Piotr Biler
email: piotr.biler@math.uni.wroc.pl

Lorenzo Brandolese
email: brandolese@math.univ-lyon1.fr

Andrzej Hulanicki

On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis

We establish new convergence results, in strong topologies, for solutions of the parabolic-parabolic Keller-Segel system in the plane, to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.

Introduction

We consider two related nonlinear parabolic systems which are frequently used as models for a description of chemotactic phenomena, including the aggregation of microorganisms caused by a chemoattractant, i.e. a chemical whose concentration gradient governs the oriented movement of those microorganisms. The parabolic character of the systems comes from the diffusion described by the Laplacians. A version of the system (PE) below is also used in astrophysics as a model of the evolution of a cloud of self-gravitating particles in the mean field approximation.

The first one is the classical parabolic-elliptic Keller-Segel system     

u t = ∆u -∇ • (u∇ϕ), ∆ϕ + u = 0, u(0) = u 0 .
x ∈ R 2 , t > 0, (PE)

Here, u = u(x, t), ϕ = ϕ(x, t) are either functions or suitable (tempered) distributions. When u ≥ 0, ϕ ≥ 0, they may be interpreted as concentrations (densities) of microorganisms and chemicals, respectively. The second one is the parabolic-parabolic system     

u t = ∆u -∇ • (u∇ϕ), τ ϕ t = ∆ϕ + u, u(0) = u 0 , ϕ(0) = 0, x ∈ R 2 , t > 0, (PP)
where τ > 0 is a fixed parameter. Each of the models can be considered as a single nonlinear parabolic equation for u with a nonlocal (either in x or in (x, t)) nonlinearity since the term ∇ϕ can be expressed as a linear integral operator acting on u. In the latter model, the variations of the concentration ϕ are governed by the linear nonhomogeneous heat equation, and therefore are slower than in the former system, where the response of ϕ to the variations of u is instantaneous, and described by the integral operator (-∆) -1 whose kernel has a singularity. Thus, one may expect that the evolution described by (PE) might be faster than that for (PP), especially for large values of τ when the diffusion of ϕ is rather slow compared to that of u. Moreover, the nonlinear effects for (PE) should manifest themselves faster than for (PP).

The theory of the system (PE) is relatively well developed, in particular when this is studied in a bounded domain in R d , d = 1, 2, 3, with the homogeneous Neumann conditions for u and ϕ at the boundary of the domain. One of the most intriguing properties of (PE) considered for positive and integrable solutions u in d = 2 case is the existence of a threshold value 8π of mass M ≡ u(x, t) dx, see the pioneering work [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] and [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles, III[END_REF][START_REF] Biler | Local and global solvability of parabolic systems modelling chemotaxis[END_REF]. Namely, if u 0 ≥ 0 is such that u 0 (x) dx > 8π, then any regular, positive solution u of (PE) cannot be global in time. We refer the reader for a fine description of the asymptotic behaviour of integrable solutions of (PE) in the subcritical case M < 8π to [START_REF] Blanchet | Two dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions[END_REF] and to [START_REF] Blanchet | Infinite time aggregation for the critical Patlak-Keller-Segel model in R 2[END_REF] for the limit case M = 8π. See also [START_REF] Biler | The 8πproblem for radially symmetric solutions of a chemotaxis model in a disc[END_REF][START_REF] Biler | The 8πproblem for radially symmetric solutions of a chemotaxis model in the plane[END_REF] in the radially symmetric case. The higher dimensional versions of (PE) have been also extensively studied, cf., e.g., [START_REF] Biler | The Cauchy problem and self-similar solutions for a nonlinear parabolic equation[END_REF][START_REF] Biler | Global regular and singular solutions for a model of gravitating particles[END_REF][START_REF] Biler | Global existence versus blow up for some models of interacting particles[END_REF], and [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles, III[END_REF] for blow up phenomena.

The doubly parabolic system (PP) has been a bit less studied. For instance, it is known that if for the initial data u 0 one has M < 8π, then positive solutions are global in time, see [START_REF] Calvez | The parabolic-parabolic Keller-Segel model in R 2[END_REF] in the case of a bounded planar domain, and also [START_REF] Biler | Local and global solvability of parabolic systems modelling chemotaxis[END_REF]Theorem 5]. However, it is not known whether M ≤ 8π is, in general, a necessary condition for the existence of global in time solutions. That is, it is not known whether the blow up occurs for solutions, except for a specific example in [START_REF] Herrero | A blow-up mechanism for a chemotaxis model[END_REF] of a particular blowing up solution for a system close to (PP). Even, it is an open question what is the exact range of M guaranteeing the existence of integrable self-similar solutions. For the system (PE) it is proved that M ∈ [0, 8π), and the self-similar solutions (unique for a given M ∈ [0, 8π)) describe the generic asymptotic behaviour of global in time, positive and integrable solutions of (PE). Concerning (PP), it is known that M < M(τ ) with M(τ ) linear in τ , is a necessary condition for the existence of self-similar solutions, cf. [START_REF] Biler | A note on the paper of Y. Naito[END_REF]. For (PP) with small M such special solutions are also important in the study of space-time decay of general solutions, see [START_REF] Naito | Asymptotically self-similar solutions for the parabolic systems modelling chemotaxis[END_REF]. The analysis if any M > 8π may correspond to a self-similar solution is under way, see [7]. For a different point of view about self-similar solutions for higher dimensional models of (PP), see also [START_REF] Kozono | Keller-Segel System of parabolicparabolic type with initial data in weak L n/2 (R n ) and its applications to self-similar solutions[END_REF]. Usual proofs of a blow up for (PE) involve calculations of moments of a solution and then symmetrization, cf. [START_REF] Blanchet | Two dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions[END_REF][START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles, III[END_REF]. These methods seem do not work for (PP), hence another approach is needed to show a blow up for that system. For a numerical insight on blow-up issues we refer, e.g., to [START_REF] Filbet | A finite volume scheme for the Patlak-Keller-Segel chemotaxis model[END_REF].

A nice result in [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF] shows that the solutions of the systems (PP) and (PE) enjoy a kind of stability property as τ ց 0: solutions of (PP) converge in a suitable sense to those of (PE). It had been an old question raised by J. J. L. Velázquez and D. Wrzosek, recently solved in [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF]. However, this result obtained for suitably small solutions in quite a big functional space of pseudomeasures, gives no indication on the behaviour of possible ("large") blowing up solutions.

The solvability of the systems (PE), (PP) has been studied in various classes of functions and distributions, like Lebesgue, Morrey, Besov, etc., with an immediate motivation to include the a priori strongest possible critical singularities of either solutions or initial data which appear to be point measures in the two-dimensional case and the multiples of |x| -2 function in the higher dimensional case. In particular, "vast" functional spaces suitable for analysis of the two-dimensional systems include measure and pseudomeasure spaces, cf. [START_REF] Biler | The Cauchy problem and self-similar solutions for a nonlinear parabolic equation[END_REF][START_REF] Biler | Local and global solvability of parabolic systems modelling chemotaxis[END_REF][START_REF] Biler | Global regular and singular solutions for a model of gravitating particles[END_REF][START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF].

We show in this paper a result on the existence of (in general, nonintegrable) solutions in a class X of functions with natural space-time decay properties, see Theorem 2.1 and 3.1. Here, the space E of admissible initial conditions also contains Dirac measures. The corresponding solutions may be positive and "large" in the sense of their nonintegrability. Nevertheless, they are defined globally in time. Unlike the paper [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF], we work in (x, t) space, when [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF] has dealt with the Fourier variables ξ, cf. the formulation (36) below. Such results are obtained by an extension and refinement of techniques used in [START_REF] Biler | Global existence versus blow up for some models of interacting particles[END_REF] for (PE) in higher dimensions, but neither for (PP), nor in the two-dimensional case of (PE) which often requires a specific treatement. Moreover, the function spaces that we employ here allow us to deal with data that can be more singular than those considered in [START_REF] Biler | Global existence versus blow up for some models of interacting particles[END_REF]. The spaces X and E defined in the next section are, in a sense, critical for that analysis, and have been already considered, in slightly different forms, e.g. in the studies of the Navier-Stokes system in [START_REF] Cazenave | Chaotic behavior of solutions of the Navier-Stokes system in R N[END_REF][START_REF] Miyakawa | On space time decay properties of nonstationary incompressible Navier-Stokes flows in R n[END_REF].

