρ 3/2 (x) dx .
Therefore, we define:

1.3 DEFINITION (Entropy dissipation functional). For any density ρ of total mass 8π such that ρ 1/4 has a square-integrable distributional gradient, we define the entropy dissipation functional D[ρ] by

D[ρ] = 8 R 2 |∇ρ 1/4 (x)| 2 dx - R 2 ρ 3/2 (x) dx ,
and we define D[ρ] = ∞ in all other cases.

1.4 LEMMA (Dissipation of H λ ). For all densities ρ of mass M = 8π,

D[ρ] ≥ 0 ,
and moreover, there is equality if and only ρ is a translate of ̺ λ for some λ > 0.

Proof: Let f = ρ 1/4 and note that R 2 f 4 (x) dx = 8π. Multiplying D[ρ] through by π, the claim follows directly from Lemma 1.2.

The main results on the PKS equation

The formal result (1.14) may now be written as

H λ [ρ(T )] + T 0 D[ρ(t)] dt ≤ H λ [ρ 0 ] .
Since H λ [ρ(T )] ≥ 0, this suggests at the very least that lim

T →∞ 1 T T 0 D[ρ(t)] dt ≤ lim T →∞ 1 T H λ [ρ 0 ] = 0 ,
and then Lemma 1.4 suggests that for all large t, ρ(t) must be close to ̺ µ for some µ > 0. However, an easy calculation, see Remark 2.5, using the fact that

R 2 |x| 2 ̺ λ (x) dx = ∞, shows that H λ (̺ µ ) = ∞ for µ = λ. Therefore, since H λ [ρ(t)
] is non-increasing, one expects that µ = λ. In short, the formal calculations made so far suggest that for solutions ρ of the PKS system with initial data

ρ 0 satisfying H λ [ρ 0 ] < ∞, lim t→∞ ρ(x, t) = ̺ λ (x).
We now make one more definition, and then state our main results on the PKS equation:

1.5 DEFINITION (Properly dissipative weak solutions of the PKS equation). Let Let ρ 0 be any density on R 2 with mass 8π, such that for some λ > 0,

H λ [ρ 0 ] < ∞. Let ρ : [0, ∞) → L 1 (R 2 ) satisfy:
(1.5.1) For each t ≥ 0, ρ(t) is a continuous curve of densities of mass 8π in the sense that for each bounded and globally Lipschitz function

ψ on R 2 , t → R 2 ψ(x) ρ(t, x) dx is continuous with ρ(0) = ρ 0 .
(1.5.2) For each T > S ≥ 0, and each smooth and compactly supported function

ψ on R 2 , R 2 ψ(x)ρ(T, x) dx = R 2 ψ(x)ρ(S, x) dx + T S R 2 ∆ψ(x)ρ(t, x) dx dt - 1 4π T S R 2 ×R 2 ρ(t, x) (∇ψ(x) -∇ψ(y)) • (x -y) |x -y| 2 ρ(t, y) dx dy dt .
(1.5.3) For each T > 0,

H λ [ρ(T )] + T 0 D[ρ(t)] dt ≤ H λ [ρ 0 ] , (1.15)
so that ρ satisfies the entropy-entropy dissipation inequality expected for solutions of the PKS equation.

Then ρ is a properly dissipative weak solution of the PKS equation (1.1) with initial data ρ 0 .

1.6 THEOREM (Existence and regularity of properly dissipative weak solutions). Let ρ 0 be any density on R 2 with mass 8π, such that F PKS [ρ 0 ] < ∞, and for some λ > 0, H λ [ρ 0 ] < ∞.Then there exists a properly dissipative solution of the PKS equation (1.1) with initial data ρ 0 . Moreover, the solutions we construct have additional regularity properties, including:

(1.6.1) For any S > 0 and any p with 1 < p < ∞, there is a constant C depending only on S, p, λ and H λ [ρ 0 ] such that for all t ≥ S, ρ(t) p ≤ C.

(1.6.

2) The distributional gradient of ρ 1/4 is square integrable over [0, ∞) × R 2 , and in fact,

∞ 0 R 2 |∇ρ 1/4 (t, x)| 2 dx dt ≤ H λ [ρ 0 ] .
(1.6.3) F PKS [ρ(t)] is well defined for each t, and is monotone decreasing:

F PKS [ρ(t)] ≤ F PKS [ρ(s)] for all 0 ≤ s < t.
1.7 THEOREM (Basins of attraction). Let ρ 0 be any density on R 2 with mass 8π, such that F PKS [ρ 0 ] < ∞, and for some λ > 0, H λ [ρ 0 ] < ∞. Let ρ be any properly dissipative weak solution of the PKS equation (1.1) with initial data ρ 0 satisfying the additional regularity properties (1.6.1), (1.6.1) and (1.6.3) of Theorem 1.6. Then

lim t→∞ F PKS [ρ(t)] = F PKS [̺ λ ] and lim t→∞ ρ(t) -̺ λ 1 = 0 .
Let B λ denote the set of densities ρ 0 of mass 8π for which

F PKS [ρ] < ∞, and H λ [ρ] < ∞.
According to Theorem 1.7, B λ is a basin of attraction for ̺ λ under the PKS evolution in the sense that any properly dissipative weak solution with initial data in B λ , and the regularity produced here, converges strongly to ̺ λ in L 1 (R 2 ).

Controlled concentration inequalities

The proof of the additional regularity in Theorem 1.6, and then Theorem 1.7, might at first appear to be possible by a standard application of entropy-entropy dissipation methods, given the entropy-entropy dissipation inequality (1.15). However, this is not the case. The essential point is that D[ρ] is not a convex function of ρ, and even worse, it is a difference of two functionals of ρ that can each be arbitrarily large even when D[ρ] is very close to zero. Indeed, for M = 8π and each λ > 0,

D[̺ λ ] = 0 while lim λ→0 ̺ λ 3/2 = ∞ , lim λ→0 ∇̺ λ 1/4 2 = ∞ , and lim λ→0 ̺ λ = 8πδ 0 .
the point mass of 8π at 0. It follows that the level sets of D cannot be weakly compact in L 1 (R 2 ). Likewise, F PKS [̺ λ ] = 8π(log 8 -1) for all λ > 0 while lim λ→0 ̺ λ = 8πδ 0 . Thus, a family of densities of mass 8π on which F PKS is uniformly bounded need not be weakly compact in

L 1 (R 2 )
In these examples of non-compactness for level sets of D and F PKS , we have a family of densities, which, in the limit, concentrate all of their mass at single point. We shall show here that this is essentially the only way compactness can fail for a family of densities of mass 8π on which D or F PKS is uniformly bounded.

Compactness of level sets of H λ fails for more mundane reasons: A glance at (1.11) is enough to see that for all h > 0, one can construct a sequence of {ρ k } small perturbations of ̺ λ lying in {ρ : H λ [ρ] ≤ h} such that for some ǫ > 0 depending on h, lim inf k→∞ {|x|<1/k} ρ k (x) dx ≥ ǫ. Thus, level sets of H λ are not uniformly integrable, and thus not even weakly compact in L 1 (R 2 ).

However, the densities in level sets of H λ do have a crucial property: They must have "thick tails" and for this reason, they cannot concentrate more than a limited fraction of their mass on any given small set. Thus, the examples of non-compact subsequences in level sets of D and F PKS that we exhibited above do not lie in any level set of H λ , and as we shall show, neither do any other non-compact subsequences. In Section 3 we prove:

1.8 THEOREM (Thick Tails). Let ρ be any density of mass M such that H λ [ρ] < ∞. Then for η * := 1 5 e -1/5 and any s > 1

|x| 2 ≥λs 2 ρ(x) dx ≥ η * e -4 √ πM λ H λ [ρ] |x| 2 ≥λs 2 ̺ λ (x) dx = M η * 1 + s 2 e -4 √ πM λ H λ [ρ] .
Though the statement of this theorem makes no reference to optimal mass transportation, the proof we give in Section 3 relies heavily on the optimal mass transportation results we present in Section 2, including a new Talagrand type inequality, Theorem 2.4, involving H λ .

Using the very mild control on concentration provided by the Thick Tails Theorem, which says that densities ρ with H λ finite for some λ cannot possibly concentrate most of their mass near any one point, we prove two "compactness via controlled concentration" theorems. Of course, some sort of concentration control is inherent in any compactness theorem for densities, but the point here is that given only the very mild limit on concentration provided by the Thick Tails Theorem, we are able, in Section 3, to prove compactness for the level sets of F PKS and D. The first of these theorems concerns F PKS .

1.9 THEOREM (Concentration control for F PKS ). Let ρ be any density with mass M = 8π, with H λ [ρ] < ∞ for some λ > 0. Then there exist positive computable constants γ 1 and C CCF depending only on λ and H λ [ρ] such that

γ 1 R 2 ρ log + ρ dx ≤ F PKS [ρ] + C CCF .
Our second 'compactness via controlled concentration" theorem concerns D:

1.10 THEOREM (Concentration control for D). Let ρ be any density with mass 8π, F PKS [ρ] finite, and H λ [ρ] finite for some λ > 0. Then there exist positive computable constants γ 2 and C CCD depending only on λ, H λ [ρ] and

F PKS [ρ] such that γ 2 R 2 |∇ρ 1/4 | 2 dx ≤ πD[ρ] + C CCD .
Moreover, the same conclusion holds if we replace the assumption that F PKS [ρ] finite with the assumption that the entropy E[ρ] is finite, except that now the constant depends on the bound on E[ρ] instead of the bound on F PKS [ρ]. Theorems 1.9 and 1.10 give us the "vertical control" needed for a compactness result. The horizontal control is provided by H λ alone. Not only does a bound on H λ [ρ] ensure that ρ has thick tails, it also ensures that the tails are not too thick: A bound on H λ [ρ] provides a bound on all moments of ρ up to but not including order 2. Unlike the Thick Tails Theorem, this result is elementary:

1.11 LEMMA (Localization). For all densities ρ with mass M and all λ > 0, and all 0 < q < 2, there is an explicitly computable constant C depending only on q, λ and M so that

R 2 |x| q ρ(x) dx ≤ C (1 + H λ [ρ]) q/2 .
Proof: By the Cauchy-Schwarz inequality

R 2 |ρ -̺ λ |̺ -1/4 λ dx ≤ R 2 | √ ρ - √ ̺ λ |̺ -1/4 λ | √ ρ + √ ̺ λ | dx ≤ H λ [ρ] √ ρ + √ ̺ λ 2 ≤ 2M H λ [ρ] . Since ̺ 3/4 λ
is integrable, there is a constant C, depending only on λ and M , whose explicit form is easily worked out, for which

R 2 ̺ -1/4 λ ρ dx ≤ R 2 ̺ -1/4 λ ̺ λ dx + 2M H λ [ρ] ≤ C (1 + H λ [ρ]) 1/2 .

Now repeat the Cauchy-Schwarz inequality, but this time with

̺ -3/8 λ in place of ̺ -1/4 λ
, then there is a (different) constant C, depending only on λ and M such that:

R 2 |ρ -̺ λ |̺ -3/8 λ dx ≤ R 2 | √ ρ - √ ̺ λ |̺ -1/4 λ | √ ρ + √ ̺ λ |̺ -1/8 λ dx ≤ H λ [ρ] √ ρ̺ -1/8 λ + √ ̺ λ ̺ -1/8 λ 2 ≤ H λ [ρ] C (1 + H λ [ρ]) 1/2 . Since ̺ 5/8 λ
is integrable, then by changing the constant C accordingly, whose explicit form is easily worked out, we deduce

R 2 ̺ -3/8 λ ρ dx ≤ R 2 ̺ -3/8 λ ̺ λ dx + C (1 + H λ [ρ]) 3/4 .
The obvious iteration of this argument leads to

R 2 ̺ -r λ ρ dx ≤ R 2 ̺ -r λ ̺ λ dx + C (1 + H λ [ρ]) 2r
for each r of the form 1/2 -(1/2) k for k ∈ N, and then by interpolation, for all 0 ≤ r < 1/2, where of course C depends on r as well as λ and M . To conclude, note that ̺ -r λ (x) ∼ |x| 4r for large |x|.

Lemma 1.11 shows in particular that when

H λ [ρ] < ∞, then ρ log(e + |x| 2 ) ∈ L 1 (R 2 )
, so that the Newtonian potential of ρ is well defined. Also, Lemma 1.11 together with Theorem 1.9 shows, via the Dunford-Pettis Theorem that the intersections of level sets of H λ and F PKS are at least weakly compact in L 1 (R d ), and stronger conclusions follows for the intersections of level sets of H λ , F PKS and D.

The "compactness via controlled concentration" provided by H λ and its dissipation D through Theorem 1.10 and Lemma 1.11 is the core of our proof of Theorem 1.7. However this is not the only use we shall make of compactness via controlled concentration: It is absolutely essential to our construction of properly dissipative weak solutions.

Indeed, in many problems in which one seeks to prove an entropy-entropy-dissipation inequality such as (1.15), both the entropy functional H and its dissipation D would be weakly lower semicontinuous, often due to some convexity property. Then, if {ρ n } n∈N is a sequence of nice or approximate solutions of the evolution equation converging weakly to a weak solution ρ, one would have

H[ρ(T )] ≤ lim n→∞ H[ρ n (T )]
and

T 0 D[ρ(t)] dt ≤ lim n→∞ T 0 D[ρ n (t)] dt ,
which is very helpful if one is trying to prove something like (1.15). While in our case H λ is convex and lower semicontinuous, D is the difference of two non-comparable convex functions and has no lower semicontinuity. Therefore, we need new tools to prove (1.15), and as we shall see, it is once again the compactness via controlled concentration that does the trick.

Further developments

One can build on the regularity results obtained here to prove additional regularity. Indeed, if ρ is one of the solutions we have constructed here, it is easy to prove that for any a > 0, ∇c(x, t) is bounded and continuous on (a, ∞) × R 2 , only using the continuity properties on ρ in t, the uniform control on first moments, and the fact that ρ(t) is uniformly bounded in both L 1 and L 3 for all t > a. Thus "freezing" b := ∇c, ρ is seen to be a weak solution of the linear parabolic equation ∂ρ ∂t = ∆ρdiv(bρ) , with b bounded and continuous. Parabolic regularity theory may now be applied. In fact, the arguments developed in [START_REF] Alikakos | Lp bounds of solutions of reaction-diffusion equations[END_REF][START_REF] Kowalczyk | Preventing blow-up in a chemotaxis model[END_REF][START_REF] Calvez | Volume effects in the Keller-Segel model: energy estimates preventing blowup[END_REF] can be applied to get L ∞ -bounds of the density for all positive times based on the L p -bounds, 1 ≤ p < ∞, obtained in Theorem 1.6. A further development that requires new tools is to bound the rate of convergence to the equilibrium ̺ λ in our convergence theorem.

An interesting problem whose solution would lead to rate information is to characterize the stability of the GNS inequality that we have used. That is, we know that D[ρ] = 0 if and only if ρ is a translate of ̺ λ for some λ > 0, since, as we have seen, this is simply a restatement of a sharp GNS inequality of Del Pino and Dolbeault. A stability result for this inequality would be a result stating that, for any ǫ > 0, if D[ρ] is sufficiently small, then the distance, in some metric, from ρ to some translate of some ̺ λ , λ > 0, is no more than ǫ. It would also be useful to quantify the qualitative stability result for the Log-HLS inequality that we prove and use in Section 5. Work in this direction is underway.

Other equations with a second Lyapunov functional

The second Lyapunov functional H λ is more useful to us than the primary Lyapunov functional F PKS , which actually drives the evolution, because of its convexity properties, especially its displacement convexity, as explained in Section 3.

There is a "canonical way" to produce gradient flow evolution equations that have a convex second Lyapunov functional that has been investigated in [START_REF] Matthes | A family of nonlinear fourth order equations of gradient flow type[END_REF]. Indeed, both the PKS equation and the fast diffusion equation are gradient flow systems where the gradient is computed using the 2-Wasserstein metric, as we recall in Section 3. To keep things simple here, let us explain the mechanism studied in [START_REF] Matthes | A family of nonlinear fourth order equations of gradient flow type[END_REF] in the finite dimensional Euclidean case.

Let V be a smooth convex function on R n . Let W be the smooth function on R n defined by W (x) = |∇V (x)| 2 . Now consider the evolution equation d dt

x(t) = -∇W (x(t)) .
Then of course, for any solution x,

d dt W (x(t)) = -|∇W (x(t))| 2 ≤ 0 ,
and so W is monotone decreasing along the evolution. It is the primary Lyapunov function for this flow. Next, note that since

∇W = 2[HessV ] ∇V , d dt V (x(t)) = -[∇V • ∇W ] (x(t)) = -2 {∇V • [HessV ] ∇V } (x(t)) ≤ 0 ,
since the Hessian of V is positive. Thus, V is a second Lyapunov function for the gradient flow driven by W . An example in [START_REF] Matthes | A family of nonlinear fourth order equations of gradient flow type[END_REF] concerns a porous medium equation on the line, which is gradient flow in the 2-Wasserstein metric for a certain entropy functional. With this entropy functional playing the role of V , the gradient flow equation for the functional corresponding to W is a certain fourth order equation of thin-film type.

The fact that the entropy for the porous medium equations is a second Lyapunov functional for this fourth order thin film equation had been discovered earlier in [START_REF] Carrillo | Intermediate asymptotics for strong solutions of the thin film equation[END_REF] and exploited as the key to understanding the long time behavior of the latter equation. Again in this case, the second Lyapunov function is strictly and uniformly displacement convex, while the primary Lyapunov functional is not displacement convex at all.

In the case studied here, the second Lyapunov functional does not arise through the mechanism studied in [START_REF] Matthes | A family of nonlinear fourth order equations of gradient flow type[END_REF], or any other evident natural mechanism, and we have no "explanation" of why one should expect H λ to decrease along the PKS flow. However, as explained in [START_REF] Matthes | A family of nonlinear fourth order equations of gradient flow type[END_REF], once one knows this, it is a consequence, formally at least, that F PKS decreases along the fast diffusion flow. This has interesting consequences that are investigated in [START_REF] Carlen | Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF].

The motivation for doing the computation to check the monotonicity is twofold: First, both evolution equations have the same steady states, which is certainly necessary, but not at all sufficient, for the computation to work out. Second, there are many sharp inequalities that have negative powers of 1 + |x| 2 as their cases of equality, so there are tools available to try to prove the positivity of the dissipation.

A brief outline of the rest of the paper

The rest of the paper is organized as follows. Section 2 begins with a brief summary of some results concerning the 2-Wasserstein metric and gradient flows with respect to it. In particular, we recall a discrete variational scheme due to Jordan, Kinderlehrer and Otto [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF] for constructing solutions of a class of equations including both the PKS equation and the critical fast diffusion equation. We also recall McCann's [START_REF] Mccann | A convexity principle for interacting gases[END_REF] notion of displacement convexity, and explain how this should, at least formally, lead to the entropy-entropy dissipation inequality that we seek. Making the formal calculation rigorous will then be reduced to rigorously proving certain consequences of displacement convexity for H λ , and this will be facilitated by the "robustness" of displacement convexity.

The latter half of Section 2 is more novel. As we have noted earlier, H λ is formally a sum of displacement convex terms, however, for the densities that concern us, each of the terms is divergent. Thus, we are forced to introduce a regularization of H λ . While there are many tools available to regularize functions that are convex in the usual sense (e.g. infimal convolution), there is no general approach to regularizing functionals while preserving, or at least not severely damaging, their formal displacement convexity properties. The regularization developed in the second half of Section 2 is one of the cornerstones of the paper.

In Section 3 we prove the controlled concentration results that have been stated and discussed in previous subsections.

In Section 4, we lay the ground work for the proof of Theorem 1.6 on the existence of properly dissipative weak solutions. These will be constructed using a variant of the Jordan, Kinderlehrer and Otto [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF] scheme, which constructs the evolutions by solving a sequence of variational problems, as in di Giorgi's "minimizing steps" method.

In this method, the Euler-Lagrange equation for the variational problem solved at each step often provides essential a-priori regularity on the minimizing density ρ. Once again, at this point in our problem, we encounter difficulties due to potential cancelation of infinities. To resolve these, we are forced to regularize F PKS . The discrete scheme provides a very convenient framework in which to impose and control the regularization: We use a different degree of regularization at each discrete time step. Because of the regularization, we will at least know that at each time step, ∇ √ ρ is square integrable, but we shall have no useful quantitative bound on ∇ √ ρ 2 . Still, this gives us enough regularity to make some crucial integrations by parts, and then eventually through the use of Theorem 1.10, we shall obtain a useful quantitative bound on ∇ρ 1/4 2 .

In Section 5, we pass to the continuous time limit, and provide the proofs of Theorems 1.6 and 1.7. Here, the flexibility of choosing the degree of regularization at each time step is crucial to cope with the errors committed in the displacement convexity of the regularized functional and to get the right dissipation in the limit. The convergence in L 1 needs a qualitative control of the error in the log-HLS inequality: if F PKS [ρ] is small enough, the densities are closed to some ̺ µ in L 1 norm. This together with the fact that H λ [ρ] is non increasing proves that µ = λ.

