G Dubois

J Cesbron
email: julien.cesbron@ifsttar.fr

H P Yin
email: yin@lami.enpc.fr

F Anfosso-Lédée

Numerical evaluation of tyre/road contact pressures using a multi-asperity approach

Keywords: Contact mechanics, Numerical methods, Tyre/road contact, Road roughness

Highlights

> Numerical study of frictionless tyre/road contact. > Image processing to partition a real road surface. > Macro and micro-scale approach for contact between real road surface and elastic half-space. > Present approach gives a rather good accuracy and a drastically computation time reduction.

PAPER IJMS-10545

Response to the reviewer In the following lines, the comments are reminded in italic. Then the arial font is used for the new text in the revised paper.

The 3 points suggested by the reviewer 1 have been taken into account in the revised version of the article:

Reviewer 1:

1) The paper describes a numerical method for the calculation of normal contact forces during the contact of a rubber tread and a rough road. The method is based on two steps -the macro-scale algorithm that provides initial estimates for the local contact forces and a micro-algorithm that uses these estimates for the calculation of a fine pressure distribution. In between, the load penetration relationship (eqs. 8 and 15) is required and is obtained by fitting analytical equations to numerical contact results, obtained by a direct matrix inversion method. Although all this is described fine in the text, due to the complexity of the method, I think that the paper would benefit from a global flow chart describing all processes involved, e.g. estimation of analytical functions > estimation of contact loads > calculation of fine distribution. Since the whole approach is efficiency-driven, it would be nice to show collectively the relative computational effort in each part and also indicate which computations are one-off, e.g. the estimation of parameters for eq. 15.

It is a very good idea and we have introduced a new figure (Fig. 3) which describes all processes involved and these following sentences in the beginning of the section 3:

"The global flow chart of the proposed method is depicted in Fig. 3. The right-hand part of the figure corresponds to the existing multi-asperity approach as described in section 2. The left-hand part corresponds to the new preconditioning part which will be introduced in section 3.1 and 3.2. This new part will make the final micro-scale computation possible for a real tyre/road contact patch. The numbers in percent are the relative computational effort in each part. For a given surface, the preconditioning part is a one-off computation. The relative computational effort between both parts cannot be indicated as it depends on the size and roughness characteristics of the contact area.".

The global flow chart clarifies our method. In this figure, we show the relative computational effort in percent in each part: preconditioning (one-off computation) and contact model. The relative computational effort between both parts cannot be indicated as it depends on the size and roughness characteristics of the contact area. However, examples of calculation time are provided later in the text.

2) In figure 11 the colour key is labelled as p/pref. If that was the case all pressures on the left should be black and on the right almost black. Did the authors mean p/(prefmax)?

It is correct and we apologize for the mistake. In figure 12 the colour key is labelled as p/p ref-max . Then the figure was modified in consequence.

3) Finally, in section 4.2 an initial and final pressure distribution are mentioned. It would be beneficial to the reader to better define the initial distribution. I assume it is the one after one iteration? Could it be the one corresponding to the preliminary macro-stage? If that is the case how do the authors move from the calculated loads to the actual pressure? This requires clarification. Indeed, because the initial and the final pressure distributions are not defined, it leads to confusion. These following sentences have been added in the section 4.2:

"The initial pressure distribution p 0 is an approximation of the contact pressure. It is calculated using the method in section 2.2.1 from the macro-scale forces P k 0 , i.e. a classical matrix inversion method for each individual asperity. The final pressure distribution p final is the pressure distribution of the final iteration of the algorithm in section 2.2.2 with full interaction between all the asperities.".

Introduction

The interaction between automotive tyres and road surfaces determines many physical phenomena such as rolling resistance, skid resistance, wear and noise generation which are of great importance to safety or environmental issues.

The road texture is a significant parameter of this interaction for each phenomena mentioned above [START_REF] Deraad | The influence of road surface texture on tire rolling resistance[END_REF][START_REF] Rimondi | Tire contribution in the context of automobile noise reduction[END_REF]. This paper focuses on a tyre/road contact model for application to tyre/road noise, but the principles and the tools developed could be used for other purposes related to the tyre/road interaction. Road traffic noise pollution is a major environmental problem, which is predominated by the generation of tyre/road noise [START_REF] Sandberg | Tyre/road noise : myths and realities[END_REF]. A great part of noise emission is caused by radial tyre vibrations whose origin is tyre/road contact [START_REF] Sandberg | Tyre/road noise reference book[END_REF]. The interaction between the tyre and the road is a complex contact mechanics problem involving multiple scales. The description of the road at a fine scale of macro-texture is often neglected in models for rolling noise prediction while its effect on the interaction is important [START_REF] Andersson | Time domain contact model for tyre/road interaction including nonlinear contact stiffness due to small-scale roughness[END_REF].

Since they are the main source of radial tyre vibrations, several approaches were proposed in the literature for modelling normal tyre/road contact forces for road macro-texture. The first is based on boundary element methods in two [START_REF] Clapp | Development and validation of a method for approximating road surface texture-induced contact pressure in tire-pavement interaction[END_REF][START_REF] Klein | An envelopment procedure for tire/road contact[END_REF] or three dimensions [START_REF] Wullens | A three-dimensional contact model for tyre/road interaction in rolling conditions[END_REF]. The second uses a Winkler bedding model in two [START_REF] Kropp | Ein Modell zur Beschreibung des Rollgeräusches eines unprofilierten Gürtelreifens auf rauher Strassenoberfläche[END_REF] or three dimensions [START_REF] Hamet | Road texture and tire noise[END_REF]. A Winkler model with non-linear contact stiffness due to small-scale roughness was introduced by Andersson and Kropp [START_REF] Andersson | Time domain contact model for tyre/road interaction including nonlinear contact stiffness due to small-scale roughness[END_REF]. These last approaches take into account the vibration of the tyre via Green's theory. Finally, many other approaches are modelling the tyre tread by an elastic half-space in contact with a perfectly rigid road surface, which are based on Boussinesq [START_REF] Boussinesq | Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques[END_REF] potential theory. According to this theory on the contact between an elastic half-space and a rigid surface, several models use a multiasperity approach to describe the road surface in two [START_REF] Fujikawa | Definition of road roughness parameters for tire vibration noise control[END_REF] or three dimensions [START_REF] Sameur | Modèle de contact pneumatique/chaussée pour la prévision du bruit de roulement[END_REF][START_REF] Cesbron | Influence de la texture de chaussée sur le bruit de contact pneumatique/chaussée[END_REF]. For three dimensional models, the results are still limited to relatively large elements [START_REF] Wullens | A three-dimensional contact model for tyre/road interaction in rolling conditions[END_REF], or to asperities of simple shapes using Hertz [START_REF] Hertz | Über die Berührung fester elastischer Körper[END_REF] theory or results of Sneddon [START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF] in Sameur [START_REF] Sameur | Modèle de contact pneumatique/chaussée pour la prévision du bruit de roulement[END_REF]. Moreover the model proposed in Cesbron et al. [START_REF] Cesbron | Influence of road texture on tyre/road contact in static conditions[END_REF] only uses the emergent part of the surface texture.

In this paper, the two-scale multi-asperity approach of macro-texture of Cesbron et al. [START_REF] Cesbron | Influence of road texture on tyre/road contact in static conditions[END_REF] is used. A new preconditioning is introduced in order to evaluate tyre/road contact pressures at the full contact patch scale of the tyre with a real road surface. This study presents a partitioning method within the framework of tyre/road noise prediction for using the whole measured surfaces of the road. Moreover, a new load/penetration relation on a single road asperity is introduced. After presenting the tyre/road contact model, numerical results will be given and discussed. Then comparisons with a reference method on a small sample will be assessed. Finally, results in real size of tyre/road contact area are carried out.

A multi-asperity approach for the Boussinesq problem on a real road macro-texture surface

Within the framework of tyre/road noise prediction, the tyre tread is approximated by an incompressible elastic half-space (Fig. 1). The road surface is assumed to be a perfectly rigid rough surface. Since noise generated by vibrations is mainly influenced by the radial acceleration at the surface of the tyre, only normal contact forces are studied, i.e. friction is not taken into account in this paper. Under these assumptions, the contact problem reduced to Boussinesq theory is governed by the following relations:

∀M ∈ Σ, u(M ) = Σ T (M, S)p(S)dΣ (1)
∀(M (x, y), S(x ′ , y ′ )) ∈ Σ 2 , T (M, S) = 1 -ν 2 πE (x -x ′ ) 2 + (y -y ′ ) 2 (2)
where Σ is the surface of the half-space, u is the displacement at the surface of the half-space, p is the normal contact pressure and T is the influence function with the Young modulus E and the Poisson's coefficient ν of the half-space. The conditions of unilateral contact are given by the following relations:

∀M ∈ Σ c , u(M ) = z r (M ) -δ -z t (M ) and p(M ) > 0, Contact (3) ∀M ∈ Σc , u(M ) > z r (M ) -δ -z t (M ) and p(M ) = 0, Separation (4)
where Σ c is the contact area, Σc is the non-contact area (Σ = Σ c ∪ Σc ), δ is the global penetration between the contacting bodies, z r describes the height of the road surface and z t the height of the tyre. Eq. (3) contains two relations which have to be fulfilled within the contact area: the first one describes the condition of non-penetration and the second the condition of compressive contact. Eq. ( 4) contains two relations which have to be fulfilled outside the contact area. If the global penetration δ is known, the problem can be described by relations (1), ( 3) and ( 4) and the unknowns are the contact area Σ c and the pressure distribution p. Else, if only the total load P applied to the tyre is known, the equilibrium condition in statics gives the additional equation:

-P = Σ p(S)dΣ (5)
Solving the Boussinesq contact problem in the full tyre print can be very time consuming with direct matrix inversion methods [START_REF] Johnson | Contact mechanics[END_REF][START_REF] Cesbron | Contact analysis of road aggregate with friction using a direct numerical method[END_REF]. Thus Cesbron et al. [START_REF] Cesbron | Influence of road texture on tyre/road contact in static conditions[END_REF][START_REF] Cesbron | Numerical and experimental study of multi-contact on an elastic half-space[END_REF] have proposed to solve the problem in two steps, as illustrated in Fig. 2. The first step consists in limiting the Boussinesq problem at the macro-scale in order to obtain local contact forces acting at the summits of the asperities.