Our main results are contained in Section 4, where we address the problem of the convergence as τ ց 0 of solutions u τ of the system (PP) to the corresponding solutions u of (PE), in the space X , arising from small data in E. Mathematically, our stability result is not included in, and does not imply, that of [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF]. However, it seems to us that the use of the natural (x, t) variables provides a more immediate physical interpretation. Furthermore, our method looks more flexible, and can be used to prove the stability of the system with respect to stronger topologies. For example, we establish also the convergence in the L ∞ t (L 1 x )-norm for data belonging to E ∩ L 1 , and in the L ∞ t,x -norm for data in E ∩ L ∞ . Motivated by [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF], we will also address this issue in the more general setting of shift invariant spaces of local measures. The main difficulty for obtaining the convergence u τ → u in strong norms is that ∇ϕ τ enjoys some kind of instability as τ → 0, in particular in weighted spaces.

Moreover, we give a nonexistence (blow up) result for solutions of (PP) in R d , d ≥ 1, with the positive Fourier transform of u 0 in the spirit of [START_REF] Montgomery-Smith | Finite time blow up for a Navier-Stokes like equations[END_REF], see Theorem 5.1. These are complex valued solutions with no straightforward physical/biological interpretation. However, such a result tells us that there is no hope to prove the global existence of solutions to (PP) and similar models for arbitrarily large data relying only on size estimates.

The parabolic-elliptic system

In order to study the systems (PE) and (PP) we introduce the Banach space X of functions u = u(x, t) and the Banach space E of tempered distributions u 0 ∈ S ′ (R 2 ) by defining the norms

u X = ess sup t>0,x∈R 2 (t + |x| 2 ) |u(x, t)|, (1)
and

u 0 E = e t∆ u 0 X . (2)
Here, e t∆ denotes the heat semigroup defined by the Gaussian kernel g t , g t (x) = 1 4πt e -1 4t |x| 2 . For example, the Dirac mass u 0 = δ in R 2 is an element of E. Notice that, by the definition, E is continuously embedded into the weak Hardy space H 1 w , which is the space consisting of all tempered distributions f such that sup t>0 |e t∆ f | belongs to the Lorentz space L 1,∞ (R 2 ). See [START_REF] Miyakawa | Application of Hardy space techniques to the timedecay problem for incompressible Navier-Stokes flow in R n[END_REF].

Let us define the bilinear form B 0 by

B 0 (u, v)(t) ≡ t 0 e (t-s)∆ ∇ • (u∇(-∆) -1 v)(s) ds. (3)
Here, (-∆) -1 is the convolution operator on functions defined on R 2 with the kernel K(x) = -1 2π log |x|. With this notation, the equivalent integral (mild) formulation to (PE), called also the Duhamel formula, reads

u(t) = e t∆ u 0 -B 0 (u, u). (4)
We begin by establishing the following simple result Theorem 2.1 There exist two absolute constants ǫ, β > 0 with the following property. Let u 0 ∈ E be such that u 0 E < ǫ. Then there exists a unique (mild) solution u ∈ X of (PE) such that u X ≤ ǫβ.

The proof of Theorem 2.1 will follow from a series of lemmata.

First, we have the following estimate of the leading term in ∇ϕ Lemma 2.1 Let u ∈ X and ϕ be such that ∆ϕ + u = 0. Then

∇ϕ(x, t) = c 0 x |x| 2 |y|≤|x|/2 u(y, t) dy + R(x, t) (5)
with c 0 = -1 2π and the remainder R satisfying

|R(x, t)| ≤ C u X t 1 2 + |x| -1
.

Proof. Indeed, let us represent the partial derivatives of ϕ, for j = 1, 2, as

∂ j ϕ = c 0 x j |x| 2 * u ≡ I 1 + I 2 + I 3 ,
where . On the other hand, we can rewrite

I 1 = |y|≤|x|/2 c 0 (x j -y j ) |x -y| 2 u(y,
I 1 as c 0 x j |x| 2 |y|≤|x|/2 u(y, t) dy + R 1 (x, t).
An application of the Taylor formula shows that the above bound holds also for R 1 .

We immediately deduce from ( 5) the following useful estimate Lemma 2.2 Let u ∈ X and ∆ϕ + u = 0. Then

∇ϕ(t) L ∞ ≤ C u X t -1 2 .
The last lemma that we need is the following

Lemma 2.3 Let u, v ∈ X . Then, for some constant C 0 independent of u, v B 0 (u, v) X ≤ C 0 u X v X .
Proof. We can assume, without any restriction, that u

X = v X = 1. Lemma 2.2 implies |u∇(-∆) -1 v|(x, t) ≤ C|x| -3 2 t -3 4 , (6a)
and also

|u∇(-∆) -1 v|(x, t) ≤ C|x| -2 t -1 2 . ( 6b
)
We denote the gradient of the heat semigroup kernel g t by

G(•, t) ≡ ∇ x 1 4πt e -1 4t | • | 2 . ( 7
)
Then we may represent B 0 as

B 0 (u, v)(x, t) = t 0 G(x -y, t -s)(u∇(-∆) -1 v)(y, s) dy ds ≡ J 1 + J 2 ,
where

J 1 = t 0 |y|≤|x|/2 . . . and J 2 = t 0 |y|≥|x|/2 . . . . Using the estimate |G(x -y, t -s)| ≤ C|x -y| -5 2 (t -s) -1 4
and inequality (6a), we get the bound

|J 1 (x, t)| ≤ C|x| -2 . ( 8
)
Another possible estimate is

|J 1 (x, t)| ≤ C|x| -3 2 t -1 4 , (9)
which is obtained using the bound

|G(x -y, t -s)| ≤ C|x -y| -2 (t -s) -1 2 .
On the other hand, from the property

G(•, t -s) L 1 = c(t -s) -1 2 (10)
and inequality (6b), we obtain

|J 2 (x, t)| ≤ C|x| -2 . ( 11
)
As before, we have also the bound

|J 2 (x, t)| ≤ C|x| -3 2 t -1 4 . ( 12
)
This second estimate is deduced from (6a). Then, using ( 8), [START_REF] Blanchet | Two dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions[END_REF], we obtain the space decay estimate

|B 0 (u, v)|(x, t) ≤ C|x| -2 (13)
and from ( 9), ( 12) -a provisory (not optimal) time decay estimate

B 0 (u, v)(t) L 4 3 ,∞ ≤ Ct -1 4 .
But we may represent B 0 as

B 0 (u, v)(t) = e t 2 ∆ B 0 (u, v)(t/2) + t t/2 G(t -s) * (u∇(-∆) -1 v)(s) ds.
Thus, applying the weak Young-type inequality for convolutions in Lorentz spaces L 4 3 ,∞ * L 4,1 ⊂ L ∞ , see [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF], and the equality obtained from the scaling laws in Lorentz spaces,

1 4πt e -1 4t | • | 2 L 4,1 = ct -3 4 , (14)
we finally get

B 0 (u, v)(t) L ∞ ≤ Ct -1 + C t t/2 (t -s) -1 2 u∇(-∆) -1 v(s) L ∞ ds ≤ Ct -1 . (15)
Combining inequalities ( 13) and ( 15) we get B 0 (u, v) ∈ X , together with its continuity with respect to u and v.

Proof. Note that (using the duality S -S ′ ) we have e t∆ u 0 → u 0 in S ′ as t → 0. The conclusion of Theorem 2.1 follows in a standard way (cf., e.g., [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF][START_REF] Biler | The Cauchy problem and self-similar solutions for a nonlinear parabolic equation[END_REF][START_REF] Biler | Global regular and singular solutions for a model of gravitating particles[END_REF]) from the contraction fixed point theorem.

The parabolic-parabolic system

Let τ > 0 be a fixed parameter. We consider the system (PP) whose equivalent integral formulation reads

u(t) = e t∆ u 0 - t 0 ∇ • e (t-s)∆ u(s) 1 τ ∇ s 0 e 1 τ (s-σ)∆ u(σ) dσ ds. ( 16
)
We introduce for all τ ≥ 0 the bilinear form B τ (recall that G is defined by the expression ( 7))

B τ (u, v)(x, t) ≡ t 0 G(x -y, t -s) u W τ (v) (y, s) dy ds, (17)
where W τ (v) is the linear operator acting on v

W τ (v)(x, t) = t 0 1 τ G t -σ τ * v(σ) (x, σ) dσ for τ > 0, (18a)
with a natural convention

W 0 (v)(x, t) = ∇(-∆) -1 v (x, t). (18b)
In this way, the system (PP) is also rewritten in a compact form (cf. ( 4)) as u = e t∆ u 0 -B τ (u, u).

