Infinite Time Aggregation for the Critical Patlak-Keller-Segel model in R 2

Transcription

1 Infinite Time Aggregation for the Critical Patlak-Keller-Segel moel in ADRIEN BLANCHET CRM (Centre e Recerca Matemàtica), Universitat Autònoma e Barcelona, E Bellaterra, Spain. blanchet@ceremae.auphine.fr, Internet: blanchet/ JOSÉ A. CARRILLO ICREA (Institució Catalana e Recerca i Estuis Avançats) an Departament e Matemàtiques, Universitat Autònoma e Barcelona, E Bellaterra, Spain. carrillo@mat.uab.es, Internet: carrillo/ AND NADER MASMOUDI Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, NY 10012, U.S.A. masmoui@cims.nyu.eu, Internet: Abstract We analyze the two-imensional parabolic-elliptic Patlak-Keller-Segel moel in the whole Eucliean space. Uner the hypotheses of integrable initial ata with finite secon moment an entropy, we first show local in time existence for any mass of free-energy solutions, namely weak solutions with some free energy estimates. We also prove that the solution exists as long as the entropy is controlle from above. The main result of the paper is to show the global existence of free-energy solutions with initial ata as before for the critical mass 8π/χ. Actually, we prove that solutions blow-up as a elta Dirac at the center of mass when t keeping constant their secon moment at any time. Furthermore, all moments larger than 2 blow-up as t if initially boune. c 2000 Wiley Perioicals, Inc. Contents 1. Introuction 2 2. Characterization of the Maximal Time of Existence 7 3. Asymptotic behaviour in the case χ M = 8π 16 Bibliography 27 Communications on Pure an Applie Mathematics, Vol. 000, (2000) c 2000 Wiley Perioicals, Inc.

2 2 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI 1 Introuction The Patlak-Keller-Segel (PKS) moel escribes the collective motion of cells which are attracte by a self-emitte chemical substance. A moel organism for this type of behavior is the ictyostelium iscoieum which segregates cyclic aenosine monophosphate attracting themselves in starvation conitions. It is observe that after the appearance of a suitable number of mixamoebae, they aggregate to form a multi-cellular organism calle pseuo-plasmoi. This ruimentary phenomenon coul be a step towar the unerstaning of cell ifferentiation. There have been many mathematical moelling sources for chemotaxis. Historically, the first mathematical moel was introuce in 1953 by C. S. Patlak in [47] an E. F. Keller an L. A. Segel in [32] in Here, we focus on the minimal Patlak-Keller-Segel moel-type introuce by V. Nanjuniah in [44], which is: (1) n t (x,t) = n(x,t) χ (n(x,t) c(x,t)) x R2, t > 0, c(x,t) = n(x,t) x, t > 0, n(x,t = 0) = n 0 0 x. Here (x,t) n(x,t) represents the cell ensity, an (x,t) c(x,t) is the concentration of chemo-attractant. The first equation takes into account that the motion of cells is riven by the steepest increase in the concentration of chemo-attractant while following a Brownian motion ue to external interactions. In fact, this equation is a stanar rift-iffusion equation obtaine from the unerlying stochastic ynamical system. The secon equation takes into account that cells are proucing themselves the chemo-attractant while this is iffusing onto the environment. In fact, this secon equation has an aitional time erivative of c that in this moel has been neglecte assuming that the relaxation of the concentration is much quicker than the time scale of cell movement. The constant χ > 0 is the sensitivity of the bacteria to the chemo-attractant. Mathematically, it measures the non-linearity of the system. Since the total mass of cells is assume to be preserve, it is usual to impose no-flux bounary conitions if these equations were pose in boune omains. Here, we are not intereste in bounary effects an for this reason we are going to consier the system in the full space without bounary conitions. The imension 2 is critical when we consier the problem in L 1 because in the Green kernel associate with c(x,t) = n(x,t) has a logarithmic singularity. In R, for > 2, the critical space is L /2 (R ), see [20, 21]. In [19], S. Chilress an J. K. Percus conjecture that the aggregation or chemotactic collapse, if any, shoul procee by the formation of a elta Dirac at the center of mass of cell ensity. Concerning the unerstaning of this phenomena we refer to the seminal work of M. A. Herrero an J. J. L. Velázquez, [27] an subsequent works of the last author [54, 55, 56]. Numerical evience of this collapse has also

3 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 3 been reporte in [39]. The literature on these moels is huge an it is out of the scope of this paper to give exhaustive references. For further bibliography on the Patlak-Keller-Segel system an relate moels we refer the intereste reaer to the surveys [28, 29, 48]. Since the solution of the Poisson equation c = n is given up to a harmonic function, we will efine the concentration of the chemo-attractant irectly by c(x,t) = 1 2π log x y n(y,t) y. Hence, for n(t) M + ( ), the space of positive boune total variation measures, c(t) L 2, ( ), where L 2, ( ) is the Lorenz space given by f L 2, ( ) if an only if t > 0, meas{ f (x) > t} C t 2. Thanks to this remark the Patlak-Keller-Segel system (1) can be viewe as a parabolic equation with a non-local interaction term. Initial ata are assume to verify (2) (1 + x 2 )n 0 L 1 +( ) an n 0 logn 0 L 1 ( ). If we assume the existence of smooth fast-ecaying at infinity non-negative L 1 ([0,T ] ) solutions n, for all T > 0, to the Patlak-Keller-Segel system (1) with initial ata satisfying assumptions (2), then the solutions satisfy the formal conservations of the total mass of the system M := n 0 (x) x = n(x,t) x an its center of mass M 1 := xn 0 (x) x = xn(x,t) x. Due to the conservation of the center of mass an the translational invariance of (1), we may assume M 1 = 0 without loss of generality. There is a competition between the tenency of cells to sprea all over by iffusion an the tenency to aggregate because of the rift inuce by the chemoattractivity. The balance between these two mechanisms happens precisely at the critical mass χ M = 8π. In fact, in the case χ M > 8π, uner assumptions (2) on n 0, it is easy to see, using moment estimates, that global classical solutions to the Patlak-Keller-Segel system (1) cannot exist an that they blow-up in finite time. J. Dolbeault an B. Perthame announce in [22] that if χ M < 8π there is global existence of solution for the Patlak-Keller-Segel system (1) in a weak sense. This result was further complete an improve in [13] where the existence of free-energy solutions is prove uner the hypothesis (2) for the subcritical case χ M < 8π. Furthermore, the asymptotic behavior in the subcritical case is shown to be given by unique selfsimilar profiles of the system. We also refer to [43] for raially symmetric results concerning self-similar behavior.