Then, these forces are used in a second step to compute the pressure distribution in the whole contact area. In these two steps the whole road surface is considered as a partition of N asperities such as:

∀l ∈ [1, N ], Σ = N l=1 Σ l (6)
where Σ l is the surface of asperity l.

The macro-scale approach of the Boussinesq problem

Considering that the interaction between two asperities k and l is described by a constant coefficient T kl , which is the inter-summit asperity interaction1 , the local penetration δ k on asperity k can be written as:

δ k = z s r,k -δ -z s t,k - N l=1 l =k T kl P l (7)
where z s r,k is the height of the summit of the k th asperity and z s t,k is the height of the tyre relative to z s r,k . The local penetration δ k corresponds to the displacement of the half-space at the summit of the k th asperity. If only asperity k acts at the surface of the half-space, then δ k is equal to z s r,k -δ -z s t,k . In the multi-asperity case, each force P l that acts at the summit of another asperity l will induce a displacement on asperity k given by T kl P l .

In a multi-asperity approach, the contact forces P k at the summit of the asperities can be written as follows:

∀k ∈ [1, N ], P k = f k (δ k ) if δ k > 0 0 if δ k ≤ 0 and P + N k=1 P k = 0 (8)
where f k is a continuous and differentiable function on ]0, +∞]. This contact law f k of a single asperity reflects the displacement on its top induced by the contact pressure distribution on the asperity. Usually, f k is a non-linear function.

Introducing Eq. ( 7) in Eq. ( 8) leads to a non-linear system of N +1 equations with N + 1 unknowns {P 1 , . . . , P N , δ}. It is solved using the Newton-Raphson iterative method and gives the contact forces P 0 k and global penetration δ 0 which will be used as inputs for the second steps at micro-scale.

The micro-scale resolution method of the Boussinesq problem

According to Cesbron et al. [START_REF] Cesbron | Numerical and experimental study of multi-contact on an elastic half-space[END_REF], the micro-scale resolution method of the Boussinesq problem is described by the following equations.

The surface of the elastic half-space is divided into n identical square elements with coordinates (x i , y i , z i ) of the center of it and size h x by h y . A uniform pressure is assumed on each square element. Then Eqs. ( 1) and ( 3) are combined to give the vectorial equation:

Ap = b (9)
A global influence matrix A is defined and its coefficients are calculated using the analytical results of Love [START_REF] Love | Stress produced in a semi-infinite solid by pressure on part of the boundary[END_REF]. The unknown global pressure vector is denoted p = {p i } T i∈ [1,n] and the second member vector is defined by b = {z r,i -δ -z t,i } T i∈ [1,n] . The vectorial Eq. ( 9) can be solved using a direct matrix inversion method, but this can be very time consuming for a real road surface due to the very large size of the matrix. In the multi-asperity approach, the matrix A is organized by blocks as follows:

        A 1 • • • A 1k • • • A 1N . . . . . . . . . . . . . . . A k1 • • • A k • • • A kN . . . . . . . . . . . . . . . A N 1 • • • A N k • • • A N                        p 1 . . . p k . . . p N                =                b 1 . . . b k . . . b N                (10)
where A k is the local influence matrix, p k is the local unknown pressure vector and b k is the local displacement vector on the k th asperity. The extra-diagonal block A kl is the part of A relative to the interaction of asperity l on asperity k. From Eq. ( 10), the Boussinesq problem can be solved by a succession of local matrix inversion. Then the contact problem is solved using a non-linear block Gauss-Seidel like algorithm. The non-linearity is due to the contact conditions.

Initial pressure distribution

From the macro-scale forces P 0 k , an initial approximation of the contact pressure p 0 = {p 0 1 , . . ., p 0 N } T is calculated using the classical matrix inversion method of Johnson [START_REF] Johnson | Contact mechanics[END_REF], with p 0 k such as:

     A k11 • • • A k1n k 1 . . . . . . . . . . . . A kn k 1 • • • A kn k n k 1 1 • • • 1 0               p 0 k1 . . . p 0 kn k δ + u k          =            z r,k1 -z t,k1 . . . z r,kn k -z t,kn k - P 0 k hxhy            (11)
where A kij are the elements of matrix A k , n k is the number of points on the asperity k and

u k = N l=1 l =k
T kl P l is the displacement induced by the contact forces on the others asperities computed by the macro-scale approach.

Non-linear block Gauss-Seidel like algorithm

Then, the contact problem is solved using a non-linear block Gauss-Seidel like algorithm, which starts from the initial pressure distribution p 0 . At step m + 1, while the pressure distribution on each asperity k, noted p m+1 k , has negative values, the contact problem is solved by inverting the following local linear problem, starting with the total number of points n k on the surface of the asperity:

A k p m+1 k = z r,k -δ m -z t,k - k-1 l=1 A kl p m+1 l - N l=k+1 A kl p m l (12)
When vector p m+1 k satisfies the contact conditions, the local inversion matrix procedure is repeated on the other asperities. Once the new vector p m+1 has been computed, the following convergence criteria is checked:

p m+1 -p m p m ≤ ε with x = n i=1 |x i | 2 ( 13
)
where ε is the convergence tolerance. The procedure is stopped when the condition in Eq. ( 13) is satisfied. Else the global penetration δ m is updated using the following relation:

δ m = δ m -ρ - P h x h y - n i=1 p i (14)
Eq. ( 14) satisfies the condition of equilibrium in statics given by Eq. ( 5). The parameters ε and ρ are given by the operator and fix the accuracy and the convergence of the method. Finally, this multi-asperity approach gives the solution to the global contact problem given by Eqs. ( 1), ( 3), ( 4) and ( 5) using a two-scale procedure which is less time-consuming (6 times faster) than usual direct methods.

Description of the tyre/road contact data

The global flow chart of the proposed method is depicted in Fig. 3. The right-hand part of the figure corresponds to the existing multi-asperity approach as described in section 2. The left-hand part corresponds to the new preconditioning part which will be introduced in section 3.1 and 3.2. This new part will make the final micro-scale computation possible for a real tyre/road contact patch. The numbers in percent are the relative computational effort in each part. For a given surface, the preconditioning part is a one-off computation. The relative computational effort between both parts cannot be indicated as it depends on the size and roughness characteristics of the contact area.

Partitioning a real road macro-texture surface

In the multi-asperity approach of macro-texture introduced in section 2, partitioning of the road surface is needed (Eq. ( 6)). The partitioning method proposed in this paper for a real road surface is illustrated in Fig. 4. The method consists in using two classical image processing methods (binarization and segmentation) in order to obtain a partitioned road surface, which is then corrected to be used for tyre/road contact.

First, the binarization of the image of the measured surface consists in an iterative labelling as described in Cesbron et al. [START_REF] Cesbron | Influence of road texture on tyre/road contact in static conditions[END_REF]. The result of this step is the identification of the emergent part of the asperities with respect to the road macro-texture scale. However a part of the road surface is also missing for the study of tyre/road contact and can induce several differences and problems on numerical results. Therefore, an algorithm for partitioning the whole measured surface was introduced. Then a watershed segmentation method is applied to the results of the iterative labelling. This method considers a gray-level image as a topographic relief. The objective is then to calculate the watersheds of this topographic relief, which are the contours of the asperity for a measured road image. In this paper, the watershed segmentation method used is based on immersion simulations of Vincent and Soille [START_REF] Vincent | Watersheds in digital spaces: an efficient algorithm based on immersion simulations[END_REF]. Finally, the segmented image is partitioned and the result is corrected relative to tyre/road contact. For example, the partitioning of a sample of 200 mm x 190 mm of Dense Asphalt Concrete 0/10 (DAC 0/10, i.e. with aggregate size of maximum 10 mm) is illustrated in Fig. 5. For clarity, the results are given in a small area of 45 mm x 40 mm (right), but the partitioning is performed on the whole measured surface (left) with a resolution h x = h y = 0.4 mm. The partitioning of the measured surface (239 000 pixels) with a standard PC takes about twenty minutes. This is performed only once and the results are saved for use as input data in contact calculations.

Load/penetration relation on a single road asperity 3.2.1. Definition of the contact law on a single road asperity

In the multi-asperity approach, the load/penetration relation for each asperity is needed at macro-scale as presented before in Eq. ( 8). The analytical contact law f k proposed in this paper is illustrated in Fig. 6 and is defined as follows:

∀k ∈ [1, N ], P k =      0 if δ k ≤ 0 Non-contact, C k E * δ γ k k if 0 < δ k < d k Power law, K k E * (δ k -d k ) + C k E * d γ k k if d k ≤ δ k
Linear. ( 15) where E * = E/(1 -ν 2 ) is the equivalent Young's modulus, C k and γ k are constants depending on the shape and the size of the asperity. These constants are analytically known for axisymmetric punches [START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF], like flat-ended, spherical or conical punches. The constant d k can be considered as a critical depth above which the contact law becomes linear. Beyond this displacement, the contact area does not change any more. Thus these bodies can be considered as a single one which behaviour is intrinsically linear. The constant K k E * corresponds to a linear stiffness.

