Martin Burger
email: martin.burger@fau.de

Philippe Laurençot
email: philippe.laurencot@math.univ-toulouse.fr

Ariane Trescases
email: ariane.trescases@math.univ-toulouse.fr

Delayed Blow-Up for Chemotaxis Models with Local Sensing

Keywords: Chemotaxis Models, Blow-Up, Nonlinear Cross-Diffusion Systems, Degenerate Parabolic Equations

The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same, however we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time.

Our observation is made rigorous from a mathematical point via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller-Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the (H 1 ) ′ -norm of the solutions, which can only grow with a square-root law in time. This additional (H 1 ) ′bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case.

Besides, regularity and uniqueness of solutions are also studied.

Introduction

The purpose of this paper is to study the well-posedness and some qualitative properties of the following system arising in the modeling of chemotaxis:

∂ t u = ∆(γ(v)u), in (0, ∞) × Ω, (1.1a)
∂ t v = ǫ∆v -βv + u, in (0, ∞) × Ω, (1.1b)
on a bounded domain Ω of R d (d ≥ 1) supplemented with no-flux boundary conditions

∂ ν [γ(v)u] = ∂ ν v = 0, on (0, ∞) × ∂Ω, (1.1c)
where ν = ν(x) is the outward unit normal vector field at the boundary ∂Ω, and nonnegative initial data u(0, x) = u 0 (x) ≥ 0, v(0, x) = v 0 (x) ≥ 0, in Ω.

(1.1d)

Here, u = u(t, x) ≥ 0 is the density of cells (or other organisms) producing a certain chemoattractant (or other field), its concentration being given by v = v(t, x) ≥ 0. The cell motility γ is given by γ(s) = e -s , for s ≥ 0, (1.1e) while the parameters ǫ and β are two positive constants representing the diffusivity of the chemoattractant and its degradation rate, respectively. Mathematical models of chemotaxis have been studied extensively in the last decades, based on several variants of the classical Patlak-Keller-Segel model (cf. [START_REF] Keller | Model for chemotaxis[END_REF][START_REF] Patlak | Random walk with persistence and external bias[END_REF]), and we refer to [START_REF] Bellomo | Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues[END_REF][START_REF] Hillen | A user's guide to PDE models for chemotaxis[END_REF][START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF][START_REF] Othmer | Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks[END_REF][START_REF] Perthame | Transport equations in biology[END_REF] for an overview of the modelling and basic results. The system (1.1a)-(1.1b) is a particular case of the well-studied system introduced by Keller and Segel in their series of three seminal papers (cf. [START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF][START_REF] Keller | Model for chemotaxis[END_REF][START_REF] Keller | Traveling bands of chemotactic bacteria: A theoretical analysis[END_REF]). In [START_REF] Keller | Model for chemotaxis[END_REF], the equation they propose to model the evolution of the cell density is the following

∂ t u = ∇ • (µ(v)∇u -uχ(v)∇v), in (0, ∞) × Ω,
where the cell diffusivity µ and the chemosensitivity χ are linked through

χ(v) = (α -1)µ ′ (v).
The coefficient α is a parameter related to the distance between the receptors in the cells with a suitable scaling. Equation (1.1a) is recovered when α is taken equal to zero. This choice corresponds to the case where there is a single receptor in the cell. Therefore, the cell is not able to perceive a gradient in the concentration by comparing the concentrations at two different spots (this is gradient sensing), it can only sense the local concentration at the point where it is located (this is local sensing). Let us mention the discussion of different sensing mechanisms in [START_REF] Othmer | Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks[END_REF] related to microscopic descriptions of gradient and local sensing, which lead to the derivation of the continuum models mentioned above. Let us also mention the models for direct aggregation of animals (cf. [START_REF] Burger | Individual based and mean-field modeling of direct aggregation[END_REF]) and social segregation dynamics (cf. [START_REF] Burger | Mean field models for segregation dynamics[END_REF]), which are based on similar paradigms. The system (1.1a)-(1.1b) (more precisely a more general version, possibly adding a logistic growth term in the cell equation) has been recently used in the Biophysics literature (cf. [START_REF] Fu | Stripe formation in bacterial systems with density-suppressed motility[END_REF]) in order to describe the formation of stripe patterns in colonies of bacteria observed in experiments (cf. [START_REF] Liu | Sequential establishment of stripe patterns in an expanding cell population[END_REF]). In this study, the cell motility γ is taken as a sharply degenerating function of the concentration (more precisely, a regularized version of step function decreasing to zero). Beyond simplicity, this choice is motivated by the experimental data showing that the cell motility gets close to zero above a certain threshold value of the chemoattractant concentration.

To illustrate the difference between the local sensing model and classical models, let us focus on the spatial dimension two. A key property of the minimal Patlak-Keller-Segel model is the ability to describe a mass-dependent threshold phenomenon: Cell aggregates only form if there is sufficient mass in the system, which is mathematically described by a blow-up of the solution. This was studied in particular for the model

∂ t u = ∇ • (∇u -u∇v), in (0, ∞) × Ω, ∂ t v = ǫ∆v -βv + u, in (0, ∞) × Ω, (1.2)
and for its quasi-stationary version (with the second equation replaced by its elliptic counterpart). For these models, there is global existence and boundedness of solutions for small mass, whereas finite time blow-up could be shown as soon as the mass exceeds a critical threshold (cf. [START_REF] Biler | Existence and nonexistence of solutions for a model of gravitational interaction of particles[END_REF][START_REF] Herrero | A blow-up mechanism for a chemotaxis model[END_REF][START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF][START_REF] Nagai | Global existence and blow-up of radial solutions to a parabolicelliptic system of chemotaxis[END_REF]). It is hence an interesting question to investigate whether the system (1.1a), (1.1b) has the same behaviour. This question is even more motivated from the fact that both models share the same entropy (or Lyapunov functional), namely

E(u, v) := ( ǫ 2 |∇v| 2 + β 2 |v| 2 -uv + u log u -u + 1) dx. (1.
3)

The minimal Keller-Segel model (1.2) is a gradient flow of the form

∂ t u = ∇ • (u∇∂ u E), in (0, ∞) × Ω, ∂ t v = -∂ v E, in (0, ∞) × Ω,
for which the entropy has the dissipation (formally)

d dt E(u(t), v(t)) = - Ω u|∇ log(ue -v )| 2 dx - Ω |ǫ∆v + u -βv| 2 dx.
The system (1.1a), (1.1b) is of the form

∂ t u = ∆(e ∂uE -1), in (0, ∞) × Ω, ∂ t v = -∂ v E, in (0, ∞) × Ω,
and a simple computation (formally) gives the dissipation

d dt E(u(t), v(t)) = - Ω 4|∇ √ e -v u| 2 dx - Ω |ǫ∆v + u -βv| 2 dx.
This generalized gradient structure also motivates the specific choice of γ in (1.1e). The natural assumptions the motility γ should satisfy from the point of view of the modeling is that it should be typically a nonnegative decreasing function decaying to zero, the above generalized gradient structure is obtained only for an exponential dependence, which also corresponds to the use of Boltzmann statistics in the microscopic local sensing models.

Based on the properties of the entropy, which is bounded below only for the case of subcritical mass (which is related to logarithmic functional inequalities, cf. [START_REF] Dolbeault | Optimal critical mass in the two-dimensional Keller-Segel model in R 2[END_REF]), it is not surprising that there is a similar threshold behaviour for (1.1a), (1.1b), in particular the existence of bounded solutions in the subcritical regime. The latter has recently been established in terms of classical solutions in [START_REF] Jin | On the Keller-Segel system with signal-dependent motility[END_REF] and will be further developed for weaker solutions below. The surprising result we obtain in this paper is however a global existence result for arbitrary masses (which actually holds for all dimensions). This implies that the blow-up is delayed compared to the minimal Keller-Segel model, it arises in the infinite time limit instead of finite time. From a modelling point of view it confirms the slower evolution via local sensing compared to the gradient sensing mechanism. From the point of view of mathematical analysis, the form (1.1a), sometimes referred to as Laplace form, allows the use of some duality techniques which give an estimate that is crucial for the global existence. More precisely, this duality estimate allows us to derive a lower bound for the entropy (for finite time). Such estimate and its interplay with the entropy will be explained at the formal level in the next section.

Infinite-time blow-up has been identified in other variants of the classical Keller-Segel system, but is then due to different mechanisms. For instance, in [START_REF] Cieślak | Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions[END_REF][START_REF] Cieślak | New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models[END_REF], the first equation is nonlinear in u and linear in v; that is, the motility and the chemosensitivity are densitydependent (but independent on the chemical concentration), which models a volume-filling effect. This volume-filling effect slows down the aggregation and, if strong enough and suitably tuned, can finally prevent finite-time blow-up without implying boundedness of the solution. In our study, the nonlinearity in v (together with the specific laplacian-form) accounts for the nonlinear chemosensitivity of the organism, which models the saturation of the receptor(s) of the organism. In particular there is no volume-filling effect. Note also that our model has the same entropy as the original Keller-Segel model, it can be viewed as the natural counterpart when replacing the mechanism of gradient sensing by local sensing. Another example of delayed blow-up may be found in [START_REF] Ph | Global bounded and unbounded solutions to a chemotaxis system with indirect signal production[END_REF][START_REF] Tao | Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production[END_REF] and is due in that case to the indirect production of the chemoattractant; that is, the chemoattractant is no longer produced by the cells, but by an intermediate species.

Main results

Our first two results concern the well-posedness of the system in any dimension. For this we first introduce some notations and the notion of solution that we use.

Notations. In the following, we consider a bounded domain Ω in R d , that we assume of measure one (|Ω| = 1) for simplicity. We denote by 1 the constant function on Ω, and for a function space V on Ω such that 1 is in the predual space of V , we denote by V ⋄ the subspace of elements of V with mean value zero, i.e.

V ⋄ = {v ∈ V | v, 1 = 0}.
For an element v ∈ V we denote its mean value by v = v, 1 . This way the notation is also defined for duals of Sobolev spaces such as the dual space (H 1 ) ′ (Ω) of H 1 (Ω). We denote by K = (-∆) -1 : (H

Ω ∇w • ∇ϕ dx = f, ϕ , ∀ ϕ ∈ H 1 ⋄ (Ω).
We assume that ∂Ω is sufficiently regular so that K maps continuously L 2 ⋄ (Ω) into H 2 ⋄ (Ω). We still denote by K its extension from D(-∆) ′ ⋄ to L 2 ⋄ (Ω), where D(-∆) := {ϕ ∈ H 2 (Ω) : ∂ ν ϕ = 0 on ∂Ω} is the domain of the Laplace operator with homogeneous Neumann boundary conditions.

