Isabelle Tristani

LANDAU DAMPING FOR THE LINEARIZED VLASOV POISSON EQUATION IN A WEAKLY COLLISIONAL REGIME

Keywords: 2010 Mathematics Subject Classification. 35B35, 35Q83, 35Q20, 35Q84 Landau damping, Vlasov-Poisson equation, Fokker-Planck operator, linear Boltzmann operator

In this paper, we consider the linearized Vlasov-Poisson equation around an homogeneous Maxwellian equilibrium in a weakly collisional regime: there is a parameter ε in front of the collision operator which will tend to 0. Moreover, we study two cases of collision operators, linear Boltzmann and Fokker-Planck. We prove a result of Landau damping for those equations in Sobolev spaces uniformly with respect to the collision parameter ε as it goes to 0.

Introduction

In this paper, we are interested in the phenomenon of Landau damping for a linear Vlasov Poisson equation in a weakly collisional regime. The physical motivation for this problem comes from plasma theory. Indeed, in kinetic theory, the Landau equation, which can be derived from the Boltzmann one, allows to describe collisions in plasmas. However, this description is only valid for large times. The Vlasov model, on the contrary, completely neglects collisions in plasmas. Between these two models, is an intermediate one which is a more realistic physical model and which can be written as follows:

(1.1)

∂ t f + v • ∇ x f + F (t, x) • ∇ v f = ε Q L (f, f )
where f = f (t, x, v) is the distribution function in phase space, the position x lies in the torus of size 2π denoted T d := R d /(2πZ) d , the velocity v in R d . Note that one can reduce to the case where the size of the box is 2π by a rescaling argument. The operator Q L is the Landau collision operator (that we do not describe precisely because we won't deal with this general model). The mean-field electrostatic force F (t, x) is given by

F (t, x) = -∇W * x ρ f (t, x) - T d
ρ f (t, y) dy with ρ f (t, x) := R d f (t, x, v) dv. The potential W (x) describes the mean-field interaction between particles, we consider here the case of Coulomb repulsive interactions (which occur between electrons in plasmas) for which

(1.2) F = e 0 ∇ x ∆ -1 x ρ f and W (k) = e 0 |k| -2 , k ∈ Z d \ {0}
where e 0 > 0 is the electron charge-to-mass ratio.

In classical plasma physics, the coefficient ε equals (log Λ)/(2πΛ) where Λ is the plasma parameter which is very large. It is thus coherent to study this type of model in the limit ε → 0. We expect the behaviour of solutions to (1.1) to be complex and depend on the considered scales of time. When ε > 0 is fixed, one can expect the hypocoercive effect of collisions to be dominant over large time O(1/ε), which should induce an exponential convergence towards the Maxwellian equilibrium. In the Fokker-Planck case, it would also be interesting to analyse the hypoellipticity of the equation, which would provide a regularization effect. Those phenomena are relevant for fixed ε and their study is a challenging issue and a very interesting perspective. In the sequel, we are interested only in the phenomenon of Landau damping and in giving a result which is uniform with respect to ε in the limit ε → 0 (we recall that when ε = 0, Landau damping occurs in time of order O(1)).

As pointed out by Villani in [START_REF] Villani | Ir)reversibility and entropy[END_REF], the study of this problem is of great interest for several reasons: as we mentioned above, from a physical viewpoint, the model is more realistic than the pure Landau or Vlasov ones; it also permits quantification with respect to ε of the Landau damping; and finally, one could hope to bypass the obstacle of Gevrey regularity that faces the non collision model thanks to regularizing effects induced by collisions in the case of fixed ε. Let us underline the fact that the last aspect is only significant in the nonlinear case that we won't explore in this paper. Our study, which is restricted to the linearized problem with some linear collision operators, can thus be seen as the starting point of a long-range program.

1.1. The model. As mentioned above, we only deal with collision operators denoted C(f ) which are linear. To simplify the problem, we also suppose that

(1.3) Ker(C) = Span(M )
where M is the Maxwellian distribution

(1.4) M (v) := (2π) -d/2 e -|v| 2 /2 .
We consider the model

(1.5) ∂ t f + v • ∇ x f + F (t, x) • ∇ v f = ε C(f ).
We write the solutions of (1.5) as perturbations of the Maxwellian distribution M :

f (t, x, v) = M (v) + h(t, x, v)
where h is a mean-zero perturbation if

T d ×R d f in (x, v) dx dv = T d ×R d M (v) dx dv,
an assumption that we will suppose to hold throughout the paper. Let us emphasize here that we consider a linearization around the Maxwellian distribution M and not around some homogeneous distribution or other as it is usually done for the Vlasov-Poisson equation. It is explained by the fact that, under the assumption (1.3), M is also the unique equilibrium of the collision operator C satisfying R d M (v) dv = 1. Then h satisfies the following equation (where we denote ρ(t, x) := R d h(t, x, v) dv the perturbation density):

(1.6)

     ∂ t h + v • ∇ x h + F (t, x) • ∇ v h + F (t, x) • ∇ v M = ε C(h) F (t, x) = -∇W * x ρ(t, x) h(0, x, v) = h in (x, v).
where we have used that C(M ) = 0.

In this paper, we only work on the linearized problem associated to (1.6) which reads:

(1.7)

     ∂ t h + v • ∇ x h + F (t, x) • ∇ v M = ε C(h) F (t, x) = -∇W * x ρ(t, x) h(0, x, v) = h in (x, v).
We end the description of the model we are concerned with by the definition of the collision operators C that we shall consider. It can be a linear Boltzmann operator of type:

(1.8) C(h) := ρ M -h
or a Fokker-Planck operator defined through:

(1.9)

C(h) := ∆ v h + div v (vh).
These are classical operators of statistical physics that allow to describe the interaction between particles and a fixed background. Note that in the case of the collisional Vlasov equation (without the term coming from the mean-field interaction between particles):

∂ t h + v • ∇ x h = ε C(h)
for fixed ε > 0, it is clear that the hypocoercive effect is dominant for large times. We can for example apply the results of Dolbeault, Mouhot and Schmeiser [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF] to obtain a result of decay at infinity with a rate of type e -ελt for some λ > 0 in some Hilbert space.

Main results and known results.

Main results and comments. In this paper, we focus on the limit ε → 0 and our first objective is to obtain a result of Landau damping in the limit ε → 0 uniformly with respect to ε ∈ [0, ε 0 ] for some ε 0 > 0 for the equation (1.7). Before stating our theorems, we introduce the Sobolev spaces in which we are going to develop our analysis. We fix for the rest of the paper ℓ > d/2 and introduce the space L 2 x H n v , n ∈ N associated to the following norm:

f 2 L 2 x H n v := |α|≤n T d ×R d v 2ℓ |∂ α v f | 2 dx dv
where we have used the notation v := (1 + |v| 2 ) 1/2 . The dependency of L 2 x H n v with respect to ℓ is not included in the notation since ℓ is definitely fixed.

Here is our main result:

Theorem 1.1. Consider a mean-zero distribution h in ∈ L 2 x H n v for n ∈ N, n ≥ 3.
There exist ε 0 > 0, λ 0 > 0 such that for all ε ∈ [0, ε 0 ], the density ρ = ρ(t, x) of the solution h = h(t, x, v) of (1.7) satisfies the following estimate:

∀ t ≥ 0, ρ(t, •) L 2 x ≤ C t n , the solution h = h(t, x, v) satisfies itself ∀ t ≥ 0, ∀ ξ ∈ R d , | h(t, 0, ξ)| ≤ C e -ελ 0 t and ∀ t ≥ 0, ∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , | h(t, k, ξ)| ≤ C(ξ) t n-2 where C(ξ) := C ξ n-2 and C is a positive constant depending on h in L 2 x H n v . Remark 1.2.
In the Fokker-Planck case, we have a better estimate which provides us an exponential dissipation (see Theorem 3.1) for ε fixed, which is coherent since it comes from the Fokker-Planck structure, which disappears for ε = 0. One could also expect that for ε fixed, enhanced dissipation occurs over large times with a rate of dissipation of type e -εt 3 thanks to the diffusion part of the Fokker-Planck operator. However, we are not able to prove such a result, due to the presence of the drift term and the reaction term, the study is more intricate.

Remark 1.3. For ε = 0, it is well-known that ∀ ξ ∈ R d , ∀ t ≥ 0, h(t, 0, ξ) = h in (0, ξ),
it is not anymore the case for ε > 0 due to the presence of the collision operator.

Let us underline that our study fits into a Sobolev framework. We do not work with analytic or Gevrey regularity as in [START_REF] Mouhot | On Landau damping[END_REF] or [START_REF] Bedrossian | Landau damping: paraproducts and Gevrey regularity[END_REF] in which more regularity is needed to handle the nonlinear problem. However, the study of the linearized problem could have been done in Sobolev regularity, this is what we are doing since we only deal with the linearized problem. To do that, we take advantage of the paper by Faou and Rousset [START_REF] Faou | Landau Damping in Sobolev Spaces for the Vlasov-HMF Model[END_REF] which consists in the analysis of Landau damping for the (nonlinear) Vlasov-HMF equation in a Sobolev setting. For this equation, the nonlinear study does not require analytic (or Gevrey regularity), indeed, due to the fact that the interaction kernel has finite support in Fourier space, the control of the nonlinear effect "plasma-echo" is made easier. Finally, let us mention that our result can easily be adapted to an analytic framework. Also, for the nonlinear equation (1.6), it would be particularly interesting to investigate if the gain of regularity induced by collisions between particles could allow us to relax the hypothesis on the initial datum.