Obtaining the analytical contact law parameters

The procedure used to get analytical contact law parameters (C k , γ k , d k , K k ) is a fitting between the contact law functions of Eq. ( 15) and numerical results from classical matrix inversion method [START_REF] Johnson | Contact mechanics[END_REF]. As an example, the sample used here is a DAC 0/10 partitioned by the process introduced in the section 3.1, composed of 1087 asperities. For each asperity, a local Boussinesq contact problem is solved by matrix inversion method, for 25 displacements imposed at the summit of the asperity z s r,k . The Young's modulus of the half-space used to model the tyre tread is E = 2.5 MPa and the Poisson's coefficient ν is set to 0.5, a common value for incompressible rubber-like materials. The contact problems on individual asperities are parallelized in a 4-cores PC with 6 Go of RAM. The entire calculation takes around 6 hours, depending on the surface texture (the number and the shape of asperities). Once all the numerical pairs (δ k , P k ) are known, the analytical contact law parameters are obtained by linear regression on the logarithm of (δ k , P k ) before d k and by direct linear regression after. The constant parameter d k is obtained when the differential function of P k relative to δ k becomes constant. For example, analytical and numerical contact laws are illustrated in Fig. 7 for asperity k = 100. For all asperities, the differences between analytical and numerical laws are below 5%. Table 1 summarizes the minimum, maximum and average values of contact parameters for the sample of DAC 0/10. The average values of C k and γ k are closed to parameters of a spherical punches with a radius equal to 5.5 mm (C = 3.127 and γ = 1.5). The values in Table 1 obtained for real road asperities are coherent with the values found in the literature [START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF] for asperities of simple shape. It is interesting to study the probability density of the pair (C k , γ k ) as shown in Fig. 8. On the one hand, there is a strong probability for the γ k parameter to be between 1 and 2, which are the extreme values obtained in the case of a flat-ended punch and a conical punch respectively. There is also a strong probability for the C k parameter to be between 2 and 4, which correspond to the sizes of 2.5 mm and 9 mm for spherical asperities respectively. On the other hand, the probability of C k is not independent of γ k : the higher γ k , the smaller C k and reciprocally. This phenomenon is visible through the emergence of a maximum of the probability density within intermediate values of C k and γ k .

Description of the tyre

The surface profile of a smooth non-deformed tyre was measured in the transversal direction using a laser displacement transducer. The result is illustrated in Fig. 9 and shows a gap of 10 mm between the center and the edges of the tyre tread. Knowing that the radius of the smooth non-deformed tyre given by the manufacturer is 284 mm, the tyre surface is then generated by revolution of the transverse profile around the wheel center. The surface obtained is subsequently used in the contact model to take the curvature of the tyre into account.

Numerical Results

In order to validate the multi-asperity method at macro and micro scales for road macro-texture surfaces, a comparison was performed with a reference method on a small surface sample. The reference method is the direct matrix inversion method of Johnson [START_REF] Johnson | Contact mechanics[END_REF], fully described in Cesbron and Yin [START_REF] Cesbron | Contact analysis of road aggregate with friction using a direct numerical method[END_REF]. Then the numerical results at the full tyre/road contact scale are given to show the efficiency and the accuracy of the multi-asperity method.

Validation on a small road surface sample 4.1.1. Calculation data

The tyre is represented by an elastic half-space with a Young's modulus of 2.5 MPa and a Poisson's coefficient of 0.5. There is no curvature of the tyre. The surface in contact with the half-space is a portion of about 7 cm2 of the sample of DAC 0/10 partitioned in section 3.1. This sample contains 23 asperities. It is illustrated in Fig. 10 and composed of 4 408 square elements of 0.4 mm x 0.4 mm. The number of potential elements in contact is very high. This surface is one of the largest which can be studied with a classical matrix inversion method. At macro-scale, the surface is described by the 23 summits of the asperities and by the contact law parameters on each of them.

The total load to be imposed (P =175 N) is obtained by multiplying the inflation pressure of the tyre (here equal to 0.25 MPa) by the surface area (7 cm 2 ). Then, for the convergence of the Newton-Raphson process, the multiasperity method at macro-scale needs several iteration loads before obtaining the value at the required load. So, 280 load steps from 0 to 175 N have been used, including 15 load values directly comparable with the matrix inversion method.

Results at the macro-scale

The contact forces obtained with both methods are compared in Fig. 11 for three load values: 50 N, 112.5 N and 175 N. The overall comparison is good and there is no outlier. In addition, the distribution of contact forces is mostly similar with both methods for the three load values. The differences between both methods slightly increase with the loads. The greatest differences are observed on the same asperities in all load cases. This problem probably stems from the geometry of the asperities. As it can be seen for the label 20 in Fig. 10, this asperity has several parts and could be better partitioned. This probably induced discrepancies in the description of inter-asperity interaction at the macro-scale. The global errors ε M for each load in Fig. 11 is defined by:

ε M = 100 N k=1 |P k -P k ref | 2 N k=1 |P k ref | 2 (16)
where P k ref are the contact forces obtained by direct matrix inversion method.

For all loads, the global error does not exceed 5%, which seems acceptable for tyre/road noise prediction at low-frequency 2 . Solving the contact problem by matrix inversion method for 15 loads between 0 and 175 N took 1 h 15 min, while it took 1 s for the multi-asperity method at macro-scale for 280 loads.

However the matrix inversion method provides the pressure distribution, while only the contact forces are obtained with the multi-asperity method at macroscale. The results at macro-scale can be sufficient for tyre/road noise prediction induced by vibrations, but for others phenomena such as air-pumping the results at micro-scale are needed.

Results at the micro-scale

The multi-asperity method at micro-scale has been computed to obtain the pressure distribution which is needed for tyre/road noise at medium and high frequency. The results at the macro-scale have been used for initializing the procedure for the surface illustrated in Fig. 10. The pressure distributions obtained by both methods at the maximum imposed load (P =175 N) are illustrated in Fig. 12. No difference is detected between both methods. The parameters ε and ρ introduced in section 2.2.2 were fixed to 1.10 -6 and 1.10 -3 respectively. Thus the convergence of the multi-asperity method was reached in a maximum of 8 iterations for all loads and a maximum of 10 iterations for the matrix inversion method. For these values of ε and ρ, the convergence of the multi-asperity method taking the matrix inversion method as a reference is given in Fig. 13. The differences were estimated using the indicators ε m in percent:

ε m = 100 p -p ref p ref (17)
where p ref is the pressure distribution obtained by direct matrix inversion method.

At the third iteration, ε m is below 0.1%, which can be already acceptable for tyre/road contact noise at medium and high frequency. As previously mentioned the reference method gave the pressure distribution in 1 h 15 min for 15 loads, whereas it took 12 min with the multi-asperity method. Hence, while giving the same results, the multi-asperity method is much faster than the classical matrix inversion.

Evaluation of tyre/road contact pressures within the full contact patch

The multi-asperity method has been computed on three real tyre/road surfaces with different textures. The three surfaces are a Dense Asphalt Concrete 0/10 (DAC 0/10), a Fine Surface Dressing 0.8/1.5 (FSD 0.8/1.5) and a Porous Asphalt 0/10 (PA 0/10). They are shown in Fig. 14. The partitioning method, load/penetration laws and the curvature of the tyre are those introduced in section 3. The area of interest for the static contact between the tyre and the three textures is a 11 cm x 18 cm rectangular area, illustrated by red boxes in Fig. 14. The contact elements used are squares of side 0.4 mm. This resolution enables to clearly distinguish the contact area on each asperity. The imposed load is P = 3 000 N, which corresponds to typical tyre/road loading for passenger cars.

The contact prints obtained for the three surfaces at P = 3 000 N are given in Fig. 15 with the initial and the final pressure distributions, for a pressure scale between 0 and 3 MPa. The initial pressure distribution p 0 is an approximation of the contact pressure. It is calculated using the method in section 2.2.1 from the macro-scale forces P 0 k , i.e. a classical matrix inversion method for each individual asperity. The final pressure distribution p f inal is the pressure distribution of the final iteration of the algorithm in section 2.2.2 with full interaction between all the asperities. The differences between initial and final distributions can be seen on the global contact areas. In addition, the average and maximum pressures for the three surfaces for the initial and the final distributions are summarized in Table 2. Regarding the differences between the initial and final pressure distributions for the same texture, the contact area tends to spread out, thus lowering the average and maximum pressures. Concerning the differences between the three surfaces, the maximum contact area is obtained with the sample of DAC 0/10, while the contact areas for the other two remain almost the same. However, the values of average and maximum pressures are increasing in the following order: DAC 0/10, FSD 0.8/1.5 and PA 0/10.

Concerning the computation time, it took 1 min on a single texture to compute the initial pressure distribution using the result at macro-scale, calculation of which also took less than 1 min. Then it took about 12 h to compute in eight iterations the final pressure distribution with the parameters ε = 1.10 -6 and ρ = 2.10 -4 . It is expected that this last step for the calculation of the final pressure distribution enhances the accuracy of the results as it was demonstrated previously on a small surface by comparison with the classical matrix inversion method. However this last step increases drastically the calculation time.

Considering the cost of the computation times, it is interesting to know if the contact forces obtained at macro-scale (associated to initial pressure distribution) are sufficiently accurate for tyre/road noise prediction at medium and high frequency. So, the contact forces obtained at macro-scale and the contact forces integrated at the summit of the asperities from final pressure distribution are illustrated in Fig. 16 for the three surfaces. R is the correlation coefficient and a and b are the coefficients of the linear regression: y = ax + b. The correlation coefficients are all close to 1, indicating a good correlation between both calculations. There is a small bias for the three different textures when considering the coefficient a and b of the linear regression: the contact forces calculated at macro-scale are almost slightly underestimated.