2 Displacement convexity and the PKS system

Gradient flows in the Wasserstein metric and displacement convexity

We recall some facts concerning the 2-Wasserstein metric that will be used here. We shall be brief, aiming mainly to establish terminology and notation. For more background, see [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]. Let P(R 2 ) denote the set of probability measures in R 2 , and let P 2 (R 2 ) the subset of probability measures with finite second moments. Define the functional W 2 in P(R 2 ) × P(R 2 ) by

W 2 2 (µ, ν) = inf Π∈Γ R 2 ×R 2 |x -y| 2 dΠ(x, y) , (2.1)
where Π runs over the set Γ of all couplings of the probability measures µ and ν; that is, the set of joint probability measures in R 2 × R 2 with first marginal µ and second marginal ν. For absolutely continuous probability measures f dx and g dx we will simply write W 2 (f, g) in place of W 2 (f dx, g dx). Clearly, W 2 is finite in P 2 (R 2 ) × P 2 (R 2 ), though it takes on the value +∞ in certain pairs (µ, ν) ∈ P(R 2 ) × P(R 2 ) -for example if µ belongs to P 2 (R 2 ), but ν does not. It is easy to see that W 2 is a metric on P 2 (R 2 ); it is called the 2-Wasserstein metric, where the 2 refers to the exponent 2 on the distance |x -y|. More generally, given any ν ∈ P(R 2 ), W 2 is a metric on the subset of P(R 2 ) given by µ ∈ P(R 2 ) : W 2 (µ, ν) < ∞ . A result of Brenier [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF] as extended by McCann [START_REF] Mccann | Existence and uniqueness of monotone measure-preserving maps[END_REF], provides effective control over the minimization problem defining W 2 (µ, ν). To recall this result, let T be a measurable map R 2 → R 2 . We say that T transports µ onto ν, if for any measurable set

B ⊂ R 2 , ν(B) = µ • T -1 (B).
In this case we say that ν is the push-forward of µ by T ,

ν = T #µ. An equivalent formulation is that ν = T #µ if R 2 ζ(T (x)) dµ(x) = R 2 ζ(y) dν(y) ∀ζ ∈ C 0 b (R 2 ) . (2.2)
By the Brenier-McCann Theorem [START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF][START_REF] Mccann | Existence and uniqueness of monotone measure-preserving maps[END_REF], for any two probability measures µ and ν on R 2 not charging Hausdorff dimension 1 sets, there is an essentially unique convex function ϕ in R 2 such that ∇ϕ#µ = ν and

W 2 2 (µ, ν) = R 2 |x -∇ϕ(x)| 2 dµ(x) . (2.3)
The essential uniqueness is that if ϕ and ϕ are two such convex functions, then ∇ϕ = ∇ ϕ almost everywhere with respect to µ. In this paper we will be concerned with densities whose mass is not necessarily one. If µ and ν are two positive measures of mass M > 0, we define W 2 (µ, ν) in terms of the 2-Wasserstein distance between the probability measures µ/M and ν/M as follows:

W 2 2 (µ, ν) = M W 2 2 (µ/M, ν/M ) . (2.4)
This normalization convention has the advantage that if ∇ϕ#(µ/M ) = (ν/M ), then (2.3) is still valid for arbitrary M . Note that if (2.2) holds for µ and ν, it also holds if we change µ and ν by multiplying them by a positive constant, i.e., ∇ϕ#(µ/M ) = (ν/M ) if and only if ∇ϕ#µ = ν.

In Section 5 we shall also use the p-Wasserstein distance, 1 ≤ p < 2, especially for p = 1, on account of a useful description of compact sets for this metric. For two probability measures µ and ν on R 2 , p-Wasserstein distance W p (µ, ν) is defined by (2.1) where 2 is substituted by p. For two positive measures of mass M , we define

W p (µ, ν) = √ M W p (µ/M, ν/M
). This normalization is chosen taking into account (2.4) to extend the standard ordering relation for the W p -metrics on probability measures; that is, by Hölder's inequality, we have for any

1 ≤ p < 2 W p (µ, ν) ≤ W 2 (µ, ν) . (2.5)
A fundamental insight of Otto [START_REF] Otto | Doubly degenerate diffusion equations as steepest descent[END_REF] is that the 2-Wasserstein metric is useful when considering any evolution equation on densities ρ that can be written in the form

∂ρ ∂t (t, x) = div ρ(t, x)∇ δG[ρ(t)] δρ (t, x) (2.6)
for some functional G. The prime example of (2.6) considered in [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF] is the Fokker-Planck equation for probability densities for which

G[ρ] = R 2 ρ(x) log ρ(x) dx + 1 2 R 2 |x| 2 ρ(x) dx . (2.7)
In [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF] rescaled porous medium equations were also included. Otto's insight [START_REF] Otto | Doubly degenerate diffusion equations as steepest descent[END_REF][START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF] is that the equation (2.6) is gradient flow for the functional G with respect to the 2-Wasserstein metric. This is true for a large class of equations of the form (2.6), see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF][START_REF] Carlen | Solution of a Model Boltzmann Equation via Steepest Descent in the 2-Wasserstein Metric Arch[END_REF][START_REF] Carlen | Localization, smoothness and convergence to equilibrium for a thin film equation[END_REF][START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF][START_REF] Carrillo | Contractions in the 2-Wasserstein length space and thermalization of granular media[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF]. The "gradient flow in the 2-Wasserstein metric" point of view is useful to us here for two reasons:

• It provides the means for constructing well-behaved solutions of the equation in question through the solution of a sequence of variation problems; the Jordan-Kinderlehrer-Otto (JKO) scheme.

• It provides the means for studying the rate at which solutions of (2.6) converge to minimizers of the functional G, at least when the functional G has a certain convexity property.

The convexity property referred to in the second point is McCann's notion of displacement convexity [START_REF] Mccann | A convexity principle for interacting gases[END_REF], which turns out to be convexity in the "Riemannian metric" associated to the 2-Wasserstein metric; see [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF]. If the functional G is uniformly displacement convex, then there are automatically a family of functional inequalities that govern the convergence of solutions of (2.6) to minimizers of G. In concrete terms, the functional G is said to be displacement convex in case the following is true: For any two densities ρ 0 and ρ 1 of the same mass M , let ϕ be the essentially unique convex function such that ∇ϕ#ρ 0 = ρ 1 . For 0 < t < 1, define

ϕ t (x) = (1 -t) |x| 2 2 + tϕ(x) and ρ t = ∇ϕ t #ρ 0 .
The displacement interpolation between ρ 0 and ρ 1 is the path of densities t → ρ t , 0 ≤ t ≤ 1. Let γ be any real number. To say that G is γ-displacement convex means that for all such densities ρ 0 and ρ 1 , and all 0

≤ t ≤ 1, (1 -t)G[ρ 0 ] + tG[ρ 1 ] -G[ρ t ] ≥ γt(1 -t)W 2 2 (ρ 0 , ρ 1 )
. G is simply displacement convex if this is true for γ = 0, and G is uniformly displacement convex is this is true for some γ > 0. Let us recall the characterization of displacement convexity given by McCann in [START_REF] Mccann | A convexity principle for interacting gases[END_REF] for functionals of the form

G Φ [ρ] := R d Φ(ρ(x)) dx , (2.8)
where

Φ : [0, ∞) → [0, ∞) with Φ(0) = 0. Then McCann's Theorem says that if s → s d Φ(s -d
) is convex nonincreasing on (0, +∞) then the functional G Φ is displacement convex, and this condition is essentially necessary.

A much simpler result, also from [START_REF] Mccann | A convexity principle for interacting gases[END_REF], is that if V is any real valued function on R 2 such that for all x 0 , x 1 ∈ R 2 and all 0

≤ t ≤ 1, (1 -t)V (x 0 ) + tV (x 1 ) -V ((1 -t)x 0 + tx 1 ) ≥ γt(1 -t)|x 0 -x 1 | 2 , then the functional V [ρ] = R 2 V (x) ρ(x) dx is γ-displacement convex.
Using these results, one readily checks that in the case of the Fokker-Planck equation, the functional (2.7) is indeed 2-displacement convex. The consequent inequalities that govern the long time behavior of solutions are Gross's logarithmic Sobolev inequality and the Talagrand inequality for Gaussian measures. Our analysis of long time behavior for the PKS system falls outside the scope of previous work in this direction since the functional F PKS is not displacement convex. The key reason that it is useful to bring the second formal Lyapunov functional H λ into the analysis of the PKS system is that it is displacement convex. In the next section we prove the displacement convexity of H λ , and study its consequences.

The critical fast diffusion equation as gradient flow of a uniformly displacement convex entropy

The equation ∂u ∂t

(t, x) = ∆u m (t, x) + κ div(x u(t, x)) , (2.9)
where κ is a non-negative constant and m > 0, is called the porous medium equation with κ = 0 and m > 1 while for κ = 0 and 0 < m < 1 is called the fast diffusion equation. When κ > 0, there is a restoring drift. In case m = 1, (2.9) is of course the heat equation for κ = 0, and the linear Fokker-Planck equation for κ > 0. Equation (2.9) can be written in the gradient flow form

∂u ∂t (t, x) = div u(t, x)∇ δG δu with G[u] = R 2 1 m -1 u m (x) + κ |x| 2 2 u(x) dx , (2.10)
which shows that the evolution equation (2.9) is gradient flow for G with respect to the Wasserstein metric.

The value m = 1/2 for this equation in R 2 is critical in the sense that the functional G in (2.10) is strictly displacement convex for m ≥ 1/2, but is not displacement convex for m < 1/2. There are many of other "critical" values of m between 0 and 1/2 at which other things happen, see [START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF] for instance. But since displacement convexity plays a crucial role in our work, it is natural to refer to the m = 1/2 case as critical here. Indeed, by the criteria of McCann introduced above since

G[u] = V [u] + G φm [u] with V (x) = κ|x| 2 /2 and φ m (s) = s m /(m -1), then V [u] is uniformly displacement convex, and for m ≥ 1/2, G φm [u] is displacement convex.
As might be expected, some difficulties arise at the critical value m = 1/2. Since (2.9) is gradient flow for G, one might hope to find stable steady states by finding the minimizers ū of G. Computing the Euler-Lagrange equation we find m/(m -1)ū m-1 + κ|x| 2 /2 = C, where C is a Lagrange multiplier for the constraint M := R 2 u(x) dx, which is conserved. In the case m = 1/2 and choosing

κ = κ M,λ := 2 π M λ we find ū(x) = M π λ (λ + |x| 2 ) 2 = ̺ λ (x) .
One readily checks that ū = ̺ λ is a steady state solution to (2.9) with κ = κ M,λ and so the family of stationary solutions of the PKS system which we are investigating are also stationary solutions of the critical fast diffusion equation for different drifts κ = κ M,λ . However, as neither √ ̺ λ (x) nor |x| 2 ̺ λ is integrable, these functions are not in the domain of definition of G, and so are not minimizers of

G[u] = R 2 -2 u(x) + κ M,λ |x| 2 2 u(x) dx , (2.11)
the m = 1/2 version of (2.9) with κ = κ M,λ as above.

The cure is a simple renormalization as introduced in [START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF][START_REF] Lederman | On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass[END_REF]: Consider instead the functional

H λ [u] defined by u → R 2 -2 u(x) -̺ λ (x) dx + κ M,λ R 2 |x| 2 2 [u(x) -̺ λ (x)] dx . (2.12)
Then, at least as long as u has the same behavior at infinity as does ̺ λ , the integrals will converge. The counter terms that we have subtracted off from our functional do not depend on u, and hence they do not affect δG/δu. This is the key idea used in the improvements of rates of convergence for the fast diffusion equation, see [START_REF] Carrillo | Fine asymptotics for fast diffusion equations[END_REF][START_REF] Lederman | On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass[END_REF].

Since κ M,λ |x| 2 = 2/ √ ̺ λ -λ κ M,λ
, the functional in (2.12) can be written in the following simpler form, which we take to be the definition of the critical fast diffusion entropy:

H λ [u] := R 2 √ u - √ ̺ λ 2 √ ̺ λ dx.
It is easy to check that for m = 1/2, (2.9) can be written in the general form (2.6) with G = H λ . As noted above the displacement convexity of H λ is formally obvious from the fact that where u(x), ̺ λ (x) and |x| 2 u(x) are integrable, H λ [u] would differ from the right hand side of (2.11) by a constant. We provide a rigorous proof in the next subsection.

Regularization of the critical fast diffusion entropy

To show that u → H λ [u] is displacement convex, and more generally, to make rigorous computations involving critical fast diffusion entropy, H λ [u], we introduce a regularized version of the critical fast diffusion entropy:

2.1 DEFINITION (Regularized fast-diffusion relative entropy functional). For δ > 0, and u a density with mass M , define H λ,δ [u] by

H λ,δ [u] = R 2 √ u + δ - √ ̺ λ + δ 2 √ ̺ λ + δ dx .

PROPOSITION (Displacement convexity of relative entropy functionals). For any density

u ∈ L 1 + (R 2 ) of mass M , δ → H λ,δ [u]
is monotone increasing as δ decreases to zero, and

lim δ→0 H λ,δ [u] = H λ [u] .
(2.13)

Furthermore, let u 0 and u

1 belong to L 1 + (R 2 ) of total mass M such that W 2 (u 0 , u 1 ) < ∞, and let u t , 0 ≤ t ≤ 1 be their displacement interpolation. Then for each δ > 0, (1 -t) H λ,δ [u 0 ] + t H λ,δ [u 1 ] -H λ,δ [u t ] ≥ t (1 -t) K δ (u 0 , u 1 ) , where K δ (u 0 , u 1 ) satisfies lim δ→0 K δ (u 0 , u 1 ) = κ M,λ W 2 2 (u 0 , u 1 ) , (2.14)
and

K δ (u 0 , u 1 ) ≥ γ δ W 2 2 (u 0 , u 1 ) with γ δ < 0. Consequently, the maps u → H λ,δ [u] are γ δ -displacement convex and the map u → H λ [u] is strictly uniformly displacement convex: (1 -t) H λ [u 0 ] + t H λ [u 1 ] -H λ [u t ] ≥ κ M,λ t (1 -t) W 2 2 (u 0 , u 1 ) . Proof: As δ → √ u + δ - √ ̺ λ + δ 2 is non-increasing. Then, as δ decreases, √ u + δ - √ ̺ λ + δ 2 √ ̺ λ + δ increases to √ u - √ ̺ λ 2 √ ̺ λ .
By the monotone convergence theorem and (2.13), the monotonicity in δ follows. Next,

√ u + δ - √ ̺ λ + δ 2 √ ̺ λ + δ = u √ ̺ λ + δ -2 √ u + δ - √ δ + δ √ ̺ λ + δ + ̺ λ + δ -2 √ δ .
Where by the mean value theorem

u √ ̺ λ + δ ≤ u √ δ , √ u + δ - √ δ ≤ u 2 √ δ and δ √ ̺ λ + δ + ̺ λ + δ -2 √ δ ≤ ̺ λ √ δ .
These three terms are integrable and

H λ,δ [u] = (I) + (II) + const. (2.15) where (I) := R 2 1 √ ̺ λ + δ u dx and (II) := 2 R 2 √ δ - √ u + δ dx .
The criterion (2.8) are easily checked for Φ(u) = √ δ -√ u + δ, and thus (II) is displacement convex. The term (I) in (2.15) is unfortunately not displacement convex in general. In fact, we will show that is γ δdisplacement convex with a explicit computable constant. In order to check the γ δ -displacement convexity of the regularized functional, notice that (̺ λ + δ) -1/2 is a function of |x| 2 . Thus, the functional (I) is of the general form: [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] ensures that this is implied by

u → R 2 V δ (x) u(x) dx with V δ (x) = (̺ λ + δ) -1/2 . The characterization of γ δ -displacement convexity
D 2 V δ ≥ γ δ I 2 . Let us compute the hessian of the potential V δ (x). Define the function f δ on [0, ∞) by f δ (|x| 2 ) := V δ (x) = 1 ̺ λ (x) + δ .
We compute

f ′ δ (s) = A [A + δ(λ + s) 2 ] 3/2 and f ′′ δ (s) = - 3 A δ(λ + s) [A + δ(λ + s) 2 ] 5/2 , with A = M λ/π. Therefore D 2 V δ (x) = 2f ′ δ (|x| 2 )δ ij + 4f ′′ δ (|x| 2 )(x ⊗ x), and taking into account that f ′′ δ (s) ≤ 0, then ξ • D 2 V δ (x) • ξ T ≥ 2f ′ δ (|x| 2 ) + 4f ′′ δ (|x| 2 )|x| 2 |ξ| 2 := F δ (|x| 2 )|ξ| 2 (2.16)
for all x, ξ ∈ R 2 , where the function F δ is given by

F δ (s) = 2f ′ δ (s) + 4sf ′′ δ (s) = 2A 2 + 2Aδλ 2 -8Aλδs -10Aδs 2 [A + δ(λ + s) 2 ] 5/2 .
It is obvious that the function F δ converges point-wise to the constant κ M,λ as δ → 0 in [0, ∞). Moreover, since for each δ > 0, the function F δ (s) → 0 as s → ∞ and it is clear that is negative for s large enough since the denominator is positive and the numerator has a negative dominant term, then F δ attains its maximum and minimum in [0, ∞).

Then, we can choose its minimum value as γ δ < 0 and the γ δ -displacement convexity is proved.

In order, to show the limiting uniform displacement convexity, we need to refine our arguments. For that, we come back to the definition of convexity. Let ψ be the essentially unique convex function such that ∇ψ#u

0 = u 1 . For 0 ≤ t ≤ 1, define η δ (t) := R 2 V δ (x) [(1 -t) u 0 (x) + t u 1 (x) -u t (x)] dx = R 2 [(1 -t) V δ (x) + t V δ (∇ψ(x)) -V δ (x + t(∇ψ(x) -x))] u 0 (x) dx .
We seek a lower bound on η δ of the form η δ (t) ≥ t (1t) K δ (u 0 , u 1 ). Since η δ (0) = η δ (1) = 0, it suffices for this purpose to show that η ′′ δ (t) ≥ 2 K δ (u 0 , u 1 ) for all 0 ≤ t ≤ 1. By denoting y := ∇ψ(x)x, we have

η ′′ δ (t) = R 2 y • D 2 V δ (x + ty) • y T u 0 (x) dx .
Using (2.16), we readily obtain that

η ′′ δ (t) ≥ R 2 F δ (|x + ty| 2 )|y| 2 u 0 (x) dx ≥ K δ (u 0 , u 1 ) , with K δ (u 0 , u 1 ) := min 0≤t≤1 R 2 F δ (|x + ty| 2 )|y| 2 u 0 (x) dx.
Now, let us observe that the function

F δ is bounded in [0, ∞) uniformly in δ. For that, note that f ′ δ is decreasing and thus f ′ δ (s) ≤ f ′ δ (0) ≤ A -1/2
. On the other hand, by the geometric-arithmetic mean inequality, we get

|sf ′′ δ (s)| ≤ 3δ 1/2 s A + δs 2 ≤ 3 2 √ A . (2.17)
As a consequence, we get

F δ L ∞ (0,∞) ≤ 8 √ A ,
and thus,

|F δ (|x + ty| 2 )||y| 2 u 0 (x) ≤ 8 √ A |∇ψ(x)| 2 u 0 (x) ∈ L 1 (R 2 ) ,
for all 0 ≤ t ≤ 1. Thus, the dominated convergence theorem guarantees that

lim δ→0 R 2 F δ (|x + ty| 2 )|y| 2 u 0 (x) dx = κ M,λ R 2 |y| 2 u 0 (x) dx = κ M,λ W 2 2 (u 0 , u 1 ) ,
uniformly in 0 ≤ t ≤ 1, which together with the definition of K δ (u 0 , u 1 ) implies the uniform displacement convexity of the limiting functional

H λ [u].
Continuing with the notation of Proposition 2.2, define the function

h δ on [0, 1] by h δ (t) = H λ,δ [u t ] - K δ (u 0 , u 1 )t 2 . Then by Proposition 2.2, (1 -t) h δ (0) + t h δ (1) -h δ (t) ≥ 0, so that h δ is convex. Therefore, for all t ∈ (0, 1), h δ (1) -h δ (0) ≥ h δ (t) -h δ (0) t .
This in turn implies that

H λ,δ [u 1 ] -H λ,δ [u 0 ] ≥ lim sup t→0 H λ,δ [u t ] -H λ,δ [u 0 ] t + K δ (u 0 , u 1 ) .
To compute the lim sup of the right hand side, we treat the two non-constant terms (I) and (II) in (2.15) separately. As we have noted (II) is displacement convex, and by well known theorems on the sub-gradients of displacement convex functions [2, Chapter 10], this part contributes

R 2 ∇u 0 (x) 2(u 0 (x) + δ) 3/2 • (∇ψ(x) -x) u 0 (x) dx ,
as long as the integrand satisfies mild regularity properties; in particular whenever u 0 is bounded below on every compact set by some strictly positive number, and √ u 0 has a square integrable distributional gradient. We shall show that both of these conditions hold in our application. Given that they do, then by the Cauchy-Schwarz inequality,

R 2 |∇u 0 | (u 0 + δ) 3/2 |∇ψ(x) -x| u 0 dx ≤ R 2 |∇u 0 | 2 (u 0 + δ) 3 u 0 dx R 2 |∇ψ(x) -x| 2 u 0 dx ≤ 1 3δ R 2 |∇u 0 | 2 u 0 dx W 2 (u 0 , u 1 ) = 2 √ 3δ R 2 |∇ √ u 0 | 2 dx W 2 (u 0 , u 1 ) .
The contribution of (I) in (2.15) can be treated by appealing to the general results in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] since this functional is γ δ -displacement convex, in the notation of [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], and thus, this part contributes 2

R 2 f ′ δ (|x| 2 ) x • (∇ψ(x) -x) u 0 dx . Which is finite because s → √ s f ′ δ (s)
is a rational functional which tends to 0 when s goes to infinity. Thus we have:

2.3 LEMMA (First-order characterization of displacement convexity). Let u 0 and u 1 be two densities of total mass M such that W 2 2 (u 0 , u 1 ) < ∞, and such that u 0 is uniformly bounded below on compact subsets of R 2 by a strictly positive number, and that √ u 0 has a square integrable distributional gradient. Let ∇ψ be the unique gradient of a convex function ψ in R 2 so that ∇ψ#u 0 = u 1 . Then

H λ,δ [u 1 ] -H λ,δ [u 0 ] ≥ R 2 2Ax [A+δ(λ+|x| 2 ) 2 ] 3 2 + ∇u 0 2(u 0 +δ) 3 2 (∇ψ(x) -x) u 0 dx + K δ (u 0 , u 1 ) (2.18)
where K δ (u 0 , u 1 ) is defined in Proposition 2.2, and the integrand in (2.18) is integrable.

One might be tempted to take the limit δ → 0 at this stage and to conclude

H λ [u 0 ] ≤ H λ [u 1 ] - R 2 κ M,λ x + ∇u 0 2u 3/2 0 • (∇ψ(x) -x) u 0 dx -κ M,λ W 2 2 (u 0 , u 1 ) ,
but without further information about ∇ψ(x)x, it is not possible to do this, or to justify the convergence of the integral. In our applications, it will be simpler to use the specific information that we obtain on ∇ψ(x)x, then to do some integrations by parts, and then take the limit δ → 0. Let us finally deduce as an application of the uniform displacement convexity of the functional H λ [u], an interesting functional inequality of Talagrand type. Actually, generalized Log-Sobolev-type inequalities lead formally to generalized Talagrand-type inequalities for this functional by repeating arguments due to Otto and Villani [38, Theorem 1, Proposition 1] in the linear case and generalized in [START_REF] Carrillo | Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates[END_REF]Theorem 2.1]. Here, we are able to show it in full rigor by the previous approximation argument.