Definition 1.1 ((Very) weak solution). Let T > 0 and let

u ∈ L ∞ (0, T ; L 1 (Ω)) ∩ L ∞ (0, T ; (H 1 ) ′ (Ω)), v ∈ L ∞ (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; (H 1 ) ′ (Ω))
We call (u, v) a weak solution of System (1.1) on (0, T ) if e -v u ∈ L 2 ((0, T ) × Ω),

K∂ t u = -e -v u + e -v u in L 2 ((0, T ) × Ω), (1.4)
∂ t v, ϕ + Ω (ǫ∇v • ∇ϕ + βvϕ) dx = u, ϕ , ∀ϕ ∈ H 1 (Ω), a.e. in (0, T ), (1.5)
and, setting m := u 0 ,

K(u(0, •) -m) = K(u 0 -m) in L 2 (Ω), v(0, •) = v 0 in (H 1 ) ′ (Ω). (1.6)
Notice that we interpret Equation (1.1a) in a very weak sense, reminiscent to the treatment of the porous medium or similar nonlinear diffusion equations (cf., e.g., [START_REF] Vázquez | The porous medium equation[END_REF]). The initial condition (1.6) makes sense since ∂ t K(u-m) = K∂ t u ∈ L 2 ((0, T )×Ω) by (1.4) and

K(u-m) ∈ L ∞ (0, T ; H 1 (Ω)).
Definition 1.2 (Weak-strong solution). Let T > 0. A weak-strong solution of System (1.1) on (0, T ) is a weak solution (u, v) which satisfies additionally

∂ t v = ǫ∆v -βv + u in L 2 ((0, T ) × Ω).
For u and v defined on (0, +∞) × Ω, we call (u, v) a global weak-strong solution of System (1.1) if it is a weak-strong solution on (0, T ) for all times T > 0.

Our main result of existence of global solution is the following.

Theorem 1.3 (Existence of global solution). Let ǫ > 0, β ≥ 0, and let Ω ⊂ R d (d ≥ 1)
be a smooth bounded domain of measure one. Assume that the initial conditions satisfy the following:

u 0 ∈ L 1 (Ω) ∩ (H 1 ) ′ (Ω), u 0 ≥ 0, m := Ω u 0 dx, Ω u 0 log u 0 m dx < ∞, (1.7
)

and v 0 ∈ H 1 (Ω), v 0 ≥ 0. (1.8)
Then there exist two nonnegative functions u, v with

u ∈ L ∞ (0, T ; L 1 (Ω)) ∩ L ∞ (0, T ; (H 1 ) ′ (Ω)), v ∈ L ∞ (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)),
for all times T > 0, such that (u, v) is a global weak-strong solution of (1.1). Furthermore it satisfies, for all t > 0,

Ω u(t) dx = m, u(t) -m 2 (H 1 ) ′ (Ω) + 2 t 0 Ω e -v u 2 dx ds ≤ u 0 -m 2 (H 1 ) ′ (Ω) + 2 m 2 t,
and

Ω ( ǫ 4 |∇v| 2 + β 4 |v| 2 + u log u -u + 1) dx + 4 t 0 Ω |∇ √ e -v u| 2 dx ds + t 0 Ω |ǫ∆v + u -βv| 2 dx ds ≤ E(u 0 , v 0 ) + 1 ǫ u 0 -m 2 (H 1 ) ′ (Ω) + 2 m 2 t + m 2 β .
Remark 1.4 (Regularity of the domain). The regularity required for the boundary of the domain is the minimal regularity such that (I -∆) -1 (the self-adjoint solution operator for the homogeneous Neumann problem) maps H m (Ω) into H m+2 (Ω) for all m ∈ {0, . . . , 2d -1}. This property is used in the approximation procedure in Section 3.

Remark 1.5 (The parabolic-elliptic problem). For ǫ > 0, β ≥ 0, Ω ⊂ R d (d ≥ 1) a smooth bounded domain of measure one, and an initial condition u 0 satisfying (1.7), we consider the parabolic-elliptic problem

∂ t u = ∆(e -v u), in (0, ∞) × Ω, 0 = ǫ∆v -βv + u, in (0, ∞) × Ω. (1.9)
We expect that our methods of proof of existence hold for this problem with minor adaptations, so that there exists a global weak solution of (1.9) with no-flux boundary conditions and initial data u 0 . We give in Section 3.4 some elements in this direction.

Having verified the global existence of solutions it is natural to consider the question of uniqueness. Indeed we are able to prove uniqueness by the following result, with the restriction that we need to assume minimal additional regularity of u.

Theorem 1.6 (Uniqueness of weak-strong solution). Let ǫ > 0, β ≥ 0, and let Ω ⊂ R d (d ≥ 1) be a smooth bounded domain of measure one. There is at most one weak-strong solution

(u, v) to (1.1) such that u ∈ L ∞ (0, T ; L q (Ω)) for all T > 0 , with q > 2 when d ∈ {1, 2} and q = d when d ≥ 3.
Due to the additional regularity needed for the uniqueness proof it is natural to investigate additional regularity, possibly with further conditions on the initial values. The results we can obtain in this case depend on the dimension however, so we will distinguish different cases. We start with spatial dimension one for completeness and then discuss the two-dimensional case, which appears to be the most interesting for applications and the expected mass-dependent blow-up.

Proposition 1.7 (Regularity, dimension 1). Let d = 1 and, in addition to assumptions of Theorem 1.3, u 0 ∈ L r (Ω) for r ≥ 1. Then, for any T > 0, u ∈ L ∞ (0, T ; L r (Ω)) and v ∈ L ∞ ((0, T ) × Ω). In particular, for u 0 ∈ L r (Ω) with r > 2, the solution is unique.

In the following we consider the case d = 2, the main result we prove is again a uniform bound on u and v for finite time, given the initial values are bounded:

Proposition 1.8 (Regularity, dimension 2). Let d = 2 and, in addition to assumptions of Theorem 1.3, u 0 ∈ L ∞ (Ω), v 0 ∈ L ∞ (Ω). Then, for any T > 0, u ∈ L ∞ ((0, T ) × Ω) and v ∈ L ∞ ((0, T ) × Ω).
As a consequence the solution is unique.

A striking consequence of this result, compared to the minimal Keller-Segel model, is the absence of finite-time blow-up (in L ∞ ) in dimension 2. Indeed, Proposition 1.8 rules out finite time blow-up in L ∞ for arbitrary initial data and extends [START_REF] Jin | On the Keller-Segel system with signal-dependent motility[END_REF]Theorem 1.1], where the existence of global solutions is established only when m ∈ (0, 4πǫ). Under this additional condition, it is also shown in [20, Theorem 1.1] that (u, v) ∈ L ∞ ((0, ∞) × Ω; R 2 ), see also [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF]Theorem 2.3]. Also, combining Proposition 1.8 and [20, Theorem 1.1] guarantees that there are initial data with m > 4πǫ for which the corresponding solution is global and unbounded, see also [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF]Theorem 2.3]. Finally, Proposition 1.8 also provides an alternative proof of [13, Theorem 2.1] for the specific choice γ(s) = e -s , s ≥ 0.

Litterature on System (1.1)

The mathematical analysis of System (1.1a)-(1.1b) has been tackled in some very recent works. In [START_REF] Tao | Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system[END_REF] the authors choose the motility function γ to be smooth and both upper and lower bounded by some positive constants. Under these assumptions, they obtain weak solutions in all spatial dimensions d ≥ 2, and, in dimension d = 2, assuming furthermore that the domain Ω is convex, they obtain smooth solutions (and also in dimension d ≥ 3 under an appropriate smallness assumption on the initial data). In [START_REF] Desvillettes | A logarithmic chemotaxis model featuring global existence and aggregation[END_REF], the motility γ is taken slowly algebraically decreasing (that is, with a constraint on the algebraic rate which depends on the spatial dimension). Weak solutions are obtained in spatial dimensions d = 1, 2, 3 (actually smooth solutions in dimension d = 1). The long-time behaviour is also explored through a numerical and linear stability analysis. We also mention the work [START_REF] Yoon | Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion[END_REF] where smooth solutions are obtained in the two-dimensional case for a motility γ taken sufficiently small. Both works [START_REF] Desvillettes | A logarithmic chemotaxis model featuring global existence and aggregation[END_REF][START_REF] Tao | Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system[END_REF] rely on duality and energy techniques (but not on an entropy in the strict sense of the term). Shortly afterwards, in [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF], smooth solutions are obtained in dimension d = 2 for a very generic motility (positive and decreasing to zero). Furthermore, the authors obtain uniformly bounded smooth solutions if the motility γ(v) decays at most algebraically in dimension d = 2, and at most in 1/v in dimension d = 3. Their method relies on the introduction of an adequate auxiliary elliptic problem that enjoys a comparison principle.

When it comes to the case where the motility has precisely the exponential decay (1.1e), it has been studied specifically in two-dimensional domains in [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF][START_REF] Jin | On the Keller-Segel system with signal-dependent motility[END_REF]. Both works focus on smooth solutions and the question of the possibility of blow-up. They show the existence of a critical mass such that, in the subcritical case, there exist uniformly bounded smooth solutions, and convergence to a single stationary solution is further shown in [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF]. In the supercritical case, they obtain the blow-up in L ∞ , which is furthermore proved to happen only in infinite time in [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF].

Compared to these results, one added value of the present work is first that we obtain global solutions in any dimension d ≥ 1. Furthermore, our solutions are weak (more precisely weak-strong) solutions, allowing to consider more general initial data. Note that one new difficulty that we encounter when dealing with weak solutions is the construction of an adequate approximation procedure. A crucial step is to define an appropriate approximate system that conserves both the entropy and the duality structures, which are of very different nature. Note that we can connect our weak solutions with the strong solutions obtained in [START_REF] Fujie | Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities[END_REF][START_REF] Jin | On the Keller-Segel system with signal-dependent motility[END_REF] thanks to our regularity and uniqueness results.

Finally, we refer to [START_REF] Ahn | Global well-posedness and stability of constant equilibria in parabolic-elliptic chemotaxis systems without gradient sensing[END_REF][START_REF] Fujie | Global existence for a kinetic model of pattern formation with density-suppressed motilities[END_REF] for studies of the quasi-stationary model, where the second equation is replaced by its elliptic counterpart.

Organization of the paper

In Section 2 we discuss the basic properties of the model and present some a priori estimates, which we make rigorous in Section 3. The latter is devoted to the global existence of weak solutions, which we derive by an approximation argument that preserves the generalized gradient structure and only modifies the cross-term in the entropy. In Section 4 we investigate further regularity and uniqueness. Finally we conclude with some remarks and outlooks in Section 5.

Structure of the model and a priori estimates

In the following we discuss the mathematical structure of the model and formally derive a priori estimates, which will be made rigorous in existence proofs in Section 3.

Since we are looking for solutions having a finite entropy E as defined in (1.3), we restrict ourselves to meaningful initial values that lead to E(u 0 , v 0 ) < ∞, namely we assume that u 0 and v 0 satisfy (1.7) and (1.8), respectively.

Basic a priori estimates

As mentioned in the introduction the system has some formal gradient structure and natural mass preservation, so we can formally derive estimates from these structures. We will assume sufficient regularity to justify the computations in this section and make the results rigorous later.