Another aspect that we point out is that, to the best of our knowledge, it is the first paper in which such a problematic is investigated for the Vlasov-Poisson equation in a weakly collisional regime (in the limit ε → 0). It should be relevant to compare this kind of question with the one studied by Bedrossian, Masmoudi and Vicol [START_REF] Bedrossian | Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier-Stokes Equations Near the Two Dimensional Couette Flow[END_REF] about the two-dimensional Euler equation where the equivalent of ε should be the inverse of the Reynolds number. Concerning this kind of problematic of uniform analysis of large time behaviour with respect to a small parameter, we can also mention the work of the author with Mischler [START_REF] Mischler | Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations[END_REF] in which an analysis of large-time behaviour is developed for several homogeneous Fokker-Planck equations uniformly with respect to a small parameter which allows to go from a model to another.

Strategy of the proof. To deal with this problem, we use the classical strategy to analyse Landau damping. We use method of characteristics and Fourier transform to obtain a closed equation on ρ(t, k) for each k ∈ Z d . The presence of the collisional operator raises additional difficulties. Indeed, in this equation, there is a term of the form

t 0 K ε (t -s) ρ(s, k) ds
which is typical. However, K ε splits into two parts, one coming from the Vlasov-Poisson term and the other coming from the collision term. In order to obtain our final result uniformly with respect to ε, we have to sharply estimate the Fourier transform in time of K ε and this is the main technical issue of this paper. We are then able to check that a Penrose type criterion is satisfied uniformly with respect to ε for ε small enough. Taking advantage of the analysis developed in [START_REF] Faou | Landau Damping in Sobolev Spaces for the Vlasov-HMF Model[END_REF] in a Sobolev framework, we end up with an estimate on ρ(t, •) L 2

x . We then go back to the kinetic distribution h and estimate each mode of it separately thanks to the result obtained previously on ρ.

State of the art. The behaviour of solutions to the equation (1.7) has been largely studied for some fixed ε.

If ε = 0, we are in the framework of Mouhot and Villani [START_REF] Mouhot | On Landau damping[END_REF], they prove Landau damping (linear and nonlinear) in analytic or Gevrey regularity. We also mention the work of Bedrossian, Masmoudi and Mouhot [START_REF] Bedrossian | Landau damping: paraproducts and Gevrey regularity[END_REF] in which a new simplified proof is given in Gevrey regularity. Let us also specify that some earlier results were obtained in [START_REF] Caglioti | Time asymptotics for solutions of Vlasov-Poisson equation in a circle[END_REF] by Caglioti and Maffei and in [START_REF] Hwang | On the existence of exponentially decreasing solutions of the nonlinear Landau damping problem[END_REF] by Hwang and Velázquez. We refer to the works of Ryutov [START_REF] Ryutov | Landau damping: half a century with the great discovery[END_REF], Mouhot and Villani [START_REF] Mouhot | On Landau damping[END_REF], Villani [START_REF] Villani | Landau damping. Notes de cours[END_REF] for a detailed historical background on Landau damping and references therein.

If ε > 0 is fixed, one can expect that the collisional term induces a hypocoercive effect. Some results have been obtained on linear and/or nonlinear frameworks for the Vlasov-Poisson-Fokker-Planck equation. About existence results, we mention the papers of Victory and O'Dwyer in two dimensions [START_REF] Victory | On classical solutions of Vlasov-Poisson Fokker-Planck systems[END_REF], Rein and Weckler [START_REF] Rein | Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions[END_REF] in three dimensions for small data and Bouchut [START_REF] Bouchut | Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions[END_REF] in three dimensions. Concerning the long-time behaviour, we point out the papers of Bouchut and Dolbeault [START_REF] Bouchut | On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials[END_REF], Carrillo, Soler and Vásquez [START_REF] Carrillo | Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system[END_REF], Dolbeaut [START_REF] Dolbeault | Free energy and solutions of the Vlasov-Poisson-Fokker-Planck system: external potential and confinement (large time behavior and steady states)[END_REF] and Hérau and Thomman [START_REF] Hérau | On global existence and trend to the equilibrium for the vlasovpoisson-fokker-planck system with exterior confining potential[END_REF], in the latter, both Cauchy problem and long-time behaviour have been studied.

As far as the case of the linear Boltzmann is concerned, the literature is more scarce and we were not able to find any reference on it. However, let us emphasize the fact that a large literature is devoted to the study of the Vlasov-Poisson-Boltzmann equation with a general Boltzmann collision operator. On this topic, we call attention on the paper of Dolbeault and Desvillettes [START_REF] Desvillettes | On long time asymptotics of the Vlasov-Poisson-Boltzmann equation[END_REF] which deals with the large time behaviour of solutions. On this matter, we also have to refer to a series of papers of Guo in which a robust energy method is developed to treat the global stability of global Maxwellians in a perturbative framework. There are many variants based on Guo's works and its energy method which allow to analyse the Cauchy problem and the large time behaviour of solutions. Let us mention two papers of Guo on the Vlasov-Poisson-Boltzmann equation [START_REF] Guo | The Vlasov-Poisson-Boltzmann system near vacuum[END_REF][START_REF] Guo | The Vlasov-Poisson-Boltzmann system near Maxwellians[END_REF] the first one falls in a near-vacuum regime, the second one in a near Maxwellian setting. Let us give other references which are concerned with the Cauchy problem in a perturbative setting and/or the large time behaviour of solutions and the rates of convergence, among others, we mention the papers of Duan and Strain [START_REF] Duan | Optimal time decay of the Vlasov-Poisson-Boltzmann system in R 3[END_REF], Duan, Yang and Zhao [START_REF] Duan | The Vlasov-Poisson-Boltzmann system in the whole space: the hard potential case[END_REF], Duan and Liu [START_REF] Duan | The Vlasov-Poisson-Boltzmann system without angular cutoff[END_REF], Xia, Xiong and Zhao [START_REF] Xiao | The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials[END_REF]. From another viewpoint, an analysis of the spectrum has been carried out in [START_REF] Li | Spectrum analysis for the Vlasov-Poisson-Boltzmann system[END_REF] by Li, Yang and Zhong. There is another phenomenon which could occur when the initial datum of the equation approaches distinct global Maxwellians at far fields, the solution to the Cauchy problem does not converge anymore to a constant equilibrium state in large time but we do not enter into details in this regard and refer to the recent paper of Duan and Liu [START_REF] Duan | Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system[END_REF] and the references given therein. 1.3. Notations and tools. We here define some notations and state an elementary lemma on Fourier transform that we will make good use in what follows.

Fourier transform. For a function f = f (x), x ∈ T d , we define its Fourier transform as follows:

f (k) = 1 (2π) d/2 T d f (x) e -ix•k dx, k ∈ Z d .
Similarly, for a function f = f (v), v ∈ R d , we define its Fourier transform by:

f (ξ) = 1 (2π) d/2 R d f (v) e -iv•ξ dv, ξ ∈ R d .
Let us notice that the Maxwellian distribution M defined in (1.4) satisfies M = M , an equality that we will use all along the paper.

Finally, if f = f (x, v), (x, v) ∈ T d × R d
, we define its Fourier transform through the following formula:

f (k, ξ) = 1 (2π) d R d f (x, v) e -ik•x-iv•ξ dx dv, (k, ξ) ∈ Z d × R d .
We shall also use the Fourier transform in time, if

f = f (t), t ∈ R, we denote (1.10) f (τ ) = R f (t) e -itτ dt, τ ∈ C.
Elementary lemma. We here state a simple result which links Fourier transform and regularity of a function (one can check the proof in [14, Lemma 2.1] in dimension 1).

Lemma 1.4. For any n ∈ N, we have:

∀ k ∈ Z d , ∀ ξ ∈ R d , | f (k, ξ)| ≤ C(n, ℓ) ξ -n f L 2 x H n v
where C(n, ℓ) is a positive constant depending only on ℓ and n.

Notation. Finally, let us specify that we shall use the same notation C for positive constants that may change from line to line, when C is a positive constant depending only on fixed number.

1.4. Outline of the paper. In Section 2, we deal with the behaviour of solutions of (1.7) with the linear Boltzmann operator (1.8) as a collision operator. In Section 3, we handle the problem with collisions given by a Fokker-Planck operator (1.9).

Acknowledgments. The author would like to thank Daniel Han-Kwan for enlightened discussions, his help and his suggestions. The author has been supported by the Fondation Mathématique Jacques Hadamard.

Linear Boltzmann type collisions

In this section, we deal with the following model:

(2.1)

     ∂ t h + v • ∇ x h + F (t, x) • ∇ v M = ε (ρ M -h) F (t, x) = -∇W * x ρ(t, x) h(0, x, v) = h in (x, v).
where we recall that ρ(t, x) = R d h(t, x, v) dv. The main theorem reads:

Theorem 2.1. Consider a mean-zero distribution h in ∈ L 2 x H n v for n ∈ N, n ≥ 3.
There exists ε 0 > 0 such that the solution h = h(t, x, v) of (2.1) and its associated density ρ = ρ(t, x) satisfy the following estimates:

(2.2) ∀ ε ∈ [0, ε 0 ], ∀ t ≥ 0, ρ(t, •) L 2 x ≤ C t n , and
(2.3)      ∀ ε ≥ 0, ∀ ξ ∈ R d , ∀ t ≥ 0, | h(t, 0, ξ)| ≤ C e -εt , ∀ ε ∈ [0, ε 0 ], ∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , ∀ t ≥ 0, | h(t, k, ξ)| ≤ C(ξ) t n-2 ,
where

C(ξ) := C ξ n-2 and C is a positive constant depending on h in L 2 x H n v .
2.1. The Penrose criterion. Before going into the proof of Theorem 2.1, we start by giving a lemma which allows us to apply a result from [START_REF] Faou | Landau Damping in Sobolev Spaces for the Vlasov-HMF Model[END_REF]. To do that, we introduce the following notations: for (t, k) ∈ R × Z d , we define for ε ≥ 0, (2.4)

       K 0 ε (t, k) := ε e -εt M (kt) 1 t≥0 K 1 ε (t, k) := -W (k) e -εt M (kt) |k| 2 t 1 t≥0 K ε (t, k) := K 0 ε (t, k) + K 1 ε (t, k).
Lemma 2.2. There exists ε 0 > 0 such that the kernel K ε defined in (2.4) satisfies the following condition:

(H) ∃ κ > 0, ∀ ε ∈ [0, ε 0 ], inf k∈Z d \{0} inf Im τ ≤0 |1 -K ε (τ, k)| ≥ κ
where we recall that K ε (τ, k) stands for the Fourier transform in time of K ε defined in (1.10).