THEOREM (Talagrand's inequality). Whenever

u ∈ L 1 + (R 2 ) of mass M with H λ [u] < ∞, then W 2 (u, ̺ λ ) ≤ 2H λ [u] κ M,λ .
Proof: Using Lemma 2.3 with u 0 = ̺ λ and u 1 = u, we obtain that

H λ,δ [u] ≥ K δ (̺ λ , u). for all δ > 0, since H λ,δ [̺ λ ] = 0 and 2Ax [A + δ(λ + |x| 2 ) 2 ] 3/2 + ∇̺ λ 2(̺ λ + δ) 3/2 = 0.
Thus, passing to the limit δ → 0 taking into account (2.14), the desired inequality is obtained.

2.5 Remark (Basins of attraction). Theorem 2.4 tells us that the 2-Wasserstein distance of our initial data to the stationary state ̺ λ is finite provided H λ [u 0 ] < ∞. Moreover, each of the equilibrium solutions ̺ λ are infinitely far apart in the W 2 metric: We can easily check that with ϕ(x) = λ/µ|x| 2 /2, one has ∇ϕ#̺ µ = ̺ λ . Thus, the uniqueness part of Brenier-McCann Theorem ensures

W 2 2 (̺ µ , ̺ λ ) = 1 2 R 2 λ µ x -x 2 ̺ µ (x) dx = +∞
since the equilibrium densities ̺ λ all have infinite second moments. In particular, H λ [̺ µ ] = +∞ for µ = λ.

3 Proof of the concentration controlled inequalities

Concentration control for F PKS

To prepare the way for the proof of Theorem 1.9, it is useful to give an elementary demonstration of a crude form of the log HLS inequality, without sharp constants, but which would nonetheless provide bounds on E[ρ] for all M < 8π.

3.1 LEMMA (Bounds on the entropy). Let ρ be a density of mass M on R 2 such that ρ log ρ and |x|ρ are in L 1 (R 2 ). Then, for any α > 1/(8π), there exists a constant C(M, α, λ) > 0 only depending on M , α and λ such that

1 2 (G) + * ρ(x) ≤ α R 2 ρ log ρ dx + C(M, α, λ) + 4 α M log R 2 λ + |x| 2 ρ dx for al x ∈ R 2 .
For α and λ fixed, C(M, α, λ) is monotone increasing in M .

Proof: Recall the following Young type inequality: For all s, t > 0, st ≤ s log s + e t-1 . Then, for any α > 0, we have

st = α [s (t/α)] ≤ αs log s + αe t/α-1 . (3.1)
We now apply this to

1 2 (G) + * ρ(x) = 1 4π R 2 (log |x -y|) -ρ(y) dy = |x-y|<1 1 4π (-log |x -y|) ρ(y) ̺ λ (y) ̺ λ (y) dy ,
under the integral sign with weight ̺ λ , and with

s = ρ(y) ̺ λ (y) and t = - 1 4π log |x -y| .
Since ̺ λ is bounded above by M/(λ π), this yields

1 2 (G) + * ρ(x) ≤ α |y-x|<1 ρ ̺ λ log ρ ̺ λ ̺ λ dy + α M λπe |z|≤1 1 |z| 1/(4πα) dz . (3.2)
The second integral on the right converges as long as α > 1/8π, in which case, doing the integral explicitly, we find

α M λ π e |z|≤1 1 |z| 1/(4πα) dz = M λe 8πα 2 8πα -1 for 8πα > 1 . (3.3)
To relate the first integral to E[ρ], use the fact that s → s log s is bounded below by -1/e to conclude that

|y-x|<1 ρ ̺ λ log ρ ̺ λ ̺ λ dy ≤ R 2 ρ ̺ λ log ρ ̺ λ ̺ λ dy + M e ≤ R 2 ρ log ρ dy - R 2 ρ log ̺ λ dy + M e .
By Jensen's inequality for the concave function log in L 1 ((ρ/M ) dx),

R 2 ρ log ̺ λ dx = M log λM π -4 R 2 log λ + |x| 2 ρ dx ≥ M log λM π -4M log 1 M R 2 λ + |x| 2 ρ dx . (3.4)
Using (3.3) and (3.4) in (3.2), we obtain

1 2 (G) + * ρ(x) ≤ α R 2 ρ log ρ dy + M λe 8πα 2 8πα -1 + α M 1 e -log λ π + 3 log M + 4 log R 2 λ + |x| 2 ρ dx .
The result follows with

C(M, α, λ) := α M 1 λe 8πα 8πα -1 + 1 e -log λ π + 3(log M ) + .
To apply this, let ρ be any density on R 2 of mass 8π or less. For any number R > 1, define

ρ 1 (x) := 1 {|x|≥R} ρ(x) and ρ 2 (x) := 1 {|x|<R} ρ(x) .
Also define

ρ 1 (x) := 1 {|x|≥R-1} ρ(x) and ρ 2 (x) := 1 {|x|<R+1} ρ(x) .
Then since the support of (G) + has unit radius,

R 2 ρ (G) + * ρ dx = R 2 ρ (G) + * ρ 1 dx + R 2 ρ (G) + * ρ 2 dx = R 2 ρ 1 (G) + * ρ 1 dx + R 2 ρ 2 (G) + * ρ 2 dx . (3.5)
Now suppose it is possible to choose R > 1 so that for some 0 < a < 8π,

|x|>R-1 ρ dx = R 2 ρ 1 dx ≤ 8π -a and |x|<R+1 ρ dx = R 2 ρ 2 dx ≤ 8π -a . (3.6)
Then choosing α = (8πa/2) -1 , and applying the pointwise bounds from Lemma 3.1 in (3.5), we obtain that

1 2 R 2 ρ (G) + * ρ dx ≤ 16π -2a 16π -a R 2 ρ log ρ dx + 2C(8π, α, λ) + 32π α log R 2 λ + |x| 2 ρ dx .
It follows from this and the obvious fact that

- R 2 ρ G * ρ dx ≥ - R 2 ρ (G) + * ρ dx, that F PKS [ρ] ≥ a 16π -a R 2 ρ log ρ dx -2C(8π, α, λ) -32π α log R 2 λ + |x| 2 ρ dx . (3.7)
If the mass M is less that 8π, the estimate (3.7) gives us an upper bound on the entropy of ρ in terms of F PKS [ρ] and the first moment of ρ. This would suffice, in place of the sharp logarithmic Sobolev inequality, to prove that no blow-up occurs or mass less than 8π as it was done in [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF]. This cutting-in-pieces argument is similar but simpler than the one in [6, Lemma 3.1] used for the bounded second initial moment case.

If the mass is equal to 8π, one needs additional information to find an 0 < a < 8π and an R > 1 for which (3.6) is true. The additional information in case the second initial moment is bounded was given by contradicting the convergence to a Delta Dirac, see [START_REF] Blanchet | Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2[END_REF]Lemma 3.1]. Here, we need to localize the mass and quantify the tails by using a bound on H λ [ρ]. One of the bounds is easy: 3.2 LEMMA (Solid core). Let ρ be a density of mass M such that for some λ > 0,

H λ [ρ] < ∞. Then {|x|≥4 √ λ+4(λ/Mπ) 1/4 √ H λ [ρ]} ρ dx ≤ M 2 . (3.8)
Proof: We start by reminding the bound

R 2 |x| ρ dx < R 2 λ + |x| 2 ρ dx ≤ 2 √ λ M + 2M 3/4 (λ/π) 1/4 H λ [ρ] ,
proved in Lemma 1.11, here written with explicit constants. The result is a direct consequence of this bound since:

if |x|≤r ρ dx ≤ M/2 , then R 2 |x|ρ dx ≥ rM/2 . Choosing r = 4 √ λ + 4(λ/M π) 1/4 H λ [ρ],
we get a contradiction unless (3.8) is satisfied.

Now define

R := 4 √ λ 1 + H λ [ρ] √ M πλ + 1 .
and then using this value of R define ρ 1 , ρ 2 , ρ 1 and ρ 2 as above. By Lemma 3.2,

R 2 ρ 1 dx ≤ M 2 ≤ 4π .
Thus, the left inequality in (3.6) is valid for this choice of R, and any a ≤ a 1 := 4π. The inequality on the right in (3.6) requires more work to achieve. The key is the following result on "thick tails":

3.3 THEOREM (Thick tails). Let ρ be a density of mass M such that W 2 2 (ρ, ̺ λ ) < ∞.
Then for η * := 1 5 e -1/5 and any s > 1,

|x| 2 ≥λs 2 ρ(x) dx ≥ η * e -4 M λ W 2 2 (ρ,̺ λ ) |x| 2 ≥λs 2 ̺ λ (x) dx = M η * 1 + s 2 e -4 M λ W 2 2 (ρ,̺ λ ) (3.9)
and

|x| 2 ≥λs 2 ρ(x) dx ≥ η * e -4 √ πM λ H λ [ρ] |x| 2 ≥λs 2 ̺ λ (x) dx = M η * 1 + s 2 e -4 √ πM λ H λ [ρ] . (3.10)
Proof: Let 0 < η * < 1/5 to be fixed later. Fix any s > 1 and define η by

η := |x| 2 ≥λs 2 ρ(x) dx |x| 2 ≥λs 2 ̺ λ (x) dx . (3.11)
We may assume that η < η * , or else there is nothing to prove for this s.

Let ∇ϕ be the gradient of a convex function such that ∇ϕ#̺ λ = ρ. Define r := √ λs, and define

A := { x : |x| ≥ r and |∇ϕ(x)| < r } . Then since on A, |∇ϕ(x) -x| 2 ≥ (|x| -r) 2 , W 2 2 (ρ, ̺ λ ) = R 2 |∇ϕ(x) -x| 2 ̺ λ dx ≥ A (|x| -r) 2 ̺ λ dx .
We now claim that

A ̺ λ dx ≥ (1 -η) |x|≥r ̺ λ (x) dx . (3.12)
Assuming (3.12) for the moment,

W 2 2 (ρ, ̺ λ ) ≥ inf A (|x| -r) 2 ̺ λ dx : A ⊂ {|x| ≥ r} , A ̺ λ dx ≥ (1 -η) |x|≥r ̺ λ (x) dx . (3.13)
By the "bathtub principle", the infimum is achieved by choosing A to be the level set of (|x|r) 2 in {|x| ≥ r} that has the minimal mass, that is, A = {x : r ≤ |x| ≤ r } where r is such that r≤|x|≤ r

̺ λ dx = (1 -η) |x|≥r ̺ λ (x) dx , which is equivalent to |x|≥ r ̺ λ (x) dx = η |x|≥r ̺ λ (x) dx .
By direct computation, this means M λ/(λ + r 2 ) = ηM λ/(λ + r 2 ), and then since η < 1, this means that r 2 > r 2 /η. In fact, since η < 1/5, r > 2r, and so the optimal set A in (3.13) contains {x : 2r ≤ |x| ≤ r/ √ η}. Therefore, combining the last estimates with (3.13),

W 2 2 (ρ, ̺ λ ) ≥ 2r≤|x|≤r/ √ η (|x| -r) 2 ̺ λ dx ≥ 1 4 2r≤|x|≤r/ √ η |x| 2 ̺ λ dx . Now recalling that r = √ λs, explicitly calculating the integral yields W 2 2 (ρ, ̺ λ ) ≥ λM 4 log 1 + s 2 /η 1 + 4s 2 + 4 - 1 η s 2 (1 + s 2 /η)(1 + 4s 2 ) .
Remembering that s > 1 and η < 1 5 ,

1 + s 2 /η 1 + 4s 2 ≥ s 2 /η 5s 2 = 1 5η and 4 - 1 η s 2 (1 + s 2 /η)(1 + 4s 2 ) ≥ 4 - 1 η η 5(1 + η) ≥ - 1 5 .
Therefore, fixing η * := e -1/5 /5 < 1/5, we get

η ≥ η * e -4 λM W 2 2 (̺,̺ λ ) .
Combining this with (3.11) yields (3.9). Thus, to prove (3.9), it suffices to prove (3.12). By definition,

A ̺ λ dx = {|x|>r}∩{|∇ϕ(x)|<r} ̺ λ dx ≥ |x|>r ̺ λ dx - |∇ϕ(x)|≥r ̺ λ dx = |x|>r ̺ λ dx - |y|≥r ρ dy = |x|>r ̺ λ dx -η |x|≥r ̺ λ dx ,
and this proves (3.12). Finally, (3.10) follows from (3.9) and the Talagrand-type inequality, Theorem 2.4. Now note that, by the definition of R, R 2 /λ > 1, and hence Theorem 3.3 implies that

|x|≥R+1 ρ(x) dx ≥ η * e -4 √ πM λ H λ [ρ] |x|≥R+1 ̺ λ (x) dx . Then since |x|≥R+1 ̺ λ (x) dx ≥ 8πλ (R + 1) 2 .
Thus, with R as above and

a 2 := 8πλη * e -4 √ πM λ H λ [ρ] 4 √ λ 1 + H λ [ρ] √ M πλ + 2 -2
< 8π , the second inequality in (3.6) is also satisfied.

Proof of Theorem 1.9: Most of the work is now done since (3.6) is satisfied by choosing a = min(a 1 , a 2 ). We need a final argument to take care of the control for the negative contribution of the entropy in terms of the localization of the mass of the distribution known as Carleman-type estimate.

LEMMA (Control on the negative part of the entropy). For any density

ρ ∈ L 1 + (R 2 ), if the moment R 2 m(x) ρ(x) dx is bounded with e -m(x) ∈ L 1 (R 2 ) and m : R + 0 -→ R + 0 , then R 2 ρ(x) log -ρ(x) dx ≤ R 2 m(x)ρ(x) dx + 1 e R 2 e -m(x) dx . Proof: Let ρ := ρ χ {ρ≤1} and M = R 2 ρ(x) dx ≤ R 2 ρ(x) dx = M . Then R 2 ρ(x) (log ρ(x) + m(x)) dx = R 2 [U (x) log U (x)]µ dx -M log Z
where U := ρ/µ, µ(x) = e -m(x) /Z with Z = R 2 e -m(x) dx. The Jensen inequality yields

R 2 [U (x) log U (x)]µ dx ≥ R 2 U (x)µ dx log R 2 U (x)µ dx = M log M and - R 2 ρ(x) log -ρ(x) dx = R 2 ρ(x) log ρ(x) dx ≥ M log M -M log Z - R 2 m(x) ρ(x) dx ≥ - Z e - R d m(x) ρ(x) dx .
To control the negative part of the entropy in (3.7), use Lemma 3.4 with m(x) = λ + |x| 2 and Lemma 1.11:

- a 16π -a R 2 ρ log -ρ dx ≥ - R 2 ρ log -ρ ≥ - R 2 m(x)ρ(x) dx - 1 e R 2 e -m(x) dx ≥ -2 √ λM -2M 3/4 (λ/π) 1/4 H λ [ρ] - 1 e R 2 e -m(x) dx .
This gives the final control on the positive part of the entropy from (3.7):

F PKS [ρ] ≥ a 16π -a R 2 ρ log + ρ dx -2C(8π, α, λ) -32π α log R 2 λ + |x| 2 ρ dx -2 √ λM -2M 3/4 (λ/π) 1/4 H λ [ρ] - 1 e R 2 e -m(x) dx . (3.14)
Finally, we choose

γ 1 := a 16π -a
where a is given just above and an explicit expression for C CCF follows from (3.14).

Concentration control for D

Proof of Theorem 1.10. While in the proof of Theorem 1.9, we used a "horizontal splitting" of ρ, here we use a "vertical splitting": Let f := ρ 1/4 . For β > 0, define f β := min{f, β 1/4 } and h β := ff β . We have

π D[ρ] = 8 π R 2 |∇f | 2 dx -π R 2 f 6 dx . (3.15) Defining A β = {x : f (x) ≥ β 1/4 } = {x : ρ(x) ≥ β}, we get R 2 f 6 dx = R 2 \A β f 6 β dx + A β (h β + β 1/4 ) 6 dx = R 2 f 6 β dx -β 3/2 |A β | + A β (h β + β 1/4 ) 6 dx . (3.16)
By the convexity of x → x 6 , for any η ∈ (0, 1)

A β (h β + β 1/4 ) 6 dx ≤ β 3/2 η 5 |A β | + 1 (1 -η) 5 A β h 6 β dx . (3.17)
By the inequality f 6 β ≤ √ βf 4 , and plugging (3.17) and (3.16) into (3.15), we obtain

π D[ρ] ≥ 8 π R 2 |∇f | 2 dx - π (1 -η) 5 R 2 h 6 β dx -8 π 2 β -π β 3/2 1 η 5 -1 |A β | . (3.18)
By the GNS inequality, Lemma 1.2, applied to h β :

- π (1 -η) 5 R 2 h 6 β dx ≥ - R 2 |∇h β | 2 dx 1 (1 -η) 5 R 2 h 4 β dx . (3.19)
By definition of f β and h β , ∇f β = 0 in the support of h β so that

R 2 |∇f | 2 dx = R 2 |∇f β | 2 dx + R 2 |∇h β | 2 dx . ( 3
π D[ρ] ≥ 8 π R 2 |∇f β | 2 dx + 8π - 1 (1 -η) 5 R 2 h 4 β dx R 2 |∇h β | 2 dx -8 π 2 β -π β 3/2 1 η 5 -1 |A β | . (3.21)
We obtain a result of the type we seek under any conditions that ensure the second term on the right is positive. Our first approach uses Theorem 1.9, and so requires that both F PKS [ρ], and H λ [ρ], for some λ > 0, be finite.

By (3.1) once more and Theorem 1.9, we get for all α > 0,

R 2 h 4 β dx = R 2 1 A β ρ dx ≤ α R 2 ρ log ρ dx + e (1/α)-1 |A β | ≤ α F PKS [ρ] + C CCF γ 1 + αe (1/α)-1 |A β | . (3.22)
This sort of estimate is frequently used in large deviations problems. Now choose η = 1/2 in (3.21), and then choose α so that the first term in the right hand side in (3.22) is π/8, we have, for this choice of η and α,

8π - 1 (1 -η) 5 R 2 h 4 β dx ≥ 4π -αe (1/α)-1 |A β | . By Chebychev's inequality, |A β | ≤ 8π
β , and so we can choose β so that for any γ ∈ (0, 4π),

8π - 1 (1 -η) 5 R 2 h 4 β dx ≥ γ ,
as was to be shown with C CCD := 248π 2 √ β. Though we have explained how to compute β, we shall not write down a formula. This proves the first part of the theorem.

As for the second, note that we used the bound on F PKS only to obtain a bound on the entropy which was used in (3.22). However, if we have by other means a bound on the entropy, we can use that in (3.22) in place of the bound on F PKS . This proves the final part of the theorem.

Analysis of the discrete time variational scheme for the critical mass PKS system

From now on, we will assume that the mass is 8π.

4.1 The Jordan-Kinderlehrer-Otto scheme for the critical mass PKS system

The Jordan-Kinderlehrer-Otto (JKO) scheme for constructing solutions to (2.6), as described in Section 2, would be to fix a time step τ > 0, and inductively define the sequence {ρ k } k∈N by setting ρ 0 to be the initial density, and then for k ≥ 0,

ρ k+1 ∈ argmin W 2 2 (ρ, ρ k ) 2τ + G[ρ] . (4.1)
In other words, ρ k+1 is some minimizer of the functional

ρ → W 2 2 (ρ, ρ k )/(2τ ) + G[ρ].
Only existence of the minimizer is an issue, and not uniqueness, although in many examples that have been investigated a strict convexity argument furnishes the uniqueness. The key point is existence of a minimizer, since that provides a solution of the Euler-Lagrange equation for the minimization problem in (4.1). Then, as shown in [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], the fact that each ρ k+1 satisfies this Euler-Lagrange equation means that, under certain conditions on G, if one defines ρτ by

ρτ (t, x) = ρ k (x)
for kτ ≤ t < (k + 1)τ , with ρτ (0, x) = ρ 0 , there is a sequence of values of τ tending to zero along which ρτ tends to a solution of (2.6) in a suitable weak sense. This scheme of constructing weak solutions of the PKS system for M < 8π was developed in [START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF]. However, for M = 8π we can not proceed in a very direct manner. Our problem lies outside the scope of previous applications of the JKO scheme, since at the critical mass M = 8π, (1.5) provides no upper bound on E[ρ], and hence, it is not even clear that minimizers exist for the variational problem in (4.1) when G = F PKS and M = 8π. Our controlled concentration inequalities could be used to solve this problem, but other more thorny issues arise when we would try to analyze the Euler-Lagrange equation.

To circumvent these difficulties, we introduce a regularized functional. In fact, for reasons that will become evident later on, we shall even be forced to choose a different degree of regularization at each time step.

t > 0, x ∈ R 2 : ∂ ∂t γ √ t = 1 2 ∆γ √ t . It follows that ∂ ∂t G * γ √ t = 1 2 ∆G * γ √ t = - 1 2 γ √ t ,
and thus,

∂ ∂ǫ G ǫ = -2ǫγ 2ǫ . (4.2)
The right hand side is strictly negative everywhere. From this we deduce a useful monotonicity property:

ǫ 1 < ǫ 2 ⇒ G ǫ1 (x) > G ǫ2 (x) (4.3) for all x ∈ R 2 . Let us point out that ∂ 2 ∂ǫ 2 G ǫ = -2γ 2ǫ -8ǫ 2 ∆γ 2ǫ = 2x • ∇γ 2ǫ + 2γ 2ǫ = 2 1 - |x| 2 4ǫ 2 γ 2ǫ , (4.4)
since γ ǫ satisfies ǫ 2 ∆γ ǫ + div (xγ ǫ ) = 0.

4.1 LEMMA (First properties of G ǫ ). Let G ǫ be defined as above then:

(i) For all x ∈ R 2 , G ǫ (x) ≤ G(x).
(ii) There exists C > 0 such that for all x ∈ R 2 , G ǫ (x) ≤ Cǫ -2 .

(iii) For all (x, y) ∈ R 4 ,

G ǫ (x -y) ≥ - 1 4π 4 + log e + |x| 2 + log e + |y| 2 .
Proof: (i) As γ is radially symmetric and subharmonic in R 2 so that, by the mean value property the first item holds.