4 4 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI The critical case χ M = 8π has a family of explicit stationary solutions [54] of the form (3) n b (x) = 8b χ(b + x 2 ) 2 with b > 0. All of these stationary solutions have critical mass an infinite secon moment an they playe an important role in the matche-asymptotic expansions one by J.J.L. Velázquez in [54] to show the existence of blowing-up solutions with elta Dirac formation for the supercritical case χ M > 8π an subsequent improvements [55, 56]. In the case χ M = 8π, P. Biler, G. Karch, P. Laurençot an T. Nazieja [9] prove the existence of global raially symmetric solutions to (1) for initial ata with finite or infinite secon moment. They also show that all the stationary solutions n b given in (3) have an attraction region for raial symmetric solutions with initial ata of infinite secon moment in a suitable sense [9, Proposition 3.6]. However, the asymptotic behavior of raially symmetric solutions with initial ata of finite secon moment was left open an the raial stationary solution given by the 8π χ δ 0 was not prove to attract finite secon moment raial solutions. The main aim of this paper is to show the existence of global in time freeenergy solutions, efine below, for the Patlak-Keller-Segel system (1) with critical mass χ M = 8π an initial ata satisfying assumptions (2). Moreover, we will show that these solutions keep their secon moment constant at any subsequence time. Furthermore, they blow-up in infinite time converging towars a elta Dirac istribution at the center of mass. The main tool for the proof of global existence is the free energy functional: t F [n](t) := n(x,t)logn(x,t) x χ n(x,t)c(x,t) x. 2 The free energy functional has a long history in kinetic moelling an its iffusive approximations, see [1] for a moern perspective on the use of free energy or entropy functionals in nonlinear iffusion an kinetic moels. In fact, it is important to point out that the PKS moel has also been consiere for a long time by the kinetic theory community as a iffusive limit of gravitational kinetic moels with the name of gravitational rift-iffusion-poisson or Smoluchowsky-Poisson system, see [3, 46, 10, 57, 11, 23] for instance. Free-energy functionals were introuce for chemotactic moels by T. Nagai, T. Senba an K. Yoshia in [41], by P. Biler in [7] an by H. Gajewski an K. Zacharias in [25]. We also refer to [50, 53] an references therein. Moreover, the free energy functional an relate functional inequalities playe an essential role in improving the range of existence of global in time solutions up to the critical mass in [22, 13] for the PKS system (1), for nonlinear iffusion an chemotactic moels in [33, 14] an for relate moels [15]. Let us finally mention that the PKS system has been erive from kinetic moels of chemotaxis [52, 45, 17].

5 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 5 We will work with improve weak solutions to the PKS system (1). We first remin that the notion of weak solutions n in the space C 0( [0,T );Lweak 1 (R2 ) ), with fixe T > 0, using the symmetry in x, y for the concentration graient was introuce in [49]. This notion is capable of hanling measure solutions. We shall say that n C 0( [0,T );Lweak 1 (R2 ) ) is a weak solution to the PKS system (1) if for all test functions ψ D( ), t ψ(x)n(x,t)x = ψ(x)n(x,t)x χ [ ψ(x) ψ(y)] 4π x y x y 2 n(x,t)n(y,t)xy hols in the istributional sense in (0,T ) an n(0) = n 0. We now efine the concept of free-energy solution that we consier in this paper: Definition 1.1 (Free-energy solution). Given T > 0, the function n is a free-energy solution to the Patlak-Keller-Segel system (1) with initial ata n 0 on [0,T ] if (1 + x 2 + logn )n L ((0,T ),L 1 ( )), n satisfies (1) in the above weak sense an t F [n](t) + n(x,s) (logn(x,s)) χ c(x,s) 2 xs F [n 0 ] 0 for almost every t (0,T ). The concept of free-energy solutions is necessary to apply entropy methos to analyze the asymptotic behaviour of solutions. Our first result shows local in time existence of free-energy solutions for all masses, an characterizes the maximal time of existence of free-energy solutions. This result extens the known existence theory of free-energy solutions obtaine in [13] for subcritical mass. Uniqueness of free energy solutions is an open issue in this setting. Proposition 1.2 (Maximal Free-energy Solutions). Uner assumptions (2) on the initial ata n 0, there exists a maximal time T > 0 of existence of a free-energy solution to the PKS system (1). Moreover, if T < then lim t T n(x,t)logn(x,t) x = +. The main result of this paper escribes the behaviour of the solution to the Patlak-Keller-Segel system (1) with critical mass M = 8π/χ. Theorem 1.3 (Infinite Time Aggregation). If χ M = 8π, uner assumptions (2) on the initial ata n 0, there exists a global in time non-negative free-energy solution of the Patlak-Keller-Segel system (1) with initial ata n 0. Moreover if {t p } p N as p, then t p n(x,t p ) converges to a elta Dirac of mass 8π/χ concentrate at the center of mass of the initial ata weakly-* as measures as p.

6 6 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI As mentione before, the main tool is the free energy F [n](t) which is relate to its time erivative, the Fisher information, in the following way: consier a non-negative solution n C 0 ([0,T ),L 1 ( )) of the Patlak-Keller-Segel system (1) such that n(1 + x 2 ), nlogn are boune in L ((0,T ),L 1 ( )), n L 1 ((0,T ),L 2 ( )) an c L ((0,T ) ). Then (4) t F [n](t) = n(x,t) logn(x,t) χ c(x,t) 2 x. We will make a funamental use of the Logarithmic Hary-Littlewoo-Sobolev inequality. Proposition 1.4 (Logarithmic Hary-Littlewoo-Sobolev inequality). [16, 4] Let f be a non-negative function in L 1 ( ) such that f log f an f log(1 + x 2 ) belong to L 1 ( ). If R2 f x = M, then (5) f log f x + 2 f (x) f (y)log x y xy C(M), M with C(M) := M(1 + logπ logm). Let us point out that the Logarithmic Hary-Littlewoo-Sobolev inequality remains true in boune omains just by multiplying f by the corresponing characteristic function of the omain. Note that, in the case χ M = 8π, as a irect consequence of the Logarithmic Hary-Littlewoo-Sobolev inequality, the free energy F [n](t) is boune from below. The Logarithmic Hary-Littlewoo-Sobolev inequality will be the key ingreient to characterize the blow-up profile, either at finite or infinite time, of solutions as the elta Dirac at the center of mass. Another crucial point in our arguments is to show that the secon moment of free energy solutions is kept constant at all times. This is one by arguments base on De la Vallée-Poussin type arguments [36, 34] commonly use in kinetic theory. This fact allows us to show that solutions cannot blow-up in finite time as a elta Dirac at the center of mass. On the other han, this information is also use to select the only possible stationary state to converge since all stationary states in (3), extremals of the Logarithmic Hary-Littlewoo- Sobolev inequality, o not posses finite secon moments. Using the evolution of the secon moment to obtain information in this kin of systems goes back to [40] being wiely use since then. Finally, let us make aitional comments on the literature. Here, we show that all solutions for the critical case with finite secon moments globally exist blowing-up as a elta Dirac at infinite time. It is an open question if the other equilibria in (3) attract solutions with initial infinite secon moment an what are the conitions that may select one of them at infinity. The stability result for equilibria of the form (3) in the raial case proven in [9] is an interesting step in this irection. Finally, let us point out that results of blowing-up at infinite time have been reporte in similar systems in [30].