Finally, we can consider that the macro-scale accuracy is sufficient for tyre/road noise at low-frequency generated by vibration mechanisms and that the initial micro-scale accuracy is enough for tyre/road noise at medium and high frequency generated partially by air-pumping. The final pressure distribution can be useful for other applications linked to the contact of rough surfaces.

Conclusion

In this paper, a multi-asperity method for computing the Boussinesq contact problem was introduced and implemented, allowing calculations on real size of tyre/road surfaces. First a new method for partitioning a measured road surface was introduced. Several classical image processing methods were used, such as iterative labelling and watershed segmentation. Then a new load/penetration relation was proposed for a single road asperity. These analytical laws were fitted from numerical data, with less than 5% of differences.

A numerical validation of the multi-asperity method was performed by comparison with a reference matrix inversion method on a small road surface sample. At macro-scale results between both methods are fairly close with global errors on the forces below 5%. For use in tyre/road prediction at low frequency, results at this scale thought to be acceptable especially when considering that the developed method is far less time-consuming. At micro-scale, the results of both methods are identical, and calculations at the full tyre/road contact prints are possible in a reasonable calculation time only with the multi-asperity method.

Finally the pressure distribution on real road surfaces was evaluated on different road textures. Within the framework of tyre/road noise, the contact forces should be sufficient for predicting low-frequency noise generated by vibrations. The initial pressure distribution could be used to estimate the surface deformation of the tyre during rolling, which influences air-pumping. In addition, the possibility to obtain the pressure distribution with a very good accuracy opens up interesting prospects for introducing the phenomena of friction at the origin of high frequency noise such as stick and slip noise. The model could also be used for other tyre/road contact purposes such as skid resistance or rolling resistance. Fig. 3: Global flow chart of the method to evaluate numerically tyre/contact pressures using a multi-asperity approach (the numbers in percent are the relative computational effort in each part). Fig. 11: Comparison of contact forces at the summit of each asperity between a reference method (matrix inversion method) and the multi-asperity method at macro-scale.

Fig. 12: Comparison of the pressure distribution obtained with the reference method (matrix inversion method) and the multi-asperity method at micro-scale.

Fig. 13: Convergence of multi-asperity method at micro-scale taking the matrix inversion method as the reference.

Fig. 1 :

 1 Fig.1: Contact between an elastic half-space and a road surface.

Fig. 2 :

 2 Fig. 2: Schematic view of the two-step iterative method, (a) macro-scale, (b) micro-scale.

Fig. 4 :

 4 Fig. 4: Schematic view of the partitioning method for a real road surface.

Fig. 5 :

 5 Fig. 5: Example of partitioning for the DAC 0/10.

Fig. 6 :

 6 Fig. 6: Analytical contact law proposed for a single road asperity.

Fig. 7 :

 7 Fig. 7: Analytical and numerical contact laws (a) of a single road asperity of label 100 in the sample of DAC 0/10 (b).

Fig. 8 :

 8 Fig. 8: Probability density of the pair (C k , γ k ) for the whole area of the sample of DAC 0/10.

Fig. 9 :

 9 Fig. 9: Transverse profile measured on a real slick tyre and resulting geometry.

Fig. 10 :

 10 Fig. 10: Illustration of the small surface sample taken from the DAC 0/10.

Fig. 14 : 10 Fig. 16 :

 141016 Fig. 14: Upper view of the three surfaces used for the calculations.

Table 1 :

 1 The minimum, maximum and average values of parameters describing the analytical laws for all asperities of the sample of DAC 0/10.

		C k	γ k	K k (N/mm) d k (mm)
	Minimum	1.003 1.152	1.553	0.337
	Maximum	7.812 2.367	11.51	3.490
	Average	3.159 1.571	6.265	1.544
	Standard deviation 0.925 0.208	1.490	0.542

Table 2 :

 2 Global contact parameters obtained for the three surfaces used for the calculations for initial and final results: differences in percent are given in parentheses.

∀l ∈ [1, N ], ∀(M, S) ∈ Σ k × Σ l , T (M, S) = T (x s k , y s k ; ξ s l , η s l ) ≡ T kl , where M s k (x s k , y s k )is the position of the summit of the k th asperity.

The correlations found by Cesbron et al.[START_REF] Cesbron | Experimental study of tyre/road contact forces in rolling conditions for noise prediction[END_REF] showed that a global error of 6% on the contact forces produces a 0.5 dB error on tyre/road noise level.

List of Figures

Contact between an elastic half-space and a road surface. . . . . 16 2 Schematic view of the two-step iterative method, (a) macro-scale, (b) micro-scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Global flow chart of the method to evaluate numerically tyre/contact pressures using a multi-asperity approach (the numbers in percent are the relative computational effort in each part). . . . . . 18 4 Schematic view of the partitioning method for a real road surface.


Thomas Lepoutre
email: thomas.lepoutre@inria.fr

Nicolas Meunier
email: nicolas.meunier@parisdescartes.fr

Nicolas Muller
email: nicolas.muller@parisdescartes.fr

Cell polarisation model : the 1D case

Keywords: Cell polarisation, global existence, blow-up, asymptotic convergence, entropy method, Keller-Segel system, logarithmic Sobolev inequality, HWI inequality

come

Introduction

In this paper we improve the analysis of a one-dimensional non-linear and non-local convection-diffusion equation introduced in a previous paper [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF]      ∂ t n(t, x) -∂ xx n(t, x) = n(t, 0)∂ x n(t, x), x > 0, ∂ x n(t, 0) + n(t, 0) 2 = 0, n(0, x) = n 0 (x), x > 0.

(

The boundary condition ensures mass conservation:

∞ 0 n(t, x)dx = ∞ 0 n 0 (x)dx = M. (2)
1

The previous model [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF] comes from the models given in [START_REF] Hawkins | Rebuilding cytoskeleton roads: active transport induced polarisation of cells[END_REF] and [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF] under a radial symmetry assumption. These latter models describe cell polarisation. Cell polarisation is a major step involved in several important cellular processes such as directional migration, growth, oriented secretion, cell division, mating or morphogenesis. When a cell is not polarised proteins Cdc42 are uniformly distributed on the membrane while polarisation is characterized by a concentration of proteins in a small area of the cell membrane.

In [START_REF] Hawkins | Rebuilding cytoskeleton roads: active transport induced polarisation of cells[END_REF] and [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF], in dimension higher than two, a class of non-linear convectiondiffusion models were designed, and studied, for cell polarisation. In these models there is a coupling between the evolution of proteins and the dynamics of the cytoskeleton: the proteins diffuse and they are actively transported along tubes or filaments towards the membrane. The advection field is obtained through a coupling with the concentration of markers on the membrane. The resulting motion is a biased diffusion regulated by the markers themselves.

Of special interest is the fact that solutions of (1) may become unbounded in finite time (so-called blow-up). Blow-up of solution of (1) means that convection wins over diffusion. In such a situation markers concentrate on the membrane of the cell. We recall this result from [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF].

Theorem 1.1. Assume M > 1. Any weak solution of equation (1) (in the sense of Definition 2.1) with non-increasing initial data n 0 blows-up in finite time.

In this paper, we state a so-called concentration-comparison principle on the equation obtained after space integration. This principle together with the use of a self-similar supersolution gives estimates from above on n(t, 0) and a comparison with a suitable heat equation allow obtaining estimates from below on n(t, 0) and extending the blow-up result to any initial data above the critical mass. On the other hand, global existence and asymptotic behaviour in the sub-critical and critical cases, M ≤ 1, were established in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF].

Theorem 1.3. [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF] Assume that the initial datum n 0 satisfies both n 0 ∈ L 1 ((1 + x)dx) and ∞ 0 n 0 (x)(log n 0 (x)) + dx < +∞. Assume in addition that M ≤ 1, then there exists a global weak solution that satisfies the following estimates for all T > 0, sup t∈(0,T ) ∞ 0 n(t, x)(log n(t, x))

+ dx < +∞ , T 0 ∞ 0 n(t, x) (∂ x log n(t, x)) 2 dxdt < +∞ .
In the sub-critical case M < 1 the solution strongly converges in L 1 towards the self-similar profile G α given by [START_REF] Kullback | On the convergence of discrimination information[END_REF] in the following sense:

lim t→+∞ n(t, x) - 1 √ 1 + 2t G α x √ 1 + 2t L 1 = 0 .
In the critical case M = 1, assuming in addition that the second momentum is finite ∞ 0 x 2 n 0 (x)dx < +∞, the solution strongly converges in L 1 towards the unique stationary state ν α (x) := α exp(-αx), where α -1 = ∞ 0 xn 0 (x)dx.

In [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF], in the sub-critical case, the asymptotic result was obtained through the convergence to zero of a suitable Lyapounov functional L defined by [START_REF] Muller | Mathematical and numerical studies of nonlinear and nonlocal models involved in biology[END_REF].

Here, using a logarithmic Sobolev inequality with a suitable function, we obtain an exponential decay to equilibrium in self similar variables replacing in particular the former result by Proposition 1.4. Under the assumptions of theorem 1.3, we have

n(t, x) - 1 √ 1 + 2t G α x √ 1 + 2t L 1 ≤ C (1 + 2t) 3/2 .
Finally, in the critical case M = 1, we actually improve the result given in Theorem 1.3 by precising the speed of convergence. Firstly, we give a rate. Secondly, using the HWI inequality, we improve the rate of convergence when the third momentum is initially finite.

We end this introductory Section with one open question that we are not able to resolve: obtain a blow-up profile for large initial datum.