(ii) Since log -|x| is locally integrable in R 2 , for any

x ∈ R 2 G * γ ǫ (x) = |y|≤ǫ G(x -y) γ ǫ (y) dy ≤ 1 2π |y|≤ǫ log -|x -y| γ ǫ (y) dy ≤ C ǫ 2 .
since γ is bounded. Thus, we get

G ǫ (w) ≤ C ǫ 2 R 2 γ ǫ (z) dz = C ǫ 2 .
(iii Therefore,

)
G(z -w) ≥ - 1 2π (2 + log |z| + log |w|) .
Integrating both sides against γ ǫ (xz) γ ǫ (yw), and using Jensen's inequality, we find

G ǫ (x -y) ≥ - 1 2π 2 + log R 2 |z| γ ǫ (x -z) dz + log R 2 |w| γ ǫ (y -w) dw ≥ - 1 2π 2 + log |x| + R 2 |z| γ ǫ (z) dz + log |y| + R 2 |w| γ ǫ (w) dw ≥ - 1 4π 4 + log e + |x| 2 + log e + |y| 2 ,
at least for ǫ small enough so that R 2 |z|γ ǫ (z) dz is small enough.

One of the main uses that we will make of the regularization of the self interaction functional is that it provides a regularized density for the chemical attractant: given a mass density ρ, we define the regularized chemical attractant density c ǫ by c ǫ (x) = G ǫ * ρ(x).

4.2 LEMMA (Uniform estimate regularized chemoattractant). For all ǫ > 0 and all densities ρ with mass 8π, the regularized chemical attractant density

c ǫ = G ǫ * ρ satisfies ∇c ǫ ∞ ≤ 4 C HLS ǫ γ 2 4/3 and |x|∇c ǫ ∞ ≤ 8C HLS γ 4/3 |x|γ 4/3 + 4 + C HLS 2πǫ γ 2 4/3 |x|ρ 1 . (4.5)
Here C HLS denotes the constant of the sharp Hardy-Littlewood-Sobolev (HLS) inequality [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] for the special case p = q = 4/3:

R d ×R d f (x) 1 |x -y| g(y) dx dy ≤ C HLS f 4/3 g 4/3 . (4.6)
Though the explicit value of C HLS is simple enough, see [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF], our bounds and their proofs will perhaps be easier to read if leave C HLS unevaluated in them, as a marker of the use of the HLS inequality.

Proof: By the Young inequality, we have

∇c ǫ ∞ = ∇G ǫ * ρ ∞ ≤ 8 π ∇G ǫ ∞ .
And by the HLS inequality,

∇G ǫ ∞ ≤ 1 2π R 2 ×R 2 γ ǫ (x -z) 1 |z -w| γ ǫ (w -y) dz dw ≤ C HLS 2π γ ǫ 2 4/3 = C HLS 2πǫ γ 2 4/3 .
Using the triangle inequality |x| ≤ |x -z| + |z -w| + |w -y| + |y| we have

||x|∇c ǫ | ≤ 2 |(|x|γ ǫ ) * ∇G * γ ǫ * ρ| + |γ ǫ * ρ| + |γ ǫ * ∇G * γ ǫ * |x|ρ| .
Using the Young and HLS inequalities, see (4.6), we obtain

|x|∇c ǫ ∞ ≤ 8 C HLS |x|γ ǫ 4/3 γ ǫ 4/3 + 4 + C HLS 2π γ ǫ 2 4/3 |x|ρ 1 ,
and the second part of the result is obtained by using γ ǫ 4/3 = ε -1/2 γ 4/3 .

Using the regularized Green's function G ǫ , we introduce the regularized self-interaction functional W ǫ :

W ǫ (ρ) = R 2 ×R 2 ρ(x) G ǫ (x -y) ρ(y) dx dy .
4.3 LEMMA (Continuity of the regularized interaction energy). Let ρ 1 and ρ 2 be any two densities in R 2 of mass 8π bounded in L 1 (R 2 , log(e + |x| 2 )). Then, for all 0 < ǫ ≤ 1,

|W ǫ [ρ 1 ] -W ǫ [ρ 2 ]| ≤ 3 π + 2Cǫ -2 ρ 1 L 1 (R 2 ,log(e+|x| 2 ) dx) ρ 1 -ρ 2 L 1 (R 2 ,log(e+|x| 2 ) dx) (4.7)
Moreover, let {ρ n } n∈N be a sequence of densities all bounded in

L 1 (R 2 , log(e + |x| 2 )) uniformly in n. If {ρ n } n∈N converges weakly in L 1 (R 2 ) to ρ, then for each ǫ > 0, W ǫ [ρ] ≤ lim inf n→∞ W ǫ [ρ n ] .
Proof: By definition of G ǫ :

W ǫ [ρ] = W 0 [ρ * γ ε ] = R 2 |∇G * ρ * γ ε | 2 (z) dz ≥ 0 .
Using ρ := ρ 1ρ 2 we write

W ǫ [ρ 1 ] -W ǫ [ρ 2 ] = W ǫ [ ρ] + 2 R 2 ρ 1 * G ǫ (x) ρ(x) dx ≥ 2 R 2 [ρ * G ǫ ] ρ dx . (4.8)
Then combining Lemma 4.1 and (4.8), we obtain

W ǫ [ρ 1 ] -W ǫ [ρ 2 ] ≥ 2 ρ>0 ρ(y) G ǫ (x -y) ρ(x) dy dx + 2 ρ≤0 ρ(y) G ǫ (x -y) ρ(x) dy dx ≥ - 1 2π ρ>0 ρ(y) 4 + log e + |x| 2 + log e + |y| 2 ρ(x) dy dx + 2 C ǫ 2 ρ≤0 ρ(y) ρ(x) dy dx ≥ - 3 π + 2 C ǫ 2 ρ 1 L 1 (R 2 ,log(e+|x| 2 ) dx) ρ L 1 (R 2 ,log(e+|x| 2 ) dx) .
Now swapping the roles of ρ 1 and ρ 2 , we obtain (4.7). By Lemma 4.1

{ρ n } n∈N bounded in L 1 (R 2 , log(e + |x| 2 ) dx) uniformly in n implies that ρ n * G ǫ is bounded in L ∞ (R 2 ) uniformly in n. Since {ρ n } n∈N converges to ρ weakly in L 1 (R 2 ), then ρ n * (χ R G ǫ ) → ρ * (χ R G ǫ ) point-wise
for given any cut-off function χ R with support in B(0, R) and thus ρ * G ǫ ∈ L ∞ (R 2 ). Therefore, applying (4.8) with ρ 1 =: ρ and ρ 2 := ρ n , we have

lim inf n→∞ (W ǫ [ρ n ] -W ǫ [ρ]) ≥ lim n→∞ 2 R 2 [ρ * G ǫ ] (ρ n -ρ) dx = 0 ,
where we have used the weak convergence on the right hand side.

We are now ready to introduce our regularized free energy functional.

4.4 DEFINITION (Regularized free energy functional). For all 0 < ǫ ≤ 1, define

F ǫ PKS [ρ] := R 2 ρ(x) log ρ(x) dx - 1 2 R 2 ×R 2 ρ(x) G ǫ (x -y) ρ(y) dx dy
on the set of densities ρ of mass 8π such that ρ ∈ L 1 (R 2 , log(e + |x| 2 ) dx) and ρ log ρ is integrable.

The following lemma is an immediate consequence of (4.3):

4.5 LEMMA (Monotonicity in ǫ). For all densities ρ of mass 8π such that ρ ∈ L 1 (R 2 , log(e + |x| 2 ) dx) and ρ log ρ is integrable, ǫ → F ǫ PKS [ρ] is monotone decreasing in ǫ. Note that by Lemma 4.1 (ii) and (iii), ρ (G ǫ * ρ) is integrable for ρ ∈ L 1 (R 2 , log(e + |x| 2 ) dx). Moreover, by Lemma 4.1 (i)

F ǫ PKS [ρ] ≥ F PKS [ρ] . (4.9)
In particular, by the sharp log HLS inequality, see Lemma 1.1

F ǫ PKS [ρ] ≥ -C(8π) = 8π(-1 + log 8) . (4.10)
By Lemma 4.1 (iii), we have the upper bound independent of ǫ:

F ǫ PKS [ρ] ≤ R 2 ρ(x) log ρ(x) dx + 32 π + 2 ρ L 1 (R 2 ,log(e+|x| 2 ) dx) . (4.11)
4.6 LEMMA (Error estimate for regularized free energy). For all ρ ∈ L 1 + ∩ L 3/2 (R 2 ) with mass 8π, and all ǫ < (2

√ e) -1 , F ǫ PKS [ρ] -F PKS [ρ] ≤ C γ ρ 3/2
3/2 ǫ . Proof: We use Hölder's inequality and Young's inequality for convolutions to get

F ǫ PKS [ρ] -F PKS [ρ] ≤ ρ 2 4/3 G ǫ -G 2 .
Hölder's inequality gives

F ǫ PKS [ρ] -F PKS [ρ] ≤ √ 8π ρ 3/2 3/2 G ǫ -G 2 . Since G ∈ L 1 loc (R 2 ), then G ǫ (x) → G(x) a.e. in R 2 as ǫ → 0. By Lemma 4.1, G ǫ ≤ G ǫ ≤ G for 0 < ǫ < ǫ. This implies that G ǫ -G 2 is
non decreasing in ǫ and it has zero limit as ǫ → 0 by monotone convergence theorem. Moreover, the same arguments also show that

lim ǫ→0 G ǫ -G ǫ 2 = G ǫ -G 2 .
On the other hand, using (4.2) and (4.4) for |x| > 2ǫ, we get for any fixed

x ∈ R 2 G ǫ(x) -G ǫ (x) ≤ (ǫ -ǫ) ∂ ∂ǫ G ǫ (x) ≤ 2ǫ 2 γ ǫ (x)
where second-order Taylor expansion at ǫ of the function G ǫ (x) for 0 < ǫ < ǫ was used and the second order term is nonpositive due to (4.4). Taking the limit ǫ → 0 and integrating, we deduce that

|x|>2ǫ (G(x) -G ǫ (x)) 2 dx 1/2 ≤ 2ǫ γ 2 .
Since G ǫ ≤ G, we can thus directly compute

|x|<2ǫ (G(x) -G ǫ (x)) 2 dx ≤ 4 |z|≤2ǫ |G(z)| 2 dz = 16π ǫ 2 1 2 -log(2ǫ) + (log(2ǫ)) 2 .
Finally, simple computations show that when | log(2ǫ)| > 1/2, the term in parentheses on the right in no greater than 5| log(2ǫ)| 2 . Collecting all together leads to the result with C γ explicitly computable.

Existence and first properties of the JKO scheme minimizers

Let S denote the set :

S := {ρ ∈ L 1 (R 2 ) : R 2 ρ(x) dx = M, W 2 (ρ, ̺ λ ) < ∞, E[ρ] < ∞, R 2 |x|ρ(x) dx < ∞} .
By (4.10), the functional

ρ → W 2 2 (ρ, ρ 0 ) 2τ + F ǫ PKS [ρ],
is bounded from below on S. The next lemma asserts that it has minimizers, and begins the task of their analysis. We state this lemma for a single step since we shall be changing the value of ǫ from step to step.

THEOREM (Existence of minimizers).

Let λ > 0, 0 < τ ≤ 1 and 0 < ǫ ≤ 1. For all ρ 0 ∈ S

arg min ρ∈S W 2 2 (ρ, ρ 0 ) 2τ + F ǫ PKS [ρ] (4.12)
is not empty, and each minimizer ρ belongs to S. Moreover, there exists K 1 > 0 depending only on E[ρ 0 ] and

R 2 |x|ρ 0 dx such that F ǫ PKS [ρ] ≤ K 1 .
Proof: Let {ρ (k) } k∈N be a minimizing sequence i.e. such that

lim k→∞ W 2 2 (ρ (k) , ρ 0 ) 2τ + F ǫ PKS [ρ (k) ] = inf ρ∈S W 2 2 (ρ, ρ 0 ) 2τ + F ǫ PKS [ρ]
.

By what we have noted just above, the infimum on the right hand side is finite. The following observation is the starting point for obtaining all of the bounds we need: Considering the trial function ρ = ρ 0 itself, one sees that we may suppose

W 2 2 (ρ (k) , ρ 0 ) 2τ + F ǫ PKS [ρ (k) ] ≤ W 2 2 (ρ 0 , ρ 0 ) 2τ + F ǫ PKS [ρ 0 ] = F ǫ PKS [ρ 0 ]
for all k. Consequently, for all k,

F ǫ PKS [ρ (k) ] ≤ F ǫ PKS [ρ 0 ] and W 2 2 (ρ (k) , ρ 0 ) ≤ 2τ F ǫ PKS [ρ 0 ] -F ǫ PKS [ρ (k) ] . (4.13)
We first bound W 2 (ρ (k) , ̺ λ ) uniformly in k. Since ρ 0 ∈ S, (4.11) ensures that F ǫ PKS [ρ 0 ] < ∞, and provides a bound depending only on E[ρ 0 ] and R 2 |x|ρ 0 dx. Then (4.10) provides a universal lower bound on F ǫ PKS [ρ (k) ], and thus by (4.13), there is a finite constant K 1 depending only on E[ρ 0 ], and R 2 |x|ρ 0 dx such that for all k,

F ǫ PKS [ρ (k) ] ≤ K 1 and W 2 2 (ρ (k) , ρ 0 ) ≤ K 1 . (4.14)
In particular, by the triangle inequality, for all k, W 2 (ρ

(k) , ̺ λ ) ≤ √ K 1 + W 2 (ρ 0 , ̺ λ ) < ∞.
We next bound the first moments of ρ (k) uniformly in k. Let ∇ϕ be the optimal transportation plan ∇ϕ#ρ (k) = ̺ λ . Then since |x| ≤ |x-∇ϕ(x)|+|∇ϕ(x)| for all x, integrating against ρ (k) and using the Cauchy-Schwarz inequality yields

R 2 |x| ρ (k) (x) dx ≤ √ 8 π W 2 (ρ (k) , ̺ λ ) + R 2 |x| ̺ λ (x) dx ≤ √ 8 π [ K 1 + W 2 (ρ 0 , ̺ λ )] + R 2 |x| ̺ λ (x) dx .
The right hand side is finite and independent of k.

We next bound E[ρ (k) ]. By part (ii) of Lemma 4.1, there is a constant C such that

R 2 ρ (k) (x) log ρ (k) (x) dx ≤ F ǫ PKS [ρ (k) ] + C (8π) 2 ǫ 2 ≤ F ǫ PKS [ρ 0 ] + C (8π) 2 ǫ 2 ≤ K 1 + C (8π) 2 ǫ 2 ,
where we have used (4.14) once more. Again the right side is finite and independent of k.

The last two uniform bounds show that {ρ (k) } k∈N is uniformly integrable. Hence, by the Dunford-Pettis theorem, there exist a weakly in L 1 convergent sub-sequence whose limit we shall denote by ρ.

By a standard weak lower semicontinuity argument (see e.g. [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for the weak lower semicontinuity of W 2 2 ), ρ satisfies each of the three bound that we have proved uniformly for {ρ (k) } k∈N , and thus ρ ∈ S.

It remains to prove that the functional F ǫ PKS is lower semi-continuous on L 1 (R 2 ). For the entropy part, this is standard. For the self interaction part, this follows from Lemma 4.3. So that

F ǫ PKS [ρ] ≤ lim inf F ǫ PKS [ρ (k) ] < K 1 . (4.15)
Finally, the weak limit ρ is a minimizer.

PROPOSITION (Strict positivity of the minimizers).

Let ρ 0 satisfies the conditions of Theorem 4.7. Then any minimizer ρ of (4.12) is uniformly bounded below on compact sets, i.e. for all R > 0, there exists s > 0 such that ρ(x) ≥ s almost everywhere in D R := {x : |x| ≤ R} . (4.16)

Moreover, s does not depend on the chosen minimizer of Problem (4.12) in case of non-uniqueness.

4.9 Remark (Idea of the proof). Let us try to quantify this simple statement: Entropy abhors a vacuum. The functional derivative of E[ρ] is log ρ. On any set where ρ is very close to zero, log ρ is very negative, and we can lower E[ρ] by transporting some mass from where ρ is relatively large to this spot where it is very small. This will lower the entropy by a very large multiple of the transported mass. On the other hand, if we do not have to transport the mass too far, the effects on W 2 2 (ρ, ρ 0 ) and W ǫ (ρ) will be relatively small.

Proof: For any s > 0, let

A R (s) := {x ∈ D R : ρ(x) ≥ s} and C R := AR(2/R 2 ) ρ(x) dx .
For any s > 0, let α R (s) := {x ∈ D R : ρ(x) ≤ s}, and let |α R (s)| denote its Lebesgue measure. By Theorem 4.7, |x|ρ(x) is integrable, and hence

DR ρ(x) dx = R 2 ρ(x) dx - |x|>R ρ(x) dx ≥ 8 π - R 2 |x| R ρ(x) dx ≥ 4 π , as long as 4πR ≥ R 2 |x|ρ(x) dx.
If |α R (s)| = 0 for some s > 0, there is nothing to prove: ρ is bounded below uniformly by s on D R . Therefore, suppose that |α R (s)| > 0 for all s > 0. Pick some small positive numbers δ and s, and define a new density ρ by transporting a mass δ C R |α R (s)| from A R (2/R 2 ) to α R (s), distributing it uniformly there, which raises the density there by δ C R . In formulas, choose s < 2/R 2 to have α R (s) ∩ A R (2/R 2 ) = ∅, and define a new density ρ by

ρ(x) =        (1 -δ |α R (s)|) ρ(x) x ∈ A R (2/R 2 ) , ρ(x) + δ C R x ∈ α R (s) , ρ(x) otherwise .
In order to ensure positivity, we have to impose δ|α R (s)| ≤ δ π R 2 ≤ 1/2. In this way, it is easy to check that ρ is a density.

Note that ρρ 1 ≤ 2 δ |α R (s)|C R , and since all the modifications take place on D R ,

ρ -ρ L 1 (R 2 ,log(e+|x| 2 ) dx) ≤ log(e + R 2 ) 2 δ |α R (s)| C R .
It now follows from the bounds on ρ derived Theorem 4.7 and from Lemma 4.3 that there is a constant K depending only on R, ǫ, E[ρ 0 ] and H λ [ρ 0 ] such that

W ǫ [ ρ] ≤ W ǫ [ρ] + δ |α R (s)| K . (4.17)
Using Taylor's expansion of x → x log x, that log x is increasing and assuming s ≤ δ C R , we obtain

R 2 [ ρ log ρ -ρ log ρ] dx = AR(2/R 2 ) {(1 -δ |α R (s)|) ρ log [(1 -δ |α R (s)|) ρ] -ρ log ρ} dx + αR(s) [(ρ + δ C R ) log (ρ + δ C R ) -ρ log ρ] dx ≤ -δ |α R (s)| AR(2/R 2 ) ρ log [(1 -δ |α R (s)|) ρ] dx + δ |α R (s)| C R log(2δ C R ) ≤ δ |α R (s)| C R -log 1 R 2 + log(2δ C R ) , (4.18)
where δ |α R (s)| ≤ 1/2 and x ∈ A R (2/R 2 ) were used in the last estimate.

To estimate the difference W 2 2 ( ρ, ρ 0 ) -W 2 2 (ρ, ρ 0 ), let Π denote the optimal coupling of ρ and ρ 0 , and use it to define a non-optimal coupling Π of ρ and ρ 0 . To do this, let µ be the measure supported on A R (2/R 2 ) with density ρ, and hence total mass C R . Let ν be the dx-uniform distribution on α R (s) with total mass C R . Let ∇ψ be the optimal transportation plan with ∇ψ#µ = ν, and define the map T : R 2 → R 2 by

T (x) = ∇ψ(x) x ∈ A R (2/R 2 ) , x otherwise .
Then Π, given by Π = (1 -δ|α R (s)|)Π + δ|α R (s)|(T ⊗ Id)#Π is a coupling of ρ and ρ 0 , and hence

W 2 2 ( ρ, ρ 0 ) ≤ R 2 ×R 2 |x -y| 2 d Π(x, y) = (1 -δ|α R (s)|) W 2 2 (ρ, ρ 0 ) + δ|α R (s)| R 2 ×R 2 |T (x) -y| 2 d Π(x, y) .
Then, since |T (x) -y| 2 ≤ 2|T (x) -x| 2 + 2|x -y| 2 , and |T (x) -x| ≤ 2R, since all of the transportation induced by ∇ψ takes place inside D R , it follows that

W 2 2 ( ρ, ρ 0 ) ≤ (1 + δ|α R (s)|) W 2 2 (ρ, ρ 0 ) + δ|α R (s)|(8π) 2 8R 2 .
By bounds on ρ derived in the proof of Theorem 4.7, there is a constant K depending only on E[ρ 0 ] and H λ [ρ 0 ] such that W 2 2 (ρ, ρ 0 ) ≤ Kτ . Finally then, there is a constant depending only on R, τ , E[ρ 0 ] and

H λ [ρ 0 ] such that W 2 2 ( ρ, ρ 0 ) ≤ W 2 2 (ρ, ρ 0 ) + δ|α R (s)|K . ( 4
) 2τ + F ǫ PKS [ρ] ≤ W 2 2 (ρ, ρ 0 ) 2τ + F ǫ PKS [ρ] + δ |α R (s)| C R -log 1 R 2 + log(2δC R ) + K ′ ,
with a given constant K ′ . If |α R (s)| > 0 for all s > 0, then choosing δ small enough such that

-log 1 R 2 + log(2δ C R ) + K ′ < 0
contradicts the optimality of ρ. For instance, choosing s R = δ/C R , the procedure described above can be carried out, and we conclude that ρ is bounded below by s R on D R . This proves (4.16).

We now continue the analysis of the minimizers ρ begun in Theorem 4.7. We obtained ρ ∈ S and the lower bound (4.16) directly from the variational principle, but to proceed, we need the Euler-Lagrange equation for the variational problem (4.12).

By the Brenier-McCann Theorem, there is a a lower semi-continuous convex function ϕ in R 2 such that ∇ϕ#ρ = ρ 0 , and ∇ϕ is uniquely determined on the support of ρ, which is all R 2 by (4.16) . The Euler-Lagrange equation for (4.12) relates ρ, ρ 0 and ∇ϕ: 4.10 LEMMA (Euler-Lagrange equation). Let ρ 0 satisfy the conditions of Theorem 4.7 and ρ be any minimizer for the variational problem in (4.12), and let ∇ϕ be the unique gradient of a lower semi-continuous convex function such that ∇ϕ#ρ = ρ 0 . Then the distributional gradient of ρ satisfies

-∇ρ + ρ∇c ǫ = id -∇ϕ τ ρ (4.20)
where c ǫ = G ǫ * ρ. In particular, since c ǫ is differentiable everywhere, and ϕ is differentiable almost everywhere, ρ is differentiable almost everywhere.