7 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 7 Section 2 is evote to some remarks on the necessary conition of existence of free-energy solution to the Patlak-Keller-Segel system (1). We prove the local in time existence of free-energy solution to the Patlak-Keller-Segel system (1) for any mass an the characterization of the maximal time of existence, see Proposition 1.2. Section 3 is evote to the critical mass case χ M = 8π. We prove first that if the blow-up occurs the blow-up profile is a elta Dirac of mass 8 π/χ concentrate at the center of mass, then that blow-up cannot happen in finite time an finally that blow-up oes happen in infinite time as a elta Dirac of mass 8π/χ concentrate at the center of mass. 2 Characterization of the Maximal Time of Existence First, we introuce a regularize system an prove that it is enough to show that the entropy of the regularize cell ensity n ε, S [n ε ] := n ε (x,t) logn ε (x,t)x is boune from above uniformly in ε on [0,T ) to prove that there exists a freeenergy solution to the Patlak-Keller-Segel system (1) in [0, T ). Section (2.3) is evote to the proof of the existence of maximal free-energy solutions to the Patlak- Keller-Segel system (1) for any mass (Proposition 1.2). 2.1 Regularize system We introuce the truncate convolution kernel K ε to be such that ( ) z K ε (z) := K 1 1 ε 2π logε where K 1 is a raial monotone non-increasing smooth function satisfying K 1 ( z ) = 1 log z if z 2, 2π K 1 ( z ) = 0 if z 1 2. Moreover, we assume that (1) K 1 (z) 1 2π z, K 1 (z) 1 2π log z an K 1 (z) = φ(z) for any z, with φ(z) boune an integrable since φ = 0 outsie the ball of raius 2. Since K ε (z) = K 1 ( z /ε), we also have (2) K ε (z) 1 2π z z.

8 8 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI We consier the following regularize version of (1) n ε t (x,t) = nε (x,t) χ (n ε (x,t) c ε (x,t)) (3) c ε (x,t) = (K ε n ε )(x,t) n0 ε(x) := min{n 0,ε 1 }(x) x, t > 0, written in the istribution sense. For any fixe positive ε, uner assumptions (2) on the initial ata, it is prove in [13, Proposition 2.8] that there exists a global solution in L 2 ([0,T ],H 1 ( )) C([0,T ],L 2 ( )) for the regularize version of the Patlak-Keller-Segel system (3). The proof of this result uses the Schauer s fixe point theorem an the Lions-Aubin s compactness metho [2, 38, 51]. 2.2 On the Local in time Existence Proof An attentive reaing of the proof of Theorem 1.1 in [13] permits to see that the existence of free-energy solution is vali as long as S [n ε ](t) an the secon momentum are uniformly boune from above in ε an t [0,T ). Since the proof follows the same lines as in [13, Theorem 1.1], we will just sketch the proof by unerlining the most relevant aspects for reaer s sake. Proposition 2.1 (Criterion for Local Existence). Let n ε be the solution of (3) an T > 0. If S [n ε ](t) is boune from above uniformly in ε an t (0,T ), then the cluster points of {n ε } ε>0, in a suitable topology, are non-negative free-energy solutions of the Patlak-Keller-Segel system (1) with initial ata n 0 on [0,T ). Proof. We will make use of the Gagliaro-Nirenberg-Sobolev inequality: (4) u 2 L p ( ) C GNS(p) u 2 4/p L 2 ( ) u 4/p L 2 ( ) u H 1 ( ), p [2, ). In a first step we show a priori estimates on the solution n ε to the regularize version of the Patlak-Keller-Segel system (3). These estimates shall allow us to prove in the secon step the existence of solutions in the istribution sense to the Patlak-Keller-Segel system (1) by passing to the ε 0 limit. To prove that we get a free-energy solution as efine in Definition 1.1, we nee to prove further regularity in the thir step. Step 1.- A priori estimates on n ε an c ε. This step correspons to Lemma 2.11 in [13]. We start by estimating the secon moment of solutions inepenently of ε. Since K ε is a raial non-increasing function satisfying (2), we have x 2 n ε (x,t)x=4m+2χ n ε (x,t)n ε (y,t)(x K ε (x y))xy t R 2 (5) =4M+χ n ε (x,t)n ε (y,t)((x y) K ε (x y))xy 4M, from which (1 + x 2 )n ε L ((0,T ),L 1 ( )) uniformly in ε.

9 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 9 The following usual notation will be aopte. For any real-value function f = f (x), we efine its positive an negative part as f + (x) = max{ f (x),0} an f (x) = ( f ) + (x), so that f = f + f. Lemma 2.2 (Control of the Negative Part of the Entropy). For any g such that (1 + x 2 )g L+(R 1 2 ), we have glog g L 1 ( ) an g(x)log g(x)x 1 x 2 g(x)x + log(2π) g(x)x e. Proof. Let u := g1l {g 1} an m = ux M = R2 gx. Then ( u logu x 2) x = U logu µ mlog(2π) where U := u/µ, µ(x) = µ(x)x an µ(x) = (2π) 1 e x 2 /2. By Jensen s inequality, ( ) ( ) U logu µ U µ log U µ = mlogm, R 2 m ) ulogux mlog( 1 x 2 ux 1 2π 2 e M log(2π) 1 x 2 ux 2 completing the proof. The previous lemma implies that ( n ε (x,t) logn ε (x,t) x n ε (x,t) logn ε (x,t)+ x 2) x+2log(2π)m+ 2 e, an thus, the entropy term verifies n ε logn ε L ((0,T ),L 1 ( )) uniformly in ε whenever (1+ x 2 )n ε L ((0,T ),L 1 ( )) uniformly in ε an S [n ε ](t) is boune from above uniformly in ε an t (0, T ). Using that n ε logn ε L ((0,T ),L 1 ( )), that c ε (x,t) = (K ε n ε )(x,t) an applying Young inequality, we euce that c ε (x,t) C +C log(1+ x ) uniformly in ε. Inee, we have c ε 1 1 (6) n(y) log n(y)log x y y (7) (8) (9) 2π [ 1 2π + 1 2π x y 1 x y y + 1 2π n(y)log(1 + n(y)) + x y 1 x y 1 C +C log(1 + x ). x y 1 C x y ] y ( n(y) log(1 + x ) + log(1 + y ) As a consequence, we euce that the map (10) t n ε (x,t)c ε (x,t)x is boune uniformly in ε in L (0,T ). ) y

10 10 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI The main a priori estimate is the L 2 ((0,T ) ) estimate on n ε c ε. First, we prove that n ε in boune in L 2 ((0,T ) ). For the sake of simplicity we show only the formal computations for the solutions of the Patlak-Keller-Segel system (1), referring to [13] for etails. The main ifficulty, when proving the same estimate for the solutions to the regularize Patlak-Keller-Segel problem (3), is that we o not have c = n. Nevertheless, K ε converges to a elta Dirac as ε 0, an thus, the result remains true for ε small enough up to technical computations. Given K > 1, by the Gagliaro-Nirenberg-Sobolev inequality (4): (n K) 2 + x CGNS 2 2 (n K)+ x (n K) + x. The left han sie can be mae as small as esire by taking K large enough an using (n K) + x 1 (n K) + logn x C =: η(k), logk logk where C is the boun on n logn L ((0,T ),L 1 ( )) obtaine above. We ifferentiate in time S [n](t). By using an integration by parts, the Gagliaro-Nirenberg- Sobolev s inequality (4) an the equation for c, we obtain: t S [n](t) = 4 n(t) 2 x + χ n(t) 2 x R 2 4 n(t) 2 x + 2χMK + 2χ (n(t) K) 2 + x ( 4 + 2χ η(k)c 2 GNS) n(t) 2 x + 2χMK. The factor ( 4 + 2χ η(k)cgns) 2 can be mae non-positive for K large enough an therefore n is boune in L 2 ((0,T ) ). This iea will be use in the forthcoming proof of Proposition 1.2 an the intereste reaer can refer to it for full etails applie to the regularize problem. Now, we assume that we have erive a uniform in ε estimate for n ε in L 2 ((0,T ) ). As a consequence of the L 2 ((0,T ) )-estimate on n ε an of the computation t S [nε ](t) = 4 then the map (11) t n ε c ε x n ε 2 x + χ n ε ( c ε )x, is boune uniformly in ε in L 1 (0,T ). A computation shows that 1 n ε c ε x = n ε c ε x + χ n ε c ε 2 x, 2 t