The plan of this work is the following. First, we state a concentrationcomparison principle and the finite time blow-up for supercritical mass. Then, we give a quantitative argument to the systematic blow-up for supercritical mass. In a third step we study the sub-critical mass and we give a rate to the self similar decay. Finally, we study the critical mass, by using first a Lyapounov approach and then the HWI inequality.

A brief account of some useful facts

We first recall some facts concerning [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF], see [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF] for more details. Let n be a classical solution of (1) on (0, T ), a straightforward computation of the evolution of the entropy yields

d dt ∞ 0 n(t, x) log n(t, x)dx = - ∞ 0 (∂ x n(t, x) + n(t, 0)n(t, x)) ∂ x n(t, x) n(t, x) dx = - ∞ 0 n(t, x) (∂ x log n(t, x)) 2 dx + n(t, 0) 2 . ( 3
)
Moreover a proper definition of weak solutions, adapted to our context is Definition 2.1. [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF] We say that n(t, x) is a weak solution of (1) on (0, T ) if it satisfies:

n ∈ L ∞ (0, T ; L 1 + (R + )) , ∂ x n ∈ L 1 ((0, T ) × R + ) ,
and n(t, x) is a solution of (1) in the sense of distributions in D ′ (R + ).

Since the flux ∂ x n(t, x) + n(t, 0)n(t, x) belongs to L 1 ((0, T ) × R + ), the solution is well-defined in the distributional sense under the assumptions of definition (2.1). In fact we can write

T 0 n(t, 0)dt = - T 0 ∞ 0 ∂ x n(t, x)dxdt.
From the maximum principle [START_REF] Evans | Partial differential equations[END_REF] it follows that n ≥ 0 if n 0 ≥ 0. Furthermore weak solutions in the sense of Definition 2.1 are mass-preserving:

M = ∞ 0 n 0 (x)dx = ∞ 0 n(t, x)dx.
Throughout the paper, for n ≥ 0 ∈ L 1 (R + ), we define the generalized distribution function in the following way:

N (x) = x 0 n(y)dy,
and for any weak solution n in the sense of definition 2.1, the first momentum is defined as

J(t) = ∞ 0 xn(t, x)dx.
Let u(x)dx and ν(x)dx be measures with smooth densities on R + , we note the relative entropy

H(u|ν) = ∞ 0 u(x) log u(x) ν(x) dx
and the Fisher information

I(u|ν) = ∞ 0 u(x) ∂ x log u(x) ν(x)
We also define the quadratic Wasserstein distance between two probability measures µ and ν on R + with finite second momentum as in [START_REF] Villani | Topics in optimal transportation[END_REF],

W 2 (µ, ν) = inf π∈Π(µ,ν) R + ×R + |x -y| 2 dπ(x, y). (4)
where Π(µ, ν) denotes the set of probability measures on R + × R + with marginals µ and ν, i.e. such that for all test functions φ and ψ in a suitable class of test functions,

R + ×R + (φ(x) + ψ(y))dπ(x, y) = ∞ 0 φ(x)dµ(x) + ∞ 0 ψ(y)dν(y).
In particular, if ν is a Dirac measure δ a with a ∈ R + we have

W 2 (µ, δ a ) 2 = ∞ 0 |x -a| 2 dµ(x). (5)
A simple computation on the first momentum (if it is initially finite) leads to

d dt J(t) = n(t, 0)(1 -M ). (6)
We recall the three situations:

• when M = 1, there is a balance between drift and diffusion and we expect a convergence to a steady state (see section 5),

• when M < 1, the diffusion drives the equation and we have a self similar behaviour see section 4,

• when M > 1, solution blows up in finite time see section 3.

Remark 1. Such a critical mass phenomenon (global existence versus blowup) has been widely studied for the Keller-Segel system (also known as the Smoluchowski-Poisson system) in two dimensions of space, see [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF][START_REF] Perthame | Transport equations in biology[END_REF] e.g. and the references therhein. The equation [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF] represents in some sense a caricatural version of the classical Keller-Segel system in the half-line (0, +∞).

Remark 2. There is a strong connection between the equation under interest here (1) and the one-dimensional Stefan problem. The later writes [START_REF]Singularity formation in the one-dimensional supercooled Stefan problem[END_REF]:

∂ t u(t, x) = ∂ xx u(t, x) , t > 0 , x ∈ (-∞, s(t)) , lim x→-∞ ∂ x u(t, x) = 0 , u(t, s(t)) = 0 , ∂ x u(t, s(t)) = -s ′ (t) .
The temperature is initially non-negative: u(0, x) = u 0 (x) ≥ 0. By performing the following change of variables: φ(t, x) = -u(t, s(t) -x), we get an equation that is linked to (1) by n(t, x) = ∂ x φ(t, x). This connection provides some insights concerning the possible continuation of solutions after blowup [START_REF]Singularity formation in the one-dimensional supercooled Stefan problem[END_REF]. This question has raised a lot of interest in the past recent years [START_REF] Herrero | Chemotactic collapse for the Keller-Segel model[END_REF][START_REF] Dolbeault | The two-dimensional Keller-Segel model after blow-up[END_REF]. It is postulated in [START_REF]Singularity formation in the one-dimensional supercooled Stefan problem[END_REF] that the one-dimensional Stefan problem is generically non continuable after the blow-up time.

3 Concentration-comparison principle and finite time blow up for supercritical mass

The main result of this section is a concentration-comparison principle given by Proposition 1.2 that we first prove. Then, we use this principle to compare solutions of equation [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF], rewritten as an equation on N , to both suband supersolutions. Finally we provide a quantitative blow-up argument by estimating the blow-up time T * defined by J(T * ) = 0.

Proof of Proposition 1.2

We actually prove the following stronger lemma:

Lemma 3.1. Let N, N , N be nondecreasing (in space) functions in C 1 (0, T ; C 2 (R + )) satisfying ∂ x N, ∂ x N , ∂ x N ∈ L ∞ (0, T ; L ∞ (R + )) and                ∂ t N (t, x) -∂ xx N (t, x) -∂ x N (t, 0)∂ x N (t, x) = 0, ∂ t N (t, x) -∂ xx N (t, x) -∂ x N(t, 0)∂ x N (t, x) ≥ 0, ∂ t N (t, x) -∂ xx N (t, x) -∂ x N(t, 0)∂ x N (t, x) ≤ 0, N (t, 0) = N (t, 0) = N (t, 0) = 0. (7)
Assume that N (0, .) ≤ N (0, .) ≤ N (0, .), then for 0 < t ≤ T we have that

N (t, .) ≤ N (t, .) ≤ N (t, .).
Proof. It is obviously sufficient to prove the comparison N ≥ N . Firstly, we assume that

∂ x N (0, 0) > ∂ x N (0, 0) (≥ 0).
Integrating equation ( 1) in space, we see that δN = N -N satisfies the parabolic equation

∂ t δN (t, x) -∂ xx δN (t, x) -∂ x N (t, 0)∂ x δN (t, x) = (∂ x N(t, 0) -∂ x N (t, 0))∂ x N (t, x), N i (t, 0) = 0.
Since we supposed ∂ x N (0, 0) > ∂ x N (0, 0), it remains true at least until a time T ′ ∈]0, T [, we choose the maximal T ′ possible. On the time interval [0, T ′ ] we have, since

∂ x N ≥ 0, ∂ t δN (t, x) -∂ xx δN (t, x) -∂ x N (t, 0)∂ x δN (t, x) ≥ 0, N i (t, 0) = 0. Hence N > N on ]0, T ′ [×R *
+ by strong maximum principle [START_REF] Evans | Partial differential equations[END_REF]. Furthermore, by Hopf Lemma (see [START_REF] Evans | Partial differential equations[END_REF]), we also have

∂ x δN (T ′ , 0) = ∂ x N (T ′ , 0) -∂ x N (T ′ , 0) > 0.
As T ′ is maximal we immediately conclude that T ′ = T . To treat the case ∂ x N (0, 0) = ∂ x N (0, 0), we use the following Lemma 3.2. Let N be a supersolution as in lemma 3.

1, denote K = sup [0,T ] ∂ x N , then N ǫ (t, x) = N (t, x) + ǫe 4K 2 t (1 -e -4Kx ) is a supersolution on [0, T ] in the sense of lemma 3.1 for 0 ≤ ǫ < 1 2 e -4K 2 T .
Proof. It is straightforward that ∂ x N ǫ ≥ 0 and N ǫ (t, 0) = 0. We essentially have to prove the inequation

I ǫ = ∂ t N ǫ -∂ xx N ǫ -∂ x N ǫ (t, 0)∂ x N ǫ ≥ 0. We denote N ǫ (t, x) = N (t, x) + β(t)(1 -e -αx ).
Conditions of the lemma ensure β(t) ≤ 1 2 for 0 ≤ t ≤ T . Since N is a supersolution, we have

I ǫ ≥ β ′ (t)(1-e -αx )+α 2 β(t)e -αx -β(t)α∂ x N -β(t)αe -αx ∂ x N (t, 0)-α 2 β 2 e -αx .
This leads to, by definition of K,

I ǫ ≥ β ′ -βαK + βe -αx - β ′ β + α 2 (1 -β) -αK
and the right hand side is nonnegative. We choose the β(t) = ǫe αKt , so that

I ǫ ≥ βe -αx α 2 (1 -ǫe αKT ) -2αK ,
and finally, choosing α = 4K and ǫ ≤ e -4K 2 T

2

we have I ǫ ≥ 0. End of the proof of lemma 3.1. Thanks to the first step of the proof, since for ǫ > 0 we have

∂ x N ǫ (t, 0) > ∂ x N (t, 0) ≥ ∂ x N (t, 0), we can compare N ǫ (t, x) ≥ N (t, x) for 0 ≤ t ≤ T and ǫ small enough. Letting ǫ → 0 we can conclude N (t, x) ≥ N (t, x).