Proof of Lemma 2.2. We start by giving an estimate on |1-K 1 0 (τ, k)|. We recall that since all marginals of M are increasing/decreasing, one can check (see [START_REF] Villani | Landau damping. Notes de cours[END_REF]) that the gaussian distribution M satisfies the following form of Penrose stability criterion [START_REF] Penrose | Electrostatic instability of a uniform non-Maxwellian plasma[END_REF] (given in [START_REF] Mouhot | On Landau damping[END_REF] in this form):

∀ k ∈ Z d \ {0}, ∀ w ∈ R s.t. (M k ) ′ (w) = 0, W (k) p.v. R (M k ) ′ (r) r -w dr < 1
where M k denotes the marginals of M along k ∈ Z d \ {0}:

M k (r) := kr/|k|+k ⊥ M (w) dw.
Consequently, we know (see [START_REF] Mouhot | On Landau damping[END_REF] and [START_REF] Bedrossian | Landau damping: paraproducts and Gevrey regularity[END_REF] for a complete proof) that there exists λ † > 0 and κ 0 > 0 such that

(2.5) ∀ ξ ∈ C, ℜe ξ < λ † , inf k∈Z d |L(k, ξ) -1| ≥ κ 0
where L is defined through ( ξ is the complex conjugate of ξ)

L(k, ξ) = - ∞ 0 e ξ|k|t M (kt) W (k) |k| 2 t dt, (k, ξ) ∈ Z d × R d ,
note that this integral is well defined because it is absolutely convergent due to the decay property of M = M . Then, one can check that if we denote

τ = λ + iζ with (λ, ζ) ∈ R 2 , K 1 0 (τ, k) = L k, ζ + iλ |k|
and thus using (2.5), we deduce that for any k ∈ Z d \ {0},

(2.6) ∀ τ ∈ C, Imτ ≤ 0, | K 1 0 (τ, k) -1| ≥ κ 0 .
Still using the notation τ = λ + iζ, we also have

| K 0 ε (τ, k)| = ε ∞ 0 e -itτ e -εt M (kt) dt ≤ ε ∞ 0
e ζt e -εt M (kt) dt.

We deduce for any

k ∈ Z d \ {0}, (2.7)
∀ τ ∈ C, Imτ ≤ 0, | K 0 ε (τ, k)| ≤ ε ∞ 0 M (kt) dt ≤ ε |k| ∞ 0 M (t) dt ≤ C ε.
Then, it remains to estimate the difference coming from K 1 ε -K 1 0 . We have:

| K 1 ε (τ, k) -K 1 0 (τ, k)| = ∞ 0 e -itτ (K 1 ε (t, k) -K 1 0 (t, k)) dt = e 0 ∞ 0 e -itτ (1 -e -εt ) M (kt) t dt ≤ C ε ∞ 0 e ζt t 2 e -|k| 2 t 2 2 dt
where we used that 1 -e -εt ≤ ε t for any t ≥ 0. Performing a change of variable in the last integral, we get for any k ∈ Z d \ {0},

(2.8) ∀ τ ∈ C, Imτ ≤ 0, | K 1 ε (τ, k) -K 1 0 (k, τ )| ≤ C |k| 3 ε ≤ C ε.
We now go back to our full problem which consists in estimating the difference

|1 -K ε (τ, k)| with K ε = K 0 ε + K 1 0 + K 1 ε -K 1 0 . Gathering (2.6), (2.7) and (2.8), we conclude that for any k ∈ Z d \ {0}, for any τ ∈ C, Imτ ≤ 0, | K ε (τ, k) -1| ≥ | K 1 0 (τ, k) -1| -| K 1 ε (τ, k) -K 1 0 (k, τ )| -| K 0 ε (τ, k)| ≥ κ 0 -c 0 ε,
for some positive constant c 0 > 0 which only depends on a fixed number. We then choose ε 0 > 0 such that κ 0 -c 0 ε 0 > 0. Denoting κ := κ 0 -c 0 ε 0 , we obtain

∀ ε ∈ [0, ε 0 ], inf k∈Z d \{0} inf Im τ ≤0 | K ε (τ, k) -1| ≥ κ,
from which we conclude that (H) is fulfilled.

2.2. Proof of Theorem 2.1. We can now handle the proof of Theorem 2.1, we divide it into two parts, the first one being devoted to the proof of estimate (2.2) and the second one to (2.3).

Proof of the estimate (2.2). The first thing that we have to notice is that, due to the conservation properties of the operator

-v • ∇ x h -F (t, x) • ∇ v M + ε (ρ M -h),
we know that for all times, ρ(t, 0) = (2π) d/2 h in (0, 0) = 0 since h in is a mean-zero distribution.

We then focus on the study of ρ(t, k) for k ∈ Z d \ {0}. In the rest of the proof, we consider ε ∈ [0, ε 0 ] where ε 0 is given in Lemma 2.2. Using the method of characteristics, we have

h(t, x, v) = h in (x -vt, v) e -εt + ε t 0 e -ε(t-s) ρ(s, x -v(t -s)) M (v) ds - t 0 e -ε(t-s) (∇ v M • F )(s, x -v(t -s), v) ds
We then take the Fourier transform in both variables (x, v) of the previous equality:

h(t, k, ξ) = e -εt T d ×R d h in (x -vt, v) e -ik•x-iv•ξ dx dv + ε (2π) d t 0 T d ×R d e -ε(t-s) ρ(s, x -v(t -s)) M (v) e -ik•x-iv•ξ dx dv ds - 1 (2π) d t 0 T d ×R d e -ε(t-s) (∇ v M • F )(s, x -v(t -s), v) e -ik•x-iv•ξ dx dv ds.
Performing change of variables, we obtain

h(t, k, ξ) = e -εt h in (k, ξ + kt) + ε (2π) d t 0 T d ×R d
e -ε(t-s) ρ(s, x) M (v) e -ik•x-iv•(ξ+k(t-s)) dx dv ds

- 1 (2π) d t 0 T d ×R d e -ε(t-s) (∇ v M • F )(s, x, v) e -ik•
x-iv•(ξ+k(t-s)) dx dv ds, from which we deduce (2.9)

h(t, k, ξ) = e -εt h in (k, ξ + kt) + ε t 0 e -ε(t-s) ρ(s, k) M (ξ + k(t -s)) ds - t 0 e -ε(t-s) k • (ξ + k(t -s)) W (k) ρ(s, k) M (ξ + k(t -s)) ds.
Taking ξ = 0 in (2.9), we obtain the closed equation on ρ(t, k):

ρ(t, k) = e -εt h in (k, kt) + ε t 0 e -ε(t-s) ρ(s, k) M (k(t -s)) ds - t 0 e -ε(t-s) k • (k(t -s)) W (k) ρ(s, k) M (k(t -s)) ds.
that we can sum up in

ρ(t, k) = e -εt h in (k, kt) + t 0 K ε (t -s) ρ(s, k) ds
where we recall that K ε is defined in (2.4). We can notice here that we are almost in the same framework as the one of [14, Lemma 2.3] for ε ∈ [0, ε 0 ]. Indeed, we have

• 2 M ∈ L 2
x H n v for any n ∈ N and from Lemma 2.2, we know that K ε satisfies the following condition:

(H) ∃ κ > 0, ∀ ε ∈ [0, ε 0 ], inf k∈Z d \{0} inf Im τ ≤0 |1 -K ε (τ, k)| ≥ κ.
The only difference is that the proof in [START_REF] Faou | Landau Damping in Sobolev Spaces for the Vlasov-HMF Model[END_REF] is done only for the case k = ±1. But it is uncomplicated to extend the result for |k| > 1. We then obtain that for any γ ≥ 0, for any T ≥ 0:

(2.10)

∀ k ∈ Z d \ {0}, sup t∈[0,T ] kt γ | ρ(t, k)| ≤ sup t∈[0,T ] kt γ e -εt | h in (k, kt)| ≤ sup t∈[0,T ] kt γ | h in (k, kt)|.
Using Lemma 1.4, there holds,

(2.11) | h in (k, kt)| ≤ C(n) kt -n h in L 2
x H n v . Combining (2.10) and (2.11) with γ = n, we obtain for any T ≥ 0,

∀ k ∈ Z d \ {0}, sup t∈[0,T ] kt n | ρ(t, k)| ≤ C(n) h in L 2 x H n v .
from which we deduce (2.12)

∀ k ∈ Z d \ {0}, ∀ t ≥ 0, | ρ(t, k)| ≤ C(n) kt n h in L 2 x H n v ,
this yields the final result (2.2). We now concentrate on the proof of estimate on the solution h = h(t, x, v). Proof of the estimate (2.3). The first part concerns the moment of order 0 in x. Contrary to the case of the linear Vlasov-Poisson equation, the latter is not preserved due to the presence of the collision operator. Indeed, noticing that since h is mean-zero, we have:

∂ t h(t, 0, ξ) = -ε h(t, 0, ξ)
we then deduce that for any fixed ξ ∈ R d ,

∀ ε ≥ 0, | h(t, 0, ξ)| = e -εt | h in (0, ξ)| ≤ C e -εt h in L 2 x H n v
where the last estimate comes from Lemma 1.4.