Mass. Let t > 0. First of all, the system preserves nonnegativity and the mean value of u, thus we obtain

u(t) L 1 (Ω) = ū(t) = m,
and d dt v(t) - m β = m -βv(t), which implies v(t) = m β + e -βt v 0 - m β .
In particular since v is also nonnegative

v(t) L 1 (Ω) ≤ max{ m β , v 0 L 1 (Ω) }.
Entropy dissipation. We recall the definition (1.3) of the entropy:

E(u, v) = ( ǫ 2 |∇v| 2 + β 2 |v| 2 -uv + u log u -u + 1) dx,
and emphasize that it is in general not bounded from below due to the negative and quadratic term -uv, see [START_REF] Bellomo | Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues[END_REF][START_REF] Hillen | A user's guide to PDE models for chemotaxis[END_REF][START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF][START_REF] Othmer | Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks[END_REF][START_REF] Perthame | Transport equations in biology[END_REF]. From the entropy dissipation we obtain the following a priori

identity d dt E(u(t), v(t)) = -4 Ω |∇ √ e -v u| 2 dx - Ω |ǫ∆v + u -βv| 2 dx.
Note that, from the entropy dissipation, we expect that ue -v and ǫ∆v + u -βv have increased regularity, provided the entropy is bounded from below. This gives a particular motivation for the definition of weak-strong solutions given above.

Duality estimate. Finally, we have an estimate in (H 1 ) ′ , which follows by testing Equation (1.4) with u -m. We see that

1 2 d dt u -m, K(u -m) = -u -m, e -v u -e -v u = - Ω e -v u 2 dx + m e -v u.
Owing to the nonnegativity of u and v,

e -v u = Ω e -v u dx ≤ Ω u dx = m,
so that, for all T > 0,

1 2 sup [0,T ] u -m 2 (H 1 ) ′ (Ω) + T 0 Ω e -v u 2 dx dt ≤ 1 2 u 0 -m 2 (H 1 ) ′ (Ω) + m 2 T. (2.1)

Entropy on (H 1 ) ′ -bounded sets

A key property of the entropy is that, due to the nonlinear negative term -uv, it is in general not bounded from below, except when d = 1 or d = 2 in the subcritical case. This feature leads to the occurrence of finite time blow-up for the minimal Keller-Segel model in the supercritical case when d = 2 and in higher spatial dimensions d ≥ 3. Since the model (1.1a), (1.1b) has the same entropy functional, a similar behaviour is to be expected. However, the key difference is that the changed dynamics results in a bound on the (H 1 ) ′ -norm of u for finite time, and the entropy is bounded below if there is such an additional (H 1 ) ′ -bound. Indeed, suppose that two nonnegative functions u and v defined on Ω satisfy

u -m (H 1 ) ′ (Ω) ≤ C,
for some positive constant C. Then we can estimate

- Ω uv dx = - Ω (u -m)v dx - Ω mv dx = Ω v ∆K(u -m) dx - Ω mv dx = - Ω ∇K(u -m) • ∇v dx - Ω mv dx ≥ - C 2 ǫ - ǫ 4 Ω |∇v| 2 dx - m 2 β - β 4 Ω |v| 2 dx.
As a consequence, we can estimate the entropy from below by

E(u, v) = Ω ǫ 2 |∇v| 2 + β 2 |v| 2 -uv + u log u -u + 1 dx ≥ - C 2 ǫ - m 2 β + Ω ǫ 4 |∇v| 2 + β 4 |v| 2 + u log u -u + 1 dx,
where the last integral is clearly nonnegative. Thus, boundedness of the entropy functional implies an upper bound on the H 1 -norm of v and on the logarithmic entropy of u for every finite time. This holds true even in the supercritical case, hence if blow-up occurs, then it has to be postponed to the infinite time limit.

Global existence of weak solutions

In the following we prove the existence of global solutions (Theorem 1.3) by an appropriate approximation procedure. Since we have several a priori estimates based on the entropy dissipation and the structure of the system ((H 1 ) ′ estimates in the first equation), it is important to keep these basic structures in the approximation. To this end, for ν ∈ (0, 1), we introduce the system

∂ t u = ∆(e -Lν v u), in (0, ∞) × Ω, (3.1a)
∂ t v = ǫ∆v -βv + L ν u, in (0, ∞) × Ω, (3.1b)
together with no-flux boundary conditions

∂ ν [e -Lν v u] = ∂ ν v = 0, on (0, ∞) × ∂Ω, (3.1c)
and initial data (u 0 , v 0 ). Here, L ν = Λ d ν (with d the spatial dimension) and Λ ν = (I -ν∆) -1 is the positivity-preserving self-adjoint (positive definite) solution operator for the homogeneous Neumann problem with small diffusion coefficient ν, defined from (H 1 ) ′ (Ω) onto H 1 (Ω). Note that with our regularity conditions on the domain Ω, Λ ν also maps continuously H m (Ω) into H m+2 (Ω) for all m ∈ {0, . . . , 2d -1}. As a consequence, L ν maps continuously (H 1 ) ′ (Ω) into H 2d-1 (Ω) ֒→ C(Ω). In the limit ν → 0, L ν → I and we expect to recover the original system.

The approximate system (3.1) preserves the gradient structure but only mollifies the entropy to

E ν (u, v) = Ω ( ǫ 2 |∇v| 2 + β 2 |v| 2 -(L ν u)v + u log u -u + 1) dx. (3.2)
Therefore we can derive rigorously at the level of the approximate system some uniform-in-ν estimates which are reminiscent of the previously described a priori estimates for the original system. The basic idea of the proof is to first establish the existence of solutions for (3.1) via a fixed-point technique and then use uniform-in-ν estimates to pass to the limit ν → 0.

Existence for the approximate system

Our first objective is to establish the following existence result for the approximate problem 3.1:

Theorem 3.1. Let T > 0. Let u 0 ∈ L 2 (Ω), v 0 ∈ H 1 (Ω) both be nonnegative with m := u 0 > 0.
Then, there exists a weak solution

u ν ∈ L 2 (0, T ; H 1 (Ω)) ∩ L ∞ (0, T ; L 2 (Ω)) ∩ H 1 (0, T ; (H 1 ) ′ (Ω)), v ν ∈ C([0, T ]; H 1 (Ω)) ∩ C 1 ([0, T ]; (H 1 ) ′ (Ω))
of (3.1) with initial condition u 0 , v 0 . Moreover, u and v are nonnegative almost everywhere on (0, T ) × Ω and, for all t ∈ [0, T ],

u ν (t) = m, u ν (t) -m (H 1 ) ′ (Ω) ≤ u 0 -m 2 (H 1 ) ′ (Ω) + 2 m 2 T .
The proof is based on an application of Schauder's fixed point theorem and a comparison principle to establish the additional nonnegativity. We consider the fixed-point map

F = I • G, with G : C([0, T ]; (H 1 ) ′ (Ω)) → L 2 (0, T ; H 1 (Ω)) ∩ L ∞ (0, T ; L 2 (Ω)) ∩ H 1 (0, T ; (H 1 ) ′ (Ω)) ũ → u,
where (u, v) solves

∂ t u = ∆(e -Lν v u) = ∇ • (e -Lν v ∇u -ue -Lν v ∇L ν v), (3.3)
∂ t v = ǫ∆v -βv + L ν ũ (3.4)
with no-flux boundary conditions and initial condition (u 0 , v 0 ). The operator I is simply the embedding operator

I : L 2 (0, T ; H 1 (Ω)) ∩ L ∞ (0, T ; L 2 (Ω)) ∩ H 1 (0, T ; (H 1 ) ′ (Ω)) → C([0, T ]; (H 1 ) ′ (Ω)). Lemma 3.2. Let u 0 ∈ L 2 (Ω) be nonnegative, v 0 ∈ H 1 (Ω).
Then the operator G is well-defined and continuous. Moreover, the solution u is nonnegative almost everywhere in (0, T ) × Ω and the same holds for v if v 0 and ũ are nonnegative.

Proof. Let ũ ∈ C([0, T ]; (H 1 ) ′ (Ω)). First of all, L ν ũ ∈ C([0, T ]; H 2d-1 (Ω)). Since v 0 ∈ H 1 (Ω), classical solvability of Eq. (3.4) yields a unique solution v in C([0, T ]; H 1 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)). By (3.4), ∂ t v ∈ C([0, T ]; (H 1 ) ′ (Ω)), so that v ∈ C([0, T ]; H 1 (Ω)) ∩ C 1 ([0, T ]; (H 1 ) ′ (Ω)).
Moreover, if ũ and v 0 are nonnegative, then so are L ν ũ and v by the comparison principle.

Next,

L ν v ∈ C([0, T ]; H 1+2d (Ω)) and thus L ν v ∈ C([0, T ]; W 1,∞ (Ω)).
Consequently, e -Lν v is an element of L ∞ ((0, T )×Ω) which is positive and bounded away from zero, while ∇e -Lν v = -e -Lν v ∇L ν v is also an element of L ∞ ((0, T ) × Ω). Thanks to these properties and the nonnegativity of u 0 , the existence and uniqueness of a nonnegative weak solution

u ∈ L 2 (0, T ; H 1 (Ω)) ∩ L ∞ (0, T ; L 2 (Ω)) ∩ H 1 (0, T ; (H 1 ) ′ (Ω))
of Eq. (3.3) follow from [START_REF] Aronson | Non-negative solutions of linear parabolic equations[END_REF].

The continuous dependence of u on L ν v and hence on v follows from a standard difference technique using the bounds of solutions in terms of the coefficients in [START_REF] Aronson | Non-negative solutions of linear parabolic equations[END_REF].

Using the compact embedding of L 2 (Ω) in (H 1 ) ′ (Ω), an application of a version of the Aubin-Lions-Simon lemma (cf. [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 4]) further yields: Lemma 3.3. The embedding operator I is compact.

As a consequence of the two above lemmas, we see that the operator F is compact and continuous from C([0, T ]; (H 1 ) ′ (Ω)) into itself. In order to apply Schauder's fixed point theorem it remains to verify that it maps a closed bounded convex set into itself, which is the purpose of the following lemma. Lemma 3.4. Let u 0 ∈ L 2 (Ω) be nonnegative with m = u 0 and let v 0 ∈ H 1 (Ω) be nonnegative. For R ≥ m > 0, we introduce the bounded, closed, and convex sets

M R := {u ∈ C([0, T ]; (H 1 ) ′ (Ω)) | u(t) -m (H 1 ) ′ (Ω) ≤ R and u(t) = m, for t ∈ [0, T ]} and M + R := {u ∈ M R | u(t) ≥ 0 in D ′ (Ω), for t ∈ [0, T ]}. Then for R ≥ m sufficiently large, the operator F maps M + R into itself.
Proof. Let ũ ∈ M + R for some R ≥ m to be specified later. We show that u = F(ũ) belongs to M + R . Since the solution u of (3.3) is nonnegative, and since Equation (3.3) is of divergence form with no-flux boundary conditions, we readily obtain

u(t) = m, for all t ∈ [0, T ].
Owing to the regularity of u and v established in the proof of Lemma 3.2, Equation (3.3) may be equivalently written K∂ t u = -e -Lν v u + e -Lν v u, so that, using the nonnegativity of L ν v, we may proceed as in the derivation of the estimate (2.1) and obtain

u -m C([0,T ];(H 1 ) ′ (Ω)) ≤ u 0 -m 2 (H 1 ) ′ (Ω) + 2 m 2 T .
Thus, u lies in

M + R provided R ≥ u 0 -m 2 (H 1 ) ′ (Ω) + 2 m 2 T ,
and the proof is complete.