11 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 11 proving finally that n ε c ε L 2 ((0,T ) ) where the bouneness of the maps (10) an (11) were use. In this way, we have obtaine estimates on the two terms appearing in the issipation of the free energy in (4). Step 2.- Passing to the limit. We will use the Aubin-Lions compactness metho, (see [38], Ch. IV, 4 an [2], an [51] for more recent references). A simple statement goes as follows: Take T > 0, p (1, ) an let ( f n ) n N be a boune sequence of functions in L p (0,T ;H) where H is a Banach space. If ( f n ) n N is boune in L p (0,T ;V ), where V is compactly imbee in H an f n / t is boune in L p (0,T ;W) uniformly with respect to n N where H is imbee in W, then ( f n ) n N is relatively compact in L p (0,T ;H). Boun on n ε L 2: As n ε logn ε L ((0,T ),L 1 ( )) uniformly in ε the first equation in (3) has the hyper-contractivity property [13, Theorem 3.5]. It means that for any p (1, ), there exists a continuous function h p on (0,T ) such that for almost any t (0,T ), n ε (,t) L p ( ) h p (t). Hence n ε L ((δ,t ),L p ( )), p (1, ), for any δ (0,T ). Boun on n ε L 2: Recall the Hary-Littlewoo-Sobolev inequality: For all f L p (R ), g L q (R ), 1 < p, q < such that 1 p + 1 q + λ = 2 an 0 < λ <, there exists a constant C = C(p,q,λ) > 0 such that R R 1 x y f (x)g(y) xy λ C f L p (R ) g L q (R ). As a consequence, for any (q, p) (2,+ ) (1,2) with 1 q = 1 p 1 2, there exists a constant C > 0 such that (12) c ε (t) L q ( ) C n ε (t) L p ( ). An using Höler s inequality, we can write n ε (t) c ε (t) L 2 ( ) n ε (t) L q/(q 1) ( ) cε (t) L q ( ) C n ε (t) L q/(q 1) ( ) nε (t) L p ( ). Hence n ε c ε is boune in L ((δ,t ),L 2 ( )). Now the following computation n ε 2 x = 2 n ε 2 x + 2χ n ε ( n ε c ε ) x t shows that X := n ε L 2 ((δ,t ) ) satisfies the estimate 2X 2 2 χ n ε c ε L ((δ,t ),L 2 ( )) X 2 n ε 2 L ((δ,t ),L 2 ( )). This implies that n ε is boune in L 2 ((δ,t ) ).

12 12 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI V is relatively compact in H: We take H = L 2 ( ), V = H 1 ( ) {n x n 2 L 1 ( )} an W = H 1 ( ). Using Höler s inequality, we get ( ) 1/2 ( ) 1/2 x n ε 2 (x,t)x x 2 n ε (x,t)x n ε 3 (x,t)x. This boun allows to consier only compact sets, on which compactness hols by Sobolev s imbeings. Aubin-Lions Lemma applies to prove that n ε is relatively compact in L 2 ((δ,t ) )). Let us enote n the limit of the sub-sequence {n ε k} k. Note that the L 4/3 ((δ,t ) )-boun of n ε an (12) with q = 4 implies that n ε k c ε k converges to n c in the istribution sense. Step 3.- Free energy estimates. By convexity of n n 2 x, see [5, 6], an weak semi-continuity we have n 2 xt liminf n ε k 2 xt, (δ,t ) k (δ,t ) R 2 n c 2 xt liminf n ε k c ε k 2 xt. (δ,t ) k (δ,t ) Moreover, it can be prove as in [13, Lemma 4.6] that the regularize entropy converges to the limiting entropy for almost every t > 0, i.e., (13) S [n ε ](t) S [n](t) as ε 0. This proves the free energy estimate using the strong convergence of {n ε k} in L 2 ((δ,t ) ) an n (logn) χ c 2 xt = 4 n 2 xt (δ,t ) (δ,t ) + χ (δ,t 2 n c 2 xt 2χ n 2 xt. ) (δ,t ) This ens the proof of Proposition 2.1. Full etails of these final arguments are given in [13, Corollary 3.6]. 2.3 Proof of Proposition 1.2 In the next lemma, we characterize the maximal time of existence of free-energy solutions. Lemma 2.3 (Maximal Free-energy Solutions). Uner the assumptions (2) on the initial ata, if there exists a time t 0 such that S [n ε ](t) is boune uniformly in ε an 0 t t, then there is τ > 0 inepenent of ε for which S [n ε ](t) is uniformly boune in ε for t [t,t + τ). Moreover, the free-energy solution of the Patlak-Keller-Segel system (1) constructe above can be extene up to t + τ. Here, τ epens only on the uniform estimate on [1 + x 2 + logn ε (t ) ]n ε (t ) in L 1 ( ).

13 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 13 Proof. Let n ε be the global solution of the regularise version of the Patlak-Keller- Segel system (3). We compute t S [nε ](t) = 4 n ε (x,t) 2 x + χ [(I) + (II) + (III)] with (I) := {n ε K} n ε (x,t) [K ε n ε (x,t)], (II) := n ε (x,t) [K ε n ε (x,t)] (III) an (III) = R n ε (x,t) 2, where the factor R will be chosen below. Let us remin (1) that the regularize kernel verifies 1 ( ) ε 2 φ = K ε, ε with φ positive, boune an integrable with integral R fixing the parameter in the efinition of (III). The term (I) is easy to estimate using ( ) 1 x y (I) K ε 2 φ n ε (y,t) yx M RK. ε {n ε K} We have by using that the integral of φ is R that (II) n ε (x,t) [ K ε n ε (x,t)] x R n ε (x,t) n ε (x,t) 2 x [n ε (x εy,t) n ε (x,t)] φ(y)yx R 2 [ n ε (x,t) n ε (x εy,t) ] n ε (x,t) φ(y) [ n ε (x εy,t) n ε (x,t) + 2 n ε (x,t)] φ(y)yx. Using the Cauchy-Schwarz inequality an (a + 2b) 2 2a 2 + 8b 2 we obtain (II) [ [ n ε (x,t) φ L ( ) 1 [ By the Poincaré s inequality we have 2 y 2 n ε (x εy,t) n ε (x,t) 2 y ] 1/2 2 n ε (x εy,t) ] n ε (x,t) 2 1/2 + 8n ε (x,t) φ(y)y] x. (II) φ 1/2 L ( ) C P n ε (x,t) L 2 ( ) n ε (x,t) 2[ φ 1/2 L ( ) C P n ε (x,t) L 2 ( ) + 2 ] n ε (x,t) x,