We now deal with the case k ∈ Z d \ {0} and consider ε ∈ [0, ε 0 ]. From (2.9), we have

h(t, k, ξ -kt) = e -εt h in (k, ξ) + ε t 0 e -ε(t-s) ρ(s, k) M (ξ -ks) ds - t 0 e -ε(t-s) k • (ξ -ks) W (k) ρ(s, k) M (ξ -ks) ds =: T 1 + T 2 + T 3 . Since h in ∈ L 2 x H n v , we have from Lemma 1.4 that (2.13) ∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , |T 1 | ≤ C ξ -n .
To estimate T 2 and T 3 , we use the estimate (2.12) on ρ that we obtained previously combined with the decay property of M = M which implies for example

M (ξ -ks) ≤ C ξ -ks 2-n
Concerning T 2 , we thus have

|T 2 | ≤ C t 0 ks -n ξ -ks 2-n ds ≤ C t 0 ks -n ξ 2-n ks n-2 ds
where we have used the estimate ξ -ks 2-n ≤ C ξ 2-n ks n-2 , we deduce

|T 2 | ≤ C ξ 2-n t 0 ds |k|s 2 .
Performing a change of variable, we obtain

(2.14) ∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , |T 2 | ≤ C 1 |k| ξ 2-n ≤ C ξ 2-n .
We treat T 3 in a similar way, noticing that |k| W (k) = O(1), and that we have

|ξ -ks| M (ξ -ks) ≤ C ξ -ks 2-n .
We conclude this proof as for T 2 and we obtain

(2.15) ∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , ∀ t ≥ 0, |T 3 | ≤ C ξ 2-n .
Combining (2.13), (2.14) and (2.15), we obtain

∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , ∀ t ≥ 0, | h(t, k, ξ -kt)| ≤ C ξ n-2 .
In particular, we have for any fixed ξ ∈ R d ,

∀ k ∈ Z d \ {0}, ∀ t ≥ 0, | h(t, k, ξ)| ≤ C ξ + kt n-2 ≤ C(ξ) t n-2
where we have used that ξ + kt 2-n ≤ C ξ n-2 kt 2-n ≤ C ξ n-2 t 2-n and this concludes the proof of (2.3).

Fokker-Planck type collisions

In this section, we deal with the following model:

(3.1)      ∂ t h + v • ∇ x h + F (t, x) • ∇ v M = ε (∆ v h + div v (vh)) F (t, x) = -∇W * x ρ(t, x) h(0, x, v) = h in (x, v)
where we recall that ρ(t, x) = R d h(t, x, v) dv.

Theorem 3.1 (Fokker-Planck operator). Consider a mean-zero distribution

h in ∈ L 2 x H n v for n ∈ N, n ≥ 3.
There exists ε 0 > 0 such that the solution h = h(t, x, v) of (3.1) and its associated density ρ = ρ(t, x) satisfy the following estimates:

(3.2) ∀ ε ∈ [0, ε 0 ], ∀ t ≥ 0, ρ(t, •) L 2 x ≤ C e -εt t n ≤ C t n , and
(3.3)      ∀ ε ≥ 0, ∀ ξ ∈ R d , ∀ t ≥ 0, | h(t, 0, ξ)| ≤ C e -ελ * t , ∀ ε ∈ [0, ε 0 ], ∀ k ∈ Z d \ {0}, ∀ ξ ∈ R d , ∀ t ≥ 0, | h(t, k, ξ)| ≤ C(ξ) e -εt t n-2 ≤ C(ξ) t n-2 ,
where λ * > 0 is the rate of decay for the homogeneous Fokker-Planck equation in L 2 ( v ℓ ) (see Lemma 3.3) and C(ξ) := C ξ n-2 and C is a positive constant depending on h in L 2

x H n v .

3.1. The Penrose criterion. As in Section 2, we introduce a kernel K ε which is going to be useful in the proof of Theorem 3.1. For (t, k) ∈ R × Z d and ε ∈ (0, 1/12], we define

(3.4) K ε (t, k) := -e εt exp -ε t 0 |χ ε (σ) k| 2 dσ k • (χ ε (t) k) W (k) M (χ ε (t) k) 1 t≥0 ,
where we denoted for ε > 0,

χ ε (t) := 1 -e -εt ε , t ∈ R.
We also introduce a kernel K 0 to which we shall compare K ε :

(3.5) K 0 (t, k) := -W (k) |k| 2 M (kt) t 1 t≥0 .
The aim of the next part is to prove that K ε satisfies a Penrose type criterion such as we did in Section 2, Lemma 2.2.

Lemma 3.2.

There exists ε 0 > 0 such that the kernel K ε defined in (3.4) satisfies the following condition:

(H) ∃ κ > 0, ∀ ε ∈ [0, ε 0 ], inf k∈Z d \{0} inf Im τ ≤0 |1 -K ε (τ, k)| ≥ κ
where we recall that K ε (τ, k) stands for the Fourier transform in time of K ε defined in (1.10).

Proof of Lemma 3.2. First, we recall that we have shown in the proof of Lemma 2.2 (see estimate (2.6)) that for any k ∈ Z d \ {0},

(3.6) ∀ τ ∈ C, Imτ ≤ 0, | K 0 (τ, k) -1| ≥ κ 0
for some κ 0 > 0.

We then estimate the difference K ε (/tau, k) -K 0 (τ, k) for k ∈ Z d \ {0} splitting it into several parts. For any τ = λ + iζ, Imτ = ζ ≤ 0, we have:

| K ε (τ, k) -K 0 (τ, k)| ≤ e 0 ∞ 0 |χ ε (t) M (kχ ε (t)) -t M (kt)| 1 0≤t≤3/ε e εt exp -ε t 0 χ 2 ε (σ) |k| 2 dσ |e -itτ | dt + e 0 ∞ 0 e εt exp -ε t 0 χ 2 ε (σ) |k| 2 dσ -1 t M (kt) 1 0≤t≤3/ε |e -itτ | dt + e 0 ∞ 0 e εt exp -ε t 0 χ 2 ε (σ) |k| 2 dσ |χ ε (t) M (kχ ε (t)) -t M (kt)| 1 t≥3/ε |e -itτ | dt + e 0 ∞ 0 e εt exp -ε t 0 χ 2 ε (σ) |k| 2 dσ -1 t M (kt) 1 t≥3/ε |e -itτ | dt =: I 1 + I 2 + I 3 + I 4 .
Concerning the term I 1 , we have since ζ = Im τ ≤ 0 and e εt ≤ e 3 for t ≤ 3/ε,

I 1 ≤ C 3/ε 0 |χ ε (t) M (kχ ε (t)) -t M (kt)| e ζt dt ≤ C 3/ε 0 |χ ε (t) -t| M (kt) dt + C 3/ε 0 |M (kχ ε (t)) -M (kt)| χ ε (t) dt =: I 11 + I 12 .
First, we perform a straightforward analysis of the function t → t -χ ε (t) -εt 2 /2 to prove that

(3.7) ∀ t ∈ R + , |χ ε (t) -t| = t -χ ε (t) ≤ ε t 2 2 .
We deduce that (3.8)

I 11 ≤ C ε 3/ε 0 t 2 M (kt) dt ≤ C ε |k| 2 3/ε 0 |k| 2 t 2 M (kt) dt ≤ C ε |k| 3 3|k|/ε 0 t 2 e -t 2 /2 dt ≤ C ε ∞ 0 t 2 e -t 2 /2 dt ≤ C ε
where we performed a change of variable. Concerning I 12 , we notice that

|M (kχ ε (t)) -M (kt)| = M 1/2 (kχ ε (t)) -M 1/2 (kt) M 1/2 (kχ ε (t)) + M 1/2 (kt) .
But, one can easily prove that χ ε (t) ≥ e -3 t for any t ∈ [0, 3/ε]. We thus have, denoting

µ := e -|•| 2 4e 6 , M 1/2 (kχ ε (t)) + M 1/2 (kt) ≤ C µ(kt).
Also, the gradient of M 1/2 is bounded in R d , we deduce that for any t ∈ [0, 3/ε],

|M (kχ ε (t)) -M (kt)| ≤ C M 1/2 (kχ ε (t)) -M 1/2 (kt) µ(kt) ≤ C |k|(t -χ ε (t)) µ(kt) ≤ C ε |k| t 2 µ(kt)
where the last inequality comes from (3.7). This implies, since χ ε (t) ≤ t for any t ≥ 0:

(3.9)

I 12 ≤ C ε 3/ε 0 |k| t 3 µ(kt) dt ≤ C ε |k| 3 6|k|/ε 0 t 3 e -t 2e 6 2 dt ≤ C ε |k| 3 ∞ 0 t 3 e -t 2e 6 2 dt ≤ C ε.
Concerning I 2 , we also split its study into two parts:

I 2 ≤ C ∞ 0 exp -ε t 0 χ 2 ε (σ) |k| 2 dσ -1 t M (kt) 1 0≤t≤3/ε e ζt dt + C ∞ 0 e εt (1 -e -εt ) exp -ε t 0 χ 2 ε (σ) |k| 2 dσ t M (kt) 1 0≤t≤3/ε e ζt dt =: I 21 + I 22 .
To deal with I 21 , we study the function t

→ 1 -exp -ε t 0 χ 2 ε (σ) |k| 2 dσ -ε |k| 2 t 3 /3 to obtain ∀ t ≥ 0, exp -ε t 0 χ 2 ε (σ) |k| 2 dσ -1 = 1 -exp -ε t 0 χ 2 ε (σ) |k| 2 dσ ≤ ε |k| 2 t 3 3 .
We deduce that for any

k ∈ Z d \ {0}, (3.10)
I 21 ≤ C ε |k| 2 ∞ 0 t 4 M (kt) e ζt dt ≤ C ε |k| 3 ∞ 0 t 4 e -t 2 /2 dt ≤ C ε,
where we have performed a change of variable to get the last inequality but one. For I 22 , we use the simple inequality ∀ t ≥ 0, 1 -e -εt ≤ ε t so that (3.11)

I 22 ≤ C ε ∞ 0 e 3 t 2 M (kt) e ζt dt ≤ C ε |k| 3 ∞ 0 t 2 e -t 2 /2 dt ≤ C ε.
We then control I 3 exploiting the term exp -ε t 0 χ 2 ε (σ) |k| 2 dσ which provides us some decay for large times since k ∈ Z d \ {0}. We compute explicitly the integral inside it:

t 0 χ 2 ε (σ) dσ = t ε 2 + 1 2ε 3 -e -2εt + 4e -εt -3 .
Since e -εt ∈ (0, 1] for any t ≥ 0, -y 2 + 4y -3 ∈ [-3, 0] for any y ∈ [0, 1] and |k| ≥ 1, we then write the following bound:

∀ t ≥ 3/(2ε), t 0 χ 2 ε (σ) |k| 2 dσ ≥ 1 ε 2 t - 3 2ε |k| 2 ≥ 1 ε 2 t - 3 2ε |k|.
As a consequence, we get (2ε) .