We are going to solve [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF] in the space X exactly as was in the parabolicelliptic case. An additional estimate, however, is needed:

Lemma 3.1 Let u ∈ X and τ > 0.
Then there exists a constant C * > 0, independent of u and τ , such that

W τ (u)(t) L ∞ ≤ C * t -1 2 u X . (20)
Proof. As usual we can and do assume u X = 1. Then, for all 1 < p ≤ ∞ we have

|u(x, σ)| ≤ |x| -2 p σ -1+ 1 p .
This implies, for 1 < p ≤ ∞,

u(σ) L p,∞ ≤ σ -1+ 1 p . Now, we represent W τ (u) = I 1 + I 2 ,
where I 1 = t/2 0 . . . and I 2 = t t/2 . . . . Evidently, we obtain the bound

I 1 (t) L ∞ ≤ C t/2 0 1 τ G t -σ τ L 2,1 u(σ) L 2,∞ dσ ≤ Ct -1 2 .
For the integral I 2 , let us begin with a rough bound

I 2 (t) L ∞ ≤ C t t/2 1 τ G t -σ τ L 1 u(σ) L ∞ dσ ≤ Cτ -1 2 t -1 2 . ( 21
)
This bound gives the required estimate, excepted when τ belongs to a neighbourhood of the origin. Thus, in the sequel, it is enough to consider the case 0 < τ < 1 2 . Now, we further decompose

I 2 ≡ I 2,1 + I 2,
I 2,1 (t) L ∞ ≤ C t-τ t t/2 1 τ G t -σ τ L 3,1 u(σ) L 3 2 ,∞ dσ ≤ Cτ 1 6 t-τ t t/2 (t -σ) -7 6 σ -1 3 dσ ≤ Ct -1 2 . ( 22
)
The last estimate is

I 2,2 (t) L ∞ ≤ C t t-τ t 1 τ G t -σ τ L 1 u(σ) L ∞ dσ ≤ Cτ -1 2 t t-τ t (t -σ) -1 2 σ -1 dσ ≤ Ct -1 2 . ( 23
)
The conclusion of Lemma 3.1 follows from ( 21), ( 22), [START_REF] Montgomery-Smith | Finite time blow up for a Navier-Stokes like equations[END_REF].

Lemma 3.1 allows us to see that, if u, v ∈ X with u X = v X = 1, then |u W τ (v)|(x, t) ≤ C|x| -3 2 t -3 4 , (24a)
and

|u W τ (v)|(x, t) ≤ C|x| -2 t -1 2 , (24b)
for some constant C > 0 independent of τ > 0. These are the analogous estimates as those we obtained in the parabolic-elliptic case (see inequalities (6a) and (6b)). Then, using exactly the same arguments as in the previous section, we arrive at the following existence result Theorem 3.1 There exist two absolute constants ǫ * , β * > 0 with the following property. Let u 0 ∈ E be such that u 0 E < ǫ * . Then there exists a unique (mild) solution u ∈ X of (PP) such that u X ≤ ǫ * β * .

Remark. The case of nonzero initial data ϕ(0) can be studied in a quite similar way.

Remark.

A closer look at the proofs of estimates for ∇ϕ in (PE) and (PP) reveals that the behaviour of ∇ϕ is a bit different in these two cases. Namely, if 0 ≡ u and u(x, t)

∼ (t + |x| 2 ) -1 (in the sense that c 1 (t + |x| 2 ) -1 ≤ u(x, t) ≤ c 2 (t + |x| 2 ) -1 for some c 1 , c 2 > 0)
, then it follows from (5) that ∇ϕ(x, t) ∼ |x| -1 log(t + |x| 2 ) for (PE), while ∇ϕ is more regular:

|∇ϕ(x, t)| ≤ c t 1 2 + |x| -1
in (PP) case. In other words, letting Y be the space of functions f = f (x, t) such that f 2 ∈ X , we have ϕ τ ∈ Y for τ > 0, but (ϕ τ ) does not converge in Y as τ → 0. However, such an instability does not prevent from the convergence of the densities u τ → u for vanishing τ .

4 Study of the τ ց 0 limit

We now study the convergence as τ ց 0 of solutions u τ of the system (PP) to the corresponding solution u of (PE). A result in this direction has been obtained recently by A. Raczyński in [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF], who established the convergence u τ → u in the norm Y α , for α ∈ (1, 2), defined as

u Yα = ess sup t>0, ξ∈R 2 1 + t 1 2 |ξ| α | u(ξ, t)|. (25)
We will obtain in subsection 4.2 a similar result using the X -norm.

Regularity properties of solutions of (PE)

In this subsection we prepare some preliminary material. The first Proposition consists of a regularity result with respect to the space variable for solutions of (PE). The second Proposition describes their regularity properties with respect to the time variable.

Proposition 4.1 For all r ∈ (1, 2) there exists a constant ǫ r , with 0 < ǫ r ≤ ǫ (the absolute constant of Theorem 2.1) such that, if u 0 E < ǫ r , then the solution of (PE) constructed in Theorem 2.1 satisfies

∇u(t) L r,∞ ≤ Ct -3 2 + 1 r , (26)
for some constant C = C(u 0 , r) independent of t.

Proof. We use a standard argument involving the subspace X r ⊂ X defined by

X r = u ∈ X , ∃ C : ∇u(t) L r,∞ ≤ Ct -3 2 + 1 r ,
and equipped with its natural norm. Recalling that the kernel c 0 x |x| 2 of the operator ∇(-∆) -1 belongs to L 2,∞ , first we deduce from the Young inequality

∇ 2 (-∆) -1 v(t) L α,∞ ≤ Ct -3 2 + 1 r v Xr , 1 α = 1 r -1 2 . ( 27
)
Next, from the Hölder inequality (noticing that u(t

) L 2,∞ ≤ Ct -1 2 u X ), u∇ 2 (-∆) -1 v(t) L r,∞ ≤ Ct -2+ 1 r u X v Xr . ( 28
)
The generalization of the classical inequalities to Lorentz spaces can be found, e.g., in [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF].

We claim that the bilinear operator B 0 introduced in (3) is boundedly defined: B 0 : X r × X r → X r . Indeed, for u Xr = v Xr = 1, we combine the estimates

∇G(t -s) 1 ≤ C(t -s) -1
and (10) with the inequality (a consequence of u ∈ X )

u(s) L r,∞ ≤ Cs -1+ 1 r , (29)
the estimate ∇(-∆) -1 v(s) ∞ ≤ Cs -1 2 obtained from Lemma 2.2, and (28). Then we arrive at

∇B 0 (u, v)(t) L r,∞ ≤ C t/2 0 (t -s) -1 u(s) L r,∞ ∇(-∆) -1 v(s) ∞ ds + C t t/2 (t -s) -1 2 ∇u(s) L r,∞ ∇(-∆) -1 v(s) ∞ + s -2+ 1 r ds ≤ Ct -3 2 + 1 r .
Moreover, for u 0 ∈ E, we have |e

t 2 ∆ u 0 (x)| ≤ C(t + |x| 2 ) -1 . Hence, e t 2 ∆ u 0 L γ,∞ ≤ Ct -1+ 1 γ for 1 < γ < ∞. We now choose β, γ ∈ (1, ∞) such that 1 + 1 r = 1 β + 1 γ .
Then, the semigroup property of the heat kernel g t , and the fact that ∇g t/2 ∈ L β,1 , imply

∇e t∆ u 0 L r,∞ ≤ Ct -3 2 + 1 r .
Now the usual the application of the contraction mapping theorem, in a closed ball of small radius in the space X r , allows us to conclude.

The following proposition is the first crucial tool for our stability result. It provides the Hölder regularity, with respect to the time variable, of solutions of (PE) in Lorentz spaces. Proposition 4.2 Let 1 < r < 2 and u 0 ∈ E, such that u 0 E < ǫ r . Then the solution u of (PE) constructed in Proposition 4.1 satisfies for all 0

< t ′ < t u(t) -u(t ′ ) L r,∞ ≤ C(t -t ′ ) 1 2 (t ′ ) -3 2 + 1 r , (30)
for some C = C(u 0 , r) independent of t and t ′ .