14 14 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI which can be written as (II) 2 φ L ( )CP 2 n ε (x,t) 2 L 2 ( ) n ε (x,t) x + 2 3/2 φ 1/2 L ( ) C P n ε (x,t) L 2 ( ) Applying the Cauchy-Schwarz inequality twice, we obtain (II) 2 φ L ( )CP 2 n ε (x,t) 2 L 2 ( ) n ε (x,t) x φ 1 2 L ( ) C P n ε (x,t) L 2 ( ) ( 10 φ L ( )C 2 P n ε (x,t) 2 L 2 ( ) 1( n (x,t)x) 2 ε [n ε (x,t)] 3/2 x. [n ε (x,t)] 2 x n ε (x,t) x+ [n ε (x,t)] 2 x an by the Gagliaro-Nirenberg-Sobolev s inequality (4), we get [n ε (x,t)] 2 x 2KM +C 2 2 {n ε GNS (n K)+ x (n K) + x. >K} Hence, with (II) + (III) 2(1 + R)KM + B n ε (x,t) 2 L 2 ( ) B := 10 φ L ( )C P 2 + (1 + R)C GNS 2. Combining the previous results we have t S [nε ](t) 4 n ε (x,t) 2 x + (2 + 3R)χ M K + χ B n ε (x,t) 2 L 2 ( ) n ε (x,t) x n ε (x,t) x. Given K > 1, using Lemma 2.2 an (5), we have n ε (x,t)x 1 n ε (x,t) logn ε (x,t)x logk 1 n ε (x,t) logn ε (x,t) x 1 ( S [n ε ](t) + C(t) ) logk logk with C(t) := x 2 n ε (x,t ) x + 4M(t t ) + 2log(2π)M + 2e 1 := 4M(t t ) + C. Hence we actually prove that t S [nε ](t) (2 + 3R)χ M K + A(t) n ε (x,t) 2 x ) 1 2

15 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 15 with [ A(t) := 4 + χ B ( S [n ε ](t) + C(t) )]. logk At time t = t, it implies that t S [nε ](t) t=t (2 + 3R)χ M K + A(t ) n ε (x,t ) 2 x. By hypothesis, there exists S > 0 such that S [n ε ](t ) S since S [n ε ](t ) is boune uniformly in ε. We can thus choose K large enough such that the coefficient A(t ) is non-positive. It implies that there exists τ ε > 0 such that S [n ε ](t) S + M K (t t ) in [t,t + τ ε ]. But in this interval t [t,t + τ ε ], [ A(t) 4 + χ B ( S + M K (t t ) + C(t) )] logk is non-positive in [t,t + τ] with τ 1 MK + 4M [ 4 logk χ B ] S C, which is inepenent from ε an positive for K large enough. Therefore, the previous proceure can be continue in the time interval [t,t + τ] showing finally that [t,t + τ] [t,t + τ ε ]. We finally make use of Proposition 2.1 to conclue the existence of a free-energy solution of the PKS system (1) in [t,t + τ]. Proof of Proposition (1.2). The local in time existence is a irect consequence of Lemma 2.3. Assume by contraiction that lim t T S [n](t) <, then there exists a sequence {t p } p N T such that S[n](t p ) is uniformly boune in p. Taking into account the convergence of regularize entropies (13) an Lemma 2.3, there exists τ > 0 inepenent from p an ε such that S [n ε ](t) is boune uniformly in ε for any t [0,t p + τ). By Proposition 2.1 it implies that the limit n is a nonnegative solution of the Patlak-Keller-Segel system (1) with initial ata n 0 in [0,t p + τ), for any p N, contraicting the choice of T. Let us finally point out the following conservations for free-energy solutions of the PKS system (1). Lemma 2.4 (Mass Centering an Variance Evolution). Let n be a free-energy solution on the time interval [0,T ) to system (1) with initial ata n 0 verifying assumptions (2). Then the center of mass is preserve M 1 = xn(x,t)x = xn 0 (x)x an the variance of the cell ensity verifies ( (14) x 2 n(x,t)x = x 2 n 0 (x)x + 4M 1 χm ) t 8π

16 16 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI for all t (0,T ). Proof. For any ε < 1, we can fin a C raial compactly supporte cut-off function ψ ε (x) such that 0 ψ ε 1, ψ ε = 1 for x ε 1, ψ ε = 0 for x ε an ψ ε C 2 ( ) C 1 for all ε 1. Taking ϕ ε (x) = x i ψ ε (x), for fixe i {1,2}, as test function in the weak solution concept (1.1), we obtain (15) t ϕ ε nx = ϕ ε nx χ 4π ( ϕ ε (x) ϕ ε (y)) (x y) x y 2 n(x,t)n(y,t)xy. Since ϕ ε is boune an ϕ ε is Lipschitz continuous uniformly in ε, we euce that ϕ ε (x)n(x,t)x c 2 n 0 (x) x, t where c 2 is some positive constant, from which ϕ ε (x)n(x,t)x c 1 + c 2 t, with c 1 suitable constant, an thus x i n(x,t)x <, t (0,T ). We can now pass to the limit ε 0 in the integral version of (15) using Lebesgue s ominate convergence theorem together with ϕ ε 0 an ( ϕ ε (x) ϕ ε (y)) 0 for almost every x,y in, eucing x i n(x,t)x = x i n 0 (x)x for almost every t > 0 an i {1,2}. The case of the secon moment is one analogously. Taking ϕ ε (x) = x 2 ψ ε (x) as test function in (1.1), we obtain (15). Proceeing as above, we can pass to the limit using Lebesgue s ominate convergence theorem eucing t ( x 2 n(x,t)x = x 2 n 0 (x)x + 4M 1 χm 0 8π an obtaining the esire result. ) t The previous result allows us to restrict our analysis to the case where the center of mass is zero by the translation invariance of the PKS system (1). 3 Asymptotic behaviour in the case χ M = 8π This section is evote to the proof of the main theorem (Theorem 1.3 for the critical mass χ M = 8π). In Section 3.1, we characterize the possible blow-up profile of the Patlak-Keller-Segel system (1) with critical mass. More precisely, if

17 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 17 the solution of the PKS system (1) blows-up, then it oes it like a elta Dirac of mass 8π/χ at the center of mass of the solution. In Section 3.2 we prove that the solution to the Patlak-Keller-Segel (1) cannot blow-up in finite time. As a consequence, we show the global in time existence of free-energy solutions to the PKS system (1) for the critical mass χ M = 8π. Finally, Section 3.3 is evote to emonstrate that the solution to the PKS system (1) concentrates as a elta Dirac of mass 8π/χ at the center of mass of the solution in infinite time. 3.1 How oes it blow-up? Let us first recall the proof of the upper boun on the entropy functional S [n] = n(x) logn(x)x, in the sub-critical case, namely the mass M < 8π/χ as one in [13]. By the monotonicity of the free energy (4) the quantity [ F [n](t) = (1 θ)s [n](t)+θ S [n](t) + χ ] n(x)n(y) log x y xy 4πθ is boune from above by F [n 0 ]. We choose θ = χ M 8π Hary-Littlewoo-Sobolev inequality (5) to get: If χ M < 8π, then θ < 1 an (1 θ)s [n](t) θ C(M) F [n 0 ]. an apply the Logarithmic S [n](t) F [n 0] + θ C(M). 1 θ However, in the case when χ M = 8π, the above argument oes not imply an upper boun on S [n](t) although F [n](t) is boune from above an from below. Nevertheless, a localization argument of the free energy an a smart use of the Logarithmic Hary-Littlewoo-Sobolev inequality (5) allows to show: Lemma 3.1 (Characterization of Blowing-up Profile). Uner hypotheses (2) on the initial ata, assume that T the maximum time of existence of the free-energy solution n to the Patlak-Keller-Segel system (1) with initial ata n 0 of critical mass M = 8π/χ is finite. If {t p } p N T when p, then t p n(x,t p ) converges to a elta Dirac of mass 8π/χ concentrate at the center of mass in the measure sense as p. The main ieas of the proof of Lemma 3.1 reas as follows: we assume by contraiction that the weak-* limit of t p n(x,t p ), namely n (x) is not a elta Dirac. Hence, there exists a ball B r1 in which the mass of n is some α such that 0 < α < M = 8π/χ. We apply the Logarithmic Hary-Littlewoo-Sobolev inequality (5) to balls an annulii (see Figure 3.1). By aing the corresponing terms, we can prove that a variation of the Logarithmic Hary-Littlewoo-Sobolev