∀ t ≥ 3/ε, exp -ε t 0 χ 2 ε (σ) |k| 2 dσ ≤ e -|k| ε (t-3 2ε ) ≤ e -|k|t/
from which we have:

(3.12) ∀ t ≥ 3/ε, e εt exp -ε t 0 χ 2 ε (σ) |k| 2 dσ ≤ e -|k|t/2 ≤ e -t/2
where we used that ε ∈ (0, 1/2]. Then,

I 3 ≤ C |k| ∞ 3/ε e -t/2 |k χ ε (t) M (kχ ε (t)) -k t M (kt)| e ζt dt ≤ C |k| ∞ 3/ε e -t/2 |k| |χ ε (t) -t| dt ≤ Cε ∞ 3/ε t 2 e -t/2 dt
where we used that the derivative of y → y M (y) (where we still denote the function M : y ∈ R → (2π) -3/2 e -y 2 /2 ) is bounded on R and the inequality (3.7). It implies (3.13)

I 3 ≤ Cε |k| 3 ∞ 0 t 2 e -t/2 dt ≤ C ε.
To deal with I 4 , we use (3.12) and the fact that 1 t≥3/ε ≤ ε t/3, we thus deduce that (3.14)

I 4 ≤ C ∞ 0 t M (kt) e ζt 1 t≥3/ε dt ≤ C ε |k| 3 ∞ 0 t 2 e -t 2 /2 dt ≤ C ε.
Gathering (3.8), (3.9), (3.10), (3.11), (3.13) and (3.14) yields

(3.15) | K ε (τ, k) -K 0 (τ, k)| ≤ c 0 ε,
for some constant c 0 > 0. We then combine (3.6) with (3.15) to conclude that

∀ τ ∈ C, Imτ ≤ 0, | K ε (τ, k) -1| ≥ κ 0 -c 0 ε.
We choose ε 0 ∈ (0, 1/12] small enough such that κ 0 -c 0 ε 0 > 0, set κ := κ 0 -c 0 ε 0 and it follows that (H) is satisfied.

3.2. The homogeneous Fokker-Planck equation. In this part, we recall some features of solutions to the homogeneous Fokker-Planck equation

(3.16) ∂ t f = ε (∆ v f + div v (vf )) f (0, •) = f 0
that will be useful for the study of the evolution of h(t, 0, ξ), t ≥ 0 and ξ ∈ R d in the proof of estimate (3.3). We introduce L 2 v (m) the weighted Lebesgue space associated to the norm

(3.17) f 2 L 2 v (m) := R d f 2 (v) m 2 (v) dv, m(v) = v ℓ
where we recall that ℓ has been fixed in Subsection 1.3 and satisfies ℓ > d/2. The behaviour of solutions to the homogeneous Fokker-Planck equation has been largely studied in several types of functional spaces thanks to Poincaré inequality or Log-Sobolev inequality for example. In our case, we will make use of a result from Gualdani, Mischler and Mouhot [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-Theorem[END_REF]Theorem 3.1] which is stated in the following lemma, the main idea is that they have enlarged the space in which the solutions to the equation decay as time goes to infinity to various kind of weighted Lebesgue spaces and in particular to the space L 2 v (m) defined above. Lemma 3.3.

Consider f 0 = f 0 (v) ∈ L 2 v (m) such that R d f 0 (v) dv = 0.
Then, there exist C ≥ 1 and λ * > 0 such that the solution f = f (t, v) to the homogeneous Fokker-Planck equation (3.16) satisfies

f (t, •) L 2 v (m) ≤ C e -ελ * t f 0 L 2 v (m) .
3.3. Proof of Theorem 3.1. We now handle the proof of Theorem 3.1 and we split it into two parts, we first look at the behaviour of the density ρ = ρ(t, x) before going back to the solution h = h(t, x, v) itself. Let us underline the fact that in the proof of Theorem 2.1, we are able to deal with the cases ε = 0 and ε > 0 in the same time, which is not the case here. However, we do not write the proof for ε = 0 since it is just a (simpler) adaptation of the case ε > 0.

Proof of the estimate (3.2). As in the proof of the estimate (2.2), we start by noticing that, due to the conservation properties of the operator -

v • ∇ x h -F (t, x) • ∇ v M + ε (∆ v h + div v (vh))
, we know that for all times, ρ(t, 0) = (2π) d/2 h in (0, 0) = 0 since h is mean-zero.

We then concentrate on the study of ρ(t, k) for k ∈ Z d \ {0}. In this part, we consider ε ∈ (0, ε 0 ] where ε 0 is given in Lemma 3.2 and t ∈ R + . We first take the Fourier transform of the equation (3.1). It provides us the following equality:

∂ t h -k • ∇ ξ h + k • ξ W (k) ρ(t, k) M (ξ) + ε ξ • ∇ ξ h = -ε |ξ| 2 h
where we have denoted h = h(t, k, ξ). We rewrite the latter equality as:

∂ t h + (ε ξ -k) • ∇ ξ h = -ε |ξ| 2 h -k • ξ W (k) ρ(t, k) M (ξ)
in order to consider (ε ξ -k) • ∇ ξ as our "new" transport operator, the term -ε |ξ| 2 h is seen as an absorption term and -k • ξ W (k) ρ(t, k) M (ξ) as a source one. Then using the method of characteristics, we end up with the following formula:

(3.18) h(t, k, ξ) = exp -ε t 0 e -ε(t-s) ξ + χ ε (t -s) k 2 ds h in k, e -εt ξ + χ ε (t) k - t 0 exp -ε t s e -ε(t-σ) ξ + χ ε (t -σ) k 2 dσ k • e -ε(t-s) ξ + χ ε (t -s) k W (k) ρ(s, k) M e -ε(t-s) ξ + χ ε (t -s) k ds.
Taking ξ = 0 in the previous formula gives

ρ(t, k) = exp -ε t 0 |χ ε (t -s) k| 2 ds h in (k, χ ε (t) k) - t 0 exp -ε t s |χ ε (t -σ) k| 2 dσ k • (χ ε (t -s) k) W (k) ρ(s, k) M (χ ε (t -s) k) ds.
and thus

e εt ρ(t, k) = e εt exp -ε t 0 |χ ε (t -s) k| 2 ds h in (k, χ ε (t) k) + t 0 K ε (t -s) e εs ρ(s, k) ds
where we recall that K ε is defined in (3.4). Then, using Lemma 3.2 and proceeding as in the proof of estimate (2.2), we obtain for any k ∈ Z d \ {0} and any T > 0:

sup t∈[0,T ] kt n e εt | ρ(t, k)| ≤ sup t∈[0,T ] kt n | h in (k, χ ε (t) k) | e εt exp -ε t 0 |χ ε (t -s) k| 2 ds ≤ sup t∈[0,T ] kt n χ ε (t)k n 1 0≤t≤3/ε e 3 + kt n e -|k|t/2 1 t≥3/ε h in L 2 x H n v
where we used the estimate (3.12). But, we recall that χ ε (t) ≥ e -3 t for any

t ∈ [0, 3/ε]. Consequently (3.19) ∀ k ∈ Z d \ {0}, sup t∈[0,T ] kt n e εt | ρ(t, k)| ≤ C(n) h in L 2 x H n v ,
this provides us the result (3.2).

We are now able to go back to the analysis of the behaviour of the solution h = h(t, x, v) thanks to the previous study on the density ρ = ρ(t, x).

Proof of the estimate (3.3). We first deal with the case k = 0 and consider ε ≥ 0. One can notice that the moment of order 0 in x is not preserved due to the presence of the collision operator as in the Proof of estimate (2.3). Indeed, h := T d h(t, x, •) dx satisfies the following homogeneous Fokker-Planck equation:

∂ t h = ε (∆ v h + div v (v h )) . Note that since h in ∈ L 2 x H n v , we have h in ∈ L 2 v (m)
where L 2 v (m) is defined through its norm (3.17). Indeed, using Jensen inequality,

R d h in 2 m 2 (v) dv = R d T d h in (x, v) dx 2 m 2 (v) dv ≤ C R d ×T d h 2 in (x, v) v 2ℓ dx dv ≤ C h in 2 L 2 x H n v .
Using Lemma 3.3, we have

∀ t ≥ 0, h (t, •) L 2 v (m) ≤ C e -ελ * t h in L 2 v (m)
where we used that h in is mean-zero. Consequently, using Cauchy-Schwarz inequality combined with the fact that m -2 = • -2ℓ ∈ L 1 v (R d ), we have

∀ ξ ∈ R d , ∀ t ≥ 0, | h(t, 0, ξ)| = | h (t, ξ)| = R d h (t, v) m(v) m -1 (v) e -iv•ξ dv ≤ h (t, •) L 2 v (m) R d m -2 (v) dv
≤ C e -ελ * t h in L 2 v (m) ≤ C e -ελ * t h in L 2

x H n v from which we readily conclude.