Estimations on n(t, 0) and local existence for supercritical mass

Classically, the concentration-comparison principle allows us comparing solutions of ( 7) to sub-and supersolutions. Since the proof is very much alike the one for the comparison principle (also based on Hopf Lemma) we do not repeat it.

The main use of this Lemma is the following:

Lemma 3.3 (Self-similar supersolutions). Let λ ∈ R and f ∈ C 2 (R + , R + ) satisfying      f ′ (x) > 0, ∀x ≥ 0, f (0) = 0, λxf ′ (x) -f ′′ (x) -f ′ (0)f ′ (x) ≥ 0, ∀x ≥ 0.
Then the function Nf,λ defined by

Nf,λ (t, x) := f x √ 1 -2λt ,
is a supersolution of (7) for all t > 0 if λ ≤ 0 and for t < 1 2λ if λ > 0.

Proof. For 1 -2λt > 0 we first notice that

d dt 1 √ 1 -2λt = λ √ 1 -2λt 3 .
Therefore, by construction, as long as 1 -2λt > 0, we see that

(1 -2λt) ∂ t Nλ,f (t, x) -∂ xx Nλ,f (t, x) -∂ x Nλ,f (t, 0)∂ x Nλ,f (t, x) = λ x √ 1 -2λt f ′ x √ 1 -2λt -f ′′ x √ 1 -2λt -f ′ (0)f ′ x √ 1 -2λt ≥ 0.
Finally, it is easy to prove that such a f exists and that, for all x ≥ 0, it verifies

f (x) ≤ f ′ (0) x 0 exp λ y 2 2 -f ′ (0)y dy. ( 8
)
Remark 3. For λ ≤ 0 we can have equality in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF]. In such a case, in the critical case M = 1, we recover the steady state and in the sub-critical case, we recover the self similar profile, see [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF]. For λ > 0, the equality yields that f (∞) = +∞ which does not correspond to a L 1 derivative. Therefore a self-similar blow-up profile doesn't seem to appear.

This result is particularly useful because it allows comparing a large class of initial data to such a supersolution. Indeed, for a good choice of f , by comparison principle, we would be able to prove that N (t, x) ≤ Nf,λ (t, x) for t < 1 2λ . This will provide an a priori bound on n(t, 0), that is

n(t, 0) ≤ 1 √ 1 -2λt f ′ (0), for t < 1 2λ
.

Recalling now the entropy 'dissipation' (3), for t < 1 2λ and for ∞ 0 n 0 (x) log n 0 (x)dx < +∞, we obtain the following a priori estimates sup

(0,t) ∞ 0 n(t, x)(log n(t, x)) + dx + t 0 ∞ 0 n(t, x)(∂ x log n(t, x)) 2 dx ≤ C(t),
which is enough to get compactness, insuring existence of weak solutions build as in [START_REF] Calvez | A one-dimensional Keller-Segel equation with a drift issued from the boundary[END_REF]. Details on existence will be carried out in [START_REF] Muller | Mathematical and numerical studies of nonlinear and nonlocal models involved in biology[END_REF]. To guarantee such a bound, we need to be able to compare, that is we need existence of λ, µ > 0 such that ∀x > 0,

x 0 µ exp λ y 2 2 -µy dy -N (0, x) ≥ 0.

Systematic blow up for supercritical mass : A quantitative argument

In [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF], the proof of the blow-up in supercritical case is based on the nonexistence of the first momentum. For the convenience of the reader, we recall it now. Assume that M > 1 and that n 0 is non-increasing, then n(t, •) is also non-increasing for any time it exists due to the maximum principle. In fact v(t, x) = ∂ x n(t, x) satisfies a parabolic type equation without any source term, it is initially non-positive, and it is non-positive on the boundary. Therefore -∂ x n(t, x)/n(t, 0) is a probability density at any time t > 0. From Jensen's inequality, we deduce that

∞ 0 x -∂ x n(t, x) n(t, 0) dx 2 ≤ ∞ 0 x 2 -∂ x n(t, x) n(t, 0) dx,
which rewrites as M 2 ≤ 2n(t, 0)J(t). Plugging this latter inequality into the evolution of the first momentum (6) yields that

J(t) ≤ J(0) + (1 -M )M 2 2 t 0 1 J(s)
ds.

Introducing next the auxiliary function

K(t) = J(0) + (1 -M )M 2 t 0 J(s) -1 ds
which is positive and satisfies

d dt K(t) = (1 -M )M 2 2 1 J(t) ≤ (1 -M )M 2 2 1 K(t)
,

we deduce that d dt K(t) 2 ≤ (1 -M )M 2 , hence a contradiction.
In this work, we also use the non-existence of decreasing first momentum beyond the blow-up time T * , given by J(T * ) = 0, but the integrated solution gives us a way to find this T * by using equation ( 6):

T * 0 ∂ x N (t, 0)dt = T * 0 n(t, 0)dt = J(0) M -1 .
We are interested in using a subsolution of the parabolic equation on N to find a blow-up time. A first lower bound on N is given by Chebyshev inequality.

Lemma 3.4 (Chebyshev). For all t ≥ 0 and for all x satisfying x ≥ J(t)/M , the following inequality holds true

N (t, x) ≥ M - J(t) x .
Proof. Chebychev inequality applied to the probability distribution n/M yields that

∞ x n(t, y) M dy ≤ 1 xM ∞ 0 yn(t, y)dy.
We next use this lower bound to define a subsolution of equation ( 7) as the solution of a particular heat equation on a bounded domain [0, L]. Firstly, recalling that ∂ x N and n(t, 0) are non negative, we deduce that the solution N of the classical heat equation on

R +      ∂ t N (t, x) -∂ xx N (t, x) = 0, in R * + , N (t, 0) = 0, N 0 = N 0 , in R + , (9)
is a subsolution of equation [START_REF] Evans | Partial differential equations[END_REF]. Next, using Lemma (3.4), we will consider a bounded domain. The advection to 0 disappears in favor of a boundary term in x = L. This latter term depends on the decreasing first momentum and this gives a weak way to transport to 0. Proposition 3.5. For n solution of equation (1) with M ≥ 1, for L ≥ J(0) M , we have

n(t, 0) ≥ M -J(0) L L   1 + 2 n∈ N (-1) n exp - n π L 2 t   .
Furthemore, for t > J(0) M π 2 log(2), we have

n(t, 0) ≥ k > 0.
Proof. First, if f solution of the bounded heat equation with Dirichlet conditions,

     ∂ t f (t, x) -∂ xx f (t, x) = 0, in (0, L), f (t, 0) = f (t, L) = 0, f 0 (x) = N 0 (x), in (0, L), (10)
then we recall that

f (t, x) = n∈N * sin n π x L exp - n π L 2 t f n ,
with the Fourier coefficients

f n = 2 L L 0 N 0 (y) sin n π y L dy.
Let L be such that L ≥ J(0) M . Since the first momentum J is decreasing, from Lemma (3.4), we deduce that N (t, L) ≥ M -J(0) L for all t ≥ 0. We now define the following problem

     ∂ t Ñ (t, x) -∂ xx Ñ (t, x) = 0, in (0, L), Ñ (t, 0) = 0, Ñ (t, L) = M -J(0) L , Ñ 0 (x) = N 0 (x), in (0, L). (11)
We extend the solution Ñ on [L, +∞] by

N * (t, x) = Ñ (t, x), if x ≤ L, M -J(0) L , if x > L.
From Lemma 3.4 together with the fact that the solution N of the heat equation on R + is a subsolution of equation [START_REF] Evans | Partial differential equations[END_REF] we deduce that N * is a subsolution of equation [START_REF] Evans | Partial differential equations[END_REF]. Using Fourier series to solve [START_REF]Singularity formation in the one-dimensional supercooled Stefan problem[END_REF], we obtain that

Ñ (t, x) = x Ñ (t, L) L + n∈N * sin n π x L exp - n π L 2 t g n ,
with the Fourier coefficients defined by

g n = 2 L L 0 N 0 (y) - y Ñ (t, L) L sin n π y L dx = 2 L L 0 N 0 (y) sin n π y L dy+ 2 Ñ (t, L) n π (-1) n .
By strong maximum principle [START_REF] Evans | Partial differential equations[END_REF], a solution of [START_REF] Herrero | Chemotactic collapse for the Keller-Segel model[END_REF], with initial datum N 0 ≥ 0, is positive, hence

Ñ (t, x) ≥ x Ñ (t, L) L + n∈N * sin n π x L exp - n π L 2 t 2 Ñ (t, L) n π (-1) n .
Furthermore, recalling that N (t, 0) = Ñ (t, 0) = 0 and that Ñ ≤ N on [0, +∞[×[0, L], we deduce that

n(t, 0) = ∂ x N (t, 0) ≥ ∂ x Ñ (t, ≥ Ñ (t, L) L 1 + 2 n∈N * (-1) n exp - n π L 2 t , hence, for t > L π 2 log(2), it follows that n(t, 0) ≥ Ñ (t, L) L 1 -2 exp - π L 2 t = k(t) > 0.
The previous lower bound on n(t, 0) provides us an upper bound on the blow-up time T * . Corollary 3.6. For n solution of equation (1) with M > 1, we have the following upper bound on blow-up time

T * ≤ 4 J(0) 2 M 2 (M -1) 1 + M -1 6 .
Proof. From proposition 3.5 it follows that

J(0) M -1 = T * 0 n(t, 0)dt ≥ M -J(0) L L T * + 2 L 2 π 2 n∈N * (-1) n n 2 .
Moreover, recalling that n∈N * (-1)

n n 2 = -π 2 12
, for all L ≥ J(0) M , we see that

T * ≤ J(0) M -1 L M -J(0) L + L 2 6 .
For L optimal, i.e. given by L = 2J(0) M ≥ J(0) M , we obtain that

T * ≤ 4 J(0) 2 M 2 (M -1) 1 + M -1 6 .
Remark 4. The upper bound on the blow-up time T * given in the previous proof is 4 1 + M -1

6
times bigger than the one found in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF]. Indeed in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF],

with a non-increasing initial condition, it was found that T * = J(0) 2 M 2 (M -1) . In this work, convection is described by the decreasing of the first momentum, then have only used diffusion phenomenon in equation [START_REF]Singularity formation in the one-dimensional supercooled Stefan problem[END_REF]. This could explain the difference in results.