We are now able to conclude the proof of the existence theorem:

Proof of Theorem 3.1. The proof directly follows from the application of Schauder's fixed point theorem to the operator F which is possible thanks to Lemmas 3.2, 3.3 and 3.4.

Uniform estimates for the approximate system

In the following we collect the uniform-in-ν estimates we can get for the approximate solutions (u ν , v ν ) of (3.1). They will be helpful to pass to the limit ν → 0 in the next subsection.

Proposition 3.5. Let T > 0. Let u 0 ∈ L 2 (Ω), v 0 ∈ H 1 (Ω) both be nonnegative, and let (u ν , v ν ) be a weak solution of (3.1) given by Theorem 3.1. Then, (u ν , v ν ) satisfies the following estimates:

u ν (t) L 1 (Ω) = m, t ∈ [0, T ], (3.5)
u ν L ∞ (0,T ;(H 1 ) ′ (Ω)) ≤ C(T ), (3.6)
sup

t∈[0,T ] u ν (t) log u ν (t) m dx ≤ C(T ), (3.7)
v ν L ∞ (0,T ;H 1 (Ω)) ≤ C(T ), (3.8)
-ǫ∆v ν + βv ν -L ν u ν L 2 ((0,T )×Ω) ≤ C(T ), (3.9)
∂ t v ν L 2 ((0,T )×Ω) ≤ C(T ), (3.10)
e -Lν vν u ν L 2 ((0,T )×Ω) ≤ C(T ), (3.11)
K∂ t u ν L 2 ((0,T )×Ω) ≤ C(T ), (3.12)
for some constant C(T ) depending only on the initial data (more precisely on m, u 0 (H 1 ) ′ (Ω) , Ω u 0 log(u 0 /m) dx and v 0 H 1 (Ω) ) and time T , but independent of ν.

From Theorem 3.1 we already know that the solutions we obtained satisfy (3.5) and (3.6). We furthermore have the following dual estimate. Lemma 3.6. For almost every t ∈ (0, T ), we have

u ν (t) -m 2 (H 1 ) ′ (Ω) + 2 t 0 Ω e -Lν vν u 2 ν dx ds ≤ u 0 -m 2 (H 1 ) ′ (Ω) + 2 m 2 t. (3.13)
Proof. Thanks to the nonnegativity of L ν v ν , we can proceed as in (2.1) and we obtain the result.

To obtain the other bounds we rely on the entropy dissipation.

Lemma 3.7. Recall the definition of E ν in (3.2). Then for almost every t ∈ (0, T ), we have the estimate

E ν (u ν (t), v ν (t)) + 4 t 0 Ω |∇ e -Lν vν u ν | 2 dx ds + t 0 Ω |ǫ∆v ν + L ν u ν -βv ν | 2 dx ds ≤ E ν (u 0 , v 0 ). (3.14)
Proof. First of all, we use the properties v 0 ∈ H 1 (Ω) and L ν u ν ∈ L 2 ((0, T ) × Ω) to show that ∂ t v ν and ∆v ν both lie in L 2 ((0, T ) × Ω). Now we can employ a standard proof to derive entropy dissipation estimates: for δ > 0, using

log(δ + u ν ) -L ν v ν = log((δ + u ν )e -Lν vν ) ∈ L 2 (0, T ; H 1 (Ω))
as test function in (3.1a) and

-ǫ∆v ν -L ν u ν + βv ν ∈ L 2 ((0, T ) × Ω)
in (3.1b), then summing the two equalities and integrating in time we get

Ω (δ + u ν ) (log(δ + u ν ) -1) + 1 + ǫ 2 |∇v ν | 2 + β 2 |v ν | 2 -v ν L ν u ν dx + t 0 Ω ∇(e -Lν vν u ν ) • ∇(e -Lν vν (δ + u ν )) e -Lν vν (δ + u ν ) dx ds + t 0 Ω |ǫ∆v ν + L ν u ν -βv ν | 2 dx ds = Ω (δ + u 0 ) (log(δ + u 0 ) -1) + 1 + ǫ 2 |∇v 0 | 2 + β 2 |v 0 | 2 -v 0 L ν u 0 dx. (3.15)
Now,

I ν,δ = t 0 Ω ∇(e -Lν vν u ν ) • ∇(e -Lν vν (δ + u ν )) e -Lν vν (δ + u ν ) dx ds = 4 t 0 Ω ∇ e -Lν vν (δ + u ν ) 2 dx ds + t 0 Ω δ δ + u ν ∇L ν v ν • ∇(e -Lν vν (δ + u ν )) dx ds.
On the one hand, weak lower semicontinuity ensures that lim inf

δ→0 4 t 0 Ω ∇ e -Lν vν (δ + u ν ) 2 dx ds ≥ 4 t 0 Ω ∇ u ν e -Lν vν 2 dx ds.
On the other hand,

t 0 Ω δ δ + u ν ∇L ν v ν • ∇(e -Lν vν (δ + u ν )) dx ds = t 0 Ω δ δ + u ν e -Lν vν ∇L ν v ν • ∇u ν dx ds -δ t 0 Ω e -Lν vν |∇L ν v ν | 2 dx ds = t 0 Ω δ1 (0,∞) (u ν ) δ + u ν e -Lν vν ∇L ν v ν • ∇u ν dx ds -δ t 0 Ω e -Lν vν |∇L ν v ν | 2 dx ds.
Since 0 ≤ δ1 (0,∞) (u ν )/(δ + u ν ) ≤ 1 and converges to zero as δ → 0, we deduce with the help of Lebesgue's dominated convergence theorem and the L 2 (0, T ; H 1 (Ω))-regularity of u ν and v ν that We can now obtain the remaining estimates, thanks to the properties of the entropy on (H 1 ) ′ -bounded sets.

lim δ→0 t 0 Ω δ δ + u ν ∇L ν v ν • ∇(e -Lν vν (δ + u ν )) dx ds = 0. Consequently, lim inf δ→0 I ν,δ ≥ 4 t 0 Ω ∇ u ν e -Lν vν
Proof of Proposition 3.5. Reasoning as in Section 2.2, we can use estimate (3.6) to estimate the entropy from below by

E ν (u ν , v ν ) = Ω ( ǫ 2 |∇v ν | 2 + β 2 |v ν | 2 -u ν L ν v ν + u ν log u ν -u ν + 1) dx ≥ - C(T ) 2 ǫ - m 2 β + Ω ( ǫ 2 |∇v ν | 2 - ǫ 4 |∇L ν v ν | 2 + β 2 |v ν | 2 - β 4 |L ν v ν | 2 + u ν log u ν -u ν + 1) dx ≥ - C(T ) 2 ǫ - m 2 β + Ω ( ǫ 4 |∇v ν | 2 + β 4 |v ν | 2 + u ν log u ν -u ν + 1) dx, (3.16)
where the last inequality comes from the fact that the operator L ν is a contraction in L 2 (Ω).

Reinserting this last estimate in the entropy dissipation (3.14) and observing that

Ω (u ν log u ν -u ν + 1) dx = Ω u ν log u ν m dx + m log m -m + 1 ≥ Ω u ν log u ν m dx,
we directly get estimates (3.7), (3.8), and (3.9). Estimate (3.10) follows directly from (3.9), using (3.1b). Using furthermore 0 ≤ e -Lν vν ≤ 1 together with (3.13), we get (3.11). Rewriting Eq. (3.1a) as K∂ t u ν = -e -Lν vν u ν + e -Lν vν u ν , we deduce (3.12) from (3.5) and (3.11).

Passage to the limit in the approximate system

We are now ready to prove Theorem 1.3 by passing to the limit as ν → 0 for solutions of (3.1).

Proof of Theorem 1.3. We first suppose that u 0 ∈ L 2 (Ω), so that we can use the approximation procedure previously built. We employ the estimates (3.6)-(3.12) to extract a weakly-star convergent subsequence (u νn , v νn ) n∈N with the following properties:

u νn * ⇀ u in L ∞ (0, T ; (H 1 ) ′ (Ω)), u νn ⇀ u in L 1 ((0, T ) × Ω), K(u νn -m) ⇀ * K(u -m) in L ∞ (0, T ; H 1 (Ω)), ∂ t K(u νn -m) ⇀ ∂ t K(u -m) in L 2 ((0, T ) × Ω)), v νn * ⇀ v in L ∞ (0, T ; H 1 (Ω)), ∂ t v νn ⇀ ∂ t v in L 2 ((0, T ) × Ω), e -Lν n vν n u νn ⇀ w in L 2 ((0, T ) × Ω).
Moreover, from the Aubin-Lions-Simon theorem (cf. [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 4]), we can immediately deduce that up to the extraction of a subsequence (v νn ) n∈N and (K(u νn -m) n∈N ) converge strongly to v and K(u -m) in C([0, T ]; L 2 (Ω)), respectively. The properties of L ν also imply that L νn v νn → v in L 2 ((0, T ) × Ω) and L νn u νn ⇀ * u in L ∞ (0, T ; (H 1 ) ′ (Ω)). Then we can pass to the limit in the second equation (3.1b), which is linear and also in the left-hand side of the first equation K∂ t u νn = -e -Lν n vν n u νn + e -Lν n vν n u νn , which is obtained from (3.1a). Finally, we use that (e -Lν n vν n ) n∈N converges pointwise towards e -v and is uniformly bounded (since |e -Lν n vν n | ≤ 1), and that (u νn ) n∈N converges weakly in L 1 ((0, T )×Ω) to pass in the weak limit in the product e -Lν n vν n u νn and identify w = e -v u (see for example [START_REF] Fonseca | Modern methods in the calculus of variations: L p spaces[END_REF]Proposition 2.61]). This shows that (u, v) is a weak-strong solution of (1.1) on (0, T ), and by Cantor's diagonal argument, we can build a global (in time) weak-strong solution of (1.1), which completes the proof of Theorem 1.3 for initial values u 0 ∈ L 2 (Ω) and v 0 ∈ H 1 (Ω). Now, if u 0 / ∈ L 2 (Ω), since estimates (3.5)-(3.12) do not depend on u 0 L 2 (Ω) , but only on m, Ω u 0 log(u 0 /m)dx and u 0 (H 1 ) ′ (Ω) , we can replace u 0 by a more regular (L 2 ) approximation u 0,ν in the approximation procedure and pass to the limit in the same way. This completes the proof of the existence of a global weak-strong solution of (1.1).

Finally, we can pass to the limit in estimates (3.13), (3.14), and (3.16) and obtain the estimates listed in Theorem 1.3.