We now deal with k ∈ Z d \ {0} and consider ε ∈ (0, ε 0 ] where ε 0 is given in Lemma 3. Then using the following bound:

(1 -e -εt ) where the last inequality holds for t ≥ ln(2)/(2ε) and in particular for t ≥ 6/ε and where we used that -y 2 + 4y -5 ≥ -5 for any y ∈ [0, 1]. From this, we have:

∀ t ≥
|T 1 | ≤ C e -εt ξ + χ ε (t)k -n ≤ C ξ 2-n χ ε (t)k 2-n ≤ C ξ 2-n t 2-n
where we used that χ ε (t) ≥ e -6 t for 0 ≤ t ≤ 6/ε. Concerning the second one, we decompose it into two parts: To handle the first term T 21 , we use the decay property of M = M which in particularly implies: e -εs ξ -χ ε (s)k M e -εs ξ -χ ε (s)k ≤ C e -εs ξ -χ ε (s)k 2-n . and the estimate (3.2) so that for any 0 ≤ s ≤ 6/ε: e ε(t-s) | ρ(t -s, k)| e -εs ξ -χ ε (s)k M e -εs ξ -χ ε (s)k ≤ C t -s 2-n t -s -2 e -εs ξ -χ ε (s)k 2-n ≤ C t 2-n s n-2 t -s -2 ξ n-2 χ ε (s)k 2-n ≤ C t 2-n ξ n-2 t -s -2 where we used that χ ε (t) ≥ e -6 s for 0 ≤ s ≤ 6/ε to get the last inequality. As a consequence, we obtain:

2 . 0 e εs exp -ε s 0 e 2 . 2

 20022 Going back to the equality(3.18), we have:(3.20) e εt h(t, k, ξ) = e εt exp -ε t 0 e -εs ξ + χ ε (s)k 2 ds h in k, e -εt ξ + χ ε (t)k t -εσ ξ + χ ε (σ)k 2 dσ k • e -εs ξ + χ ε (s)k W (k) e ε(t-s) ρ(t -s, k) M e -εs ξ + χ ε (s)k ds =: T 1 + TWe start by studying the terms of type exp -ε t 0 |e -εs ξ + χ ε (s)k| 2 ds . As previously, we compute explicitly the integral inside it: -e -2εt + 4e -εt -3 |k| 2 .

T 2 = t 0 1 s≤6/ε e εs exp -ε s 0 e 0 1

 000 -εσ ξ + χ ε (σ)k 2 dσ k • e -εs ξ + χ ε (s)k W (k) e ε(t-s) ρ(t -s, k) M e -εs ξ + χ ε (s)k ds + t s≥6/ε e εs exp -ε s 0 e -εσ ξ + χ ε (σ)k 2 dσ k • e -εs ξ + χ ε (s)k W (k) e ε(t-s) ρ(t -s, k) M e -εs ξ + χ ε (s)k ds =: T 21 + T 22 .

(3. 24 )T 21 ≤ 2 ∞ 0 s - 2

 2421202 C t 2-n ξ n-2 6/ε 0 t -s -2 ds ≤ C t 2-n ξ n-ds ≤ C t 2-n ξ n-2 .The second term is treated thanks to (3.21), (3.2) and the fact that y → |y|M (y) is bounded on R d :(3.25)T 22 ≤ C t 0 e -s/6 t -s -n ds ≤ C t -n ∞ 0 e -s/6 s n ds ≤ C t -n . Gathering (3.22), (3.23), (3.24), (3.25), we conclude that (3.3) holds.

  -e -2εt + 4e -εt -5 |k| 2

	we deduce:
	ε	0	t	e -εs ξ + χ ε (s)k	2 ds ≥ 2ε 2 ≥ 1 -2e -2εt 4 |ξ| 2 + t ε 1 + 1 ε t -5 ε |k| 2
					ε	2	|ξ • k| ≤	|ξ • k| ε	≤	|k| 2 ε 2 +	|ξ| 2 4	,

  6/ε, exp -ε -εs ξ + χ ε (s)k 2 ds ≤ e εt e -t/(6ε) ≤ e -t/6where we used that ε ∈ (0, 1/3]. Let us now estimate each term of (3.20). If t ≥ 6/ε, the first term is easily treated using (3.21):(3.22) |T 1 | ≤ C e -t/6 ≤ C t 2-n .

	t	e -εs ξ + χ ε (s)k	2 ds ≤ e -t/(6ε)
	0
	and thus
	(3.21) e If 0 ≤ t ≤ 6/ε, we have t ∀ t ≥ 6/ε, e εt exp -ε 0
	(3.23)


Who's afraid of banal nationalism?

Contribution to the round table « The Future of Nationalism » ASEN conference 2017. Tuesday the 28 th of March 2017 Sophie Duchesne, CNRS. Centre Émile Durkheim. s.duchesne@sciencespobordeaux.fr Good afternoon. I'm very grateful to the organizers of this great conference for having invited me to participate in this round-table; all the more that I did not have the honor to know Anthony Smith personally and I certainly don't consider myself as a specialist of nationalism. I'm a French political sociologist mainly interested in citizenship, reactions to European integration and social science methods. My interest in nationalism follows from that. It is thus quite contemporary, focused on Western European countries and on the way citizens experience it or feel about it. A set of interests, thus, that do not particularly match Anthony Smith's own interests according to the debate he had with Jon Fox and Cynthia Miller-Idriss about their "everyday nationhood agenda" [START_REF] Fox | Everyday nationhood[END_REF][START_REF]The limits of everyday nationhood[END_REF]. Indeed, what I thought I could share with you today is a series of interrogations related to this debate and to the relationship between banal nationalism, Michael Billig's book [START_REF] Billig | Banal nationalism[END_REF], and nationalism studies. But I want to point out that I don't speak on behalf of Billig, whom I don't know personally either. Each time I'll refer to his work, it will be my interpretation.

The puzzle: the recurrent misunderstanding of Banal Nationalism (1995) I've been reading recently the discussions that Banal nationalism generated since it was published more than 20 years ago. I am surprised to see how it has become considered as the founder work of "everyday nationalism" -that Eleanor Knott, on the State of Nationalism website, identifies as "a sub-field [that] refocuses attention on the 'masses' and human agency within nationalism studies". Then she adds: "The everyday nationalism approach builds on [START_REF] Billig | Banal nationalism[END_REF] work on banal nationalism but diverts in its focus on human agency" [START_REF] Knott | Everyday nationalism: a review and annotated bibliography | The State of Nationalism (SoN)[END_REF]. In my understanding of Billig's work, the everyday nationalism agenda does much more than divert from it: it denies, more or less implicitly, its thesis. Banal nationalism was written in order to reveal the constant reminder of national belonging in established democracies as well as its legitimizing effect on the international and violent order of a world divided into nations. To me, this denial of Billig's main line of argumentation might not have happened by chance and needs to be explained. I shall briefly summarize how it seems to have happened, according to what I read, and then I'll suggest explanation that will lead me to a few conclusions regarding the future of nationalism -or at least, the study of nationalism! Banal nationalism was published in 1995. Michael Billig wrote it after Talking of the Royal family [START_REF] Billig | Talking of the royal family[END_REF]where he analyzed a series of collective interviews with family members. He then found an amazingly high level of national pride. In these discussions, the division of the world into nations and national belonging appeared natural features and were never debated. Billig reports how he thus went on reading major work on nationalism -Gellner, Anderson, Anthony Smith in particular… -and found their work fascinating. But if it made really clear how nations were built, it did not explain how the national imaginary continues to be so powerful. As you know, Billig then suggested that the reason why nationalism -i.e. the perceived naturalness of an international order based on the division of the world into nations -remains so powerful in countries such as the UK or the US, long after the years of mobilization and nation building, is because people are reminded of it all the time! They are constantly reminded that they are national by public actors, medias, marketing, architecture, art, etc. in such a way that they do not pay attention to it. Butand Billig makes it very clear, in the book -nationalism, be it banal or not, is (potentially) violent. Banal nationalism maintains a strong hierarchy between countries and people in societies that consider themselves based on human rights: nations are always first and all crimes against human beings are acceptable if they are perpetrated in the name of an acknowledged nation.

A quick tour in the reception of Michael Billig's book in nationalism studies

There is no doubt that Billig's message was clearly understood from the beginning -see Monserrat Guibernau'review in Nations and nationalism in 1997 for instance; but it seems that, as the constant reminder of the nation, it's then been forgotten. [START_REF] Edensor | National identity, popular culture and everyday life[END_REF] It seems that this begins with Tim Edensor's book National Identity, Popular Culture and Everyday Life [START_REF] Edensor | National identity, popular culture and everyday life[END_REF] written in defense of popular culture and cultural studies. Here Edensor singles out Billig as the only scholar interested in the everyday dimension of nationalism -which so far is correct -but also as someone who paved the way to a bottom-up approach of nationalism. Edensor analyses the way nations are experienced in everyday life, emphasizing individual experiences. [START_REF] Fox | Everyday nationhood[END_REF] This understanding of Banal nationalism was later comforted by Fox and Miller-Idriss in the already mentioned and programmatic article on everyday nationhood. They quote Edensor and Billig together, as evidences of a "recent increased interest in the ways in which nationhood is negotiated and reproduced (…) in everyday life". As we know, Anthony Smith responded in a quite critical way to the every nationhood agenda: there is no question that his response contributed to the visibility of the article and at the same time, to the assimilation between Edensor and Billig's approaches. [START_REF] Skey | The national in everyday life: A critical engagement with Michael Billig's thesis of Banal Nationalism[END_REF][START_REF] Skey | National belonging and everyday life: the significance of nationhood in an uncertain world[END_REF] In 2009, in a paper entitled "The national in everyday life: A critical engagement with Michael Billig's thesis of Banal Nationalism" [START_REF] Skey | The national in everyday life: A critical engagement with Michael Billig's thesis of Banal Nationalism[END_REF], Michael Skey did properly address Billig's critical analysis of everyday nationalism and in particular. I will not get into the details of his criticisms, especially since Billig already replied to him [START_REF] Billig | Reflecting on a critical engagement with banal nationalism-reply to Skey[END_REF]. What I find interesting here is that in his book, published two years later [START_REF] Skey | National belonging and everyday life: the significance of nationhood in an uncertain world[END_REF], Skey actually confirms with the collective interviews he analyses, most of Billig's points . He confirms in particular the omnipresence of the nation in discussions that were not directly related to it, as well as the general belief in the naturalness of national belonging. But, instead of following Billig's worries about it, he then suggests that the omnipresence of the nation in everyday life may be of practical and psychological value as it gives people an ongoing sense of self, place and community.