Subcritical mass and self similar decay

For the case M < 1, global existence has been proved in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF]. Here, we are interested in the asymptotic behaviour and self similar decay has also been exhibited in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF]. If we perform the following change of variable:

n(t, x) = 1 √ 1 + 2t u log(1 + 2t), x √ 1 + 2t ,
then, the density u(τ, y) satisfies

∂ τ u(τ, y) -∂ yy u(τ, y) -∂ y (yu(τ, y)) -u(τ, 0)∂ y u(τ, y) = 0,
together with a zero flux boundary condition, ∂ y u(τ, 0) + u(τ, 0) 2 = 0. The additional left-sided drift contributes to confine the mass in the new frame (τ, y). It has been proved in [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF] that u converges to G α in large time, where G α is given by

G α (y) = α exp -αy - y 2 2 , ∞ 0 G α (y)dy = ∞ 0 u(0, y)dy = M, (12)
and more precisely, that the following Lyapunov functional L converges to 0,

L(τ ) = ∞ 0 u(τ, y) log u(τ, y) G α (y) dy + (J(τ ) -α(1 -M )) 2 2(1 -M ) . (13)
In Proposition 1.4 we improve this result with an exponential decay to equilibrium. The proof of this result is done as follows. In the new variables, the equilibrium state is a gaussian, hence in a linear frame the natural tool would be a logarithmic Sobolev inequality established by Gross in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] that we first recall, see [START_REF] Bobkov | From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities[END_REF] for a proof for instance. Although we are dealing here with a non linear problem this method will be fruitful. To do so, we apply this inequality to a suitable measure, namely G u 0 (y)dy. It would have been natural to apply a logarithmic Sobolev inequality to the measure G β (y)dy but a computation of the entropy dissipation with respect to the equilibrium state, G β , leads to the Fisher information expressed with respect to G u 0 (y)dy. Therefore a natural idea was to apply a logarithmic Sobolev inequality with respect to G u 0 (y)dy. Lemma 4.1 (Logarithmic Sobolev inequality). Let ν(x)dx = exp(-V (x))dx be a measure with smooth density on R + . Assume that V ′′ (x) ≥ 1 then, for

u ≥ 0 satisfying R + u(x)dx = R + ν(x)dx, we have ∞ 0 u(x) log u(x) ν(x) dx ≤ 1 2 ∞ 0 u(x) ∂ x log u(x) ν(x) 2 dx.
On the first hand, let us define the map β → C(β) by

C(β) ∞ 0 exp -βy - y 2 2 dy = M,
and

V β by V β (y) = βy + y 2 2 + log C(β). Such a function satisfies V ′′ β (y) = 1 and C(β) exp -βy - y 2 2 = exp (-V β (y)) = G β (y),
hence lemma 4.1 can be applied and this yields that

2H(u|G β ) ≤ I(u|G β ). ( 14
)
On the second hand, we recall that the evolution of the relative entropy with respect to G α , see [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF],

d dτ L(τ ) = -I(u|G u 0 ) - (J(τ ) -u 0 (τ )(1 -M )) 2 1 -M .
Furthermore, the relative entropy can be decomposed as follows

H(u|G α ) = ∞ 0 u(τ, y) log u(τ, y) G α (y) dy = ∞ 0 u(τ, y) log u(τ, y) G u 0 (y) dy+ ∞ 0 u(τ, y) log G u 0 (y) G α (

y) dy

Recalling the definition of G u 0 we deduce that

∞ 0 u(τ, y) log G u 0 (y) G α (y) dy = ∞ 0 u(τ, y) log C 0 α dy + ∞ 0 u(τ, y) log(exp((α -u 0 (τ ))y))dy = M log C 0 α + (α -u 0 (τ ))J(τ ),
where

C 0 = C(u 0 ). Therefore, 2H(u|G β ) = 2H(u|G u 0 ) + 2M log C 0 α + 2(α -u 0 (τ ))J(τ ),
from which we deduce that

d dτ L(τ ) + 2L(τ ) = 2H(u|G u 0 ) -I(u|G u 0 ) + 2M log C 0 α -2α(u 0 (τ ) -α)(1 -M ) -(u 0 (τ ) -α) 2 (1 -M ).
Recalling next a logarithmic Sobolev inequality ( 14) together with -(u 0 (τ )α) 2 (1 -M ) ≤ 0, it follows that

d dτ L(τ ) + 2L(τ ) ≤ 2M log C 0 α -2α(u 0 (τ ) -α)(1 -M ).
Now, it remains to evaluate the sign of the right-hand side term of the previous inequality, which we denote by ∆. The definition of C 0 first provides that

log C 0 α = -log   ∞ 0 exp -u 0 (τ )y -y 2 2 dy ∞ 0 exp -αy -y 2 2 dy   = -log ∞ 0 exp ((α -u 0 (τ ))y) G α (y) M dy , furthermore since -2α(u 0 (τ ) -α)(1 -M ) = 2(α -u 0 (τ )) ∞ 0 αy exp -αy - y 2 2 dy = 2M ∞ 0 log(exp ((α -u 0 (τ ))y)) G α (y) M dy,
from Jensen inequality it follows that

∆ = 2M ∞ 0 log(exp((α -u 0 (τ ))y)) G α (y) M dy -log ∞ 0 exp((α -u 0 (τ ))y) G α (y) M dy ≤ 0, hence d dτ L(τ ) + 2L(τ ) ≤ 0.
This achieves the proof of proposition 1.4.

In order to obtain a rate of convergence for the L 1 norm we will use the Csiszár-Kullback inequality, [START_REF] Csiszár | Information-type measures of difference of probability distributions and indirect observations[END_REF], [START_REF] Kullback | On the convergence of discrimination information[END_REF].

Proposition 4.2 (Csiszár-Kullback inequality). For any non-negative functions

f, g ∈ L 1 (R + ) such that R + f (x)dx = R + g(x)dx = M , we have that f -g 2 1 ≤ 2M ∞ 0 f (x) log f (x) g(x) dx. ( 15
)
Corollary 4.3. The following inequalities hold true:

H(u|G α ) ≤ L(0) exp(-2τ ), u(τ, y) -G α (y) 1 ≤ 2M L(0) exp(-τ ), n(t, x) - 1 √ 1 + 2t G α x √ 1 + 2t 1 ≤ 2M L(0) 1 (1 + 2t) 3/2 .
Remark 5. The 'degradation' of the convergence as M → 1 is not contained in the rate of convergence but in the intial value L(0). Indeed, the correction term to the entropy contains the factor (1 -M ) in the denominator. Therefore L(0) may become very large as M tends to 1. Hence, even if the rate does not depend on the mass, the evolution of L(0) as M → 1 affects the convergence quality.

Critical mass and trend to equilibrium

In the critical case, if J(0) < ∞, using (6), we first notice that the first momentum J is conserved. Moreover the stationary states to (1) are given by the one parameter family: n ∞ (0) exp(-n ∞ (0)x). Hence J ∞ = 1 n∞(0) and there is only one equilibrium state: ν α (x) = α exp(-αx) with α = 1 J(0) . We also recall the formal computation of the time evolution of the relative entropy:

d dt H(n|ν α ) = - ∞ 0 n(t, x)(∂ x log n(t, x) + n(t, 0)) 2 dx = -I(n|ν 0 ), (16)
where we have used the notation ν 0 (t, x)dx = n(t, 0) exp(-n(t, 0)x)dx.

In [START_REF] Calvez | Analysis of a nonlocal model for spontaneous cell polarization[END_REF], under the hypothesis that the second momentum is finite, it has been proved that the solution to (1) converges in relative entropy to ν α . Here, we improve the convergence result by precising the speed of convergence. First, we give a rate in the case of initially finite second momentum. Then, when the third momentum is initially finite, using the HWI inequality we improve the rate of convergence.

The Lyapunov functional approach

As we saw in the sub-critical case, a logarithmic Sobolev inequality is a powerful tool to deal with Gaussian measure. However, in the critical case, the stationary state is an exponentially decreasing measure but non-Gaussian.

In order to use a logarithmic Sobolev inequality, we consider a Lyapunov functional approach. Indeed, corrective terms will bring the construction of a targeted Gaussian measure. We give a speed of convergence for relative entropy with the following theorem.