Proof. It is enough to show that both e t∆ u 0 -e t ′ ∆ u 0 and B 0 (u, u)(t) -B 0 (u, u)(t ′ ) satisfy the required bound in the L r,∞ -norm.

From the identity e t∆ u 0 (x) -e t ′ ∆ u 0 (x) = e t ′ ∆ u 0 (x -y) -e t ′ ∆ u 0 (x) g t-t ′ (y) dy = -

1 0 ∇e t ′ ∆ u 0 (x -θy) • yg t-t ′ (y) dy dθ,
we get

e t∆ u 0 -e t ′ ∆ u 0 L r,∞ ≤ C ∇e t ′ ∆ u 0 L r,∞ yg t-t ′ 1 ≤ C(t -t ′ ) 1 2 (t ′ ) -3 2 + 1 r .
Now, we can write

B 0 (u, u)(t) -B 0 (u, u)(t ′ ) = A 1 + A 2 ,
with

A 1 ≡ t ′ 0 G(t -s) -G(t ′ -s) * u∇(-∆) -1 u (s) ds and A 2 ≡ t t ′ G(t -s) * u∇(-∆) -1 u (s) ds.
Recall that from u ∈ X we deduce (29). Combining this with the estimate of Lemma 2.2 we get

u∇(-∆) -1 u(s) L r,∞ ≤ Cs -3 2 + 1 r . (31)
This immediately yields

A 2 L r,∞ ≤ t t ′ (t -s) -1 2 s -3 2 + 1 r ds ≤ C(t -t ′ ) 1 2 t ′ -3 2 + 1 r .
The estimate of A 1 is slightly more involved. We start with the identity

G(t -s) -G(t ′ -s) = e (t-t ′ )∆ -Id G(t ′ -s).
The action of the convolution operator with the function on the right hand side is studied via the following variant of a result established in [START_REF] Meyer | Wavelets, paraproducts and Navier-Stokes equations[END_REF]. . Then, for some constant C depending only on r ∈ (1, ∞)

e t∆ -Id f L r,∞ ≤ Ct 1 2 Λf L r,∞ .
Proof. Writing f = Λ -1 Λf , we see that

e t∆ -Id f = Φ t * (Λf ), where, Φ t (ξ) = t 1 2 Φ(t 1 2 ξ), and
Φ(ξ) = (e -|ξ| 2 -1)|ξ| -1 .
It only remains to show that Φ ∈ L 1 (R2 ), which is immediate. Indeed, it is well known, and easy to check, that Ψ(ξ) = |ξ|e -|ξ| 2 defines a function Ψ ∈ L 1 (R 2 ) (for example, with the method described in [START_REF] Furioli | Molecules of the Hardy space and the Navier-Stokes equations[END_REF], one obtains

|Ψ(x)| ≤ C(1 + |x|) -3 and |∇Ψ(x)| ≤ C(1 + |x|) -4
). We conclude applying the Bochner inequality to the identity

Φ(x) = -2 ∞ 1 Ψ(η x) dη.
Using this Lemma we deduce

A 1 L r,∞ ≤ C(t -t ′ ) 1 2 t ′ 0 Λ∇g t ′ -s * u∇(-∆) -1 u (s) L r,∞ ds = C(t -t ′ ) 1 2 (A 1,1 + A 1,2 ),
where A 1,1 and A 1,2 are obtained splitting the integral at s = t ′ /2. But, as the function Ψ introduced in the proof of Lemma 4.1 satisfies ∇Ψ ∈ L 1 (R 2 ), we see by a simple rescaling that

Λ∇g t ′ -s 1 ≤ C(t ′ -s) -1 .
Combining this estimate with inequality (31), we get

A 1,1 ≤ C(t ′ ) -3 2 + 1 r , 1 < r < 2.
For treating A 1,2 , we combine the estimate

Λg t ′ -s 1 ≤ C(t ′ -s) -1 2
with the inequality (for 1 < r < 2)

(∇u) ∇(-∆) -1 u (s) L r,∞ + u ∇ 2 (-∆) -1 u (s) L r,∞ ≤ Cs -2+ 1 r ,
obtained by applying (26), Lemma 2.2, and (28) with u = v. We get as before

A 1,2 ≤ C(t ′ ) -3

The vanishing τ limit

After Proposition 4.2, the second crucial step for the study of the limit as τ ց 0 consists in the asymptotic analysis of the linear operators W τ , τ ≥ 0, introduced in (18a)-(18b). This is the purpose of the following lemma.

Lemma 4.2 Let ε = ε(τ ) be an arbitrary function, strictly increasing and continuous on [0, 1], such that ε(0) = 0. Let also 1 < r < 2 and u be a function satisfying, for 0 < t ′ < t,

u(t) L r,∞ ≤ Ct -1+ 1 r , u(t) -u(t ′ ) L r,∞ ≤ C(t -t ′ ) 1 2 (t ′ ) -3 2 + 1 r , ( 32
)
with a constant C independent of t, t ′ . Then, for all t > 0, τ ∈ [0, 1], and for another constant C, independent of t and τ ,

ε(τ ) sup t>0 t 1 2 (W τ (u) -W 0 (u)) (t) ∞ ≤ Cτ 1 r -1 2 . ( 33
)
In particular, if u 0 ∈ E is small enough (for example, u 0 E ≤ ǫ 3/2 ), then the corresponding solution u of (PE) constructed in Proposition 4.1 satisfies

lim τ →0 sup t>0 t 1 2 (W τ (u) -W 0 (u)) (t) ∞ = 0. ( 34
)
Proof. Without any restriction we can assume that 0 < ε(τ ) < 1 2 for positive τ . Define ε(τ ) such that ε = ε 1 r -1 2 . Borrowing from [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF] the idea of splitting the time integral using intervals depending on τ , we write

W τ (u) -W 0 (u) ≡ J 1 + J 2 + J 3 ,
where

J 1 (t) = t(1-ε(τ )) 0 1 τ G t -s τ * u(s) ds, next J 2 (t) = t t(1-ε(τ )) 1 τ G t -s τ * u(t) ds -W 0 (u)(t),
and

J 3 (t) = t t(1-ε(τ )) 1 τ G t -s τ * u(s) -u(t) ds.
From the first relation of (32) and the Young inequality in Lorentz space (using that, by (7), G(•, t) ∈ L r ′ ,1 (R 2 ), where r ′ is the conjugate exponent), we get

J 1 (t) ∞ ≤ Cτ 1 r -1 2 t(1-ε(τ )) 0 (t -s) -1 2 -1 r s -1+ 1 r ds ≤ C τ ε(τ ) 1 r -1 2 t -1 2 .
Notice that this estimate of J 1 is exactly what we need for (33). As for J 2 , we see from a simple computation via the Fourier transform that J 2 (t) = -g tε(τ )/τ * W 0 (u)(t).

If 1 α = 1 r -1 2
, then we deduce from the usual weak-convolution estimates that

W 0 (u) = ∇(-∆) -1 u is bounded in the L α,∞ -norm, by Ct -1+ 1
r . Applying once more the Young inequality (using now g tε(τ )/τ ∈ L α ′ ,1 ), we get, as before,

J 2 (t) ∞ ≤ C τ ε(τ ) 1 r -1 2 t -1 2 .
Applying the second of inequalities (32), we obtain immediately

J 3 (t) ∞ ≤ Cτ 1 r -1 2 t -1 2 ,
which is even better than what we need. This proves the inequality (33). Choosing, for example, r = 3 2 and ε(τ ) = 1 2 τ 1/12 proves the last claim (34) of Lemma 4.2.

We are now in the position of establishing our first main result Theorem 4.1 There exists an absolute constant ǫ ′ > 0 (a priori smaller than the constants ǫ, ǫ * > 0 in Theorems 2.1 and 3.1), such that if u 0 ∈ E, u 0 E < ǫ ′ , then denoting by u ∈ X the solution of (PE) and u τ ∈ X the solution of (PP) constructed in the previous theorems, we have as τ ց 0

u τ → u in X .
Proof. The proof follows easily from Lemma 4.2. Indeed, from the integral equations ( 4) and ( 19), the bilinearity of B τ and B 0 , and the smallness of the solutions u τ and u, we have (similarly as in [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF], where two terms in the bilinear expansion can be absorbed by the left hand side)

u τ -u X ≤ C B τ (u, u) -B 0 (u, u) X .
But, by the definition of W τ and W 0 (see (18a)-(18b)),

B τ (u, u)(t) -B 0 (u, u)(t) = t 0 G(t -s) * u(W τ (u) -W 0 (u)) (s) ds.
Argueing as in the proof of Lemma 2.3, we obtain

B τ (u, u) -B 0 (u, u) X ≤ C u X sup t>0 t 1 2 W τ (u) -W 0 (u)(t) ∞ .
If ǫ ′ > 0 is small enough, then Lemma 4.2 can be applied to the solution u of (PE), implying that the right hand side of the above inequality has a vanishing limit for small τ . This finally gives u τ -u X → 0 as τ ց 0.