McCrone & Bechhofer (2015)

This idea that national belonging fulfils human needs and has to be acknowledged and studied as such is a recurring argument for scholars like [START_REF] Mccrone | Understanding national identity[END_REF], and more generally, Scottish, Welsh, Irish and now English academics whose work on national identity in the context of the devolution, whose work has been prolific. They do disagree with Billig but they rarely relate their disagreement directly to his denunciation of nationalism: their rather oppose the emphasis put on the State by the banal nationalism thesis which clearly shows that the author's interest is focused on nationalism in nations-states [START_REF] Mccrone | Nations and Regions: In or Out of the State?[END_REF]. Which does not mean, as it is also regularly commented, that Billig ever considered that banal nationalism was to be observed only in Western or settled nations. Billig only attracted our attention towards something we were inclined not to see, or to forget: the fact that nationalism is as omnipresent in countries like Britain, France or the United States that it is in separatist or newly established countries, which are thus overtly nationalist.

Possible explanations

Billig got plenty of occasions to clarify things and he actually did [START_REF] Dodds | An interview with Professor Michael Billig[END_REF][START_REF] Haste | The thinker as arguer: An interview with Michael Billig[END_REF][START_REF] Wertsch | Book Review Symposium: Banal nationalism[END_REF]. We could discuss my interpretation of his work but this not the reason why I took you into this very quick tour of Banal nationalism's reception. The point I wanted to make, is that the recurrent understatement of Billig's clear denunciation of nationalism in nationalism studies is something I think we should pay attention to, especially in the current context of increasing political and electoral success of nationalism. I feel this has to be underlined and explained. I shall now suggest a series of possible complementary explanations that have come to my mind while I was first realizing and then documenting this.

No time to read any more

We cannot totally dismiss the idea that if the book was not correctly understood and referred, it is the hazardous consequence of the current state of social research, where we are all so busy securing funding and publishing that we cannot afford to read entire books, not to mention someone's whole work. Reputations are this partly a matter of chance, of the way you've been quoted in the beginning.

The economy of a subfield in social sciences

Another explanation could be the economy of subfields in social sciences. Here, I see two processes that could explain what I consider to be a misunderstanding of Billig's work:

• Loving one's object I tend to think that subfields are fields where scholars have much difficulty to dislike their object. As I said, I don't consider myself as been part of national studies, I come from European studies. And this is what I have experienced there: people in the field cannot but be Europhiles. If you are not involved in the future of the European Union, you're soon considered Eurosceptic and the best thing you can do is to run away -which I did eventually [START_REF] Duchesne | Citizens' Reactions to European Integration Compared[END_REF]. I wonder if there is not something of the sort going on here: I think Billig made clear that he does not believe in national identity -not as an interior and universal human disposition. He wrote: "National identities are forms of social life, rather than internal psychological states; as such, they are ideological creations, caught up in the historical processes of nationhood". (Billig, 1995, p. 24)He feels uncomfortable when he realizes that he behaves as a nationalist -and he does, as we all do. He thus considers nationalism as the very powerful and dangerous ideology, heir to the worst ideologies humanity had to cope with in the XXth century . According to what I heard during these two days, I don't think that many of us in this conference share either his antipathy for nationalism or his skepticism regarding national identity.

• Reduced disciplinary contribution -or the influence of cognitive social psychology

Not only subfields might have a tendency to attract scholars who like their object; they also are necessarily interdisciplinary. And that also come with a price. Along the years, scholars in that area tend to lose touch with some of their disciplines in the sense that they do not keep aware of these discipline's internal debates. In this case, I observe in nationalism studies the same limited relationship with social psychology, which basically, tends to be reduced to Social Identity Theory. Billig was a student of Henry Tajfel, the father of Social identity theory, from which categorization and the in/out-group dynamic have been imported and are considered the alpha and omega of the relationships between people and political identities. But Billig then took his distance, scientifically speaking, with Tajfel [START_REF] Billig | Henri Tajfel's 'Cognitive aspects of prejudice' and the psychology of bigotry[END_REF] and developed his own psycho sociological approach, which is explicitly anti scientist and anti cognitivist. I will not make a long presentation of Billig's rhetorical psychology -but I do recommend reading Arguing and thinking [START_REF] Billig | Arguing and thinking: a rhetorical approach to social psychology[END_REF]. My point here is that Banal nationalism cannot be interpreted properly without referring it to Billig's general social science approach. And this approach is quite foreign to the general position of nationalism studies regarding social psychology. This might also contribute to explaining how his work was imported in nationalism studies in an unappropriated way.

Living a life without illusion

I think a third line of explanation is plausible, which relates to the title of my contribution -which by the way is different from the one I sent Esther a couple of weeks ago: are we afraid of Banal nationalism? You might have heard of the "Who's afraid of Virginia Woolf?", or seen the film (with Elizabeth Taylor and Burt Lancaster). Its, author, Edward Albee, explained that this meant to him: "who's afraid of living life without false illusions"? 1 I think that this is the same with banal nationalism. It is difficult, as a citizen, a human being, to give up the national imaginationsomeone yesterday talk of the national fantasy -, the idea that you belong to something that gives you its power and let you enter in eternity. All the more that the nationalist ideology, which is, one more time according to Billig still, the only international ideology, reminds you of it all the time.

Conclusion

But I do think that our role, as social scientists, is to confront this disillusion. To make everything we can to dispel the fantasy. We need for that to affront complicated methodological problems but this is what social sciences are made for: find the ways to ways to bring out what is concealed by power relationships and human fragility -in this case, the current omnipotence of nationalism. If we want it to end, we are first in line.

Thank you for your attention.

1 https://www.theparisreview.org/interviews/4350/edward-albee-the-art-of-theater-no-4-edward-albee PS. As an academic abroad, at this country, I've been to the cinema quite frequently this year and I had to watch one of these videos from Army job before any film I came to see. It has become quite an obsession.

Screenshot from Army TV advert 2017 -This Is Belonging (Part 2), available at https://www.youtube.com/watch?v=cqQvpPmfbhg copyright Army Jobs. 2


Jonathan Barés

Serge Mora

Jean-Yves ��

Delenne

��� Thierry Fourcaud

Experimental observations of root growth in a controlled photoelastic granular material

We present a novel root observation apparatus capable of measuring the mechanical evolution of both the root network and the surrounding granular medium. The apparatus consists of 11 parallel growth frames, two of them being shearable, where the roots grow inside a photo-elastic or glass granular medium sandwiched between two pieces of glass. An automated system waters the plant and image each frame periodically in white light and between crossed polarisers. This makes it possible to follow (i) the root tips and (ii) the grain displacements as well as (iii) their inner pressure. We show how a root networks evolve in a granular medium and how it can mechanically stabilize it. This constitutes a model experiment to move forward in the understanding of the complex interaction between root growth and surrounding soil mechanical evolution.

Introduction

When growing, plants and trees adapt their morphology according to the mechanical and topological properties of the surrounding soil. Conversely, they change the mechanical and chemical properties of this environment [1]. Depending on the soil density or confining pressure, on the size and architecture of the voids between solid constituents of the soil, the roots will be blocked or grow farther [2][START_REF] Clark | Roots: The Dynamic Interface Between Plants and the Earth[END_REF][START_REF] Tracy | [END_REF], either by curving, changing their direction or by stressing the soil material [5,6]. Therefore, the competing effects of soil mechanical impedance (mechanical property) [7] and penetrating capacities of the plant (biological property) lead to different strategies for the root development which can be optimized, and which in return modifies differently the soil physical properties. Indeed, the root system architecture (RSA) impacts plant fitness and crop yield by facilitating efficient nutrient and water uptake from the soil. A better understanding of the effects of soil on RSA could improve crop productivity by matching roots to their soil environment [START_REF] Rogers | Plant physiology[END_REF]. Conversely, roots can change soil aeration, local packing fraction and constituent organization [START_REF] Cresswell | [END_REF]10] but also its repartition of water and cohesion [START_REF] Baets | Earth Surface Processes and Landforms[END_REF][START_REF] Danjon | [END_REF]. Hence, the mechanism of root growth in soil is a key issue to understand both how to improve plant development and how to improve grounds properties like stability for example. When a plant grows its roots have an effect on the stress development in the surrounding soil � e-mail: jb@jonathan-bares.eu �� e-mail: serge.mora@umontpellier.fr ��� e-mail: jean-yves.delenne@supagro.inra.fr ���� e-mail: thierry.fourcaud@cirad.fr considered as a granular material. Conversely the induced pressure of the granular medium on the roots can affect the root network growth. We want to understand this complex root-grains, biological and mechanical, feedback loop between plant roots and soil structure and composition [13].

At the local scale, [5,6] measured the force applied by a growing root tip on an obstacle. At the same scale, [1] have also studied the stress induced by a root when it goes through the gap between 2 bidimensional photoelastic grains. At the global scale [14] attempted to observe the root network in a packing of such particles. But root tip cannot initiate a path if there is no gap between grains, which is the case for a 2D dense packing of cylinders. From a numerical perspective, [START_REF] Baets | Earth Surface Processes and Landforms[END_REF]15] have modelled root network architectures in different soils while [16][17][18] have studied the tree anchorage and soil stabilization phenomenons.