The proof follows exactly the original procedure in [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], see also [START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF]Theorem 3.4], and we skip it here for the sake of brevity. The interested reader can see the details in the preprint version of this paper.

4.11 LEMMA (Qualitative regularity estimates). Let ρ 0 satisfy the conditions of Theorem 4.7, and let ρ be any minimizer for the variational problem in (4.12). Then √ ρ has a square integrable distributional gradient, and for any 1 < p < ∞, ρ p is integrable.

Proof: By the positivity of ρ, see Proposition 4.7, we can divide both sides of (4.20) by √ ρ, to obtain

2∇ √ ρ = ∇c ǫ - x -∇ϕ τ √ ρ ,
where ∇ϕ is such that ∇ϕ#ρ = ρ 0 . By the triangle inequality,

2 ∇ √ ρ 2 ≤ R 2 |∇c ǫ (x)| 2 ρ(x) dx 1/2 + 1 τ R 2 |x -∇ϕ(x)| 2 ρ(x) dx 1/2 ≤ R 2 |∇c ǫ (x)| 2 ρ(x) dx 1/2 + 1 τ W 2 (ρ, ρ 0 ) . (4.21)
By Lemma 4.2, ∇c ǫ ∞ is uniformly bounded, and so is the first term of (4.21). This proves that √ ρ has a square integrable distributional gradient. The integrability of ρ p is then a consequence of the following classical version of the GNS inequality valid for functions on

R 2 with p ∈ [2, ∞) R 2 |v| p dx ≤ D p R 2 |∇v| 2 dx p/2-1 R 2 |v| 2 dx applied to v = √ ρ.
4.12 Remark. Although the bounds in Lemma 4.11 are not quantitative, and would certainly be ǫ dependent if we were to extract quantitative bounds, we shall use them only to justify certain integrations by parts, and otherwise show that substraction of infinities does not invalidate computations that follow. Thus, these qualitative estimates are all we require concerning ∇ √ ρ and ρ. However, they are absolutely crucial for their purpose, and their necessity is the main reason we have had to introduce the regularized Green's function G ǫ , and along with it, the regularized chemical attractant. Without the regularization, we would only know that 2∇ √ ρ -∇c √ ρ was square integrablebut the possible cancelation effects would not allow us to conclude that ∇ √ ρ was square integrable.

A discrete form of the entropy-entropy dissipation inequality

Our main goal in this subsection is to prove a discrete version of the entropy-entropy dissipation inequality (1.15).

The key idea is to use the κ λ displacement convexity of H λ and the "above the tangent" inequality for convex functions as follows: For given initial density ρ 0 , let ρ be any minimizer for the variational problem in (4.12). Let u t , 0 ≤ t ≤ 1 denote the displacement interpolation between ρ and ρ 0 starting at ρ and ending at ρ 0 . Then u 0 = ρ and u 1 = ρ 0 . Since H λ is displacement convex, the "above the tangent" inequality for convex functions says that

H λ [ρ] + d dt H λ [u t ] t=0 + κ λ W 2 2 (ρ, ρ 0 ) ≤ H λ [ρ 0 ] .
A formal computation of the second term on the left would give, for ǫ = 0,

d dt H λ [u t ] t=0 = D[ρ] .
Indeed, assuming Lemma 2.3 holds for δ = ǫ = 0 applied to u 0 = ρ and u 1 = ρ 0 , we get

H λ [ρ] ≤ H λ [ρ 0 ] - 1 2 R 2 κ λ x + ∇ρ ρ 3/2 • (∇ϕ(x) -x) ρ dx -κ λ W 2 2 (ρ, ρ 0 ) .
Using (4.20), i.e. (∇ϕ(x)x)ρ = τ (∇ρ -ρ∇c) and expanding, we can rewrite this as

H λ [ρ] ≤ H λ [ρ 0 ] - τ 2 R 2 |∇ρ| 2 ρ 3/2 dx - R 2 ∇c • ∇ρ √ ρ dx -κ λ R 2 x • ∇c ρ dx + κ λ R 2 x • ∇ρ dx -κ λ W 2 2 (ρ, ρ 0 ) :=H λ [ρ 0 ] - τ 2 [(I) + (II) + (III) + (IV)] -κ λ W 2 2 (ρ, ρ 0 ) .
Using -∆c = ρ we have

(II) = -2 R 2 ∇c • ∇ √ ρ dx = -2 R 2 ρ 3/2 dx .
Using the symmetrization argument we obtain

(III) = κ λ 1 4π R 2 ρ dx 2 = 16πκ λ .
And by integration by parts (IV) = -16πκ λ , resulting into

H λ [ρ] ≤ H λ [ρ 0 ] -τ D[ρ] -κ λ W 2 2
(ρ, ρ 0 ). However, to do the calculation in a rigorous manner we must take into account that ǫ > 0, and we must use the regularized entropy functional H λ,δ . Before proceding with this, we point out that no such estimate can be given for F PKS since this functional is not displacement convex.

4.13 LEMMA (Convexity estimates at the regularized level). Let ρ 0 satisfy the conditions of Theorem 4.7, and let ρ be any minimizer for the variational problem in (4.12), then

H λ,δ [ρ] ≤ H λ,δ [ρ 0 ] - τ 2 R 2 |∇ρ| 2 (ρ + δ) 3/2 dx + τ R 2 ρ 3/2 dx + 16 π √ δτ + 16π √ δ(J γ + 1 + Ĉǫ )τ -16πκ λ τ + 2C ǫ |2f ′ δ -κ λ | (1 + |x|)ρ 1 τ + 16π J γ √ 2λ τ + 2τ R 2 ∇ • [x f ′ δ (|x| 2 )]ρ(x) dx -K δ (ρ 0 , ρ) , and
H λ,δ [ρ] ≤ H λ,δ [ρ 0 ] - τ 2 R 2 |∇ρ| 2 (ρ + δ) 3/2 dx + τ R 2 ρ 3/2 dx + 16 π √ δτ + 16π √ δ(J γ + 1 + Ĉǫ )τ -16πκ λ τ + 2C ǫ |2f ′ δ -κ λ | (1 + |x|)ρ 1 τ + 32π C HLS √ 2λ √ ǫ |x| γ 4/3 ρ 4/3 τ + 2τ R 2 ∇ • [x f ′ δ (|x| 2 )]ρ(x) dx -K δ (ρ 0 , ρ) ,
where K δ is defined in Proposition 2.2 and the constants J γ , Ĉǫ , and C ǫ are explicit constants.

Proof: This is an elaborate calculation in which a number of integrations by parts operations must be carefully examined for boundary behavior. It is relegated to the Appendix.

As a consequence of this lemma, letting δ go to 0, we obtain the following result concerning the dissipation of H λ in one discrete time step.

COROLLARY (Convexity estimates)

. Let ρ 0 satisfy the conditions of Theorem 4.7. If ρ is any minimizer for the variational problem in (4.12) then

H λ [ρ] ≤ H λ [ρ 0 ] -τ D[ρ] + τ A γ 4/3 -κ λ W 2 2 (ρ, ρ 0 ) , (4.22)
and

H λ [ρ] ≤ H λ [ρ 0 ] -τ D[ρ] + τ √ ǫ A ρ 4/3 -κ λ W 2 2 (ρ, ρ 0 ) . ( 4

.23)

where A := 32π(2λ) -1/2 C HLS |x| γ 4/3 .

Proof: Let us first observe that

R 2 ∇ • [x f ′ δ (|x| 2 )]ρ(x) dx = R 2 2f ′ δ (|x| 2 ) + 2|x| 2 f ′′ δ (|x| 2 ) ρ(x) dx.
Let us recall from the proof of Proposition 2.2 that 2f ′ δ (s) ր κ λ and f ′′ δ (s) → 0 as δ → 0 for all s ≥ 0. Moreover, we have that sf ′′ δ (s) is a bounded function uniformly in δ from (2.17). These properties together with the dominated convergence theorem leads easily to

|2f ′ δ -κ λ | (1 + |x|)ρ 1 → 0 and R 2 ∇ • [x f ′ δ (|x| 2 )]ρ(x) dx → 8κ λ π as δ → 0, since (1 + |x|)ρ ∈ L 1 (R 2 )
. By monotone convergence theorem, we obtain

lim δ→0 R 2 |∇ρ| 2 (ρ + δ) 3/2 dx = R 2 |∇ρ| 2 ρ 3/2 dx .
Putting together all these facts and Proposition 2.2, we can pass to the limit as δ → 0 in Lemma 4.13 to get the desired estimates (4.23) and (4.22).

One-step estimates

Neither of the one step dissipation estimates that we have so far, namely (4.23) and (4.22), are exactly what we need. The problem is the term τ A γ 4/3 in the first of these, and the term τ A ρ 4/3 in the second of these. These terms might be large compared to the other terms so that these estimates might even give only "negative dissipation".

In the first main result of this subsection, we use one and then the other of these inequalities in combination with the controlled concentration inequality of Theorem 1.10 to produce the kind of dissipation estimate that we really want. In the second main result, we show that L p norms of the densities are essentially propagated along each step of the discrete variational scheme. Again, Theorem 1.10 plays a crucial role in both proofs.

4.15 THEOREM (One-step theorem). Let ρ 0 satisfy the conditions of Theorem 4.7, ρ be any minimizer for the variational problem in (4.12), and choose any C ρ0 such that

H λ [ρ 0 ] < C ρ0 . (4.24) Define Q 0 > 0, τ ⋆ 0 > 0 by Q 0 := C ρ0 -H λ [ρ 0 ] and τ ⋆ 0 := min Q 0 2 A γ 4/3 , 1 , (4.25)
where A is the constant given in Corollary 4.14. Finally, given Q 0 and 0 < τ ≤ τ ⋆ 0 , and also any positive integer ℓ, let ǫ ℓ be given by

τ 1/3 √ ǫ ℓ 8 π 1/3 A γ -2/3 2 (π C ρ0 + τ ⋆ 0 C CCD ) 2/3 = Q 0 4 τ 2 2 -ℓ .
Then for all τ ≤ τ ⋆ 0 and all ǫ = ǫ ℓ , ρ satisfies

F PKS [ρ] < +∞ , H λ [ρ] < C ρ0 (4.26) and H λ [ρ] -H λ [ρ 0 ] ≤ -τ D[ρ] + Q 0 4 τ 2 2 -ℓ -κ λ W 2 2 (ρ 0 , ρ) . ( 4

.27)

Proof: By (4.22), our choice of τ and Q 0 in (4.25) implies that

H λ [ρ] ≤ H λ [ρ 0 ] -τ D[ρ] + Q 0 2 = C ρ0 -Q 0 -τ D[ρ] + Q 0 2 ≤ C ρ0 -τ D[ρ] . (4.28)
On one hand, the GNS inequality, see Lemma 1.2, implies D[ρ] ≥ 0 so that (4.28) implies that ρ also satisfies (4.24).

On the other hand, since H λ [ρ] cannot be negative it implies

D[ρ] ≤ C ρ0 τ .
Moreover, by the monotonicity of ε → F ε PKS [ρ], Lemma 4.5, and (4.15),

F PKS [ρ] ≤ F ε PKS [ρ] < +∞.
We can thus apply the concentration controlled inequality, Theorem 1.10 which implies

R 2 ∇ρ 1/4 2 dx ≤ 1 γ 2 [πD[ρ] + C CCD ] ≤ 1 τ 1 γ 2 [πC ρ0 + τ ⋆ 0 C CCD ] .
By the GNS inequality of Lemma 1.2, we have

R 2 ρ 3/2 dx ≤ 8 R 2 ∇ρ 1/4 2 dx ≤ 1 τ 8 γ 2 [πC ρ0 + τ ⋆ 0 C CCD ] := C 3 τ . (4.29)
Next, by Hölder's inequality,

R 2 ρ 4/3 dx = R 2 ρ 1/3 ρ dx ≤ (8π) 1/3 R 2 ρ 3/2 dx 2/3 ≤ (8π) 1/3 C 3 τ 2/3
. Now using this bound in (4.23), we obtain

H λ [ρ] -H λ [ρ 0 ] ≤ -τ D[ρ] + τ 1/3 √ ǫ A (8π) 1/3 C 2/3 3 -κ λ W 2 2 (ρ 0 , ρ).
We thus obtain the stated result by choosing ǫ = ǫ ℓ for any positive integer ℓ.

4.16 LEMMA (Propagation of the L p -norm). Let ρ 0 satisfy the conditions of Theorem 4.7. Assume additionally that ρ 0 ∈ L p (R 2 ), 2 ≤ p < ∞, and let ρ be any minimizer for the variational problem in (4.12), then there exists

K 0 > 0 which only depends on R 2 ρ| log ρ| dx such that for all K ≥ K 0 R 2 (ρ -K) p + dx ≤ R 2 (ρ 0 -K) p + dx + τ A 1 + τ A 2 D[ρ] ,
where A 1 and A 2 are universal positive constants depending on K.

Proof: The displacement convexity of the functional

ρ → R 2 (ρ -K) p + dx
with 2 ≤ p < ∞ and K > 0, is easy to check using McCann's criterion (2.8). The Euler-Lagrange equation of the variational scheme (x -∇ϕ) ρ = -τ ∇ρ + τ ρ∇c ǫ together with the standard first-order displacement convexity characterization [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] imply

R 2 (ρ -K) p + dx - R 2 (ρ 0 -K) p + dx ≤ -p R 2 ∇ (ρ -K) p-1 + (∇ϕ -x) ρ dx ≤ -(p -1)pτ R 2 |∇(ρ -K) + | 2 (ρ -K) p-2 + dx + (p -1)τ R 2 ∇ (ρ -K) p + ∇c ǫ dx + pτ K R 2 ∇ (ρ -K) p-1 + ∇c ǫ dx ≤ - 4(p -1) p τ R 2 ∇ (ρ -K) p/2 + 2 dx + (p -1)τ R 2 (ρ -K) p + (-∆c ǫ ) dx + pτ K R 2 (ρ -K) p-1 + (-∆c ǫ ) dx := τ (I 1 + I 2 + I 3 ) . (4.30)
The last two integration by parts have to be justified for any given ǫ working as in the proof of Lemma 4.13 in the Appendix. Integrating by parts on the ball of radius R, we obtain for any k ∈ {p, p -

1} |x|≤R ∇ (ρ -K) k + ∇c ǫ dx = |x|≤R (ρ -K) k + (-∆c ǫ ) dx + |x|=R (ρ -K) k + ∇c ǫ • n dσ ≤ |x|≤R (ρ -K) k + (-∆c ǫ ) dx + ∇c ǫ ∞ |x|=R ρ k dσ .
It is enough to show by dominated convergence theorem that there exists a sequence of radii {R j } j∈N such that the boundary terms tend to zero as j → ∞. Due to Lemma 4.11 with p ≥ 2, for any given natural N > 1, we can write that

∞ N =1 N N -1 |x|=r ρ k dσ dr = R 2 ρ k dx < ∞ , implying that lim N →∞ N N -1 |x|=r ρ k dσ dr = 0 ,
for k ∈ {p, p -1}, and the two integration by parts for any given ǫ are justified. We now estimate I 2 and I 3 , showing in particular that they are finite. Starting with I 2 , using -∆c ǫ = ρ ǫ where ρ ǫ := γ ǫ * ρ * γ ǫ , so that by Hölder's inequality and Young's inequality for convolutions, obtain

R 2 (ρ -K) p + (-∆c ǫ ) dx = R 2 (ρ -K) p + ρ ǫ dx ≤ (ρ -K) + p p+1 ρ ǫ p+1 ≤ (ρ -K) + p p+1 ρ p+1 (4.31)
Likewise for I 3 , we use the fact that on the support of (ρ

-K) + , K ≤ ρ. Therefore K R 2 (ρ -K) p-1 + (-∆c ǫ ) dx = K R 2 (ρ -K) p-1 + ρ ǫ dx ≤ R 2 (ρ -K) p-1 + ρ ǫ ρ dx .
Therefore, by Hölder's inequality and Young's inequality for convolutions,

K R 2 (ρ -K) p-1 + (-∆c ǫ ) dx ≤ (ρ -K) + p-1 p+1 ρ 2 p+1 . (4.32)
Applying the arithmetic-geometric mean inequality to the right side of (4.31), we have that for any ν > 0,

(ρ -K) + p p+1 ρ p+1 ≤ p p + 1 ν -(p+1)/p (ρ -K) + p+1 p+1 + 1 p + 1 ν p+1 ρ p+1 p+1 .
Making a similar estimate for the right hand side of (4.32), and combining results, we have that

I 2 + I 3 ≤ F 1 (ν) (ρ -K) + p+1 p+1 + F 2 (ν) ρ p+1 p+1 (4.33)
where F 1 (ν) is a positive linear combination of negative powers of ν, and F 2 (ν) is a positive linear combination of positive powers of ν. By Lemma 4.11, ρ p is integrable for any 1 ≤ p < ∞, and so the right side of (4.33) is finite. Then by (4.30),

R 2 ∇ (ρ -K) p/2 + 2 dx < ∞ .
From here we show that

R 2 ∇ ρ p/2 2 dx ≤ R 2 ∇ (ρ -K) p/2 + 2 dx + 16K (2p-1)/2 γ 2 [πD[ρ] + C CCD ] . (4.34) Indeed, R 2 ∇ ρ p/2 2 dx = R 2 ∇ (ρ -K) p/2 + 2 dx - {ρ<K} ∇ ρ p/2 2 dx ,
and

{ρ<K} ∇ρ p/2 2 dx ≤ 16K (2p-1)/2 R 2 ∇ρ 1/4 2 dx ≤ 16K (2p-1)/2 γ 2 [πD[ρ] + C CCD ] ,
where we applied again the concentration controlled inequality, Theorem 1.10, using (4.26). Following an idea of Jäger and Luckhaus [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF], we use the GNS inequality

R 2 v p+1 dx ≤ D p R 2 |∇v p/2 | 2 dx R 2 v dx , (4.35)
which is a consequence of the Sobolev embedding inequality

u L 2 (R 2 ) ≤ C ∇u L 1 (R 2 ) applied to u = v (p+1)/2 and Cauchy-Schwarz inequality since ∇u = p+1 p v 1/2 ∇v p/2 . Applying (4.35) to v = (ρ -K) + , we get R 2 (ρ -K) p+1 + dx ≤ M (K) R 2 ∇ (ρ -K) p/2 + 2 dx where M (K) := R 2 (ρ -K) + dx .
Then (4.33) becomes

I 2 + I 3 ≤ F 1 (ν)M (K) R 2 ∇ (ρ -K) p/2 + 2 dx + F 2 (ν) 8π R 2 ∇ρ p/2 2 dx .
We finally work with I 1 to estimate it using (4.34) as

p 2(p -1) I 1 ≤ - R 2 ∇ (ρ -K) p/2 + 2 dx - R 2 ∇ρ p/2 2 dx + 16K (2p-1)/2 γ 2 [πD[ρ] + C CCD ] .
Now choose ν 0 > 0 small enough such that 8πF 2 (ν 0 ) < 2(p -1)/p, and then K 0 < ∞ large enough such that M (K)F 1 (ν 0 ) < 2(p -1)/p. This choice of K 0 only depends on ν 0 and the bound on R 2 ρ| log ρ| dx since

M (K) = R 2 (ρ -K) + dx ≤ ρ>K ρ dx ≤ 1 log K ρ>K ρ log ρ dx ≤ 1 log K R 2 ρ log + ρ dx .
We find

R 2 (ρ -K) p + dx - R 2 (ρ 0 -K) p + dx ≤ τ 32K (2p-1)/2 (p -1) pγ 2 [πD[ρ] + C CCD ] ,
for all K ≥ K 0 . The desired result follows with

A 1 = 32K (2p-1)/2 (p -1) pγ 2 C CCD and A 2 = 32K (2p-1)/2 (p -1) pγ 2 π .
5 Proof of the main results

Approximate solutions

We now combine the single step operations described in the previous section to inductively define infinite sequences {ρ k τ } k∈N giving a discrete-time approximation to the PKS evolution (1.1). For the rest of this section, fix any λ > 0, and any density ρ 0 in R 2 with total mass 8π with ρ 0 log ρ 0 integrable, and such that there exists C ρ0 with H λ [ρ 0 ] < C ρ0 . It then follows from Lemma 1.11 that |x|ρ 0 is integrable, and from the Talagrand inequality, Theorem 2.4, that W 2 (ρ 0 , ̺ λ ) < ∞. Thus, ρ 0 satisfies the conditions of Theorem 4.7 on the existence of minimizers for our single step variational problem.

Fixing an arbitrarily small parameter τ > 0, we now inductively define the sequence of densities {ρ k τ } k∈N with ρ 0 τ := ρ 0 by solving the sequence of variational problems

ρ k τ ∈ arg min ρ∈S W 2 2 (ρ, ρ k-1 τ ) 2τ + F ǫ k PKS [ρ] (5.1)
for a sequence of regularization parameters {ǫ k } k∈N to be specified now. By Theorem 4.7, the sequence {ρ k τ } k∈N is well defined no matter how we choose 0 < τ < 1 and {ǫ k } k∈N . We moreover define

Q k := C ρ0 -H λ [ρ k τ ] > 0 for each k . (5.2)
5.1 LEMMA (Good step sizes). Let {ρ k τ } k∈N be the sequence of minimizers defined inductively using (5.1) starting from ρ τ = ρ 0 . With Q k defined as in (5.2), let A be the constant given in Corollary 4.14, and let Λ be defined by

Λ := ∞ m=1 1 - 2 -m 4 ,
and note that 1 > Λ > 0. Choose any τ > 0 satisfying

τ < min ΛQ 0 2 A γ 4/3 , 1 := τ ⋆ , (5.3)
and define ǫ k by

τ 1/3 √ ǫ k 8 π 1/3 A γ -2/3 2 (π C ρ0 + C CCD ) 2/3 = Q 0 4 τ 2 2 -k . (5.4)
Then for all k, Q k > ΛQ 0 > 0. In particular,

F PKS [ρ k τ ] < +∞ and H λ [ρ k τ ] < C ρ0 .
Note that for some constant Z, ǫ k := Z τ 10/3 4 -k .