Remark 4.1 Notice that, the smaller the norm u 0 E , the faster the convergence u τ → u as τ → 0. This is due to the fact that for very small data it is possible to apply Lemma 4.2 with r close to 1 (despite the constants in our estimates blow up as r ց 1). More precisely, our arguments show that for any 0 < δ < 1 2 , one can find a constant C > 0 and ǫ(δ) > 0 such that, for u 0 E ≤ ǫ(δ), one has u τ -u X ≤ Cτ

The parabolic-elliptic limit in stronger topologies

If u 0 ≥ 0 is small in the E-norm, and belongs to a smaller space, for example, u 0 ∈ E ∩ L 1 , then the solutions u τ and u of (PP) and (PE) will remain in L 1 , uniformly in time, during their evolution. Hence, it is natural to ask whether the convergence u τ → u holds also in the natural norm of L ∞ ((0, ∞); L 1 ). Our next theorem provides a positive answer. As the proof of this fact does not really depend on a particular topology under consideration, it seems appropriate to consider a more abstract setting.

We denote by L any shift invariant Banach space of local measures, see [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF]Ch. 4] for their definition and main properties. These are Banach spaces of distributions, continuously embedded in D ′ (R 2 ). Moreover, they are known to satisfy the following properties (for some constant C > 0 depending only on L),

1. For all f ∈ L, g ∈ L 1 (R 2 ), the convolution product f * g is well defined in L and f * g L ≤ C f L g 1 . 2. For all f ∈ L, h ∈ L ∞ (R 2 ), the pointwise product f h is well defined in L and f h L ≤ C f L h ∞ .
3. Each bounded sequence {f k } ⊂ L has a subsequence convergent in L, in the distributional sense.

Obvious examples of spaces satisfying these properties (and which are indeed shift invariant space of local measures) are the L p -spaces, 1 < p ≤ ∞, the Lorentz spaces L p,q , 1 < p < ∞, 1 < q ≤ ∞ and the space of bounded Borel measures M(R 2 ) = C 0 (R 2 ) * . In the latter case, such duality relations ensures Property 3. Other interesting examples include the Morrey-Campanato spaces M p,q , (1 < p ≤ q < ∞) and suitable multiplier spaces, see [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF]Ch. 17].

On the other hand, the space of pseudomeasures, i.e., the space of tempered distributions f such that f ∈ L ∞ ) does not fulfill the second requirement. Therefore, the stability result in the pseudomeasure topology will not be encompassed by our next Theorem, but requires a specific (and more involved) treatment, see [START_REF] Raczyński | Stability property of the two-dimensional Keller-Segel model[END_REF].

Notice that, because of the conservation of the total mass for positive solutions of (PE) and (PP), the L 1 -norm remains constant during the evolution. This observation will allow us to handle the case of data u 0 ∈ E ∩ L 1 , despite Property 3 breaks down for L 1 . Theorem 4.2 Let u 0 ∈ E ∩ L, where L is either a shift invariant Banach space of local measures, or L = L 1 . In the latter case we require either u 0 ≥ 0 or, for signed u 0 , that u 0 1 is sufficiently small. Then there exists a positive constant ǫ , depending only on L, such that, if

u 0 E < ǫ,
then the systems (PE) and (PP) possess unique solutions u and u τ , respectively, such that for an absolute constant β > 0, u X ≤ βǫ and u τ X ≤ βǫ. In addition,

sup t>0 u(t) L < ∞ and sup t>0 u τ (t) L < ∞.
Moreover, the conclusion u τ (t)-u(t) X → 0 of Theorem 4.1 is strengthened

sup t>0 u τ (t) -u(t) L → 0 as τ ց 0.
Proof. Obviously we have, for some constant C 0 > 0 independent of t e t∆ u 0 L ≤ C 0 .

Moreover, for each τ ≥ 0 (we include in this way the analysis of (PE)), we have the estimate

B τ (u, v)(t) L ≤ C sup t>0 u(t) L v X , (35)
We now discuss the stability, including also the case L = L 1 . From the bilinearity of B τ , the mixed estimate (35) and the smallness of the solutions u and u τ (allowing two terms of the bilinear expansions to be absorbed by the left hand side), we obtain

u τ (t) -u(t) L ≤ C sup t>0 B τ (u, u) -B 0 (u, u)(t) L .
Argueing as at the end of the proof of Theorem 4.1, we arrive at

B τ (u, u)-B 0 (u, u)(t) L ≤ C sup t>0 u(t) L sup t>0 t 1 2 (W τ (u)-W 0 (u))(t) ∞ ,
and the conclusion follows from Lemma 4.2.

Remark. As an application of this general result, let us observe that taking L = L ∞ , we obtain for u 0 ∈ E ∩ L ∞ , with u 0 small in the E-norm,

u τ → u as τ ց 0, uniformly in (x, t) ∈ R 2 × [0, ∞).
5 Blow up of complex valued solutions of the parabolic-parabolic system

Consider the system (PP) in the space R d with any d ≥ 1. Passing to the Fourier transforms, we may write the Duhamel formula [START_REF] Herrero | A blow-up mechanism for a chemotaxis model[END_REF] in the form u(ξ, t) = e -t|ξ| 2 u 0 (ξ)

+ t 0 s 0 R d ξ • η τ e -(t-s)|ξ| 2 e -1 τ (s-σ)|η| 2 u(ξ -η, s) u(η, σ) dη dσ ds. ( 36
)
Our goal is to construct a class of complex valued initial data, such that the corresponding solutions blow up in finite time, in any classical norm. For a ∈ R, we denote by Ḃa,∞ ∞ the homogeneous Besov space, which can also be identified with the Hölder-Zygmund space Ċa . As it is well known (see [START_REF] Lemarié-Rieusset | Recent Development in the Navier-Stokes Problem[END_REF][START_REF] Montgomery-Smith | Finite time blow up for a Navier-Stokes like equations[END_REF]), most of the classical functional spaces (including all homogeneous Triebel-Lizorkin, and thus Lebesgue or Sobolev spaces) are continuously embedded in Ḃa,∞ ∞ for some real a.

Theorem 5.1 There exists w 0 ∈ S(R d ) such that w 0 (ξ) ≥ 0, w 0 L 1 = 1.

If A is sufficiently large, then any (tempered) distributional solution of (36) with u 0 = Aw 0 (and thus any solution of (PP)) blows up in a finite time. More precisely, for some time t * < ∞ and each a

∈ R, u(t * ) Ḃa,∞ ∞ = ∞ holds.
Our approach is closely related to that in [5, Theorem 3.1] which followed the argument in [START_REF] Montgomery-Smith | Finite time blow up for a Navier-Stokes like equations[END_REF] for the "cheap" Navier-Stokes equations. We produce some estimates from below of the Fourier transform of any solution with u(0) = u 0 that can be obtained via the iteration procedure for (36) with

u k+1 = e t∆ u 0 - t 0 ∇ • e (t-s)∆ 1 τ u k (s)∇ s 0 e 1 τ (s-σ)∆ u k (σ) dσ ds,
for k = 1, 2, . . . . This recurrence relation, in general, does not preserve the positivity of the Fourier transform of u 0 . This leads us to restrict our attention to data of the form u 0 = A w 0 with w 0 ∈ S(R d ), such that supp

w 0 ⊂ ξ ∈ R d : 2 -1 ≤ ξ 1 ≤ |ξ| ≤ 1 . Define, for k = 0, 1, . . . the set E k = ξ ∈ R d : 2 k-1 ≤ ξ 1 ≤ |ξ| ≤ 2 k .
Then we see that for w k = w 2 k 0 , we have w k = (2π) -d w k-1 * w k-1 , and therefore, supp w k ⊂ E k . This implies that if, in addition, w 0 (ξ) ≥ 0, then the positivity of the Fourier transform will be preserved by the sequence u k , and so by u. Next Lemma tells us more: Lemma 5.1 For all k = 0, 1, 2, . . . , we have u(ξ, t) ≥ β k e -2 k t 1 t k ≤t<t * (t) w k (ξ), (37

)
where {β k } and {t k } are two sequences defined below in (38) and (39).