18 18 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI inequality (5) hols for the solution n(x,t p ) in, from which we obtain a uniform boun on the entropy S [n](t p ). This contraicts the choice of the maximal time of existence of the free-energy solution ue to the characterization in Proposition 1.2. Proof. We first remark that the secon moment is preserve in time ue to (14) for the critical mass χ M = 8π. This together with the conservation of mass shows that the sequence of positive integrable functions n(x,t p ) is tight for the weak-* convergence as measures preserving its mass by Prokhorov s theorem [12]. Thus, there exists a weakly-* converging subsequence towars a limiting measure n M ( ) with mass M. It is obvious that the argument can be reuce to weakly-* converging subsequences, so we will o so an we will keep the same inex for the time sequence. Let us now assume by contraiction that there is no formation of a elta Dirac of mass M = 8π/χ as t T. For any t p an r > 0, we introuce the mass ensity on balls α p (r) := n(x,t p ) x B r which is a non ecreasing function on R + an let us efine analogously the mass ensity α (r) for n as: α (r) := n (x). B r Saying that the positive measure n of mass M is not Mδ 0 is equivalent to assert that lim r 0 α (r) < 8π + χ since the function α is non-ecreasing an lim r α (r) = 8π/χ. Let us remark that α is continuous for almost every r > 0. By changing the origin if necessary, using again the translation invariance of the system, we can assume without loss of generality that α is not ientically 0 in the neighbourhoo of 0. Then, there exists r > 0 for which 0 < α (r) < M an r is a point of continuity of α. Then, for small enough η > 0, there exist r 1 an r 2 with r 1 < r < r 2 such that 0 < α (r) η < α (r 1 ) α (r) α (r 2 ) < α (r) + η < 8π χ. Since the time sequence n(x,t p ) n as p, we can assume without loss of generality that 0 < α (r) η < α p (r 1 ) α p (r) α p (r 2 ) < α (r) + η < 8π χ for p large enough. As a consequence, for all p large enough, we have α 8π (r) η n(x,t p )x, B r1 χ α (r) η n(x,t p )x an B c r 2 n(x,t p )x 2η. B r2 \B r1

19 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 19 Let ρ := (1/3)(r 2 r 1 ). We introuce the annulii S 1 := B r1 +ρ \ B r1, S 2 := B c r 2 ρ \ B c r 2 = B r2 \ B r2 ρ an S 3 := (B r2 \ B r1 ) \ (S 1 S 2 ) (see Figure 3.1 below). B r2 S 2 S 1 S 3 B r1 FIGURE 3.1. Definition of the ifferent sets In the rest C will enote several constants, not necessarily positive, epening only on the mass M, the value of the initial secon moment an η. We apply the Logarithmic Hary-Littlewoo-Sobolev inequality (5) to n p (x) := n(x,t p ) on the sets B r1 +ρ, B c r 2 ρ an B r2 \ B r1 for p large enough to obtain B r1 +ρ B c r 2 ρ n p (x)x n p (x)x B r2 \B r1 n p (x)x B r1 +ρ B c r 2 ρ B r2 \B r1 n p (x) logn p (x)x+2 n p (x) logn p (x)x+2 n p (x) logn p (x)x+2 B r1 +ρ B r1 +ρ B c r 2 ρ Bc r 2 ρ B r2 \B r1 B r2 \B r1 n p (x) n p (y) log x y xy C r1,ρ n p (x) n p (y) log x y xy C r2,ρ n p (x) n p (y) log x y xy C r1,r 2 We expan nlogn = nlog + n nlog n in the first terms an isregar the negative part contribution. Using B r1 +ρ = B r1 S 1, B c r 2 ρ = B c r 2 S 2, an B r2 \ B r1 = S 1 S 2 S 3

20 20 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI aing the terms an collecting terms to reconstruct the integral in the whole of the positive contribution of the entropy, we euce I 1 + I 2 I 3 + I 4 := K p n p (x) log R + n p (x)x + 2 n p (x) n p (y) log x y xy 2 R 2 4 n p (x) n p (y) log x y xy with an a 1 := + 2 [B r1 B c r 1 +ρ ] [(S 1 S 3 ) B c r 2 ] [S 1 S 1 ] [S 2 S 2 ] n p (x) n p (y) log x y xy C K p := max{a 1,a 1 + a 2,a 2,a 2 + a 3,a 3 } = max{a 1 + a 2,a 2 + a 3 }, B r1 +ρ n p (x)x, a 2 := n p (x)x, an a 3 := n p (x)x, (B r2 \B r1 ) B c r 2 ρ where the secon an the fourth terms in the first expression of K p are ue to the fact that S 1 an S 2 are counte twice for the positive contribution of the entropy. We estimate the thir term I 3 by using log x y logρ in the sets where the integral is efine to obtain I 3 4M 2 logρ. Since the secon moment of the solutions is constant in time by (14), we euce I 4 2 n p (x) n p (y) log + x y xy C [S 1 S 1 ] [S 2 S 2 ] [S 1 S 1 ] [S 2 S 2 ] n p (x) n p (y) (1 + x 2 + y 2 )xy C Moreover, we can estimate the factor K p as { K p K := max α (r) + 3η, 8π } χ α (r) + 3η which is positive an strictly smaller than 8π χ for η < (1/6)min{α (r); 8π χ α (r)}. Summarizing, we have obtaine K n p (x) log + n p (x)x + 2 n p (x) n p (y) log x y xy C for all p big enough with 0 < K < 8π χ. Now, taking into account Lemma 2.2 an the conservation of the secon moment in (14), we obtain n p log n p x 1 x 2 n p x + log(2π) n p x e C an thus K n p (x) logn p (x)x + 2 n p (x) n p (y) log x y xy C

21 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 21 for p big enough with 0 < K < 8π χ. Repeating the same arguments as in the introuction to this subsection, see [13] for the subcritical case, an using the estimate on the free energy, we euce S [n p ] F [n 0] + θ C(M). 1 θ with θ = χ K 8π, for all p big enough. This fact contraicts the choice of T as the maximal time of existence of a free-energy solution n since lim S [n p] = +. p This contraiction comes from the assumption lim r 0 α (r) < 8π + χ, an thus, α (r) = M or equivalently n = Mδ 0 for all converging subsequences, completing the esire result. It is very easy to verify that the previous proof also works assuming that the entropy iverges if T =. More precisely, we have: Corollary 3.2 (Infinite Time Aggregation). Uner hypotheses (2) on the initial ata, assume that the free-energy solution n to the Patlak-Keller-Segel system (1) with initial ata n 0 of critical mass χ M = 8π is global in time an that lim t n(x,t)logn(x,t) x = +. Then t n(x,t) converges to a elta Dirac of mass 8π/χ concentrate at the center of mass in the measure sense as p. 3.2 When oes it blow-up? Proposition 3.3 (Existence of global in time solution). Uner assumptions (2) on the initial ata n 0, there exists a nonnegative free-energy solution n to the Patlak- Keller-Segel system (1) on [0, ). Proof. We start by remining the reaer about a by-now stanar gain of integrability result [24, 35, 34]. Lemma 3.4 (De la Vallée-Poussin theorem). [36, 34] Let µ be a non-negative measure an F L 1 µ( ). The set F is uniformly integrable in L 1 µ( ) if an only if F is uniformly boune in L 1 µ( ) an there exists a convex function Φ 0 C ([0, )) such that Φ 0 (0) = Φ 0 (0) = 0, Φ 0 is concave, Φ 0(r) r Φ 0 (r) 2Φ 0(r) if r > 0, r 1 Φ Φ 0 (r) is concave, lim 0 (r) r r = lim r Φ 0 (r) = an sup f F Φ 0 ( f )µ <.