The parabolic-elliptic problem

As announced in Remark 1.5, we give here some elements for the proof of the global existence for the parabolic-elliptic problem (1.9). We recall the problem

∂ t u = ∆(e -v u), in (0, ∞) × Ω, (3.17) 0 = ǫ∆v -βv + u, in (0, ∞) × Ω, (3.18)
with no-flux boundary conditions and nonnegative initial condition u 0 satisfying (1.7). We have a similar gradient structure as in the fully parabolic case, but the second equation is even at equilibrium. In particular mass conservation, duality estimates, and entropy dissipation are preserved. The only difference is the bound on the time derivative of v, which needs to replaced by the time differentiated version of (3.18)

-ǫ∆∂ t v + β∂ t v = ∂ t u, which implies (I + β ǫ K)∂ t v = 1 ǫ K∂ t u ∈ L 2 ((0, T ) × Ω),
taking into account that v(t) = m/β for t ≥ 0 by (3.18). Hence, there is an a priori bound on ∂ t v that can be used with the Aubin-Lions-Simon theorem (cf. [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 4]) in order to establish strong compactness of v. Indeed, using this strategy, the existence of a solution to (1.9) can be established with the same kind of proof as above based on the approximate problem

∂ t u = ∆(e -Lν v u), in (0, ∞) × Ω, 0 = ǫ∆v -βv + L ν u, in (0, ∞) × Ω,
after passing to the limit ν → 0, using the mass, entropy and duality bounds in an appropriate way.

Uniqueness and Regularity

We now investigate uniqueness of the solutions (in any dimension d) and their regularity (in dimension 1 and 2).

Notation. In this section we use the short notation u p for the norm u L p (Ω) for p ∈ [1, ∞].

Uniqueness

In this section we prove the result of uniqueness stated in Theorem 1.6.

Proof of Theorem 1.6. Let (u i , v i ), i = 1, 2, be two solutions of (1.1) with respective initial conditions (u 0,i , v 0,i ), i = 1, 2 such that

u 0,i = m , i = 1, 2,
and assume that

u i ∈ L ∞ (0, T ; L q (Ω)) , for all T > 0 , i = 1, 2. (4.1)
Then we know from nonnegativity and mass conservation that

u 1 (t) 1 = u 2 (t) 1 = m , t ≥ 0 ,
so that U := u 1 -u 2 has zero average with respect to the space variable and KU is welldefined. Since -∆KU = U with KU ∈ L ∞ (0, T ; H 1 (Ω)) and K∂ t U ∈ L 2 (0, T ; (H 1 ) ′ (Ω)), it follows from (1.1a) and Young's inequality that 1 2

d dt ∇KU 2 2 = Ω U K∂ t U dx = Ω U -u 1 e -v 1 + u 2 e -v 2 dx = - 1 2 Ω U (u 1 + u 2 ) e -v 1 -e -v 2 + e -v 1 + e -v 2 U dx = - 1 2 Ω e -v 1 + e -v 2 U 2 dx - 1 2 Ω (u 1 + u 2 ) e -v 1 -e -v 2 √ e -v 1 + e -v 2 e -v 1 + e -v 2 U dx ≤ 1 8 Ω (u 1 + u 2 ) 2 |e -v 1 -e -v 2 | 2 e -v 1 + e -v 2 dx . Since e -X -e -Y √ e -X + e -Y ≤ e -min(X,Y ) |Y -X| √ e -X + e -Y ≤ |Y -X| , (X, Y ) ∈ [0, ∞) 2 ,
we further obtain

d dt ∇KU 2 2 ≤ 1 4 Ω (u 1 + u 2 ) 2 V 2 dx , where V := v 1 -v 2 .
Introducing p := 2q/(q -2) ∈ (2, 2 * ] (notice that p = 2 * when d ≥ 3), we infer from Hölder's inequality and Sobolev's embedding theorem that

d dt ∇KU 2 2 ≤ 1 4 u 1 + u 2 2 2p/(p-2) V 2 p ≤ C 1 u 1 + u 2 2 q V 2 H 1 (Ω) . (4.2)
Also, from (1.1b) and Young's inequality, 1 2

d dt V 2 2 + ǫ ∇V 2 2 + β V 2 2 = - Ω V ∆KU dx = Ω ∇V • ∇KU dx ≤ ǫ 2 ∇V 2 2 + 1 2ǫ ∇KU 2 2 .
Hence,

d dt V 2 2 + min{ǫ, 2β} V 2 H 1 (Ω) ≤ 1 ǫ ∇KU 2 2 . (4.3)
Now, let T > 0 and set

M := sup t∈[0,T ] { u 1 (t) + u 2 (t) q } , (4.4)
which is finite according to (4.1). Introducing C 2 := min{ǫ, 2β}/(C 1 M 2 ), we combine (4.2), (4.3), and (4.4) to obtain

d dt C 2 ∇KU 2 2 + V 2 2 ≤ C 1 C 2 M 2 V 2 H 1 (Ω) -min{ǫ, 2β} V 2 H 1 (Ω) + 1 ǫ ∇KU 2 2 = 1 ǫ ∇KU 2 2 ≤ 1 ǫC 2 C 2 ∇KU 2 2 + V 2 2 .
After integration with respect to time, we end up with C 2 ∇KU (t) 2 2 + V (t) 2 2 ≤ e t/(ǫC 2 ) C 2 ∇KU (0) 2 2 + V (0) 2 2 , t ∈ [0, T ] .

Uniqueness readily follows from the previous inequality.

Regularity in spatial dimension one

Proof of Proposition 1.7. In spatial dimension one, the embedding H 1 (Ω) ֒→ C( Ω) and Theorem 1.3 implies that v belongs to L ∞ ((0, T )×Ω) for all T > 0, so that 1 ≥ e -v ≥ 1/C(T ) > 0 for some C(T ) > 1. Then u ∈ L 2 ((0, T ) × Ω) by Theorem 1.3 and, since ǫ∆v = (ǫ∆v -u + βv) + u -βv, the L 2 -regularity of ǫ∆v -u + βv, u, and v entail that ∆v ∈ L 2 ((0, T ) × Ω). Hence, v ∈ L 2 (0, T ; H 2 (Ω)).

Thanks to this additional regularity on v, for u 0 ∈ L r (Ω), it is straightforward to derive the identity u(t) r r + r(r -1) Since s → v(s) 2 H 2 (Ω) ∈ L 1 (0, T ), Gronwall's Lemma implies the boundedness of u r r in (0, T ). Thus we have proved Proposition 1.7.

Regularity in spatial dimension two

Let T > 0. We recall that, from the entropy and duality estimates, there is C 1 (T ) > 0 such that, for t ∈ [0, T ], ∇Ku(t) which also appears for different energies, see, e.g., [START_REF] Gao | Gradient flow approach to an exponential thin film equation: global existence and latent singularity[END_REF] for the Dirichlet energy. Note that such equations possess a formal entropy dissipation of the form

d dt E(u) = - Ω e ∂uE(u) |∇∂ u E(u)| 2 dx,
as well as duality estimates for the (H 1 ) ′ (Ω)-norm, which may be crucial for their analysis.

Concerning the link to the models with gradient sensing it might be relevant to study some intermediate cases. A natural interpolation is obtained when one starts from a microscopic model based on a jump process with rate combining local and gradient sensing. To illustrate this, let V = V (x) be a given potential (or in this case a chemotactic signal) and consider for θ ∈ [0, 1] the jump process on a one-dimensional grid (h being the size of the grid) with the probability of jump from node i to node j given by

k i,j = exp [-θV (x i ) -(1 -θ)h(V (x i ) -V (x j ))] ,
if |i -j| = 1, while k i,j = 0 for |i -j| > 1. Using a standard scaling of k i,j as the grid size tends to zero, we obtain the equation ∂ t u = ∇ • e -θV (∇u -u∇V ) = ∇ • e -θV u∇ log(ue -V ) = ∇ • e (1-θ)V ∇(ue -V ) , which is the standard Fokker-Planck equation for θ = 0 and the equation for local sensing for θ = 1. For any θ ∈ [0, 1] the system has the same entropy, but duality estimates only hold for θ = 1.

2

  dx ds, and we may let δ → 0 in (3.15) to complete the proof.

e

  -v u r-2 |∂ x u| 2 dx ds = u 0 r r + r(r -1) t 0 Ω e -v u r-1 ∂ x u∂ x v dx ds,from which we deduce with the help of Young's inequality, the nonnegativity of v, and the embedding H 1 (Ω) ֒→ C(

  |∇(ln u -v)| 2 dxds ≤ C 1 (T ) , (4.7)In order to prove Proposition 1.8 we need two lemmas providing intermediate estimates: Lemma 4.1. There is C 2 (T ) > 0 such that

							2 2 +	0	t	Ω	u 2 e -v dxds ≤ C 1 (T ) ,	(4.5)
							ǫ 2	∇v(t) 2 2 +	β 2	v(t) 2 2 ≤ C 1 (T ) ,	(4.6)
	0	t	∂ t v 2 2 ds +	0	t	Ω	ue -v

Ω u(t) log u(t) -u(t) + 1 dx + u(t) 2 + v(t) ∞ ≤ C 2 (T ) , t ∈ [0, T ] .

Acknowledgements

This work has been supported by the DAAD-PROCOPE project Nonlinear Cross-Diffusion Systems. Part of this work was done while PhL and AT enjoyed the hospitality of the Department Mathematik, FAU Erlangen-Nürnberg. The authors thank Antonio Esposito (FAU Erlangen-Nürnberg) for useful suggestions on the manuscript.

Proof. Set w := ue -v/2 . It follows from (1.1a) that 1 2 . On the one hand, it follows from Gagliardo-Nirenberg's inequality that

On the other hand, we infer from (1.1a), (4.6), and Gagliardo-Nirenberg's and Calderon-Zygmund's inequalities that

Combining the above estimates leads us to

Hence, after using Young's inequality and the property w ≤ u,

(4.9)

It follows from (4.8) and (4.9) that

Gronwall's lemma then entails that

for t ∈ [0, T ] and we deduce from (4.5) and (4.7) that

We next exploit the regularizing properties of the semigroup in L 2 (Ω) associated with the linear operator -ǫ∆ + βI supplemented with homogeneous Neumann boundary conditions to derive from (1.1b) that, for t ∈ [0, T ],

Combining the above inequality and (4.10) completes the proof.

Proof. Introducing C 4 (T ) := e -C 2 (T ) > 0, we infer from Lemma 4.1 that

A further use of Hölder's inequality gives

To estimate the last term on the right hand side of (4.12), we observe that (1.1b), (

and we complete the proof with the help of (4.13) and (4.14).

Proof of Proposition 1.8. Let p > 2. We first infer from (1.1b), Lemma 4.2, and the regularizing properties of the semigroup in L p (Ω) associated with the linear operator -ǫ∆ + βI supplemented with homogeneous Neumann boundary conditions that, for t ∈ [0, T ],

Next an alternative formulation of (1.1a) reads

with D := e -v and F := -ue -v ∇v . Now, u ∈ L ∞ (0, T ; L 2 (Ω)) and D ≥ e -C 2 (T ) > 0 in (0, T ) × Ω by Lemma 4.1, while F ∈ L ∞ (0, T ; L 4 (Ω; R 2 )) by (4.15) and Lemma 4.2. We are then in a position to apply [START_REF] Tao | Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity[END_REF]Lemma A.1] to conclude that u ∈ L ∞ ((0, T ) × Ω). Since the boundedness of v follows from Lemma 4.1, the proof is complete.