Proof: We shall show by induction that for each positive integer j

Q j ≥ j m=1 1 - 2 -m 4 Q 0 , (5.5)
which is somewhat more than we need since the right hand side is larger than ΛQ 0 . We now make the inductive hypothesis that for some positive integer k, (5.5) is true for all positive integers j < k. Since Λ < 1, we may apply Theorem 4.15 with ρ k-1 τ in place of ρ 0 , and ρ k τ in place of ρ and with τ and ǫ k specified as above. Then the conclusion (4.27) can be simplified and rewritten as

H λ [ρ k τ ] ≤ H λ [ρ k-1 τ ] + τ 2 Q k-1 4 2 -k . (5.6)
Since τ < 1, this means that

Q k ≥ Q k-1 1 - 2 -k 4 .
By the inductive hypothesis, we obtain (5.5) for j = k. The proof that (5.5) is valid for j = 1 is a direct application of Theorem 4.15, in the same way, since Λ < 1.

The passage to continuous time

Throughout the rest of this section, we assume that 0 < τ < τ ⋆ , where τ ⋆ is defined in (5.3), and that ǫ k is defined by (5.4), and then that {ρ k τ } k∈N is a corresponding sequence of minimizers of (5.1). We now interpolate between the terms of the sequence {ρ k τ } k∈N to produce a function from [0, ∞) to L 1 (R 2 ) that we shall show to be, for sufficiently small τ , an approximate solution of the PKS system. For technical reasons, we shall need two distinct, but closely related, interpolations.

• The Lipschitz interpolation: For each positive integer k, let ∇ϕ k be the optimal transportation plan with

∇ϕ k #ρ k τ = ρ k-1 τ
. Then for (k -1)τ ≤ t ≤ kτ we define

ρ τ (t) = t -(k -1)τ τ Id + kτ -t τ ∇ϕ k #ρ k τ .
• The piecewise constant interpolation: For each t and each positive integer k with (k -1)τ ≤ t < kτ we define ρτ (t) = ρ k-1 τ , with ρτ (0) = ρ 0 .

For displacement convex functionals of ρ, such as

H λ [ρ], E[ρ],
or the absolute first moment, any uniform bounds on the functional along the sequence {ρ k τ } k∈N extend to ρ(t) for all t, since if G is such a functional, then for

(k -1)τ < t < kτ , G[ρ τ (t)] ≤ t -(k -1)τ τ G[ρ k-1 τ ] + kτ -t τ G[ρ k τ ] .
Of course it is evident that for any sort of functional G[ρ], displacement convex or not, a uniform bound on G[ρ] along the sequence {ρ k τ } k∈N extends to ρ(t) for all t. Some of the functionals with which we work, such as D[ρ], are not displacement convex, and this is the reason we need the second interpolation.

The uniform equicontinuity properties that we prove next explain the utility of the first interpolation, and also why we can use the two different interpolations at once. Since ρ k τ is a minimizer for (5.1), using ρ k-1 τ as trial function yields

F ǫ k PKS [ρ k τ ] + 1 2 τ W 2 2 (ρ k τ , ρ k-1 τ ) ≤ F ǫ k PKS [ρ k-1 τ ] ,
and hence,

W 2 2 (ρ k τ , ρ k-1 τ ) ≤ 2τ F ǫ k PKS [ρ k-1 τ ] -F ǫ k PKS [ρ k τ ] , (5.7)
and by the monotonicity of ε → F ε PKS [ρ] see Lemma 4.5

F PKS [ρ k τ ] ≤ F ε PKS [ρ k τ ] < +∞ . (5.8)
In standard applications of the JKO scheme, in which the functional in the variational problem does not change from step to step, one would sum both sides in (5.7) over a range of values of k, and then the sum of the terms on the right would telescope. This is not so in our case. However, for small ǫ, F ǫ PKS ≈ F PKS and we recover the telescoping sum in a useful approximate sense. The precise version of F ǫ PKS ≈ F PKS follows from (4.9), Lemma 4.6, and (5.4), which says that ǫ k = Zτ 10/3 4 -k to get

F ǫ k PKS [ρ k-1 τ ] -F PKS [ρ k-1 τ ] ≤ C γ Z ρ k-1 τ 3/2 3/2 τ 10/3 4 -k ≤ Z ρ k-1 τ 3/2 3/2 τ 3 2 -k (5.9)
for τ < τ ⋆ := min(τ ⋆ , (2Z √ e) -3/10 ) with Z := C γ Z according to Lemma 4.6. We thus deduce

W 2 2 (ρ k τ , ρ k-1 τ ) ≤ 2τ F PKS [ρ k-1 τ ] -F PKS [ρ k τ ] + 2 Z ρ k-1 τ 3/2 3/2 τ 4 2 -k .
Using (5.8) and (4.29) as in the proof of Theorem 4.15 where the concentration control inequality (1.10) is crucial, we deduce that

ρ k-1 τ 3/2 3/2 ≤ C 3 /τ to conclude that W 2 2 (ρ k τ , ρ k-1 τ ) ≤ 2τ F PKS [ρ k-1 τ ] -F PKS [ρ k τ ] + 2 ZC 3 τ 3 2 -k . (5.10)
We are almost in a position to obtain a crucial a-priori Hölder continuity estimate, but there is still one more consequence of our step dependent regularization to deal with: If for each k we had been using the functional F PKS instead of the functional F ǫ k PKS , it would be immediate that k → F PKS [ρ k τ ] would be decreasing. Since by the Log-HLS inequality, F PKS is bounded below, this would give an immediate upper bound on the sum of the right hand side of (5.10) over any range of k.

However, we have used our freedom to choose the sequence {ǫ k } k∈N of regularization parameters to converge to zero as rapidly as we may require, and hence easily obtain: 5.2 LEMMA (Uniform bounds on the free energy F PKS ). There are positive constants F0 , F1 depending only on the initial data and the regularization mollifier γ such that for each τ < τ ⋆ and each k ∈ N,

F PKS [ρ k τ ] ≤ F0 + F1 τ 2 . Proof: Directly from the variational problem (5.1) we have F ǫ k PKS [ρ k τ ] ≤ F ǫ k PKS [ρ k-1 τ
]. Then, as above from (4.9), (5.9), and (4.29) we get

F ǫ k PKS [ρ k-1 τ ] ≤ F ǫ k-1 PKS [ρ k-1 τ ] + ZC 3 τ 2 2 -k . (5.11)
This means that the free energy F PKS is almost decreasing along {ρ k τ } k∈N . A telescoping sum argument yields

F ǫ k PKS [ρ k τ ] ≤ F ǫ0 PKS [ρ 0 ] + ZC 3 τ 2
, and then one more application of (4.9) gives

F PKS (ρ k τ ) ≤ F ǫ0 PKS [ρ 0 ] + ZC 3 τ 2 ≤ E[ρ 0 ] + 32 π + 2 ρ 0 L 1 (R 2 ,log(e+|x| 2 ) dx) + ZC 3 τ 2 := F0 + F1 τ 2
, where (4.11) was used.

We are now ready to prove the Hölder continuity estimate.

LEMMA (Hölder continuity).

There is a positive constant F2 depending only on the initial data and the regularization mollifier γ such that for each τ < τ ⋆ and each k ∈ N, such that for all t > s ≥ 0,

W 2 (ρ τ (t), ρ τ (s)) ≤ F2 (t -s) 1/2 .
Proof: Let j be such that (j -1)τ ≤ s ≤ jτ and let ℓ be such that ℓτ ≤ t ≤ (ℓ + 1)τ . By the geodesic property of McCann's displacement interpolation,

W 2 (ρ τ (s), ρ j τ ) = jτ -s τ W 2 (ρ j-1 τ , ρ j τ ) and W 2 (ρ τ (t), ρ ℓ τ ) = t -ℓτ τ W 2 (ρ ℓ τ , ρ ℓ+1 τ ) .
By Lemma 5.2 and the Log-HLS inequality (4.10),

F PKS [ρ k τ ] -F PKS [ρ k-1 τ ] ≤ F0 + F1 τ 2 -8π(log 8 -1)
, and thus, by plugging into (5.10), we get

W 2 2 (ρ k τ , ρ k-1 τ ) ≤ 2τ F0 + F1 τ 2 -8π(log 8 -1) + 2 ZC 3 τ 3 2 -k ≤ τ 2 F0 + 2 F1 -16π(log 8 -1) + 2 ZC 3 := F 2 2 τ , (5.12)
since τ < 1 and k ∈ N. Therefore, we deduce

W 2 (ρ τ (s), ρ j τ ) ≤ jτ -s τ F2 √ τ and W 2 (ρ τ (t), ρ ℓ τ ) ≤ t -ℓτ τ F2 √ τ .
Adding these two estimates and using the concavity of square root,

W 2 (ρ τ (s), ρ j τ ) + W 2 (ρ τ (t), ρ ℓ τ ) ≤ F2 (t -s) -(ℓ -j)τ . (5.13)
Next, by the triangle inequality, the Cauchy-Schwartz inequality, and (5.10) and proceeding as in (5.12), we finally conclude

W 2 (ρ ℓ τ , ρ j-1 τ ) ≤ ℓ k=j W 2 (ρ k τ , ρ k-1 τ ) ≤ (ℓ -j) 1/2   ℓ k=j 2τ F PKS [ρ k-1 τ ] -F PKS [ρ k τ ] + 2 ZC 3 τ 3 2 -k   1/2 ≤[(ℓ -j)τ ] 1/2 2 F PKS [ρ j-1 τ ] -F PKS [ρ k τ ] + 2 ZC 3 τ 2 1/2 ≤ F2 [(ℓ -j)τ ] 1/2 .
Adding this to the estimate in (5.13), and using the subadditivity of the square root concludes the proof.

Weak compactness

In this subsection, we will show the compactness of the sequence of interpolating curves. We cannot proceed as usually done, for instance in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], since we want to show that the limiting curves are not only measures but rather densities for each time and also since our densities have infinite second moment. The idea is to show a compactness in metrics W p , 1 ≤ p < 2, less strong than W 2 and then, pass to the limit p ր 2 the Hölder continuity property.

The next lemma shows a compactness property of the sets {ρ τ (t) : 0 < τ < τ ⋆ } for each fixed t.

LEMMA (Uniform integrability at fixed t).

There is a finite and computable constant F3 depending only on ρ 0 and for any fixed

1 ≤ p < 2 so that for all τ < τ ⋆ E[ρ τ (t)] = R 2 ρ τ (t, x) log ρ τ (t, x) dx ≤ F3 and R 2 |x| p ρ τ (t, x) dx ≤ F3 .
Proof: By the uniform control that we have on H λ , the moment bound is immediate from Lemma 1.11. By the uniform control of F PKS and H λ in (4.26), Lemma 5.2, and by the first concentration control Theorem 1.9 we conclude that

γ 1 R 2 ρ k τ log + ρ k τ dx ≤ F0 + F1 τ 2 + C CCF (5.14)
where 0 < γ 1 ≤ 1, uniformly in k. Finally, using the bound of the absolute first moment together with (5.14), we conclude that E[ρ k τ ] is bounded uniformly in k. Then, by the displacement convexity of E, this bound extends to ρ τ (t) for all t > 0, as explained at the beginning of this subsection.

It follows immediately from (2.5) that Lemma 5.3 remains true if W 2 there is replaced by any of the weaker metrics W p , 1 ≤ p < 2. The following characterization of the convergence in W p metrics in [START_REF] Villani | Topics in optimal transportation[END_REF]Chapter 9]: convergence of the absolute p-moment plus the weak-* convergence as measures of a sequence of densities {ρ n } n∈N towards ρ is equivalent to W p convergence; implies the following compactness result.

LEMMA (Compactness for the W p metric).

For any M > 0, let K be a subset of the set of densities ρ of mass M that is uniformly integrable, and such that {|x| p ρ(x) : ρ ∈ K} is also uniformly integrable. Suppose also that K is closed in the L 1 -weak topology. Then K is compact in the W p metric.

Proof: Let {ρ n } n∈N be any sequence in K. Since K is uniformly integrable and weakly closed in L 1 , the Dunford-Pettis Theorem provides us with a ρ ∈ K and a subsequence {ρ n k } k∈N such that lim k→∞ ρ n k = ρ weakly in L 1 and thus, weakly-* as measures. It is trivial to check that weak-L 1 convergence plus the uniform integrability of

{|x| p ρ(x) : ρ ∈ K} implies that lim k→∞ R 2 |x| p ρ n k (x) dx = R 2 |x| p ρ(x) dx .
The characterization of W p convergence mentioned above then implies lim k→∞ W p (ρ n k , ρ) = 0. 5.6 THEOREM (Convergence as τ → 0). Given T > 0 and any 1 ≤ p < 2, define (M δ T , W p ) to be the metric space in which M δ T is the set of densities on R 2 satisfying E[ρ] ≤ F3 and

R 2 |x| p+δ ρ(x) dx ≤ F3 (1 + T (p+δ)/2 ) , with p < p + δ < 2.
Then there is a function ρ on [0, ∞) with values in the set of densities of mass 8π such that for all T > 0, the restriction of ρ to [0, T ] is continuous in (M δ T , W p ), and there is a sequence {τ (n) } n∈N such that for all T > 0,

lim n→∞ max 0≤t≤T W p (ρ τ (n) (t), ρ(t)) = lim n→∞ max 0≤t≤T W p (ρ τ (n) (t), ρ(t)) = 0 .
(5.15)

Moreover the sequence {τ (n) } n∈N can be chosen independently of p, i.e., such that the convergence property (5.15) holds for all 1 ≤ p < 2. Furthermore, as a consequence for all t > s ≥ 0 and all 1 ≤ p ≤ 2:

W p (ρ(t), ρ(s)) ≤ F2 (t -s) 1/2 .
(5.16)

Proof: For each T > 0, (M δ T , W p ) is a compact metric space as a consequence of Lemma 5.5. By Lemma 5.4, for each t ≤ T , and each 0 < τ < τ⋆ , the restriction of ρ τ to [0, T ] takes values in (M δ T , W p ). Next, by (2.5) and Lemma 5.3, the set of these functions for 0 < τ < τ⋆ is uniformly equicontinuous into (M δ T , W p ). Thus by the the Arzela-Ascoli Theorem, we can select a uniformly convergent sequence. Now a simple diagonal sequence argument concludes the construction of ρ and proof of (5.15). Apply the above for T = 1 to get the initial sequence. Now take T = 2 and choose a subsequence of the first sequence, and so forth. For the piecewise-interpolation sequence, note that W 2 (ρ τ (t), ρ τ (t)) ≤ C √ τ by standard properties of displacement interpolation and (5.12) in Lemma 5.3. Therefore, the limits τ (n) → 0 of both time interpolations are the same. Note that this argument does not invoke any uniqueness of the limit. Another simple diagonal sequence argument shows that the sequence can be made independent of p. Take a sequence of increasing exponents {p n } n∈N ր 2 and {δ n } n∈N ց 0 with 1 ≤ p n < p n + δ n < 2 and apply the diagonal sequence argument to the constructed sequences for each p n . Also, take into account that the sequence of distances W p (ρ τ (n) (t), ρ(t)) is increasing in p.

The last part of the claim (5.16) follows directly from (5.15) and Lemma 5.3 for all 1 ≤ p < 2. Since the constant F2 obtained in Lemma 5.3 does not depend on 1 ≤ p < 2, then we conclude (5.16) for p = 2.

L p regularity

Our goal in this section is to prove: 5.7 THEOREM (L p -regularity). For each finite a > 0 and p > 1, there is a finite and computable constant C p depending only on a, p and ρ 0 such that whenever τ < a,

R 2 ρ p τ (t, x) dx ≤ Cp for all t ≥ a .
Proof: From (4.27), we deduce that for each m,

H λ [ρ m τ ] + D[ρ m τ ] ≤ H λ [ρ m-1 τ ] + Q m 4 2 -m τ 2
, proceeding in the same way that we did in deducing (5.6), except this time we do not discard the dissipation term. Let n ≥ k be positive integers. Since Q m ≤ C ρ0 for all m, summing from m = nk to n yields

H λ [ρ (n) τ ] + n m=n-k τ D[ρ m τ ] ≤ H λ [ρ n-k-1 τ ] + C ρ0 4 τ 2 .
(5.17)

Then since 0 ≤ H λ [ρ k τ ] ≤ C ρ0 for all k, using τ < 1 and dividing by kτ , we may simplify this to

1 k n m=n-k D[ρ m τ ] ≤ 2C ρ0 kτ . (5.18)
We now choose k to be the greatest integer less than or equal to a/τ , and of course suppose that n > k. Since kτ ≤ a < (k + 1)τ , and k ≥ 1, a/2 ≤ kτ , and then the fact that averages dominate minima yields the conclusion that for some positive integer m with τ ≤ a,

D[ρ m τ ] ≤ 4C ρ0 a .
Then since H λ [ρ m τ ] ≤ C ρ0 and F PKS [ρ m τ ] < +∞ we have from Theorem 1.10 that

R 2 |∇(ρ m τ ) 1/4 | 2 dx ≤ 4πC ρ0 aγ 2 + C CCD γ 2 .
Recall the GNS inequality f q q ≤ B q ∇f q-4 2 f 4 4 , valid for locally integrable functions f in R 2 and q > 4. Applying this with q = 4p and f = (ρ m τ )

R 2 (ρ (n) τ -K) p + dx ≤ R 2 (ρ m τ -K) p + dx + A 1 kτ + A 2 n j=m τ D[ρ j τ ] . (5.20)
We bound the first term on the right in (5.20) using (5.19), the second using the fact that kτ ≤ a, and the third using (5.18). The result,

R 2 (ρ (n) τ -K) p + dx ≤ B 4p 4πC ρ0 aγ 2 + C CCD γ 2 2(p-1) 8π 1/4 + A 1 a + 2A 2 C ρ0
uniformly for all n such that nτ ≥ a. Note that the bound depends only on a, p. Since ρ 5.5 Verification that ρ = lim τ →0 ρ τ is a solution of the PKS system Let τ (n) , ρ τ (n) and ρ be given as in Theorem 5.6. Our goal in this subsection is to prove that ρ is a weak solution of the PKS system as given in item (1.5.2) of the Definition 1.5.

(n) τ p ≤ (ρ (n) τ - K) + p + K (p-1)/p (8π)
5.8 LEMMA (ρ is a weak solution of the PKS system). Let τ (n) , ρ τ (n) and ρ be given as in Theorem 5.6. Then for all smooth and compactly supported test functions ζ and all t 2 > t 1 ≥ 0,

R 2 ζ(x) [ρ(t 2 , x) -ρ(t 1 , x)] dx = - 1 4π t2 t1 R 2 ×R 2 ρ(s, x) ρ(s, y) (x -y) • (∇ζ(x) -∇ζ(y)) |x -y| 2 dy dx + t2 t1 R 2
∆ζ(x) ρ(s, x) dx ds , Proof: In order to prove Lemma 5.8, we first remind the reader an analog for the functions ρ τ :

5.9 LEMMA (Approximate weak solutions of the PKS system). For 0 < τ < 1, define ǫ(t, τ ) = ǫ k for t ∈ ((k -1)τ, kτ ], and all integers k ≥ 1. Then for all smooth and compactly supported test functions ζ and all

t 2 > t 1 ≥ 0, R 2 ζ(x) [ρ τ (t 2 , x) -ρ τ (t 1 , x)] dx = 1 2 t2 t1 R 2 ×R 2 ρ τ (s, x) ρ τ (s, y) ∇G ǫ(t,τ ) (x -y) • (∇ζ(x) -∇ζ(y)) dy dx + t2 t1 R 2 ∆ζ(x) ρ τ (s, x) dx ds + O(τ 1/2 ) .
The proof of the previous Lemma follows the same lines as in [START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF]Theorem 3.4] and we skip it here for the sake of conciseness. The interested reader can see its proof in the preprint version of this paper.

As τ → 0 along the sequence {τ (n) } n∈N , lim n→∞ W 1 [ρ τ (n) (t), ρ(t)] = 0 uniformly on [0, T ] for any finite T . Hence by the Kantorovich-Rubinstein Theorem,

lim n→∞ R 2 ∆ζ(x) ρ τ (n) (t, x) dx = R 2 ∆ζ(x) ρ(t, x) dx ,
uniformly on [0, T ]. The interaction term can be rewritten as

R 2 ∇G ǫ(t,τ (n) ) * ρ τ (n) (t)] (x) • ∇ζ(x) ρ τ (n) (t, x) dx = R 2 ∇G * γ ǫ(t,τ (n) ) * ρ τ (n) (t) (x) • γ ǫ(t,τ (n) ) * (ρ τ (n) (t) ∇ζ) (x) dx = - 1 4π R 2 ×R 2 (γ ǫ * ρ τ (n) )(t, x) (γ ǫ * τ (n) )(t, y) (x -y) • (∇ζ(x) -∇ζ(y)) |x -y| 2 dx dy + R 2 ∇G * γ ǫ(t,τ (n) ) * ρ τ (n) (t) (x) • γ ǫ(t,τ (n) ) * (ρ τ (n) (t) ∇ζ) -(γ ǫ(t,τ (n) ) * ρ τ (n) (t))∇ζ (x) dx := I 1 + I 2 .
(5.21)

As {ρ τ (n) (t)} n∈N converges weakly in L 1 (R 2 ) towards ρ(t) as n → ∞, so does {γ ǫ(t,τ (n) ) * ρ τ (n) (t)} n∈N . We then deduce that {(γ ǫ(t,τ (n) ) * ρ τ (n) (t)) ⊗ (γ ǫ(t,τ (n) ) * ρ τ (n) (t))} n∈N converges weakly in L 1 (R 2 × R 2 ) towards ρ(t) ⊗ ρ(t)
when n → ∞, see [START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF]Lemma 2.3]. As a consequence we can pass to the limit in the first term in the right-hand-side of (5.21) to obtain

lim n→∞ I 1 = - 1 4π R 2 ×R 2 ρ(t, x) ρ(t, y) (x -y) • (∇ζ(x) -∇ζ(y)) |x -y| 2 dx dy .
We must now show that I 2 disappears in the limit. We can estimate I 2 using

|γ ǫ(t,τ (n) ) * (ρ τ (n) (t) ∇ζ)-(γ ǫ(t,τ (n) ) * ρ τ (n) (t))∇ζ|(x) ≤ R 2 γ ǫ(t,τ (n) ) (x -y) |∇ζ(y) -∇ζ(x)| ρ τ (n) (t, y) dy ≤ C ζ R 2 γ ǫ(t,τ (n) ) (x -y) |x -y| ρ τ (n) (t, y) dy = C ζ ((γ ǫ(t,τ (n) ) |x|) * ρ τ (n) (t))(x) .
By the HLS inequality, (4.6),

|I 2 | ≤ C HLS 2π γ ǫ(t,τ (n) ) * ρ τ (n) (t) 4/3 γ ǫ(t,τ (n) ) * (ρ τ (n) (t) ∇ζ) -(γ ǫ(t,τ (n) ) * ρ τ (n) (t))∇ζ 4/3 .
Then by similar arguments similar to those used to prove Lemma 4.13, we get

|I 2 | ≤ 4 C HLS C ζ ρ τ (n) (t) 4/3 γ|x| 4/3 ǫ(t, τ (n) ).
In case t 1 > 0, estimating ρ τ (n) (t) 4/3 using Theorem 5.7, we obtain the result. If t 1 = 0 we can use instead (4.29) and Hölder's inequality to obtain

|I 2 | ≤ 4 C HLS C ζ (8π) 1/4 2 C 3 γ|x| 4/3 ǫ(t, τ (n) ) τ = O((τ (n) ) 7/6 ) ,
where (5.4) was used.