Proof. For k = 0, the conclusion immediately follows from u(ξ, t) ≥ Ae -t|ξ| 2 w 0 (ξ), provided we choose β 0 = A and t 0 = 0. Let k ≥ 1. Assume that the inequality of the lemma holds for k -1. Then, for all t k ≤ t < t This suggests us to set, for some δ > 0, t * -t k-1 = δτ 2 -2k+2 . With this choice, putting t 0 = 0, we have

t * = δτ, t k = δτ (1 -2 -2k ). (38)
Then, t k -t k-1 = 3δτ 2 -2k . We get, for t k ≤ t < t * , u(ξ, t) ≥ (3δτ -1 + e -4δτ )2 -2k-3 τ -1 e -δ β 2 k-1 e -2 k t w k (ξ).

This inequality is interesting only if the right hand side is positive. Therefore we will assume that 3δτ ≥ 1. We choose {β k } in such a way that β 0 = A, β k = (3δτ -1 + e -4δτ )2 -2k-3 τ -1 e -δ β 2 k-1 , k = 1, 2, . . . .

This choice leads to inequality (37).

In order to compute β k , we introduce M δ,τ , such that 2 M δ,τ = (3δτ -1 + e -4δτ )e -δ 2 -3 τ -1 .

Notice that we have β k = 2 M δ,τ -2k β 2 k-1 . We claim that, for some a, b, c ∈ R,

β k = A 2 k 2 a+bk+c2 k .
Indeed, from an easy calculation we find b = 2, a = 4 -M δ,τ and finally c = M δ,τ -4, which is needed to ensure β 0 = A. Hence, we have

β k = A 2 M δ,τ -4 2 k 2 4-M δ,τ +2k , k = 0, 1, . . . . (39)
We conclude that when A ≥ 2 4-M δ,τ , we have by (39) β k → ∞ and, in particular, u(t k ) L 1 = u(t k ) ∞ → ∞ for k → ∞. The above size condition on A can be rewritten in an equivalent form as (3δτ -1 + e -4δτ )A ≥ 2 7 e δ τ.

A further analysis of the lower bounds obtained for the Fourier transform of a candidate solution permits us to conclude, as was in [START_REF] Biler | Global existence versus blow up for some models of interacting particles[END_REF], that u(t * ) Ḃa,∞ ∞ = ∞ for each a ∈ R, so that all Besov (and also L p or Triebel-Lizorkin) norms of u blow up not later than t * . Notice that, for a blow-up at t * = 1, we need A ≥ Ce 1/τ τ , cf. (38).

Lemma 4 . 1

 41 Denote the Calderón operator by Λ = (-∆) 1 2

1 2

 1 -δ for all 0 ≤ τ ≤ 1.

  t) dy. The terms I 2 and I 3 are obtained by taking the integration domains {|x -y| ≤ |x|/2} and {|x -y| ≥ |x|/2, |y| ≥ |x|/2}, respectively, in the convolution integrals defining ∂

j ϕ. It is straightforward to prove that I 2 and I 3 can be bounded by C u X t 1 2 + |x| -1

  2k e -1 τ (t * -t k-1 )2 2k-2 β 2 k-1 e -2 k s w k (ξ) ds -t k-1 )2 2k-2 β 2 k-1 e -2 k t w k (ξ) -t k-1 )2 2k-2 β 2 k-1 e -2 k t w k (ξ) ≥ (t k -t k-1 ) -1 -e -(t * -t k-1 )2 2k 2 2k -t k-1 )2 2k-2 β 2 k-1 e -2 k t w k (ξ).

	Thus, we can bound u(ξ, t) from below as follows
	u(ξ, t) ≥ e -(t-s)2 ≥ t t k-1 (s -t k-1 ) 2 2k-3 τ t t k-1 (s -t k-1 )e -(t-s)2 2k ds 2 2k-3 τ τ (t = e -1 t -t k-1 2 2k -1 -e -(t-t k-1 )2 2k 2 4k 2 2k-3 τ e -1 τ (t 2 -3 τ e -1 τ (t
	u(ξ, t) ≥	t	s	ξ 1 η 1 τ	e -(t-s)|ξ| 2 e -1 τ (s-σ)|η| 2 β 2 k-1 e -2 k-1 s e -2 k-1 σ ×
		t k-1	t k-1 R d
					×	w k-1 (ξ -η) w k-1 (η) (2π) d	dη dσ ds
		t
	≥
		t k-1 R ) (2π) d	dη ds

* , d (s -t k-1 ) ξ 1 2 k-2 τ e -(t-s)|ξ| 2 e -1 τ (t * -t k-1 )2 2k-2 β 2 k-1 e -2 k s × × w k-1 (ξ -η) w k-1 (η* * *

+ 1 r , 1 < r < 2,and this concludes the proof of Proposition 4.2.

Acknowledgments This research has been supported by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge "Harmonic Analysis, Nonlinear Analysis and Probability" MTKD-CT-2004-013389, and by the Polish Ministry of Science (MNSzW) grant -project N201 022 32/0902.

where C > 0 depends only on L. This follows from [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] written as

for each τ ≥ 0, with the convention (18b). Therefore, using the usual estimate [START_REF] Blanchet | Infinite time aggregation for the critical Patlak-Keller-Segel model in R 2[END_REF], we have

The last factor is bounded by v X owing to Lemma 2.2 in the case τ = 0, and to Lemma 2.3 for τ > 0. This yields (35). Now, we can consider, for τ ≥ 0, the sequence of approximating solutions

When ǫ < min{ǫ, ǫ * }, we know by the proofs of Theorems 2.1 and 3.1 that the sequence u τ k converges in X to the solution u τ of (PE) or (PP). Here, of course, u = u 0 for the solutions of (PE).

On the other hand, applying recursively (35), we get u τ k (t) ∈ L for all k and sup Iterating this inequality we arrive at

where C ′ > 0 is independent on τ . Of course, the last claim remains valid in the case L = L 1 and u 0 ≥ 0 (notice that the smallness of u 0 E prevents blow up results that could occur, otherwise, when the second moment of u 0 are finite and u 0 > 8π. See the introduction and the references therein quoted). If we remove the assumption u 0 ≥ 0, we can obtain the same conclusion provided u 0 1 is sufficiently small. Indeed, we see from inequality (35) that the fixed point argument applies in the space L ∞ t (L 1 ) ∩ X .


est destinée au dépôt

Introduction

Bruno Jetin, Mia Mikic

To cite this version:

Bruno Jetin, Mia Mikic. Introduction. ASEAN Economic Community. A Model for Asia-wide Regional Integration?, Palgrave Macmillan, 2016. hal-03666971

Introduction

Much research and media attention is focused on the progress of regional integration among Southeast Asian countries. 1 Back in 1967, five of them formed Association of Southeast Nations (ASEAN) and by 1999 the group was completed with ten ASEAN Member States (AMS): Brunei Darussalam, Cambodia, Indonesia, Lao PDR, Malaysia, Myanmar, the Philippines, Thailand, Singapore and Viet Nam. Only one country from this sub-region is currently not a member to ASEAN, however, Timor-Leste formally applied to accede in 2011.