Still, no experimental studies have been carried out to directly observe root network dynamics and surrounding stress modification while imposing specific grain configurations and density or specific mechanical loadings. It is therefore of primary importance to design a model experiment to understand progressively the intricate soil-root system from a mechanical point of view [1]. In this paper, we present a novel set-up which permits to observe the development of chickpea (cicer arietinum L.) root networks in a 2D granular material constituted of (i) home made bidisperse photoelastic discs or of (ii) glass beads. The photo-elastic particles (i) are molded with a groove on their edges so that the root tips are able to initiate a path between grains. Nevertheless the groove is small enough to observe a photoelastic signal when the root get thicker.

The position of the grains, the inter-granular spacing as well as the nature of the system confinement can be imposed in this set-up. The container of the granular material can be (i) an open cell, (ii) a confined cell and even a (iii) shearable cell. In this last case a slow shear loading can be imposed during the root growth or at the end of the development to test the stabilization effect on the granular medium. In the automated experimental apparatus several root development cells are treated in parallel to increase the statistical meaning of the observations. For each cell, each hour a picture is taken with and without polarized light. Then the evolution of the root network is followed as well as the position and pressure inside each particle by mean of classical image processing [19] and photo-elastic techniques [20,21].

Experimental set-up 2.1 Growth frames

In the experimental set-up, we observe the growth of chickpea root networks in quasi-2D granular materials. Chickpea has been chosen because it is robust, the thickness of the roots (∼ 1mm) is about the depth of the 2D medium (4mm), the root network is not too dense, it grows relatively rapidly (few weeks) and the seed is large enough to feed the plant during the first cycle of the growth. In a first configuration, these chickpeas are grown in simple 15cm 2 cells between 2 pieces of glass separated by 4mm, as presented in fig. 1-A. In a second configuration, as presented in fig. 1-B the left and right edges of the cell are movable so that the whole system (grains and roots) can be sheared by mean of a stepper motor once the root network is fully developed. In both case the plants grow in deionized water.

As shown in fig. 1-A, to observe the growth, periodically (each hour) the cells are lit from the back with a flat circularly polarized light and imaged from the front with a 20Mp CCD camera 1 with and without a crossed polariser in front of the camera lens. To observe the growth of several plants at a time, in several configurations, and to increase the statistical representativity of our results, 11 cells are observed at the same time (2 of them being shearable). As presented in fig. 1-C the growth frames lay on a bar in parallel and the imaging part of the device, moved by a stepper motor, slides in front of each cell to take pictures. The whole system, including the light control is automated as well as a watering system which maintain the deionized water level constant in each cell (not shown here).

Granular material

These chickpeas are grown either in bidisperse photoelastic discs (7 and 9mm in diameter) or in bidisperse glass beads (2 and 4mm in diameter). In the first case to avoid the problem with cylindrical particles pointed out by [14], as presented in fig. 2-C, we used particles with a 0. The system is then sheared quasistatically. The root networks avoid the top grains to rearrange and make the granular medium jammed. For A to F, pictures are taken only with back-lighting while for pictures G to I the system is between crossed polarisers.

groove on the edge. Hence root tip can initiate a way between two adjacent grains and propagate deep inside the granular medium. These grooved photo-elastic particles are molded from polyurethane using the technique first introduced by [19]. As shown in fig. 2-A a counter mold is machined out of aluminium. Then, a mold is made from silicon rubber2 (see fig. 2-B) and colored polyurethane3 is molded in each hole. As shown in fig. 2-D the photoelasticity of these particles is very good and classical techniques for photo-elastic granular materials pioneered by [20,21] can by used.

Results and discussions

In fig. 3-A, B and C we show the development of a chickpea root network in a bi-disperse quasi-monolayer of glass beads. After one day, the main root is already embedded in the granular material and secondary roots appear at the basis of the seed. Two days later, the full main root network is set and has moved the grains around the seed. After one week, the RSA is at its maximum of expansion without more nutriment. Secondary roots have grown from the main network and we clearly observe the root tortuosity imposed by the granular arrangement. Experiment is stopped and root dynamics can be studied.

In fig. 3-D, E and F we show the development of two chickpea root networks in a bi-disperse layer of photoelastic particles. Compared to the glass bead case the growth is much slower and plants need to be fed: deionized water is oxygenated and enriched. In this case it takes more then 3 weeks for the RSA to be fully developed. Tortuosity is also qualitatively lower since grains are larger than glass beads. After 23 days, the system is slowly and continuously sheared of 21% during 4 days. During this process even if the granular system is opened on its top and free to rearrange, a strong force network builds up. We see in fig. 3-G, H and I that the root networks prevent the grains from rearranging and releasing stress. This create force chains forming from top left to the bottom right of the cell and from the left to the roots. This shows how the chickpea RSA can stabilize a granular system.

As presented in fig. 3-A to F the back light pictures permit to observe the root network with a high contrast. This makes the detection of the RSA easy by thresholding the pictures. Hence, fig. 4-A shows the position of the root network between photo-elastic grains once the RSA is fully developed. Just like already done in gels by [22], a homemade code as been written to detect and track the root tips during the development of the RSA. Blue and red lines in the figure are two examples of the path followed by two root tips. In both cases, we observe that the path does not follow the final root architecture. This means the root widely rearranges during its growing process either due to granular displacement or to the growing of secondary roots. Root-grain interaction is thus a dynamical process which needs to be followed on the long term. From the evolution of the root tip position, in fig. 4-C, we plot the root speed for the two selected roots of the two plants. We see that the growth speed is not constant and not equivalent for both plants. However, it is difficult to analyse if this speed variation is due to biological or mechanical parameters. We also observed root tip circumnutation [23] in 2D (not shown here), how it is affected by the surrounding granular medium, how it changes the granular local organization and how it creates the root tortuosity.

Finally, from the back light pictures (see fig. 3-F for instance), using image processing tools designed by [19], we detect the grain positions and sizes, and from the polarized pictures (see fig. 3-I for instance) we compute the grain pressures in terms of G 2 as explained in [19][20][21]. For a system with a RSA fully developed and 21% sheared, fig. 4-B shows the pressure in each grains. We observe force chains and measure the highest pressure near the root network. This is another observation of the fact that the RSA stabilizes the granular system. shows the root networks of two chickpeas grown in photo-elastic discs just before the system is sheared. Blue and red line shows the path followed by two root tips (arbitrarily chosen). We see that due to the rearrangement of the root and grains the final root does not follow exactly the root tip path. B: Pressure measurement. For the maximum strain, when the root networks are fully developed, grain position has been tracked and pressure measured in each of them as G 2 [20]. By sake of clarity, grains with pressure lower than a certain value (2) are not presented. The force chains show the stabilization effect of the root networks C: Root tip speed. For root tips presented in blue en red in A, the evolution of their speed is shown during the plant growth before shearing the system. We see that the growth speed is not constant and not equivalent for both plants.

Conclusion

We have described a novel experimental system that can observe the growth of chickpea root networks in different granular materials (namely grooved photo-elastic discs and glass beads) for different system confinements (namely open frames and sheared frames). The growth is observed both in white light and between crossed polarisers. This permits to measure the evolution of the RSA and more specifically to follow the root tips. The grain rearrangements and the evolution of their inner pressure is also recorded. This constitutes a model experiment to move forward in the understanding of the complex interaction between root growth and surrounding soil mechanical evolution.

Figure 1 .

 1 Figure1. Experimental set-up. A: Imaging principle. The plant growths sandwiched between two pieces of glass in a 4mm thick flat frame with 15cm width and 15cm height. The roots growth in a 2D granular material constituted of bidisperse glass beads or photo-elastic cylinders covered with deionized water. Periodically, the system is imaged by a front camera with a back-light with and without cross polarisers. B: Shearable growth frame. Some plants have been grown in frames equipped with a parallelogram mechanism to shear the granular medium when the growth is ended. The mechanism is actuated by a stepper motor. C: Complete set-up. The whole set-up is constituted of 11 growth frame (2 shearable, 9 simple) laying in parallel. Periodically a robotic harm slides in front of each frame to image it with and without polariser. Lighting is also controlled for pictures.

Figure 2 .

 2 Figure 2. Photo-elastic grain molding. A: Aluminium counter mold of a set of 34 bidisperse particles. Particles are 4mm thick cylinders of diameter 7mm and 9mm with a 0.5mm deep and 0.5mm wide groove on the edge (see C). B: Silicone mold made from the aluminium counter mold. Colored polyurethane is molded in each alveolus. C: Photo-elastic polyurethane particle. D: Collection of compressed particles imaged between circular cross polarisers. Colored regions of the particles show highly stressed areas.

Figure 3 .

 3 Figure 3. A, B, C: Development of a chickpea root network cicer arietinum L. in a bi-disperse quasi-monolayer of glass beads (2 and 4mm diameters). After the main root network growth (A and B), a secondary one develops (C). Root oscillates betweens the beads. C, D, and E: Development of two chickpea root networks in a bi-disperse layer of photo-elastic particles. G, H and I:The system is then sheared quasistatically. The root networks avoid the top grains to rearrange and make the granular medium jammed. For A to F, pictures are taken only with back-lighting while for pictures G to I the system is between crossed polarisers.

Figure 4 .

 4 Figure 4. A: Root tip path. The black and white back groundshows the root networks of two chickpeas grown in photo-elastic discs just before the system is sheared. Blue and red line shows the path followed by two root tips (arbitrarily chosen). We see that due to the rearrangement of the root and grains the final root does not follow exactly the root tip path. B: Pressure measurement. For the maximum strain, when the root networks are fully developed, grain position has been tracked and pressure measured in each of them as G 2[20]. By sake of clarity, grains with pressure lower than a certain value (2) are not presented. The force chains show the stabilization effect of the root networks C: Root tip speed. For root tips presented in blue en red in A, the evolution of their speed is shown during the plant growth before shearing the system. We see that the growth speed is not constant and not equivalent for both plants.

Smooth-On, MoldStar R � 15 (slow).

Smooth-On, ClearFlex R � 50 with green SoStrong R � ink.

Acknowledgements and References

This work is supported by the Labex NumEv (anr-10-labx-20).