Strong Compactness

At this point we have shown that the limit ρ = lim n→∞ ρ τ (n) posesses the properties (1.5.1) and (1.5.2) in Definition 1.5 of properly dissipative weak solutions. In this subsection, we show that (1.5.3) is also satisfied. This will complete the proof of the existence of properly dissipative solutions in Theorem 1.6. Choosing n = k in (5.17) we obtain, for all k ∈ N and all τ = τ (n) that

H λ [ρ k τ ] + k m=1 τ D[ρ m τ ] ≤ H λ [ρ 0 ] + C ρ0 4 τ 2 .
Thus, using the discrete time interpolation ρτ , we have that for any T > 0 and the positive integer N such that N τ ≤ T ≤ (N + 1)T ,

H λ [ρ τ (T )] + N τ 0 D[ρ τ (t)] dt ≤ H λ [ρ 0 ] + C ρ0 4 τ 2 .
(5. [START_REF] Dolbeault | The two-dimensional Keller-Segel model after blow-up[END_REF] We emphasize that the use of the piecewise constant interpolation is essential at this point since the functional D[ρ] is not displacement convex. Note that the L p bounds deduced in Theorem 5.7 apply to ρτ as well as to ρ τ . To make full use of these bounds, we choose any fixed a > 0, and then for all τ < a, we weaken the bound in (5.22) by increasing the lower limit of integration in t to a. Also writing b := N τ , this yields

H λ [ρ τ (T )] + 8 b a R 2 |∇ρ 1/4 τ (t, x)| 2 dt - b a R 2 ρ3/2 τ (t, x) dx dt ≤ H λ [ρ 0 ] + C ρ0 4 τ 2 .
It is legitimate to express D[ρ τ ] as the difference of two integrals since Theorem 5.7 tells us the ρ3/2 τ is integrable over [a, T ] × R 2 . We now show that passing to a further subsequence of {τ (n) } n∈N , we may arrange that for all 0 < a < b < ∞, along this subsequence,

lim n→∞ b a R 2 |ρ τ (n) (t, x) -ρ(t, x)| 3/2 dx dt = 0 (5.23) and lim n→∞ ρτ (n) (t, x) = ρ(t, x) for almost every (t, x) ∈ [a, b] × R 2 .
The following strong compactness theorem, leading to the existence of almost everywhere convergent subsequences, is the key: Proof: By the Kolmogorov Compactness Theorem [START_REF] Hance-Olsen | The Kolmogorov-Riesz compactness thoerem[END_REF][START_REF] Kolmogorov | Über Kompaktheit der Funktionenmengen bei der Konvergenz im Mittel[END_REF], also known as the Frechet-Kolmogorov theorem [9, Corollary IV.26], a set G of functions g ∈ L 2 ((a, b) × R 2 ) is strongly precompact if and only if:

(K1) G is uniformly bounded in L 2 ((a, b) × R 2 ).
(K2) For each ǫ > 0, there is an R > 0 so that for all g ∈ G, (K3) For each ǫ > 0, there is a δ > 0 so that for all g ∈ G, whenever y ∈ R 2 satisfy |y| ≤ δ, The uniform integrability is an immediate consequence of these estimates, giving the first two conditions (K1) and (K2) of the Kolmogorov compactness theorem.

Step 2.-Spatial translations (5.24): Writing ρτ (t, x) = f 4 (t, x), we have

ρτ (t, x -y) -ρτ (t, x) = -4|y| • 1 0 f 3 (t, x -sy)∇f (t, x -sb) ds ,
and then by Minkowski's inequality, and then Hölder's inequality

R 2 |ρ τ (t, x -y) -ρτ (t, x)| 2 dx ≤ 4|y| R 2 f 6 (t, x) dx 1/2 R 2 |∇f (t, x)| 2 dx 1/2 = 4|y| R 2 ρ3/2 τ (t, x) dx 1/2 R 2 |∇ρ 1/4 τ | 2 dx 1/2
.

By our uniform L p bounds, there is a constant C independent of τ so that

R 2 |ρ τ (t, x -y) -ρτ (t, x)| 2 dx ≤ |y|C R 2 |∇ρ 1/4 τ | 2 dx 1/2
, and hence

b a R 2 |ρ τ (t, x -y) -ρτ (t, x)| 2 dx dt ≤ C|y|b 1/2 b a R 2 |∇ρ 1/4 τ | 2 dx dt 1/2 .
Since the integral on the right hand side is bounded uniformly in τ , this gives us (5.24).

Step 3.-Temporal translations (5.25): We do not have any estimates on time derivatives, so we cannot obtain the bound on temporal translations in such a simple manner as we have for the spatial translations. What we do have from Lemma 5.3 is a finite constant C so that

W 2 (ρ(t -c, •), ρ(t, •)) ≤ C √ c (5.26)
holds uniformly in τ ≤ τ ⋆ and in t ≥ c. We now use an interpolation argument based on an idea of Otto, see [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF]Subsection 3.5], to combine this with the spatial regularity provided by the square integrability of ∇ρ 1/4 τ (t, x) in R 2 × R. Our task would be very much simpler if we had, for almost every t, a uniform bound on R 2 |∇ρ 1/4 τ (t, x)| 2 dx, but don't for any fixed t. The interpolation bound we need is provided by the following theorem whose proof will be provided after we conclude the proof of Theorem 5.10: have square integrable distributional gradients. Then

σ 0 -σ 1 2 2 ≤ ∇(σ 0 ) 1/4 2 + ∇(σ 1 ) 1/4 2 (2 5/2 + 2 9/2 K)(W 2 (σ 0 , σ 1 )) (4p-3)/(4p+2)
+ 16M (p-1)/p K (p+2)/2p (W 2 (σ 0 , σ 1 )) (p-1)/(2p+1) .

We now apply Theorem 5.11 with ρτ (t, •) in place of σ 0 and ρτ (tc, •) in place of σ 1 . We have a uniform bound on K in this case, for any p < ∞, and we also have the bound (5.26), so we obtain a finite constant C so that

R 2 |ρ τ (t, •) -ρτ (t -c, •)| 2 dx ≤ C R 2 ∇ρ 1/4 τ (t, •)| 2 dx 1/2 + R 2 ∇ρ 1/4 τ (t -c, •)| 2 dx 1/2 |c| 4p-3 8p+4 + |c| p-1 4p+2
. Now integrating both sides over [ã, b], and using the Cauchy-Schwartz inequality, we obtain

R 2 ×[ã, b] |ρ τ (t, •) -ρτ (t -c, •)| 2 dx dt ≤ C AT 1/2 |c| 4p-3 8p+4 + T |c| p-1 4p+2
, where

A := T t0 R 2 ∇ρ 1/4 τ (t, •)| 2 dx dt 1/2 + T t0 R 2 ∇ρ 1/4 τ (t -c, •)| 2 dx dt 1/2 .
Our results so far give us a bound on A that is uniform in τ ≤ τ ⋆ , and thus, choosing p = 2, the proof of (5.25) is complete.

It remains to prove Theorem 5.11. Before beginning the proof itself, we explain the argument in [36, Subsection 3.5] that is the basis of the proof. Let σ 0 and σ 1 be two uniformly bounded densities of mass M in R 2 . Let dΠ be the optimal coupling of σ 0 and σ 1 . For 0 < s < 1, define σ s to be the displacement interpolant between σ 0 and σ 1 . That is, for any bounded continuous function

ϕ in R 2 , R 2 ϕ(z)σ s (z) dz = R 2 ×R 2 ϕ((1 -s)x + sy) dΠ(x, y) .
As is well known (see for example [37, Lemma 3]), we have

σ s ∞ ≤ max{ σ 0 ∞ , σ 1 ∞ } .
(5.27)

Suppose furthermore that both σ 0 and σ 1 have a square integrable gradient. Then we have Otto's interpolation estimate:

σ 0 -σ 1 2 ≤ (max{ σ 0 ∞ , σ 1 ∞ }) 1/2 ( ∇σ 0 2 + ∇σ 1 2 ) W 2 (σ 0 , σ 1 ) . (5.28)
To see this, note that

σ 0 -σ 1 2 2 = R 2 ×R 2 ([σ 0 (x) -σ 1 (x)] -[σ 0 (y) -σ 1 (y)]) dΠ(x, y) = 1 0 R 2 ×R 2 [∇σ 1 (x + s(y -x)) -∇σ 0 (x + s(y -x))] • (x -y) dΠ(x, y) ds .
Now apply the Cauchy-Schwartz inequality. By (5.27), we have that for each s,

R 2 ×R 2 |∇σ 0 (x + s(y -x))| 2 dΠ(x, y) = R 2 |∇σ 0 (x)| 2 σ s (x) dz ≤ (max{ σ 0 ∞ , σ 1 ∞ }) ∇σ 0 2 2 ,
and likewise for σ 1 . Since R 2 ×R 2 |x -y| 2 dΠ(x, y) = W 2 2 (σ 0 , σ 1 ), we obtain (5.28). We would like to apply this sort of argument with ρτ (t, •) in place of σ 0 and ρτ (tc, •) in place of σ 1 . We cannot do this directly since ρτ is not bounded. However, by the results of the previous section, for t 0 > 0, we have an upper bound on ρτ (t, •) p for all finite p that is uniform in t ≥ t 0 . The next lemma allows us to approximate ρτ (t, •) and ρτ (tc, •) by uniformly bounded densities without significantly increasing the 2-Wasserstein distance between them. 5.12 LEMMA (Approximation by bounded densities). Let σ be any density of mass M on R 2 . Suppose that for some p > 1, σ belongs to L p+1 (R 2 ) with σ p+1 p+1 ≤ K. Then for all λ > 0, there exists density of mass M , σ such that σ

∞ ≤ 2λ, W 2 ( σ, σ) ≤ 2Kλ -p-1/2 , (5.29) σ -σ 2 ≤ 2K 1/2 λ -(p-1)/2 , (5.30)
and finally, σ 2 ≤ 2 σ 2 .

Proof:

Let E λ = { x : σ(x) > λ }.
Then by Chebychev's inequality,

E λ σ(x) dx ≤ 1 λ p R 2 σ p+1 (x) dx ≤ K λ p . (5.31)
Now for any h > 0 and any integers m and n, define C m,n to be the square

C m,n = {x = (x 1 , x 2 ) ∈ R 2 : mh ≤ x 1 < (m + 1)h , nh ≤ x 2 < (n + 1)h } . Define M m,n by M m,n := E λ ∩Cm,n σ(x) dx .
Next define σ m,n := 1 E λ ∩Cm,n σ and ν m,n := M m,n h 2 1 Cm,n . There are both densities of mass M m,n supported in C m,n . Let T m,n be any map from C m,n to C m,n such that T m,n #σ m,n = ν m,n . For instance, one may use the one associated to the optimal coupling. Define T : R 2 → R 2 by

T (x) =      m,n∈Z 1 Cm,n (x)T m,n (x) x ∈ E λ x x / ∈ E λ .
Notice that only one term in the sum is non-zero. By construction,

T #σ = (1 -1 E λ )σ + m,n∈Z M m,n h 2 1 Cm,n .
By (5.31), M m,n ≤ Kλ -p . We now specify h := K 1/2 λ -(p+1)/2 . Then M m,n h -2 ≤ λ, and hence with this choice of h, T #σ is uniformly bounded above by 2λ. We now define σ := T #σ. It remains to verify that σ has all of the properties claimed in the lemma.

First of all, notice that |T (x) -x| 2 = 0 on the complement of E λ , and on E λ is is bounded by 2h 2 , since each

T m,n maps C m,n into itself. Therefore, W 2 2 (σ, σ) ≤ R 2 |T (x) -x| 2 σ(x) dz ≤ 2h 2 E λ σ(x) dx ≤ 2Kλ -(p+1) Kλ -p .
This proves (5.29). Next, by (5.31) and Jensen's inequality,

σ -σ = 1 E λ σ - m,n∈Z M m,n h 2 1 Cm,n with m,n∈Z M m,n h 2 1 Cm,n 2 ≤ 1 E λ σ 2 .
(5.32) Furthermore, by Holder's inequality with the dual indices p and p ′ , .33) This proves (5.30). For the final part, note that by (5.32),

R 2 σ 2 1 E λ dx = R 2 σ 2-1/p ′ σ 1/p ′ 1 E λ dx ≤ R 2 σ p+1 dx 1/p E λ σ dx 1/p ′ ≤ (K) 1/p Kλ -p 1/p ′ ≤ Kλ 1-p . ( 5
σ 2 ≤ (1-1 E λ )σ 2 + 1 E λ σ 2 ≤ 2 σ 2 .
This completes the proof of the lemma.

Proof of Theorem 5.11: For any λ > 0, let σ 0 and σ 1 be the bounded approximation of σ 0 and σ 1 provided by Lemma 5.12. Also, define σ 0 (x) := min{σ 1 (x) , λ } and σ 1 (x) := min{σ 1 (x) , λ } .

Starting from the identity σ

0 -σ 1 2 2 = σ 0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2
, and adding and subtracting repeatedly, and using the Cauchy-Schwartz inequality, we obtain

σ 0 -σ 1 2 2 ≤ σ 0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2 + ( σ 0 2 + σ 1 2 )( σ 0 -σ 0 2 + σ 1 -σ 1 2 ) + ( σ 0 2 + σ 1 2 ) ( σ 0 -σ 0 2 + σ 1 -σ 1 2 ) .
(5.34)

The heart of the matter is the estimation of σ

0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2 .
Let Π be the optimal coupling of σ 0 and σ 1 , and for 0 < s < 1, let σ s be displacement interpolant between σ 0 and σ 1 . By what has been explained above, σ s is uniformly bounded above by 2λ. Then, by the Cauchy-Schwarz inequality once more,

σ 0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2 = R 2 ×R 2 ([σ 0 -σ 1 ](x) -[σ 0 -σ 1 ](y)) d Π(x, y) = 1 0 R 2 ×R 2 ([∇σ 0 -∇σ 1 ]((1 -s)y + sx) • (x -y) d Π(x, y) ≤ (2λ) 1/2 ∇σ 0 2 + ∇σ 1 2 W 2 ( σ 0 , σ 1 ) .
Next, for j = 0, 1, let f j := (σ j ) 1/4 . Then since ∇σ j (x) = 0 unless f j (x) ≤ λ 1/4 , ∇σ j 2 ≤ 4λ 3/4 ∇f j 2 . Thus,

σ 0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2 ≤ 2 5/2 λ 5/4 ∇(σ 0 ) 1/4 2 + ∇(σ 1 ) 1/4 2 W 2 ( σ 0 , σ 1 ) . (5.35)
By the triangle inequality and Lemma 5.12,

W 2 ( σ 0 , σ 1 ) ≤ W 2 ( σ 0 , σ 0 ) + W 2 (σ 0 , σ 1 ) + W 2 (σ 1 , σ 1 ) ≤ 2Kλ -p-1/2 + W 2 (σ 0 , σ 1 ) + 2Kλ -p-1/2 .
Combining this with (5.35), we obtain

σ 0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2 ≤ 2 5/2 λ 5/4 ∇(σ 0 ) 1/4 2 + ∇(σ 1 ) 1/4 2 W 2 (σ 0 , σ 1 ) + 2 5/2 λ 5/4 ∇(σ 0 ) 1/4 2 + ∇(σ 1 ) 1/4 2 4Kλ -p-1/2 .
At this point we specify λ := (W 2 (σ 0 , σ 1 )) -2/(2p+1) .

(5.36)

Wit this choice, we have

σ 0 -σ 1 , σ 0 L 2 -σ 0 -σ 1 , σ 1 L 2 ≤ ∇(σ 0 ) 1/4 2 + ∇(σ 1 ) 1/4 2 (2 5/2 + 2 9/2 K)(W 2 (σ 0 , σ 1 )) (4p-3)/(4p+2) .
Finally, it remains to bound the terms in the last two lines of (5.34). Note that the same argument using Chebychev's inequality in (5.31) and then Hölder's inequality in (5.33) yields σ jσ j 2 ≤ K 1/2 λ (1-p)/2 for j = 0, 1. Note also that by Hölder's inequality once more, σ j 2 2 ≤ M (p-1)/p K 1/p . Therefore, the sum of the terms in the last two lines of (5.34) is bounded above by 16M (p-1)/p K (p+2)/2p λ -(p-1)/2 . With the value of λ specified in (5.36), the contribution of these terms is 16M (p-1)/p K (p+2)/2p (W 2 (σ 0 , σ 1 )) (p-1)/(2p+1) .

Combining results, the proof is complete.

LEMMA (Further subsequence).

There is a subsequence of the sequence {τ (n) } n∈N , denoted with the same index, such that along this subsequence, (5.23) is valid for each 0 < a < b < ∞ and

lim n→∞ ρτ (n) (t, x) = ρ(t, x)
almost everywhere on (0, T ) × R 2

for any T > 0.

Proof: Let us consider any integer N ≥ N o with N o T > 1. Applying Theorem 5.10 we get a subsequence of {τ

(n) } n∈N along which {ρ τ (n) } n∈N converges to ρ strongly in L 2 ([1/N o , T ]×R 2 ) and almost surely on [1/N o , T ]×R 2 . Next, for N = N o + 1, choose a subsequence of {τ (n) } of the previous subsequence along which {ρ τ (n) } n∈N converges to ρ strongly in L 2 ([1/(N o + 1), T ] × R 2 )
and almost surely on [1/(N o + 1), T ] × R 2 , and so forth. We finish by an obvious diagonal sequence argument.

Entropy dissipation

With the strong convergence results obtained in the previous subsection, we may now establish the entropy-entropy dissipation inequality:

5.14 THEOREM (Entropy-entropy dissipation). For each T > 0 the weak solution ρ of the PKS system that we have constructed for the initial data ρ 0 satisfies

F PKS [ρ(T )] < +∞ and H λ [ρ(T )] + T 0 D[ρ(t)] dt ≤ H λ [ρ 0 ] . (5.37)
Proof: First remind that F PKS is lower semi-continuous function for the weak-L 1 convergence, see [START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF]Lemma 3.1]. So that the bound in (5.37) is a direct consequence of (5.23). Also by (5.23), it suffices to show that

H λ [ρ(T )] ≤ lim inf n→∞ H λ [ρ τ (n) (T )] ,
and that for any 0

< a < b < T < ∞, b a R 2 ∇ ρ 1/4 2 dx dt ≤ lim inf n→∞ b a R 2 ∇ ρ1/4 τ (n) 2 dx dt , (5.38)
for a suitable sequence {τ (n) } n∈N , since the rest easily follows by a monotone convergence argument for taking a to 0 and b to T . The first of these follows from the fact that H λ [ρ] is a lower semi-continuous function on L 1 with respect to the W 1 metric just by using the expression of H λ [ρ] in (1.11).

To see the second, we again make use of the almost everywhere convergence proved in the previous subsection. Let f n := ρ1/4 τ (n) and f := ρ 1/4 . Then the sequence of functions {f n } n∈N → f in L 4 ∩ L 6 ((a, T ) × R 2 ) from Lemma 5.13. From (5.22), we have that the sequence {∇f n } n∈N is bounded in L 2 ((a, T ) × R 2 ), therefore it has a weakly convergent subsequence denoted with the same index such that {∇f n } n∈N ⇀ σ weakly in L 2 ((a, T ) × R 2 ).

Due to the strong convergence of the sequence {f n } n∈N → f in L 4 ∩ L 6 ((a, T ) × R 2 ), it is simple to identify the weak limit as σ = ∇f . By standard properties of L 2 -weak convergence, we deduce that Proof of Theorems 1.6: As noted above, Theorem 5.14 provides the final step in the construction of the properly dissipative weak solutions. Theorems 5.7 and 5.14 provide the additional regularity properties (1.6.1) and (1.6.2). It remains to prove (1.6.3), the dissipation of F PKS .

We now show that F PKS [ρ(t)] ≤ F PKS [ρ(s)] for all 0 ≤ s < t. Take τ to be any element of the sequence {τ (n) } n∈N whose corresponding approximated solutions { ρτ (n) } n∈N converges to the constructed properly dissipative weak solution ρ. Let be such that (j -1)τ ≤ s ≤ jτ and let ℓ be such that ℓτ ≤ t ≤ (ℓ + 1)τ . Using (5.11) in Lemma 5.2 and (4.9), we deduce

F PKS [ρ τ (t)] ≤ F ǫ ℓ PKS [ρ ℓ τ ] ≤ F ǫ ℓ-1 PKS [ρ ℓ-1 τ ] + ZC 3 τ 2 2 -ℓ ≤ F ǫj-1 PKS [ρ j-1 τ ] + ZC 3 τ 2 ℓ k=j 2 -k ≤ F ǫj-1 PKS [ρ j-1 τ ] + ZC 3 τ 2 .
Using (5.9) and the Lemma 5.7, we can control the error term in the right-hand side by

F PKS [ρ τ (t)] ≤ F PKS [ρ τ (s)] + Z ρτ (s) 3/2 3/2 τ 3 2 -j+1 + ZC 3 τ 2 ≤ F PKS [ρ τ (s)] + 2 ZC 3 τ 2 .
Finally, the a-priori bounds uniform in τ due to Lemmas 5.6 and 5.7 together with Lemma 5.16 allow us to pass to the limit τ → 0 leading to our claim.