While ASEAN had been labeled as a political association at the time of its founding, it slowly but certainly developed interest in economic aspects of integration. Most prominently, since the establishment of ASEAN Free Trade Area (AFTA) back in 1992, much of the integration process revolved around economic and trade issues. ASEAN financial crisis and its impact on Asian countries contributed to AMS's realization of the need for a tighter regional entity to allow building more resilient economies. Thus it was not surprising to see the proclamation of ASEAN 2020 announced in 1997. Likewise, political and economic dynamics in the region and globally were aligned with the ideas of ASEAN the full realization of the AEC. In order to assist AMS in monitoring progress along the roadmap, ASEAN has introduced so-called the AEC Scorecard, a self-assessment mechanism, tracking progress of each AMS in each pillar. Original unwillingness to reveal publicly individual AMS scores has weakened but not disappeared, thus public still has no access to the most recent status. Based on the data available for 2012 scorecard, ASEAN is reaching only 82% of its final target in early 2015 so a big push will be necessary to improve this by the end of 2015. While some AMS are better positioned than others in terms of ticking items off the scorecard, 2 the actual progress will occur only when there is a critical mass of institutions to enforce these legal and regulative measures. At present, many challenges still remain at the implementation level of each pillar. This book "AEC 2015: A model for Asia-wide regional integration?" brings together scholars and researchers who have been studying ASEAN from close by or from a distance to provide their assessment of the AEC process and progress from a perspective of wider regional integration. While it was not possible to obtain a contribution for every aspect of the AEC process, we have tried to cover most important areas or those that are most relevant for the rest of Asia and globally either in terms of impacts or in terms of valuable lessons and practices which could be leaned on by those who are pursuing any type of regional integration. In what follows we provide the brief summary of the chapters. As AEC pillars themselves are not perfectly balanced in terms of areas they cover and policies they refer to in the blueprint, while not being fully independent of each other, this book is also slightly asymmetric with a bit more focus on pillars 1, 3 and 4. Chapters, however, often provide opportunity to a reader to make connections between the pillars and also over time of integration process. The summary of chapters is provided below under the headings of three parts of the book for ease of reference.

Part I -ASEAN economic integration in the context of East Asia regionalism

All reliable indicators show that the ASEAN Economic Community (AEC) will not be fully completed by its 2015 deadline. David Martin Jones in chapter 1 asks if this outcome does not test the limit of ASEAN's guiding propositions and cooperative practices -noninterference principle and consensus building, and non-confrontational bargaining -and their efficacy in integrating AMS's economies and facilitating its wider regional economic integration. In the past, these principles and practices were sufficient to achieve the main outcome to be credited to ASEAN: the achievement, until recently, of regional security and political stability. This was a necessary condition for the success of export-oriented growth and attraction of foreign investment. But a single market and production base requires that all AMS have a common interest and accept a higher degree of cooperation. This is probably what is lacking because the political elite of each AMS grants the monopoly of decisive domestic sectors to the economic elite and their interest are so intertwined that they will not accept easily the direct competition of neighboring firms. Jones also discussed what he calls the "sinification of the ASEAN way" and how it has also profoundly changed the capacity of ASEAN to deepen its integration. ASEAN's connectivity master plan will be funded in great part directly or indirectly by China to the extent that these new infrastructures serve Chinese interests which are not necessarily those of ASEAN. Worse, ASEAN capacity to provide regional security may be endangered by China's attempt to establish its domination over the South China Sea. The rise of China and the response of the other big powers in the regionthe USA, Japan, India and Australia-raise far more critical issues than the AEC can resolve.

These issues are analyzed in deeper details by the following chapters of the book.

Jean Raphaël Chaponnière and Marc Lautier also question the nature of ASEAN regional economic integration process in Chapter 2. They recall the creation of ASEAN in 1967, after several failed attempts, to be associated with the perception of rising communist threat. Twenty five years later, ASEAN launched AFTA whose objective was not to promote intra-regional trade but to enhance ASEAN countries' attractiveness for FDI. ASEAN countries have reduced trade barriers and made significant progress towards a de jure integration, while the process of de facto integration launched by the relocation of Japanese manufacturing firms in the mid-eighties continued. If one excludes intra-firm transactions and exports from free trade zones, AFTA would explain only one fifth of the intra-regional trade.

ASEAN members are now involved in two mega-regional agreements; one with a potential to protect ASEAN centrality, ASEAN+6 or Regional Comprehensive Economic Partnership (RCEP) and another, the US-led Trans-Pacific-Partnership Agreement.

In Chapter 3 Mia Mikic addresses the issues stipulated under AEC pillar 4 seeking deeper integration of ASEAN into the global economy with emphasis on ASEAN centrality and improved coherence of various agreements that may impact the operation of AEC. As of now, it appears that RCEP might be the chosen path towards seeking a necessary consolidation of all existing ASEAN's trade agreements. This will succeed only if RCEP evolves into a high-standard trade agreement and allows for rationalization of existing deals.

Upon providing current FTA landscape in Asia, this chapter examines the possible effects of the RCEP on trade not only of ASEAN+6 but also other Asian countries, taking into account the fact that all of the negotiating countries of the RCEP are already participants in other trade agreements, under implementation or under active negotiations.

Chapter 4 by Prema-chandra Athukorala examines emerging global production sharing (GPS) and trade patterns in light of the experiences of ASEAN countries which have been major and successful participants in GPS. 'Network products' (parts and components, and final assembly traded within production networks) constitute almost 2/3 of the merchandise exports of Singapore, Malaysia, and the Philippines, almost half those of Thailand, and a smaller but still significant share for Indonesia. GPS has certainly strengthened economic interdependence among ASEAN countries, and between them and China and the other major economies in East Asia, but this has not lessened the dependence on the global economy. The operation of the regional cross-border production networks depends inexorably on trade in final goods with the rest of the world. There is no evidence to suggest that forming RCEP would help them to enhance gains from the ongoing process of GPS fragmentation while reducing the dependence on the Western markets. GPS strengthen the case for unilateral and/or multilateral approach to trade reforms.

The ASEAN countries have experimented contrasted exchange rate regimes since the 1990s. The financial crisis of 2008 has given new interest to the question of monetary cooperation at the regional level. In Chapter 5, Jacques Mazier, Myonung Keun On and Nabil Aflik use a Fundamental Equilibrium Exchange Rate (FEER) approach, to estimate exchange rate misalignments and linked them to the external performances and growth of East Asian countries. They find that exchange rate misalignments are more limited in the current period than in the 1990s, in clear contrast with what is observed between European Union countries.

The economic consequences of alternative exchange rate regimes in East Asia are examined using a four-country stock flow consistent model of East Asia. The configuration of the 1990s and 2010s can be compared and alternative scenarios for the future of ASEAN integration are discussed.

Exploring further the exchange rates issues, Witada Anukoonwattaka connects those with the presence of GVCs in Chapter 6. The growing GVCs has changed ASEAN from exporting final goods to intermediates, and from exporting directly to advanced markets to exporting via downstream countries, particularly China. This chapter looks at how an analytical framework for evaluating the impacts of exchange rates on export competitiveness has been changed by the GVC phenomenon. Findings imply that entering to GVCs make a country prone to a change in the exchange rate of other countries even if they are not their direct trading partners. There seems to be compensating impacts on trade volume and trade product range. The net impact then becomes ambiguous and sector-and country-specific. Southeast Asian economies under the ASEAN Economic Community would benefit from further intensifying trade facilitation reform, among themselves but also with other Asian regions, keeping in mind that emphasis may best be placed on completing implementation of the many signed but often delayed intra-ASEAN agreements.

ASEAN, outsourcing has by and large been a catalyst of impressive economic growth, yet the thorough understanding of this issue remains limited. In Chapter 10 Aekapol Chongvilaivan explores the implications of burgeoning outsourcing activities in ASEAN on labor market development, namely the effects on labor productivity and skill premium. The findings from this analysis yield policy implications regarding how to utilize regional production networks as the impetus for labor development.

Chapter 11 by Teemu Puutio draws attention to the importance of creative economy for ASEAN. ASEAN has adopted a soft-regionalism approach to its regional integration efforts, preferring flexibility, non-interventionism and consensus based decision-making over sovereignty transfers. As a result, the regional creative economy persists to resemble a loosely knit patchwork of disparate national regimes for creativity and innovation that interact only sporadically through non-ASEAN led developments such as supply chains. Without decisive and centralized actions to harmonize institutions and bridge the resource and capability gaps, the creative economies of members with weak creative capacities and institutions will be foreshadowed by those of which have more sophisticated labor forces, stronger enabling legal frameworks and a more comprehensive network of supportive institutions. Consequently, weaker members may find themselves in "lowtechnology and creativity traps" with diminishing prospects of taking the next step upward.

Part III -Impact of regional integration on poverty, inequalities and social cohesion

Turning to addressing issues under AEC pillar 3, Marc Lautier's Chapter 12 examines social cohesion, economic resilience and prospects for long-term growth. While structural change has been the main engine of long-term catching-up processes, it increases the vulnerability of an economy to shocks. The domestic aptitude to adjust to shocks and to minimize growth losses is a major factor of development performance. Economic resilience depends mainly on social cohesion and State's effectiveness. Specific indicators for these two notions are provided for a large sample of developing countries. The comparative analysis demonstrates that, while Southeast Asian economies are a diverse group, most of them have a strong ability to sustain growth for long periods of time. As for development institutions and growth performances, the proximity between Southeast and East Asia is much stronger than between Southeast Asia and the rest of the developing world.