22 22 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI We apply De la Vallée-Poussin theorem (Lemma 3.4) to f (x) = x 2 with µ = n 0 (x) x. Hence, there exists Φ 0 satisfying the properties enumerate in De la Vallée-Poussin theorem (Lemma 3.4) such that Φ 0 ( x 2 )n 0 (x) is in L 1 ( ). It is straightforwar, base on the properties of Φ 0 in Lemma 3.4, to show for all r > 0 (1) r Φ 0(r) 2 Φ 0(r). r Now, let us look for the evolution of Φ 0 ( x 2 )n(x,t)x. Lemma 3.5. If χ M = 8π, an n is a maximal free-energy solution of the Patlak- Keller-Segel system (1) with initial ata n 0, then for any smooth convex function Φ : R [0, ) such that Φ(0) = 0 Φ ( x 2) n(x,t) x 4 x 2 Φ ( x 2) n(x,t) x t Proof. We compute for all t (0,T ) Φ ( x 2) n(x,t) x = 4 t [ Φ ( x 2) + x 2 Φ ( x 2)] n(x,t) x χ xφ ( x 2) x y n(x,t)n(y,t) xy π x y 2 [ = 4 Φ ( x 2) + x 2 Φ ( x 2)] n(x,t) x χ 2π [ xφ ( x 2) yφ ( y 2)] x y n(x,t)n(y,t) xy. x y 2 On the other han, we have for any (x,y,s,t) R R ( xφ (s) yφ (t) ) (x y) = x 2 Φ (s) (x y) Φ (s) (x y) Φ (t) + y 2 Φ (t) = 1 2 x [ y 2 Φ (s) + Φ (t) ] + 1 ( x 2 y 2)[ Φ (s) Φ (t) ]. 2 As a consequence, we obtain ( xφ ( x 2) yφ ( y 2)) x y n(x,t)n(y,t) xy x y 2 { = 1 [Φ ( x 2) + Φ ( y 2)] + x 2 y 2 [ Φ ( 2 x y 2 x 2) Φ ( } y 2)] n(x,t)n(y,t) xy

23 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 23 In particular, as Φ is convex ( x 2 y 2)[ Φ ( x 2) Φ ( y 2)] is non-negative an as χm = 8π Φ ( x 2) [ n(x,t) x 4 Φ ( x 2) + x 2 Φ ( x 2)] n(x,t) x t R 2 4 Φ ( x 2) n(x,t) x which ens the proof. Applying Lemma 3.5 to Φ 0, we have ( Φ 0 x 2 ) n(x,t) x 4 x 2 Φ ( t 0 x 2 ) n(x,t) x. Using the Remark 1 an the properties of Φ 0 in De la Vallée-Poussin theorem (Lemma 3.4), we conclue there exist c 1,c 2 > 0 such that x 2 Φ ( 0 x 2 ) 2 Φ ( 0 x 2 ) ( x 2 c 1 Φ 0 x 2 ) + c 2, an thus, (2) t Φ 0 ( x 2 ) n(x,t) x 4c 1 Φ 0 ( x 2 ) n(x,t) x + 4c 2 M. Gronwall s lemma implies that ( (3) Φ 0 x 2 ) [ n(x,t) x e c ( 1t Φ 0 x 2 ) n 0 (x) x + 4c ] 2M c 1 for all t (0,T ). Now, let us prove that the maximal time of existence cannot be finite. Assume by contraiction that T <. We first observe that in the case χ M = 8π, the secon-momentum of a free-energy solution to the Patlak-Keller-Segel system (1) is conserve ue to (14): (4) x 2 n 0 (x) x = x 2 n(x,t) x > 0. Let us take {t p } p N T. Equation (4) together with the tail-control of the set of ensities { x 2 n(x,t p )} p N ue to (3), implies the tightness of the ensities { x 2 n(x,t p )} p N in M ( ) by Prokhorov s theorem. As a conclusion, the sequence of ensities {n(x,t p )} p N converges weakly- as measures towars n M ( ) with (5) x 2 n (x) = x 2 n 0 (x) x > 0, contraicting the fact that n shoul coincie with Mδ 0 ue to Lemma 3.1.

24 24 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI 3.3 Does it blow-up? We prove in Lemma 3.3 that if n is a non-negative free-energy solution to the Patlak-Keller-Segel system (1) with initial ata n 0 on [0,T ] for any T > 0, then n satisfies (6) t F [n](t) u(x,t) logu(x,t) χ v(x,t) 2 x. Let us see that the blow-up of the cell ensity oes happen at t. Lemma 3.6 (Blow-up in infinite time). Uner assumptions (2) on the initial ata n 0, given any free-energy solution n of (1), we have 8π lim n(t) = t χ δ M 1 weakly-* as measures. The central iea for this lemma is that if we can exten (6) to the limit when t goes to infinity, we coul prove the convergence of the solution to a L 1 -stationary solution with a finite secon moment. However, all integrable stationary solutions of the PKS system (1) with critical mass have infinite secon momentum. Proof. Consier, without loss of generality, that M 1 = 0. Assume by contraiction the existence of an increasing sequence of times {t p } p N for which S [n p ] = n(x,t p ) logn(x,t p ) x is boune. This together with the conservation of the secon moment for the solutions ue to (14) shows that there exists a subsequence, enote with the same inex for simplicity, converging weakly in L 1 ( ) towars a ensity n L 1 ( ) by Dunfor-Pettis theorem. Moreover, the secon moment of the limiting ensity satisfies (7) 0 < x 2 n (x) x x 2 n 0 (x) x < since n L 1 ( ), ue to the fact that the concentration towars a elta Dirac at 0 is rule out by the uniform estimate on the entropy S [n p ]. On the other han, as previously notice, a irect consequence of the Logarithmic Hary-Littlewoo-Sobolev inequality (5) is that the free energy F [n] is boune from below: t ( ) F [n 0 ] liminf F [n](t) = lim n(x,s) logn(x,s) χ c(x,s) 2 x s. t t 0 As a consequence the Fisher information is integrable an, ( ) lim n(x,s) logn(x,s) χ c(x,s) 2 x s = 0, t t