Conclusion and outlook

We have analyzed a model for chemotactic pattern formation with a local sensing mechanism instead of the commonly used gradient sensing. We have shown that even for arbitrary large mass there exists a weak-strong solution for arbitrary times, i.e., the blow-up is delayed to infinity compared to the well-studied case of gradient sensing. Moreover, under slightly higher integrability of the initial value we obtain uniqueness of the solution and some dimensiondependent regularity results. The results of this paper indicate the interest of a more detailed investigation of processes with local sensing, respectively the corresponding gradient structure of the form ∂ t u = ∆e ∂uE(u) , (5.1)


Eve Robinson
email: eveprobinson@gmail.com

M Travanut

L Fabre

S Larréché

L Ramelli

L Pascal

A Guinard

N Vincent

C Calba

L Meurice

M A Le Thien

E Fourgere

G Jones

N Fournet

A Smith Palmer

D Brown

S Le Hello

M Pardos De La Gandara

FX Weill

N Jourdan Da Silva

Outbreak of Salmonella Newport associated with internationally distributed raw goats' milk cheese, France, 2018 Authors

Raw milk cheeses are commonly consumed in France and are also a common source of foodborne outbreaks (FBOs). Both a FBO surveillance system and a laboratory-based surveillance system aim to detect Salmonella outbreaks. In early August 2018 5 familial FBOs due to Salmonella spp. were reported to a regional health authority. Investigation identified common exposure to a raw goats' milk cheese, from which Salmonella spp. were also isolated, leading to an international product recall. Three weeks later, on 22 August, a national increase in Salmonella Newport ST118 was detected through laboratory surveillance. Concomitantly isolates from the earlier familial clusters were confirmed as S. Newport ST118. Interviews with a selection of the laboratory identified cases revealed exposure to the same cheese, including exposure to batches not included in the previous recall, leading to an expansion of the recall. The outbreak affected 153 cases, including 6 cases in Scotland. S. Newport was detected in the cheese and in milk of one of the producer's goats. The difference in the two alerts generated by this outbreak highlight the timeliness of the FBO system and the precision of the laboratory-based surveillance system. It is also a reminder of the risks associated with raw milk cheeses.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 3

Background

Raw milk cheeses are popular in France accounting for 15% of all mature French cheeses, 75% of protected designation of origin (PDO) and protected geographical indication (PGI) cheeses, and the majority of other artisanal cheeses [START_REF]Ministère de l'agriculture et de l'alimentation[END_REF]. In the most recent national nutritional survey, 32% of people reported having eaten a raw milk cheese in at least 1 of 3 non consecutive days in the previous 3 weeks [START_REF] Dubuisson | The third French Individual and National Food Consumption (INCA3) Survey 2014-2015: method, design and participation rate in the framework of a European harmonization process[END_REF]. Collaterally, raw milk cheeses have been the vehicle of 16 major Salmonella foodborne outbreaks (FBO) in France between 2008 and 2018, representing a third of Salmonella outbreaks with an identified source [START_REF] Van Cauteren | Outbreak of Salmonella enterica serotype Muenster infections associated with goat's cheese, France[END_REF][START_REF] Buyser | Implication du lait et des produits laitiers dans les maladies infectieuses d'origine alimentaire en France de 1988 à 2003 [Implication of milk products in infectious diseases of food origine in France from 1988 to 2003[END_REF][START_REF] Espié | International outbreak of Salmonella Stourbridge infection, April-July 2005: results of epidemiological, food and veterinary investigations in France[END_REF][START_REF] Dominguez | Outbreak of Salmonella enterica serotype Montevideo infections in France linked to consumption of cheese made from raw milk[END_REF][START_REF] Ung | Disentangling a complex nationwide Salmonella Dublin outbreak associated with raw-milk cheese consumption, France, 2015 to 2016[END_REF].

There are two French surveillance systems with the objective of detecting Salmonella FBOs. The syndromic FBO surveillance system is based on mandatory notification to regional health agencies of clusters of gastrointestinal illness involving at least two cases having shared a common meal, and whose cause is therefore suspected to be foodborne. Once validated as true clusters of illness, they are notified to the national public health agency, Santé publique France (SpF). In 2016, 1,455 clusters involving 13,997 ill persons were notified [START_REF]Surveillance des toxi-infections alimentaires collectives -Données de la déclaration obligatoire[END_REF]. Salmonella spp. is the most frequent causal agent amongst clusters with an identified agent, responsible for 41% of such clusters between2007 and 2016 [START_REF] Fournet | Ten years of foodborne outbreaks in France, results of the mandatory notification surveillance system, 2007-2016[END_REF]. In around half of salmonellosis clusters, the serotype is not confirmed. The second surveillance system is laboratory based. The French national reference center for Escherichia coli, Shigella and Salmonella (FNRC-ESS) located at Institut Pasteur receives isolates from a voluntary network of laboratories across France. It was previously estimated that the laboratory surveillance system captured 66% of laboratory-confirmed human Salmonella infections in France [START_REF] Da Silva | Salmonelloses en France, 2002-2010: tendances en épidémiologie humaine, émergence de la souchemonophasique, principaux aliments impliqués dans les dernières épidémies[END_REF]. On a weekly basis algorithms are applied to FNRC-ESS data to detect unusual increases by serotype.

Salmonella enterica serotype Newport (S. Newport) is an uncommon cause of sporadic salmonellosis and Salmonella outbreaks in France. It accounts for 1 to 2% of human cases per year and since 2000

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 4 has been the responsible agent in five FBOs -four linked to different raw goats' milk cheeses and one linked to horsemeat [START_REF] Da Silva | Salmonelloses en France, 2002-2010: tendances en épidémiologie humaine, émergence de la souchemonophasique, principaux aliments impliqués dans les dernières épidémies[END_REF][START_REF] Espié | An outbreak of multidrug-resistant Salmonella enterica serotype Newport infections linked to the consumption of imported horse meat in France[END_REF]. We describe a fifth outbreak of S. Newport linked to a raw goats' milk cheese which was detected separately by both of the French surveillance systems.

Outbreak detection

The first alert Between 2 and 8 August 2018, five familial clusters of gastrointestinal illness, originating from one geographic department (department 13 -Bouches-du-Rhône), were reported to the regional health agency of Provence-Alpes-Côte d'Azur (PACA) in the south of France as part of the FBO surveillance system. These five clusters included 15 ill persons with dates of symptom onset between 28 July and 8 August 2018. Stool samples from 9 of 10 tested cases were positive for Salmonella spp. and isolates were sent to the FNRC-ESS for typing. Investigation by the regional health agency identified that the consumption of a particular raw goats' milk cheese manufactured by one French producer (X) was common to all clusters. Samples of two batches of the cheese from a vendor common to two of the clusters and samples of two further batches from the producer were positive for Salmonella spp. A product recall was issued on 10 August 2018 [START_REF]L'Earl Mounier procède au retrait de la vente et au rappel du produit « Pélardons », suite à la mise en évidence de la présence de Salmonelles[END_REF]. This recall concerned cheeses produced since 10 July 2018 and placed on the marked since 21 July 2018. As the product had been distributed internationally, a notification (2018.2296) was issued on 13 August through the Rapid Alert System for Food and Feed (RASFF), a European Commission platform to rapidly exchange information when risks to public health are detected in the food chain [13].

The second alert

On 22 August 2018, a national increase in S. Newport multilocus sequence type 118 (S. Newport ST118) cases was detected through the FNRC-ESS weekly algorithms. As of 22 August, the FNRC-ESS had detected 36 isolates of S. Newport ST118 with a sampling date between 10 July and 2 August. This compared with a total of eight isolates in July and August of 2017. Two of the isolates (detected on 21 August) belonged to cases associated with the previously described familial clusters.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 5

We hypothesised that the national increase in S. Newport ST118 cases was associated with the same cheese. An investigation was conducted to confirm the hypothesis, evaluate the extent of the outbreak, and determine if further control measures were needed.

Methods

Case definitions

We defined a confirmed case as a person with a laboratory confirmed S. Newport ST118 isolate belonging to the outbreak cluster on phylogenetic analysis (i.e. core genome MLST (cgMLST) hierarchical clustering of five or less alleles or <5 single nucleotide polymorphisms (SNPs) to another isolate) in a sample taken after 1 July 2018. We defined a probable case as a person with an epidemiological link to a confirmed case and either a non-serotyped Salmonella spp. isolate in a sample taken after 1 July or the onset of gastrointestinal illness after 1 July. This probable case definition was chosen in order to be inclusive of members of clusters where investigative protocols do not require testing or serotype confirmation of all cases. We defined a possible case as a person with a laboratory confirmed S. Newport isolate in a sample taken after 1 July for whom cluster analysis was not possible and epidemiological links were not known.

Epidemiological investigations

Amongst the 36 cases of S. Newport ST118 detected by the FNRC-ESS between 1 July and 22 August, 25 (who were not known to be related to the previously investigated clusters) were selected to be interviewed using the cheese component of the national salmonellosis trawling questionnaire Interviewers were instructed not to probe about the suspected cheese unless consumption was disclosed by the case. Cases were chosen to reflect different geographic regions and sampling dates.

Seven of the cases selected had sampling dates which would correspond with illness and potential exposure prior to 21 July (the date the previously recalled products were placed on the market).

These were selected to accurately date the outbreak onset and determine if earlier product batches may also have been implicated and thus require an extension of the recall.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 6 Given the international distribution of the product, European public health authorities were alerted through the European Epidemic Intelligence Information System (EPIS), operated by the European Centre for Disease Prevention and Control (ECDC), on 24 August in order to identify cases outside of France. Sequencing data of the outbreak strain was shared when it became available.

Human microbiological investigations

Whole genome sequencing (WGS) was performed as part of routine procedures at the FNRC-ESS.

WGS was carried out at the Plateforme de microbiologie mutualisée (P2M) from the Pasteur International Bioresources network (PIBnet, Institut Pasteur, Paris, France). The MagNAPure 96 system (Roche Diagnostics, Indianopolis, IN, USA) was used for DNA extraction, libraries were prepared using the Nextera XT kit (Illumina, San Diego, CA, USA) and sequencing was done with the NextSeq 500 system (Illumina). Serotype prediction was done by in-house scripts based on MLST [START_REF] Achtman | Multilocus sequence typing as a replacement for serotyping in Salmonella enterica[END_REF], fliC and fljB gene databases (FNRC-ESS internal flagellin database, unpublished). All genomic sequences (n=209) available from S. Newport ST118 isolates received at the FNRC-ESS between 1 January 2015 and 20 October 2018 were deposited into EnteroBase (https://enterobase.warwick.ac.uk) [START_REF] Alikhan | A genomic overview of the population structure of Salmonella[END_REF]. Phylogenetic analysis was performed by two different approaches integrated into EnteroBase: SNP analysis and cgMLST hierarchical clustering, by the "Legacy cgMLST v2 + HierCC" tool in August 2018, and confirmed in August 2019 for this publication by the new tool "cgMLST V2 + HierCC V1". Raw reads of a representative outbreak isolate (201805061) have also been deposited to the European Nucleotide Archive (ENA) (http://www.ebi.ac.uk/ena), under study accession number PRJEB29425.

Product and environmental investigations

As part of the investigation of the first alert, two batches of the cheese (produced on 10 July and 19 July) which were still available at a vendor which supplied two of the clusters, were sampled by the population protection directorate (DDPP) of department 13. The environmental health authority where Producer X was located also undertook an investigation which included environmental Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 7 sampling and microbiological testing of the production facility, product samples, bulk-milk samples and milk samples from each of the supplying animals. Serotyping of positive samples was undertaken by the French Agency for Food, Environmental and Occupational Health & Safety (ANSES). The French Directorate General for Food (DGAL) undertook traceback and traceforward investigations.