Bruno Jetin reminds that The AEC is committed to poverty reduction and the wellbeing of its people thanks to inclusive growth and equitable access to opportunity of human development. He assesses such a claim in chapter 13 through the lens of social cohesion. A society is socially cohesive when it combines three components: a low social exclusion, and a high level of trust and mobility. After a review of the long-term evolution of between and within-country inequality, he examines the recent evolution of absolute and relative poverty in ASEAN countries. He then maps social cohesion in ASEAN according to its three components, namely exclusion, social capital, and social mobility. It appears that convergence between ASEAN countries is recent and limited and that within-country inequality is high and sometimes growing. Relative poverty has substituted absolute poverty in some countries putting social cohesion at risk. His chapter concludes by delineating some country member profiles of social cohesion.

Natalie Fau in turn examines the role of infrastructural investment in reducing inequalities. According to ASEAN leaders, improved connectivity, especially through transport links, is an essential condition for economic growth in Southeast Asia. Furthermore, the upgrading and the construction of infrastructure and the harmonization of the regulatory framework would significantly narrow the development gap within ASEAN. It is precisely this hypothesis that chapter 14 is questioning, by focusing especially on the Master Plan on ASEAN Connectivity (MPAC) development projects for land (road and rail) and sea transport infrastructures. After presenting the main directions taken by the MPAC and the tools used to decrease territorial inequalities regarding provision of infrastructures, this paper attempts to assess on different scales (regional, sub-regional and local) the regions that have gained or lost since the MPAC was implemented and to explain the reasons for these disparities.

The last chapter of Part III contributed by Christine Cabasset focuses on the relationship between local governance and inequality. Indonesia has considerably improved its economic and socio-economic performance at the national level, especially since the 1997 Asian financial crisis. However, some internal weaknesses are obstacles for the country to achieve real leadership in Southeast Asia and beyond. Spatial and social inequality not only subsists, it has increased since 2003, particularly within provinces, within districts and within urban as rural areas. This rise exposes the archipelago to social risk at all administrative levels, including in the "wealthiest" provinces. Chapter 15 highlights some of the main factors explaining the difficulty national and local governance have to tackle poverty and inequality issue.

The Concluding chapter proposes that the 2015 deadline for the establishment of AEC should only be seen as one more milestone on the long journey towards an objective of deep economic integration not commonly found among developing countries in Asia. It definitely should not be seen as a final destination, because numerous challenges remain. Enforcement of the AEC accord will require changes to domestic laws or even national constitutions.

These would be considerable challenges for ASEAN member states beyond 2015. One of them is to maintain the purpose and centrality of ASEAN in Asia. During decades ASEAN has been the sole purely Asian regional institution where not only Southeast Asia countries

  leaders meeting at the 9 th Summit in 2003 to call for the ASEAN Community. Ultimately, this goal was advanced to be implemented in 2015 through the establishment of the ASEAN Economic Community (AEC) at the 12 th ASEAN Summit in 2007. The other two communities: ASEAN Political-Security Community and ASEAN Socio-Cultural Community are still planned for full implementation at a later stage. The AEC Blueprint was developed in 2007 to provide a roadmap for government entities involved in AEC implementation. It has four pillars driving a transformation of ASEAN into a single market and production base, a highly competitive economic region of equitable economic development, and a region that is fully integrated into the global economy by the end of 2015. Each of the four pillars presents a demanding set of challengeswith numerous individual actions to be taken by AMS by the end of December 2015 and for

  For instance, a currency depreciation of Yuan might reduce export product range from ASEAN, but export volume of each product remaining might increase. Last chapter of Part I, Chapter 7 by Yann Duval and Emilie Feyler takes stock of the progress made by ASEAN countries in reducing intra-and extra-regional trade costs using various cross-country indicators of trade facilitation performance and a new bilateral trade cost dataset developed by ESCAP and the World Bank. Despite significant improvements over time, trade costs other than tariffs among ASEAN members remain relatively high, with in particular a wide performance gap between Cambodia-Lao PDR-Myanmar and other ASEAN members. ASEAN has lower trade costs with North-East Asia than with itself.

Google reports about 1,330,000 results to the term "ASEAM Economic Community" on 14 April

2 For instance, recent political changes in Thailand resulted in much smoother and faster of process of transforming AEC policies and measures into domestic laws, as a first but necessary step of implementing AEC.

AEC 2015: A model for Asia-wide regional integration? Bruno Jetin and Mia Mikic

Part II -Impact of regional integration on structural change, employment and inequalities

Using the fact that trade integration is a cornerstone of the ASEAN Economic Community, Kee Kim Beom, Fan Zhai and Phu Huyuh set to review the structural changes that have taken place in the past decades in ASEAN Member States in Chapter 8. They use an innovative computable general equilibrium model to assess the impact of ASEAN trade integration on labour markets. The results show that trade liberalization contributes to sizeable increases in output and employment in ASEAN Member States, but that the benefits tend to vary by country, sectors, and gender. The mixed distributional effects point to the need for concerted employment and labour market policies, including improving access to education and training for vulnerable groups, strengthening the quality, coverage and sustainability of social protection systems and monitoring and managing the gender impacts of ASEAN trade integration Francis Cripps and Naret Khurasee in turn use a macro model based on historical series for the past four decades in Chapter 9 to project trade and GDP of ASEAN countries up to 2030 and confront the outcomes with trends in population structure and employment under different assumptions about policies in member countries. The projections imply that gaps in living standards will remain wide but suggest that exchange rate management, competition policy, agricultural policy and targeted government services and infrastructure could promote more inclusive growth and provide wider opportunities for provincial and rural populations left behind by export-led industrialization and services concentrated in large cities. In the context of the ASEAN Economic Community such policies may require closer coordination than hitherto.

Outsourcing can be loosely defined as the extent to which production activities are contracted out at arm's length, as opposed to being performed in-house. In the context of


Sylvain Colomer

Nicolas Cuperlier

Guillaume Bresson

Olivier Romain

Hierarchical Sparse Dictionaries: how sparsity, topology and pooling can improve visual place recognition

Context

• Visual place recognition (VPR) for self-driving vehicles [START_REF] Espada | From Neurorobotic Localization to Autonomous Vehicles[END_REF].

• Large-scale environment and computational cost.

• Neurocybernetics approach.

Contributions

• HSD, a sparse model for visual information encoding.

• HSD+MP, a full visual neuro-inspired localization model using HSD.

HSD : Hierarchical Sparse Dictionaries

General overview:

• Our unsupervised neural network aims to compress visual information.

HSD+MP : application to visual localization

• Modification of LPMP, a bio-inspired model of VPR [START_REF] Gaussier | Perac: A neural architecture to control artificial animals[END_REF], to integrate HSD .

• HSD replaces the log-polar block to compress landmarks representation.

Experiments

Evaluation methodology: Ability of models to recognize already visited places.

• Three distances of sampling on learning dataset : 2m, 3m and 5m.

• Six configurations of HSD : five balanced and one (HSD-15/30) unbalanced.

• Comparison with a state-of-the-art VPR model named CoHog. Metrics: AUC (area under the curve) of precision/recall curves and average processing frequency of an image. Dataset: Three routes with two distinct trajectories extracted from the KITTI dataset.

Results

AUC by configuration:

• Mean AUC increases with the size of HSD. Mean AUC by place sampling:

• HSD improved the mean AUC of LPMP by 7% for all the place sampling rates.

• Give similar results to

CoHog for all the place sampling rates, with a slight advantage for HSD+MP.

Learning time by configuration:

• Learning time of HSD increases according to the size of HSD.

Conclusion

• HSD improves computation cost and localization accuracy of LPMP.

• Allows a trade-off between localization and time performance.

• Relatively quick learning.

Future work

• Optimisation of HSD to achieve maximum performance.

• Complexification of the model : more layer, different pooling size etc.

• Evaluation on a more difficult dataset (Oxfordcar dataset).

  Hierarchical Sparse Dictionaries: how sparsity, topology and pooling can improve visual place recognition Sylvain Colomer 1,2 , Nicolas Cuperlier 2 , Guillaume Bresson 1 , Olivier Romain 2 1 Institut de recherche Vedecom, France ; 2 Laboratoire ETIS, CY Cergy Paris Université, ENSEA, CNRS, France