25 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 25 which shows that, up to the extraction of sub-sequences, the limit n (s,x) of (s, x) n(x, t + s) when t goes to infinity satisfies logn χ c = 0, c = 1 2π log n, where the first equation hols in the istribution sense at least almost everywhere in the support of n. We have skippe most of the etails of this passing to the limit argument since it follows the same steps as in the proof of convergence towars self-similar behavior in the subcritical case one in [13] an in the existence theorem in Section 2. Let us point out, that this is equivalent to the fact that (n,c ) solves the nonlocal nonlinear elliptic equation (8) u = M eχ v eχ v = v, with v = 1 x 2π log u. Moreover, by [18, Theorem 1], the solutions to (8) are raially symmetric. In the case χ M = 8π, [11] an [54] characterize explicitly the family of raial stationary solutions to (8) as being n b efine in (3). For all b, the stationary solutions n b have infinite secon momentum contraicting (7). As a conclusion, we have shown that the global free-energy solutions of the PKS system (1) satisfies lim t n(x,t)logn(x,t) x = +. Now, a irect application of Corollary 3.2 gives the esire result. Remark 3.7 (Stationary states an Minimizers of Free Energy). The free energy F [n] has an interesting scaling property: introuce n λ (x) := λ 2 n(x/λ) then ( ) χ M F [n λ ] = F [n] + 2M logλ 8π 1. In the case χ M = 8π, F [n] is thus invariant by this scaling. On the other han, the extremal functions of the Logarithmic Hary-Littlewoo- Sobolev inequality (5) are given up to a conformal automorphism by n := A ( 1 + x 2) 2. Combining these two remarks we obtain for any λ > 0, C(M) = F [n] = F [n λ ]. Hence, in the case χ M = 8π, the free energy F [n] with finite secon moment achieves its minimum C(8 π/χ) = (8 π/χ)(1 + log π log(8 π/χ)) in its stationary solution the elta Dirac. Actually, one can give a sense to 8π χ δ 0 as stationary solution to system (1) as being the weak-* limit of the suitably scale as above stationary solutions in (8). Note also that when χ M < 8π there are no stationary solutions for the Patlak- Keller-Segel system (1). Inee, in the case χ M < 8π there is uniqueness of the stationary self-similar solution as prove in [13, Theorem 2.1] an [11].

26 26 A. BLANCHET, J.A. CARRILLO, N. MASMOUDI Let us finally point out that if moments larger than 2 are initially boune, then they must iverge as t. This completes the picture of the long time asymptotics of global in time solutions of the PKS sytem (1) with critical mass: solutions exists globally an concentrate towar the elta Dirac at the center of mass as time iverges, while their secon moment is preserve an larger moments become unboune, i.e., the time sequence of measures with equal mass given by the secon moments is not tight. This fact means that the small mass escaping at infinity in space is large enough to have this effect on moments. Lemma 3.8 (Blow-up of moments in infinite time). Assume the initial ata n 0 verifies (2) an x 2k n 0 L 1 ( ), with k > 1, then any free-energy solution n of (1) with initial ata n 0 satisfies lim t x 2k n(x,t) x = +. Proof. Consier, without loss of generality, that M 1 = 0. Let us start by pointing out the propagation of moments of orer 2k for all times ue to (2), i.e., for any T > 0, there exists a constant C T such that x 2k n(x,t) x C T for almost every t (0,T ). Now, assume by contraiction the existence of an increasing sequence of times {t p } p N for which x 2k n(x,t p ) is boune in L 1 ( ). This together with the conservation of the secon moment for the solutions ue to (14) shows that there exists a subsequence, enote with the same inex for simplicity, converging weakly-* as measures in M ( ) towars a ensity n M ( ) satisfying x 2k n = x 2k n 0 (x) x > 0 by Prokhorov s theorem. This fact contraicts that the limiting ensity is a elta Dirac concentrate at 0 ue to Lemma 3.6. Acknowlegment. The authors are grateful to Vincent Calvez an Philippe Laurençot for stimulating iscussions an fruitful comments. AB acknowleges the support of ESF Network Global an bourse Lavoisier. JAC acknowleges the support from DGI-MEC (Spain) project MTM AB an JAC acknowlege partial support of the Acc. Integ./Picasso program HF NM is partially supporte by National Science Founation grant DMS We thank the Centre e Recerca Matemàtica (Barcelona) for proviing an excellent atmosphere for research.

27 INFINITE TIME AGGREGATION FOR THE CRITICAL PKS MODEL IN 27 Bibliography [1] A. ARNOLD, J. A. CARRILLO, L. DESVILLETTES, J. DOLBEAULT, A. JÜNGEL, C. LED- ERMAN, P. A. MARKOWICH, G. TOSCANI AND C. VILLANI, Entropies an equilibria of many-particle systems: an essay on recent research, Monatsh. Math., 142 (2004), pp [2] J.-P. AUBIN, Un théorème e compacité, C. R. Aca. Sci. Paris, 256 (1963), pp [3] F. BAVAUD, Equilibrium properties of the Vlasov functional: the generalize Poisson- Boltzmann-Emen equation, Rev. Moern Phys., 63 (1991), pp [4] W. BECKNER, Sharp Sobolev inequalities on the sphere an the Moser-Truinger inequality, Ann. of Math. (2), 138 (1993), pp [5] R. BENGURIA, PhD Thesis, PhD thesis, Princeton University, [6] R. BENGURIA, H. BRÉZIS, AND E. H. LIEB, The Thomas-Fermi-von Weizsäcker theory of atoms an molecules, Comm. Math. Phys., 79 (1981), pp [7] P. BILER, Local an global solvability of some parabolic systems moelling chemotaxis, Av. Math. Sci. Appl., 8 (1998), pp [8] P. BILER, G. KARCH, P. LAURENÇOT AND T. NADZIEJA, The 8π-problem for raially symmetric solutions of a chemotaxis moel in a isc, Top. Meth. Nonlin. Anal., 27 (2006), pp [9], The 8π-problem for raially symmetric solutions of a chemotaxis moel in the plane, Math. Meth. Appl. Sci., 29 (2006), pp [10] P. BILER AND T. NADZIEJA, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math., 66 (1993), pp [11], Global an exploing solutions in a moel of self-gravitating systems, Rep. Math. Phys., 52 (2003), pp [12] P. BILLINGSLEY, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, [13] A. BLANCHET, J. DOLBEAULT AND B. PERTHAME, Two-imensional Keller-Segel moel: optimal critical mass an qualitative properties of the solutions. Electronic Journal of Differential Equations, 44 (2006), pp [14] V. CALVEZ AND J. A. CARRILLO, Volume effects in the Keller-Segel moel: energy estimates preventing blow-up, Journal Mathématiques Pures et Appliquées, 86 (2006), pp [15] V. CALVEZ, B. PERTHAME AND M. SHARIFI TABAR, Moifie Keller-Segel system an critical mass for the log interaction kernel. preprint [16] E. CARLEN AND M. LOSS, Competing symmetries, the logarithmic HLS inequality an Onofri s inequality on S n, Geom. Funct. Anal., 2 (1992), pp [17] F. A. C. C. CHALUB, P. A. MARKOWICH, B. PERTHAME, AND C. SCHMEISER, Kinetic moels for chemotaxis an their rift-iffusion limits, Monatsh. Math., 142 (2004), pp [18] W. X. CHEN, AND C. LI, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), pp [19] S. CHILDRESS AND J. K. PERCUS, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), pp [20] L. CORRIAS, B. PERTHAME AND H. ZAAG, A chemotaxis moel motivate by angiogenesis, C. R. Math. Aca. Sci. Paris, 336 (2003), pp [21], Global solutions of some chemotaxis an angiogenesis systems in high space imensions, Milan J. Math., 72 (2004), pp [22] J. DOLBEAULT AND B. PERTHAME, Optimal critical mass in the two-imensional Keller-Segel moel in, C. R. Math. Aca. Sci. Paris, 339 (2004), pp