Large time asymptotics

We start by identifying the large time asymptotics of the solutions in a time average sense. Proof: This follows by a standard entropy dissipation argument. Let {t n } n∈N ր +∞ be an increasing diverging sequence of times and consider σ n (t, x) = ρ(t+t n , x), for 0 ≤ t ≤ 1. By using the entropy dissipation inequality (5.37) which is true for all T > 0, we deduce that for all 1 ≤ p < ∞. Note that the sequence {σ n } n∈N satisfies the equicontinuity property (5.16) in Theorem 5.6. Summarizing, the sequence {σ n } n∈N has the same properties (5.41) and (5.42) as the sequence of approximate solutions we used in previous sections to construct the solution in Theorem 1.6. Proceeding as in Subsections 5.3-5.6, we deduce the existence of a subsequence, denoted with the same index, such that {σ n } n∈N converges towards ρ ∞ with the same convergence properties as in previous Subsections 5. 3-5.6. Here, ρ ∞ is a weak solution of (1.1) on the time interval (0, 1) in the sense of (1.5.2) in Definition 1.5. In particular, {σ n } n∈N converges to ρ ∞ in the metric space (M Now, let us identify the limit function ρ ∞ , passing to the limit using (5.40), we obtain

1 0 R 2 8 ∇ρ 1/4 ∞ 2 -ρ 3/2
∞ dx dt = 0 , which means that ρ ∞ (t) is a minimizer to the Gagliardo-Nirenberg-Sobolev inequality for all t ∈ (0, 1), see Lemma 1.2, and thus that there exists λ(t) such that ρ ∞ (t) = ̺λ (t) where ̺ λ the family of the minimizers of the Gagliardo-Nirenberg-Sobolev inequality, see Lemma 1.2. Due to (5.44) then H λ [̺λ (t) ] < ∞, we conclude that λ(t) = λ since H λ [̺ µ ] = +∞ for µ = λ. Therefore, ρ ∞ (t) = ̺ λ that together with (5.45) implies (5.39).

We now will take advantage of the other Lyapunov functional, we shall prove that lim t→∞ F PKS [ρ(t)] = F PKS [̺ λ ]. In doing this, we shall make essential use of the monotonicity of Proof: Choose any 0 < δ ′ < min(δ, 1) so that 2δ ′ /(1δ ′ ) ≤ δ. By uniform integrability arguments such as we have made above, see Step 1 of Theorem 5.10, we can find a subsequence (denoted with the same index) along which {ρ n } n∈N is weakly convergent in L 1+δ ′ (R 2 ) and along which {ρ 1+δ ′ n } n∈N is weakly convergent in L 1 . It follows as in Subsection 5.6 that {ρ n } n∈N is strongly convergent in L 1+δ ′ (R 2 ), and passing to a further subsequence, we may suppose it is also almost everywhere convergent, and strongly convergent in L 1 (R 2 ). Let ρ denote the limit. By Fatou's Lemma, ρ ∈ S C,δ .

Since for t ≥ 1, t log t ≤ (t δ -1)/δ ′ , we have for ρ ≥ 1, ρ log ρ ≤ (1/δ ′ )ρ 1+δ ′ and for ρ < 1, ρ log(1/ρ) ≤ (1/δ ′ )ρ 1-δ ′ . Since for ǫ = 2δ ′ , we get The convergence of the positive part of the interaction potential is straightforward, due to the uniform bound of {|x| δ ρ n } n∈N in L 1 (R 2 ) and a dominated convergence argument. Concerning the negative part, it follows by Young's inequality for convolutions using the convergence of {ρ n } n∈N → ρ in L 1+δ ′ (R 2 ) and the fact that log -|x| ∈ L p (R 2 ) for all 1 ≤ p < ∞.

R 2 ρ 1-δ ′ (1 + |x| 2 ) ǫ dx = R 2 ρ 1-δ ′ (1 + |x| 2 ) 2ǫ (1 + |x| 2 ) -ǫ dx ≤ R 2 ρ(1 + |x| 2 ) 2ǫ/(1-δ ′ ) dx 1-δ ′
5.17 LEMMA (Qualitative stability for F PKS ). For any ǫ > 0 and C > 0, there exists δ(ǫ, C) > 0 so that if ρ ∈ S C,δ , then F PKS [ρ] ≤ 8π(-1 + log(8π)) + δ(ǫ, C) ⇒ ρ -̺ µ 1 ≤ ǫ for some µ > 0 , and for any R > 0, there exists δ(ǫ, C, R) > 0 such that

F PKS [ρ] ≤ 8π(-1 + log(8π)) + δ(ǫ, C, R) ⇒ {|x|≤R} |( √ ρ - √ ̺ µ )(x)| 2 dx 1/2
≤ ǫ for some µ > 0 .

Proof: Given C, R > 0 fixed, suppose not. Then for some ǫ > 0, there is a sequence {ρ n } n∈N in S C,δ such that lim n→∞ F PKS [ρ n ] = 8π(-1 + log(8π)) but inf n,µ ρ n -̺ µ 1 ≥ ǫ .

However, by Lemma 5.16, there is a subsequence, still indexed by n, converging strongly in L 1 (R 2 ) to ρ ∈ S C,δ , such that 8π(-1 + log(8π)) = lim

n→∞ F PKS [ρ n ] = F PKS [ρ] .
By the cases of equality in the Log-HLS inequality, ρ = ρ µ for some µ. This is a contradiction. The second part is proved the same way, using the uniform integrability of the √ ρ, ρ ∈ S C,δ on {|x| ≤ R}.

Proof of Theorems 1.7: Recall that F PKS [ρ(t)] ≤ F PKS [ρ(s)] for all 0 ≤ s < t. We now apply this monotonicity to improve our large time asymptotic result. By (5.39) in Lemma 5.15, there is a sequence of times {t n } n∈N ր ∞ such that lim n→∞ ρ(t n ) -̺ λ 1 = 0. By our regularity results in Lemmas 5.6 and 5.7, {ρ(t n )} n∈N ⊂ S C,δ for some 0 < C, δ < ∞. Then by Lemma 5.16, there is a subsequence, still indexed by n, such that lim n→∞ Then by Lemma 5.17 it follows that given R > 0 there exists µ > 0 such that for all sufficiently large t,

ρ(t) -̺ µ 1 ≤ ǫ and √ ̺ µ - √ ρ 2,R := {|x|≤R} |( √ ρ - √ ̺ µ )(x)| 2 dx 1/2 ≤ ǫ.
However, for any R > 0, by Minkowskii's inequality and (5.37),

{|x|≤R} | √ ̺ µ - √ ̺ λ | 2 ̺ λ -1/2 dx 1/2 ≤ {|x|≤R} | √ ̺ µ - √ ρ| 2 ̺ λ -1/2 dx 1/2 + H λ [ρ] ≤ √ 8λ(λ + R 2 ) √ ̺ µ - √ ρ 2,R + C ρ0 .
Since the left hand side diverges as R increases, uniformly for |µ -λ| > δ > 0, we readily conclude that µ = λ and lim t→∞ ρ(t) -̺ λ 1 = 0.

Appendix: proof of Lemma 4.13

By Lemma 2.3 applied to u 0 = ρ and u 1 = ρ 0 We will keep the term (I) and we need to perform some integration-by-parts in the other terms: Control of (II): We can rewrite this term as

H λ,δ [ρ] ≤ H λ,δ [ρ 0 ] - R 2 2 x f ′ δ (|x| 2 ) +
(II) = 2 R 2 ∇ ρ + 2δ √ ρ + δ • ∇c ǫ dx .
Integrating by parts on the ball of radius R and noticing that -∆c ǫ = γ ǫ * ρ * γ ǫ , we obtain for all R > 0. To cope with the first boundary term, we observe that taking any natural N > 1, and considering Finally, a simple application of Hölder's inequality gives By definition of the convolution, we have

ρ f ′ δ id * γ ǫ (z) = R 2 ρ(z -x) f ′ δ (z -x) (z -x) γ ǫ (x) dx = z R 2 ρ(z -x) f ′ δ (z -x) γ ǫ (x) dx - R 2 ρ(z -x) f ′ δ (z -x) x γ ǫ (x) dx = z (ρf ′ δ * γ ǫ )(z) -(ρf ′ δ * id γ ǫ )(z) .
As a consequence, we infer (III) = (5.51)

To estimate the second term of the right hand side of (5.50), we make again use of the HLS inequality, see (4.6):

|(III) 2 | ≤ 1 4 √ 2λ π

  2 u(x) -2 u(x) + ̺ λ (x) dx .

4. 2

 2 Regularization of F PKS Let γ(x) := 1 2π e -|x| 2 /2 be the standard Gaussian probability density in R 2 . Then, for all ǫ > 0 define γ ǫ (x) = ǫ -2 γ (x/ǫ), and define the regularized Green's function G ǫ = γ ǫ * G * γ ǫ , where * denotes convolution, and G(x) = -1/(2π) log |x|. The radially symmetric, C ∞ probability density γ √ t is the fundamental solution of the heat equation satisfying for

  From the elementary inequality |z -w| ≤ |z| + |w| ≤ 2 max{|z| , |w|}, we obtain log |z -w| ≤ log 2 + log |z| + log |w| .

5. 10 THEOREM

 10 (Strong compactness for ρ2 τ (t, x)). Let 0 < a < b < ∞ be given. The family of functions{ 1 [a,b] (t)ρ τ (t, x) : τ < τ ⋆ } is precompact in the strong topology in L 2 (R 2 × (a, b)).

  x)| 2 dx dt < ǫ .

R 2 ×

 2 (a,b) |g(t, xy)g(t, x)| 2 dx dt ≤ ǫ , (5.24) and whenever c ∈ R, and [ã, b] ⊂ (a, b) with (ãc, bc) ⊂ (a, b) R 2 ×(ã, b)

5. 11 THEOREM

 11 (Interpolation bound). Let σ 0 and σ 1 be two densities of mass M in R 2 such that for some p > 2,

b a R 2 |∇f | 2

 22 dx dt ≤ lim inf n→∞ b a R 2 |∇f n | 2 dx dt which shows (5.38).

5. 15 LEMMA

 15 (Time-averaged strong convergence). Let ρ be the properly dissipative weak solution of the PKS system that we have constructed. Then lim x) -̺ λ (x)| dx dt = 0 .(5.39)

  using the uniform in time bounds(5.37) for the solution, F PKS [σ n (t)] < +∞, andH λ [σ n (t)] ≤ H λ [ρ 0 ] ≤ C ρ0 ,the concentration control inequality in Lemma 1.11 and the Gagliardo-Nirenberg-Sobolev inequality in Lemma 1.2(t, x) dx ≤ C p , (5.42)

1 / 2 1,W 1 inf n→∞ 1 0 R 2 ∇(σ n ) 1 2 σ 3 / 2 n 1 0 R 2 ρ 3 / 2 ∞ 1 0 R 2

 121121232123212 W 1 ), with the notation of Theorem 5.6, giving lim n→∞ max 0≤t≤1 (σ n (t), ρ ∞ (t)) = 0 . (5.43) Moreover, repeating the arguments in Theorems 5.14 and 5.13, we get sup 0≤t≤1 H λ [ρ ∞ (t)] ≤ C ρ0 and lim (t, x) dx dt = (t, x) dx dt . Furthermore, Theorem 5.13 implies the almost everywhere convergence in (0, 1) × R 2 of {σ n } n∈N towards ρ ∞ , that together with (5.43) implies that lim n→∞ |σ n (t, x)ρ ∞ (t, x)| dx dt = 0 . (5.45)

F 2 ρ 2 ρ

 22 PKS [ρ(t)]. Let us introduce for any C > 0 and δ > 0 the setS C,δ := ρ ∈ L 1 + (R 2 ) : R (x) dx = 8π , R 2 |x| δ ρ(x) dx ≤ C and R 1+δ (x) dx ≤ C .

5. 16 LEMMA

 16 (Convergence for F PKS ). Given any sequence {ρ n } n∈N in S C,δ there is a ρ ∈ S C,δ and a subsequence{ρ n k } k∈N such that lim k→∞ ρ n kρ 1 = 0 and lim k→∞ F PKS [ρ n k ] = F PKS [ρ] .

R 2 ( 1 + 2 ρ 2 ρ

 2122 |x| 2 ) -2 dx δ ′ , our choice of δ ′ gives the uniform integrability of {ρ 1-δ ′ n } n∈N . Then, by what we have said above, {ρ n log ρ n } n∈N is uniformly integrable, and hence lim n→∞ R n log ρ n dx = R log ρ dx .

F

  PKS [ρ(t n )] = F PKS [̺ λ By the monotonicity of F PKS [ρ(t)] it follows that lim t→∞ F PKS [ρ(t)] = F PKS [̺ λ ] = 8π(-1 + log(8π)) .

  δ) 3/2 • (∇ϕ(x)x) ρ dx -K δ (ρ, ρ 0 ) with f ′ δ (s) = 8λ 8λ + δ(λ + s) 2 -3/2. Using (4.20), i.e. x-∇ϕ(x) = τ (∇c ǫ -∇ρ/ρ) and expanding we can rewrite it asH λ,δ [ρ] ≤ H λ,δ [ρ 0 ] + τ 1 + 2 (III) + 2 (IV) -K δ (ρ, ρ 0 ) , 2 ) x • ∇c ǫ ρ dx and (IV) := -R 2 f ′ δ (|x| 2 ) x • ∇ρ dx ,

NN - 1 |x|=r√ ρ |∇c ǫ | dσ dr 2 ≤ π N 2 -(N - 1 ) 2 ∇c ǫ 2 ∞ N -1≤|x|≤N ρ dx ≤ π ∇c ǫ 2 ∞ 2 N - 1 N - 1 N -1≤|x|≤N |x| ρ dx ≤ C 2 ǫ√ 2 R 2 √

 122122211222 ρ |∇c ǫ | dσ dr = 0, and thus, there exists a sequence {R j } ր ∞ such that lim j→∞ |x|=Rj √ ρ |∇c ǫ | dσ = 0. (5.48) Plugging (5.47) and (5.48) into (5.46), we get (II) ≤ ρ (γ ǫ * ρ * γ ǫ ) dx + 32 π √ δ + 4 √ δ(J γ + 2π) .

R 2 √ 2 ρ 3 2 (

 2232 ρ (γ ǫ * ρ * γ ǫ ) dx ≤ √ ρ 3 γ ǫ * ρ * γ ǫ 3/2 ≤ R Control of (III): Remind that f ′ δ (s) ≤ (8λ) -1/2 := κ λ /2 and that 2f ′ δ (s) → κ λ as δ → 0, see Proposition 2.2. By definition of c ǫ and G ǫ and by symmetry of γ(III) = R 2 ρ(x) f ′ δ (x) x • (∇G ǫ * ρ) (x) dx = R 2 ×R 2 ρ(x) f ′ δ (x) x γ ǫ (xz) (∇G * γ ǫ * ρ) (z) dz dx = R ρ f ′ δ id * γ ǫ )(z) (∇G * γ ǫ * ρ) (z) dz .

R 2 [ 2 R 2 3 ≤

 2223 z (ρf ′ δ * γ ǫ )(z) -(ρf ′ δ * id γ ǫ )(z)] • (∇G * γ ǫ * ρ) (z) dz := (III) 1 -(III) 2 .(5.50)By the symmetrization argument just as in (1.13), the first term of the right hand side of (5.50) reads(III) 1 = 1 2 R 2 z (ρ(2f ′ δκ λ ) * γ ǫ )(z) • (∇G * γ ǫ * ρ) (z) dz -8πκ λ := (III) 11 -8πκ λ .We now control (III) 11 using the HLS and Young inequalities, see (4.6) to obtain|(III) 11 | ≤ 1 ×R 2 |z| |(ρ(2f ′ δκ λ ) * γ ǫ )(z)| 1 |x -z| |(γ ǫ * ρ)(x)| dz dx ≤ C HLS 4π |z| |(ρ|2f ′ δκ λ | * γ ǫ ) 4/3 ρ * γ ǫ 4/3 ≤ C HLS 4π (|x| ρ|2f ′ δκ λ |) * γ ǫ 4/3 + (ρ|2f ′ δκ λ |) * (|x| γ ǫ ) 4/3 ρ * γ ǫ 4/2C HLS |x| ρ|2f ′ δκ λ | 1 γ ǫ 4/3 + ρ|2f ′ δκ λ | 1 |x| γ ǫ 4/3 γ ǫ 4/3 , from which (III) 1 ≤ -8πκ λ + C ǫ |2f ′ δκ λ | (1 + |x|)ρ 1 .

R 2 ×R 2 ( 3 .√ ǫ γ 4 / 3 .2λ |x| γ 4 /3 γ 4/ 3 =∞N =3 N -1≤|x|≤N |x| ρ dx ≤ 1 2 √

 2343432 ρ * |z| γ ǫ )(z) 1 |z -y| (γ ǫ * ρ)(y) dy dz ≤ C HLS 4 √ 2λ π ρ * (|z| γ ǫ ) 4/3 γ ǫ * ρ 4/By the Young inequality, and a direct calculation, ρ * (|z| γ ǫ ) 4/3 ≤ 8π |x| γ ǫ 4/3 = 8π √ ǫ |x| γ 4/3 . and in the same way γ ǫ * ρ 4/3 ≤ 8π γ ǫ 4/3 = 8π 1 The positive and negative powers of ǫ cancel, and using (5.51), we conclude(III) ≤ -8πκ λ + C ǫ |2f ′ δκ λ | (1 + |x|)ρ 1 + 16π C HLS √ -8πκ λ + C ǫ |2f ′ δκ λ | (1 + |x|)ρ 1 + 8π J γ √ 2λ .(5.52)Let us estimate this third term in a different way that will be useful later on. Using again the Young inequality, but this time eliminating γ instead of ρ, i.e. γ ǫ * ρ 4/3 ≤ ρ 4/3 , we getρ * (|z| γ ǫ ) 4/3 γ ǫ * ρ 4/3 ≤ 64π 2 √ ǫ |x| γ 4/3 ρ 4/3 .As a consequence, we get this other control on (III) by(III) ≤ -8πκ λ + C ǫ |2f ′ δκ λ | (1 + |x|)ρ 1 + 16π C HLS √ 2λ √ ǫ |x| γ 4/3 ρ 4/3 .(5.53)Control of (IV): By integrating by parts for any R > 0, we have|x|≤R x f ′ δ (|x| 2 ) • ∇ρ(x) dx = |x|=R ρ(x) f ′ δ (|x| 2 ) x • n dσ -|x|≤R ∇ • [x f ′ δ (|x| 2 )] ρ(x) dx ,where n denotes the outward normal to the disk D R . Taking into account that 2λ R 2 |x| ρ dx < ∞

  for some m with nk ≤ m ≤ n. We now apply Lemma 4.16, in which the constant A 1 and A 2 in (5.20) are defined, to conclude that

											1/4 , we obtain
	ρ m τ	p p ≤ B 4p	4πC ρ0 aγ 2	+	C CCD γ 2	2(p-1)	8π	1/4	.	(5.19)
	Thus we have an a-priori bound on ρ m τ	p p

  1/p , we have the same type of bound on ρ , this bound immediately extends to ρ τ (t) for all t ≥ a.

	the displacement convexity of		(n) τ	p uniformly for all such that nτ ≥ a. By
	ρ →	R 2	ρ p (x) dx
	for p > 1

  Recall the first moment is controlled by H λ in Lemma 1.11, and that H λ [ρ τ (t)] ≤ C ρ0 . Also, Theorem 5.7 give us a bound on ρτ (t) 3 uniformly in t ∈ [a, b] for all sufficiently small τ . Thus, there is a constant C depending only on a and b so that for all sufficiently small τ , Even more simply, by Theorem 5.7 we have a constant C depending only on a and b so that for all sufficiently small τ , such that

	Step 1.-Uniform integrability for ρ2 τ (t, x): First, note that for each τ , ρ2 τ =	ρ1/2 τ	ρ3/2 τ . Therefore, by the Cauchy-
	Schwartz inequality
								1/2	1/2
	R 2	|x| 1/2 ρ2 τ (t, x) dx ≤		R 2	|x|ρ τ (t, x) dx	R 2	ρ3 τ (t, x) dx	.
				a	b	R 2	|x| 1/2 ρ2 τ (t, x) dx dt ≤ C .
		b						b
	a	R 2	ρ2 τ (t, x) dx dt ≤ C	and	a	R 2	ρ3 τ (t, x) dx dt ≤ C .
							|g(t -c, x) -g(t, x)| 2 dx dt ≤ ǫ .	(5.25)

  Let us deal first with the second boundary term. By(4.5) in Lemma 4.2, we have 4 √ δ |x|=R |∇c ǫ | dσ ≤ 8π √ δ 8C HLS γ 4/3 |x|γ 4/3 + 4 +

				:= 16π	√ δ(J	C HLS 2πǫ	γ 2 4/3 |x|ρ 1	(5.47)
	2	|x|≤R ∇	ρ + 2δ √ ρ + δ	• ∇c ǫ dx = 2 + 2 |x|≤R |x|=R √ ρ + 2δ ρ + δ ρ + 2δ √ ρ + δ	(γ ǫ * ρ * γ ǫ ) dx ∇c ǫ • n dσ
					≤ 2
					ρ + 2δ √ ρ + δ ρ (γ ǫ * ρ * γ ǫ ) dx + 32 π |∇c ǫ | dσ |x|=R √ |x|≤R + 2 ≤ 2 |x|=R ( √ ρ + 2 √ δ) |∇c ǫ | dσ ,	√ δ	(5.46)
	where we used twice the estimate		ρ + 2δ √ ρ + δ	≤	√ ρ + 2	√ δ .

|x|≤R √ ρ (γ ǫ * ρ * γ ǫ ) dx + 4 √ δ |x|≤R γ ǫ * ρ * γ ǫ dx + 2 γ + 1 + Ĉǫ ) ,

Acknowledgments

All authors thank CRM-Barcelona where this work was started during the visit of the first and second authors in 2007. We also thank I.P.A.M. where all three authors were guests at various times in 2008 and 2009, and finally the University of Toulouse Paul Sabatier where the second and third authors were guests in 2008, 2009 and 2010, and where work was continued and concluded. Finally, we thank the referee for helpful comments.

 [START_REF] Alikakos | Lp bounds of solutions of reaction-diffusion equations[END_REF]Work partially supported by the project EVaMEF ANR-09-JCJC-0096-01. 2 Work partially supported by U.S. National Science Foundation grant DMS 0901632. 3 Work partially supported by the projects MTM2011-27739-C04-02. DGI-MCI (Spain) and 2009-SGR-345 from AGAUR-Generalitat de Catalunya. c 2010 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

we have lim

(5.54)

The desired estimates are obtained by putting together estimates (5.49), (5.52), (5.53) and (5.54).