Results

Descriptive epidemiology

Within France, 147 cases (133 confirmed, 13 probable, 1 possible) were identified. All 9 cases from the familial clusters of alert 1 with isolates available were confirmed. Date of sampling was between 11 July and 27 September, and peaked during week 32 (August 6-7; Figure 1-D). When known (n=37), symptom onset was between 6 July and 8 August, and peaked during week 30. Fifty two percent (74/142) of cases were male and the median age was 46 years (range: 3-87 years). [START_REF] Dgal Mus | Note de service DGAL/MUS/2019-86 Publication du bilan et des statistiques des alertes dans le domaine alimentaire enregistrées par la mission des urgences sanitaires en[END_REF] cases whose clinical history was known, 13 were hospitalised. One case died, but salmonellosis was not the principal cause of death and information on food exposure was not available.

Between 24 and 25 August, 20 cases were interviewed with 15 reporting definite or likely consumption of the suspected goats' cheese prior to symptom onset. Two cases consumed the cheese of Producer X, which had been purchased directly from the farm, at the same family event on 8 July, before the production of the cheeses included in the first recall. Two other attendees of the event were also ill (probable cases). This familial foodborne cluster (cluster A) was not notified to the regional health agency. Another case had consumed the cheese of Producer X after purchasing it on 15 July, again before products involved in the first recall had been placed on the market. Table 1 summarises the clusters within the outbreak. All of the five familial clusters (C-G) reported in department 13 had at least one confirmed case. Three other familial clusters (H-J) from different departments, which were initially notified on 10, 16 and 24 of August were also found to be part of the outbreak, although the source was not initially identified as the implicated cheese. Interviews Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 8 with cases identified two other familial clusters which had not been notified. The attack rate amongst the known clusters ranged between 29 and 100%.

Through EPIS, Scottish authorities reported six cases of S. Newport ST118, with symptom onset between 20 July and 15 August, clustering within 5 SNPs on phylogenetic analysis undertaken by the Scottish Microbiology Reference Laboratories (SMiRL). Four of these cases reported consumption of the implicated cheese in Scotland where it was sold by a single vendor, one reported the consumption of a variety of unspecified cheeses while travelling in Spain during their exposure period, and for one, information on exposure to the product was not known. No other international cases were reported.

Microbiological results

SNP analysis revealed that the majority of the 2018 French S. Newport ST118 human isolates (133/166, 80.1%) clustered tightly together (Figure 2). These clustered isolates also displayed an identical type "133068" by hierarchical clustering of cgMLST data differing by 5 or less alleles ("HC5").

Other isolates on the SNP-based phylogenetic tree showed different cgMLSTHC5 cluster types (data not shown). The search for other Salmonella isolates displaying HC5|133068 cluster type in Enterobase (175,788 Salmonella genomic sequences by 28 October 2018) identified six other nonredundant human isolates submitted by the SMiRL, corresponding to the six cases previously reported by Scotland. These Scottish isolates also belonged to the same SNP cluster as the French outbreak isolates (data not shown). French outbreak isolates were pansusceptible.

Environmental and product investigation results

The implicated cheese, an artisanal product with PDO status, was the only product of producer X, who only used milk from his own herd. A batch of approximately 800 units (each weighing around 60 grams) was produced nearly every day. The cheese was distributed across France and internationally.

In Europe, the product was distributed to Belgium, Germany, Italy, Luxembourg, Spain, Switzerland, and the UK [13].

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 9

As previously mentioned, during the investigation of the first alert four separate batches (sampled from a vendor and from the producer) produced between 10 July and 4 August had tested positive for Salmonella spp., leading to the first recall. On 24 August, during the investigation of the second alert, these isolates were confirmed as S. Newport (serotyping by agglutination). Given the epidemiological results and the rarity of the serotype food isolates were not sequenced to further confirm the association. No cheese produced before 10 July was available for testing. Milk from one animal tested positive for S. Newport.

Outbreak control measures

The cases detected by the FNRC-ESS that occurred prior to 21 July (date on which batches from first recall were placed on the market; Figure 1-A), and the confirmation that some cases had purchased the cheese before this date, suggested that earlier batches were also contaminated. A second recall was issued on 24 August which concerned batches produced since 24 June 2018 [START_REF]Contamination de fromages « Pélardon » de l'entreprise Mounier par des salmonelles: élargissement du retrait-rappel suite à plusieurs cas de salmonelloses[END_REF]. As no further product was placed on the market after the first recall and the shelf life was two months, this second recall effectively covered any remaining products potentially still on the market. Both recalls were accompanied by a national press release in addition to notices at vendors. The RASFF notification was updated to reflect the expanded recall. Producer X had halted production at the time of the first alert pending investigation. The source animal was removed from the herd and the facility underwent rigorous cleaning. After resumption of production Producer X was subject to enhanced microbiological monitoring and environmental and product samples remained negative for Salmonella.

Discussion

We have described a large outbreak of pansusceptible S. Newport linked to an internationally distributed artisanal raw goats' milk cheese produced in France. The magnitude of the outbreak is likely underestimated. In France, a multiplication factor of 20 has been estimated between Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 10

Salmonella cases ascertained by the laboratory surveillance system and cases in the community [START_REF] Van Cauteren | Community incidence of campylobacteriosis and nontyphoidal salmonellosis, France, 2008-2013[END_REF].

The high attack rate seen in the familial clusters, and that over 23,000 potentially contaminated units were distributed in France and internationally, further suggests an underestimation of the magnitude both in France and internationally. No international cases were reported except for those reported from Scotland. As only 24 units implicated in the first recall were sold in Scotland, and four cases known to have been exposed to these units were identified, it seems likely that cases also occurred in other countries that received the product. However, the first recall is likely to have mitigated against a much larger outbreak.

While 5 of 6 S. Newport outbreaks identified in France have now been associated with raw goats' milk cheeses we are not aware of any particular affinity between this serotype and goats or dairy products. Elsewhere the sources in S. Newport outbreaks have been eclectic, including fresh produce (fruits, leafy, root and vine-stalk vegetables, sprouts, nuts) [START_REF] Byrne | A multi-country outbreak of Salmonella Newport gastroenteritis in Europe associated with watermelon from Brazil, confirmed by whole genome sequencing: October 2011 to[END_REF][START_REF] Angelo | Outbreak of Salmonella Newport infections linked to cucumbers--United States, 2014[END_REF][START_REF] Sivapalasingam | A multistate outbreak of Salmonella enterica Serotype Newport infection linked to mango consumption: impact of water-dip disinfestation technology[END_REF][START_REF] Bennett | Multistate foodborne disease outbreaks associated with raw tomatoes, United States, 1990-2010: a recurring public health problem[END_REF][START_REF] Bayer | An outbreak of Salmonella Newport associated with mung bean sprouts in Germany and the Netherlands, October to November 2011[END_REF][START_REF] Lienemann | Iceberg lettuce as suggested source of a nationwide outbreak caused by two Salmonella serotypes, Newport and Reading, in Finland in 2008[END_REF][START_REF] Irvine | Investigation of an outbreak of Salmonella enterica serovar Newport infection[END_REF], ground beef [START_REF] Schneider | Multistate outbreak of multidrug-resistant Salmonella newport infections associated with ground beef, October to December 2007[END_REF][START_REF] Marshall | Protracted Outbreak of Salmonella Newport infections linked to ground beef: possible role of dairy cows -21 States, 2016-2017[END_REF], and other animal derived foods [START_REF] Jackson | Outbreak-associated Salmonella enterica serotypes and food commodities, United States, 1998-2008[END_REF], suggesting varied reservoirs. Two other outbreaks linked to consumption of raw milk cheeses have been reported -both were in the US and associated with unpasteurised cows' milk cheeses [START_REF]Outbreak of multidrug-resistant Salmonella enterica serotype Newport infections associated with consumption of unpasteurized Mexican-style aged cheese--Illinois, March 2006-April 2007[END_REF][START_REF] Mccarthy | Outbreak of multidrug-resistant Salmonella Newport associated with consumption of Italian-style soft cheese[END_REF]. Interestingly, in the United States S. Newport isolates with a pansusceptible profile are often associated with fresh produce and potential environmental reservoirs while multidrug resistant isolates are associated with animal products, leading to a hypothesis that pansusceptibility is an indicator of an environmental reservoir, or an environment with less antibiotic exposure [START_REF] Jackson | Outbreak-associated Salmonella enterica serotypes and food commodities, United States, 1998-2008[END_REF][START_REF] Crim | Salmonella enterica Serotype Newport Infections in the United States, 2004-2013: Increased incidence investigated through four surveillance systems[END_REF]. Contact with amphibians and reptiles has also been shown to be associated with pansusceptible infection, in keeping with the hypothesis of the association between pansusceptibility and environmental reservoirs [START_REF] Varma | Highly resistant Salmonella Newport-MDRAmpC transmitted through the domestic US food supply: a FoodNet case-control study of sporadic Salmonella Newport infections, 2002-2003[END_REF]. Isolates in this outbreak were pansusceptible. To comply with the PDO requirements the goats producing the implicated cheese must spend the majority of time outside and feed on natural vegetation and therefore may have been exposed to an environmental reservoir.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 11

The outbreak again highlights the potential risks associated with raw milk products. Raw goats' milk can be contaminated by similar human gastrointestinal pathogens as raw cows' milk [START_REF] Efsa Biohaz Panel | Scientific opinion on the public health risks related to the consumption of raw drinking milk[END_REF][START_REF] Verraes | A review of the microbiological hazards of raw milk from animal species other than cows[END_REF]. In France, six other Salmonella outbreaks have been linked to French raw goats' milk cheeses and three Shiga toxin-producing Escherichia coli (STEC) outbreaks have been linked to mixed raw goats' and cows' milk cheeses. [START_REF] Van Cauteren | Outbreak of Salmonella enterica serotype Muenster infections associated with goat's cheese, France[END_REF][START_REF] Buyser | Implication du lait et des produits laitiers dans les maladies infectieuses d'origine alimentaire en France de 1988 à 2003 [Implication of milk products in infectious diseases of food origine in France from 1988 to 2003[END_REF][START_REF] Deschenes | Cluster of cases of haemolytic uraemic syndrome due to unpasteurised cheese[END_REF][START_REF] Espié | Escherichia coli O157 outbreak associated with fresh unpasteurized goats' cheese[END_REF]. Gastrointestinal outbreaks of STEC, Cryptosporidium parvum, Campylobacter spp., and Salmonella spp. associated with consumption of raw goats' milk or its cheese have been reported by other countries [START_REF] Verraes | A review of the microbiological hazards of raw milk from animal species other than cows[END_REF][START_REF] Rosenthal | Notes from the field: cryptosporidiosis associated with consumption of unpasteurized goat milk -Idaho, 2014[END_REF]. In addition, consumption has also been associated with other non-gastrointestinal zoonotic infections including tick borne encephalitis, brucellosis, toxoplasmosis, Q fever, Streptococcus equi subspecies zooepidemicus, Rift valley fever, and tuberculosis [START_REF] Verraes | A review of the microbiological hazards of raw milk from animal species other than cows[END_REF].

Similarly to cows' milk, contamination of goats' milk can be due to direct shedding into the milk from the udder or external contamination during or after milking, depending on the pathogen. It is, however, thought that faecal contamination is less likely than for cows' milk as goats' faeces are pelletised and drier, therefore less likely to contaminate the udder. Also, compared to grazing cows, goats' udders are less likely to come into contact with faecally contaminated mud. In 4 of the 7

documented Salmonella outbreaks in France associated with goats' milk cheese (including the current outbreak), one or more goats were found to be asymptomatic udder excretors, while in a fifth, a goat with clinical mastitis was found to be the source. In the STEC outbreaks associated with mixed raw milks the pathogen was found in the faeces of one or more animals of both species or in the production facility, but not in excreted milk.

Consumption of goats' milk products is less common than that of cows' milk, although there may be geographic variations. In the European Union, goats' milk production accounted for only 1.4% of total milk production [START_REF] Rossi | The sheep and goat sector in the EU: main features, challenges and prospects[END_REF]. In the French nutritional survey, previously mentioned, 0.4% of people reported drinking goats' milk [2; personal communication, SpF]. Fifteen percent of people reported eating any type of goats' milk cheese, and 3% reported eating raw goats' milk cheese. This compares Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 12

to the 27% of people who reported eating raw cows' milk cheese. Raw goats' milk cheeses account for about 9% of French raw milk cheese produced. While outbreaks associated with raw goats' milk in France are less frequent than cows' milk, the number seems disproportionate given the differences in consumption. Of the 18 Salmonella outbreaks linked to raw milk cheeses in France between 2004 and 2018, six have been linked to a goats' milk cheese. In addition, raw goats' milk cheeses accounted for 26% of the 101 food safety alerts (mostly non conformities in quality-controls by the producer) relating to raw milk cheeses notified to the DGAL [START_REF] Dgal Mus | Note de service DGAL/MUS/2019-86 Publication du bilan et des statistiques des alertes dans le domaine alimentaire enregistrées par la mission des urgences sanitaires en[END_REF]. The disproportionality in pathogens and disease associated with goats' milk cheeses may be because raw goats' milk cheeses are more likely to be soft cheeses, in which the high water content and pH permits survival and enhances growth conditions for pathogens [START_REF]A review of the microbiological hazards of dairy products made from raw milk[END_REF]409]. During 2018, the DGAL undertook a study which involved random sampling at production sites of raw cheeses from all species to determine the prevalence or Listeria, Salmonella and STEC in products. The results of this study will provide a clearer picture of the risk posed by raw cheeses from different species.

France produces half of the total EU production of cheese from pure goats' milk [START_REF] Rossi | The sheep and goat sector in the EU: main features, challenges and prospects[END_REF]. Twenty two percent of goats' milk produced in France is processed on the farm, mainly without pasteurisation, with 18-20,000 tonnes of artisanal cheeses produced each year. This outbreak highlights how even a relatively small scale producer can contribute to a large scale, international outbreak. A third of French goats' milk cheese is exported. The artisanal raw milk goats' cheese market is reportedly growing in other regions such as the UK and Australia. While newer non-thermal techniques to improve microbiological quality may be an option for industrial producers of raw milk cheeses, they are unlikely to be feasible for artisanal producers. Many such producers could not afford to invest in these technologies. More importantly, they are unlikely to be acceptable to the producer or to consumers, who are often keen to retain traditional production methods. Regulations around PDO/PGI products also require adherence to traditional production methods.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 13

In European countries, raw cheese production and sale is subject to EU food safety regulations.

While additional good practice guidelines have been developed in collaboration with industry in some regions [START_REF] Dila | Guide de bonnes pratiques d'hygiène -Collecte de lait cru et fabrication de produits laitiers[END_REF], artisanal producers and their suppliers, in France and elsewhere, may need particular support to undertake the required risk assessments and implement critical control procedures to maximise the microbiological safety of their product. However, while improved husbandry and production practices in combination with microbiological testing protocols can mitigate against contaminated product, the risk cannot be eliminated and consumers of all raw milk products should be aware of the potential risks, especially those at increased risk of severe illness. In June of 2019, the DGAL recommended that young children, particularly under 5 years, pregnant women and immunocompromised persons not consume raw milk cheeses [START_REF]Consommation de fromages à base de lait cru: rappel des précautions à prendre [Consumption of raw milk cheeses: reminder of precautions to take[END_REF].

The other interesting aspect of this outbreak is the time difference, of over two weeks, between its detection in the FBO syndromic surveillance system and the laboratory surveillance system, highlighting some of the strengths and weaknesses of both. The FBO surveillance system offers timeliness, as clusters should be reported before microbiological confirmation of cases. In 2016, 31% of clusters were reported on the day or the day after the onset of symptoms of the first ill person [START_REF]Surveillance des toxi-infections alimentaires collectives -Données de la déclaration obligatoire[END_REF]. This prompt reporting can facilitate the investigation and rapid implementation of control measures. In addition, in the event of an alert originating from the laboratory system, a search for clusters of the same serotype within the FBO surveillance system is undertaken, and when present can aid hypothesis generation.. A study (covering the period 1995-2000), estimated that 26% of salmonellosis clusters with microbiologically confirmed cases were captured by the FBO syndromic system [START_REF] Haeghebaert | Les toxi-infections alimentaires collectives en France, en 1999 et[END_REF]. In addition, linking reported clusters, particularly when they span multiple geographic regions, is limited by imprecise or absent information on the pathogen or suspected source. Three other familial Salmonella clusters within this outbreak, which were retrospectively microbiologically linked, were notified from different regions prior to the second alert. A meal containing cheese was the suspected vehicle for each but the implicated cheese was not specifically identified. Therefore, they were not linked to the clusters of department 13, and an opportunity for the earlier recognition

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 14 of the true extent of the outbreak was missed. Similarly, linking clusters based on a common causative pathogen can be hindered by the fact that in around half of salmonellosis clusters the serotype is not confirmed at the time of the notification [START_REF]Surveillance des toxi-infections alimentaires collectives -Données de la déclaration obligatoire[END_REF][START_REF] Fournet | Ten years of foodborne outbreaks in France, results of the mandatory notification surveillance system, 2007-2016[END_REF].

In contrast, the laboratory based surveillance offers greater sensitivity, with an estimated 66% of laboratory-confirmed infections captured [START_REF] Da Silva | Salmonelloses en France, 2002-2010: tendances en épidémiologie humaine, émergence de la souchemonophasique, principaux aliments impliqués dans les dernières épidémies[END_REF]. It also provides detailed pathogen information, an advantage which has been further enhanced since the introduction of WGS. However, this comes at the cost of timeliness, with an average analysis delay (receipt in the FNRC-ESS to validation of serotype) of 26 days compared to 6 days when serotyping was by agglutination.

Although the first recall stemming from the FBO surveillance system was inadequate as the extent of the outbreak and contamination was not fully evident, it still occurred almost 2 weeks before the outbreak was even detected through laboratory surveillance. Timeliness can be one of the limitations of the use of WGS for outbreak detection. Turnaround times will depend on demand and available resources and the high incidence of Salmonella can be one of the challenges to routine implementation of WGS in some countries [START_REF]ECDC roadmap for integration of molecular and genomic typing into European-level surveillance and epidemic preparedness, Version 2.1[END_REF]. However, delays will likely decrease in the future as methodological advances and reduced costs improve capacity. Having said that the availability of WGS enabled timely international collaboration and the confirmation of international cases during this outbreak. This outbreak highlights the value of both surveillance systems. While still advancing towards and investing in WGS surveillance, Public health authorities should continue to support such traditional systems within multifaceted surveillance programmes in order to optimise the timeliness and responsiveness of surveillance.

This results is also a reminder of the risks associated with raw milk cheeses and that measures to prevent the entry of pathogens into the artisanal raw milk cheese production chain and the optimization of surveillance aimed at the prevention of food-borne outbreaks such as in France should be extended or strengthened in other regions of Europe with tradition in the production of cheeses made from raw milk.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 21

Table 1 :

 1 Summary of clusters that were part of the outbreak.

	Cluster	Confirmed	Probable	Attack	Dates of symptom	Notified	Date of	Region	Suspected source
	ID	cases	cases	rate (%)	onset		notification	notifying	at time of
									notification
	A*	2	2	Nk	9-10/07/2018	No	Na	Na	Na
	B*	1	2	Nk	20-26/07/2017	No	Na	Na	Na
	C	2	0	50	28/07/2018	Yes	6/08/2018	PACA	Implicate cheese
	D	1	2	75	29/07/2018	Yes	2/08/2018	PACA	Implicate cheese
	E	1	1	29	29/07/2018	Yes	8/08/2018	PACA	Implicate cheese
	F	2	4	50	30/07/2018	Yes	6/08/2018	PACA	Implicate cheese
	G	2	0	50	31/07/2018	Yes	6/08/2018	PACA	Implicate cheese
	H	2	1	100	6/08/2018 -	Yes	10/08/2018	Occitanie Meal of omelette,
					8/08/2018				unspecified cheese
									and salad
	I	2	0	100	11/08/2018 -	Yes	16/08/2018	Ile de	Meal of different
					16/08/2018			France	cheeses
	J	1	1	100	Nk; date of sampling	Yes	Nk	Ile de	Meal of different
					24/8/2018			France

cheeses Nk: not known; Na: not applicable; PACA: Provence-Alpes-Côte d'Azur; * cluster identified during the investigation through case interview.

https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0950268820000904 Downloaded from https://www.cambridge.org/core. Institut Pasteur de Paris, on 20 May 2020 at 14:25:24, subject to the Cambridge Core terms of use, available at

https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0950268820000904 Downloaded from https://www.cambridge.org/core. Institut Pasteur de Paris, on 20 May 2020 at 14:25:24, subject to the Cambridge Core terms of use, available at Running title: S. Newport outbreak linked to raw goat cheese, France, 2018

Acknowledgements

We would like to acknowledge all involved in the investigation including other staff at: FNRC-ESS; regional units of SpF; the regional health agency of PACA; environmental health authorities in departments 13 and 48; ANSES, Health Protection Scotland and the Scottish Microbiology Reference Laboratories. We would also like to thank all cases who agreed to be interviewed.

Funding

This work received no specific grant from any funding agency, commercial or not-for-profit sectors.

Declaration of interests

None.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 15

Contributions

ER and NJDS led the outbreak at the national level and drafted the manuscript. MPG, SLH, FXW and LF undertook the human microbiological investigations at the FNRC-ESS and FXW and MPG drafted the microbiological sections of the manuscript. MT investigated the first alert on behalf on the regional health agency and edited the relevant section of the manuscript. LR, LP, AG, NV, CC, LM, MALT, EF conducted investigations in their respective regions on behalf of SpFrance. APS gathered the epidemiological information, and DB microbiological data on Scottish cases. SL coordinated the traceback and traceforward investigations. NF and GJ coordinated investigations between regions. All authors reviewed the manuscript.

Running title: S. Newport outbreak linked to raw goat cheese, France, 2018 23