Theorem 5.1. Assume that ∞ 0 x 2 n 0 (x)dx < +∞ and that H(n 0 |ν α ) < +∞, then

H(n|ν α ) ≤ 1 √ 1 + 2t H(n 0 |ν α ) + ∞ 0 x 2 2 n 0 (x)dx .
Proof. Let c > 0 be a differentiable function on R + . Let us consider the nonnegative Lyapunov functional

F (t) = H(n|ν α ) + c(t) ∞ 0 x 2 2 n(t, x)dx.
Using equation ( 16), we have that

d dt F (t) = -I(n|ν 0 ) + c(t) d dt ∞ 0 x 2 2 n(t, x)dx + c ′ (t) ∞ 0 x 2 2 n(t, x)dx.
We need to bring up a Gaussian measure in Fisher information in order to use a logarithmic Sobolev inequality. An easy computation shows that

d dt ∞ 0 x 2 2 n(t, x)dx = ∞ 0 x 2 2 ∂ x n(t, x)∂ x log n(t, x) ν 0 (t, x) dx = - ∞ 0 xn(t, x)∂ x log n(t, x) ν 0 (t, x) d (17)
The previous equality (17) leads us to regroup the terms as follows

d dt F (t) = - ∞ 0 n(t, x) ∂ x log n(t, x) ν 0 (t, x) 2 + c(t)x∂ x log n(t, x) ν 0 (t, x) + c(t) 2 x 2 4 dx + c ′ (t) + c(t) 2 2 ∞ 0 x 2 2 n(t, x)dx = - ∞ 0 n(t, x) ∂ x log n(t, x) ν 0 (t, x) 2 + 2 ∂ x log exp c(t)x 2 4 ∂ x log n(t, x) ν 0 (t, x) + ∂ x log exp c(t)x 2 4 2 dx + c ′ (t) + c(t) 2 2 ∞ 0 x 2 2 n(t, x)dx. (18)
We define the Gaussian measure G 0,c(t)/2 (t, x)dx by

G 0,c(t)/2 (t, x)dx = ν 0 (t, x) exp -c(t)x 2 4 ∞ 0 ν 0 (t, x) exp -c(t)x 2 4 dx dx. ( 19
)
Recalling the definition of the Fisher information together with (19), equality (18) rewrites as

d dt F (t) = -I(n|G 0,c(t)/2 ) + c ′ (t) + c(t) 2 2 ∞ 0 x 2 2 n(t, x)dx.
Using next a logarithmic Sobolev inequality (4.1) for the Gaussian measure G 0,c(t)/2 , it follows that

I(n|G 0,c(t)/2 ) ≥ c(t)H(n|G 0,c(t)/2 ), hence d dt F (t) ≤ -c(t)H(n|G 0,c(t)/2 ) + c ′ (t) + c(t) 2 2 ∞ 0 x 2 2 n(t, x)dx. ( 20
)
In order to have a rate of convergence, we want to make appear the Lyapunov functional on the right-hand side term and we use that

H(n|G 0,c(t)/2 ) = H(n|ν α ) + ∞ 0 n log ν α G 0,c(t)/2
which allows rewriting (20) as

d dt F (t) ≤ -c(t)F (t)-c(t) ∞ 0 n log ν α G 0,c(t)/2 + c ′ (t) + c(t) 2 ∞ 0 x 2 2 n(t, x)dx+c(t)
∞ 0 exp -n(t, 0)x - c(t)x 2 4 dx = log ∞ 0 exp (α -n(t, 0))x - c(t)x 2 4 ν α (x)dx ≥ (α -n(t, 0)) ∞ 0 xν α (x)dx - c(t) 4 ∞ 0 x 2 ν α (x)dx.
Recalling the conservation of the first momentum,

R + xn(t, x)dx = R + xν α (x)dx, (22) simplifies as ∞ 0 n log ν α G 0,c(t)/2 ≥ c(t) 4 ∞ 0 x 2 n(t, x)dx - ∞ 0 x 2 ν α (x)dx = c(t) ∞ 0 x 2 4 n(t, x)dx - 1 2α 2 , coming back to (21), d dt F (t) ≤ -c(t)F (t) + (c ′ (t) + c(t) 2 ) ∞ 0 x 2 2 n(t, x)dx + c(t) 2 2α 2 ,
hence, choosing c(t) to satisfy

c ′ (t) = -c(t) 2 1 + 1 α 2 ∞ 0 x 2 n(t, x)dx , (23)
we have a rate of convergence on the time evolution of

F d dt F (t) ≤ -c(t)F (t), (24)
Using next Cauchy-Schwarz inequality on the measure n(t, x)dx, we see that

∞ 0 x 2 n(t, x)dx = ∞ 0 n(t, x)dx ∞ 0 x 2 n(t, x)dx ≥ ∞ 0 xn(t, x)dx 2 = 1 α 2 .
hence a first lower bound on c(t) defined by ( 23):

c ′ (t) ≥ -2c(t) 2 , c(t) ≥ 1 1 + 2t .
For sake of simplicity, we choose c(0) = 1. Using Gronwall lemma in (24), we obtain that

F (t) ≤ F (0) exp - t 0 c(s)ds ≤ F (0) exp - 1 2 log(1 + 2t) = F (0) √ 1 + 2t , (25)
Since H(n|ν α ) ≤ F (t), this achieves the proof.

From theorem 5.1 and Csiszár-Kullback inequality (15), we immediately deduce the L 1 convergence

n -ν α 2 1 ≤ 2H(n|ν α ) ≤ C 2 0 √ 1 + 2t , where C 0 = 2F (0) = 2H(n 0 |ν α ) + ∞ 0 x 2 n 0 (x)dx.
Corollary 5.2. Assume that ∞ 0 x 2 n 0 (x)dx < +∞ and that H(n 0 |ν α ) < +∞. For any time t > 0, we have

n -ν α 1 ≤ C 0 (1 + 2t) 1 4
, where C 0 was previously defined.

During the proof of Theorem 5.1, we could have found a better lower bound for c, defined by (23), and consequently a better rate of convergence. Proposition 5.3. Assume that ∞ 0 x 2 n 0 (x)dx < +∞ and that H(n 0 |ν α ) < +∞. For all β < 2/3, there exists a constant C and a time t β such that the following inequality holds true for any time t

≥ t β n -ν α 1 ≤ C (1 + t) β 2
.

Proof. We have a better lower bound of the second moment with the following lemma.

Lemma 5.4. With the previous definition of n and ν α , the following inequality holds true

lim inf t→+∞ ∞ 0 x 2 n(t, x)dx ≥ 2 α 2 . Proof. Let (t k ) k be the following sequence lim inf t→+∞ ∞ 0 x 2 n(t, x)dx = lim k→+∞ ∞ 0 x 2 n(t k , x)dx.
Recalling Theorem 1.3, the convergence L 1 of n towards ν α holds true. Thus, we can find a sub-sequence (t kp ) p of (t k ) k such that n(t kp , .) converges towards ν α almost everywhere. We define

u p (x) = min(n(t kp , x), ν α (x)) ≤ ν α (x). Since ∞ 0 x 2 u p (x)dx ≤ 2 α 2 and x 2 u p (x) → x 2 ν α (x) for almost all x, we can use Fatou's lemma ∞ 0 x 2 ν α (x)dx = ∞ 0 lim inf p→+∞ x 2 u p (x)dx ≤ lim inf p→+∞ ∞ 0 x 2 u p (x)dx ≤ lim inf t→+∞ ∞ 0 x 2 n(t, x)dx.
This lemma provides an upper bound on the right hand side term in (23) hence, by definition of the lim sup, for all η > 3/2, we obtain that

inf s≥0 sup t≥s 1 α 2 ∞ 0 x 2 n(t, x)dx < η -1.
Finally, there exists a time t η such that for t ≥ t η with c(t η ) > 0,

c ′ (t) ≥ -ηc(t) 2 , c(t) ≥ 1 c(t η ) -1 + η(t -t η ) , t tη c(s)ds ≥ 1 η log(c(t η ) -1 +η(t-t η )).
In the same way as in (25), we use the lower bound on c to conclude: for any time t ≥ t η

F (t) ≤ F (t η ) 1 (c(t η ) -1 + η(t -t η )) 1 η
.

For all β = 1 η < 2/3, the previous inequality proves that (

1 + t) β F (t) is bounded for t ≥ t β = t η . (1 + t) β F (t) ≤ F (t η ) (1 + t) β (c(t β ) -1 + 1 β (t -t β )) β ≤ C.
We conclude this proof by using Csiszár-Kullback inequality (15).

A better rate

In the previous paragraph, in order to use a logarithmic Sobolev inequality, we have constructed a Gaussian measure with corrective terms. In this paragraph we directly use the HWI inequality firstly established in [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF] (see also [START_REF] Villani | Topics in optimal transportation[END_REF]), which is adapted for exponentially decreasing measure, and we improve the speed of convergence by controlling the second momentum. As we have seen in the Lyapunov functional approach, the rate of convergence could have been improved by using better estimations on the second momentum. The Wasserstein distance appears in the HWI inequality and we can control this distance with the second momentum. We need a third momentum uniformly bounded in time for such control on the second momentum. This is the following result.

Theorem 5.5. Assume that sup t≥0 ∞ 0 x 3 n(t, x)dx < +∞, then, there exist t 0 > 0, C 1 > 0 and C 2 > 0 such that for any time t ≥ t 0

H(n|ν α ) ≤ 1 C 1 + C 2 t .
Proof. We break the proof into several lemmas. We start by recalling the HWI inequality obtained in [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF], see [START_REF] Villani | Topics in optimal transportation[END_REF] for instance. This inequality binds both Fisher information, entropy and Wasserstein distance and can be applied to exponentially decreasing measure.

Lemma 5.6 (HWI inequality). Let ν(x)dx = exp(-V (x))dx be a measure with smooth density on R + , with V ′′ (x) ≥ 0 and V (x) exp(-V (x))dx < +∞. Then, for n ≥ 0 with finite momentums up to order 2, we have H(n|ν) ≤ W 2 (n, ν) I(n|ν).

In the particular case of an exponential distribution ν 0 (t, x)dx = n(t, 0) exp(-n(t, 0)x)dx, the HWI inequality reads as However, even if we do not control the second momentum, we can describe the behaviour of its Cesaro mean. We can then end with a final lemma Lemma 5.8. Assume that sup t≥0 ∞ 0 x 3 n(t, x)dx < +∞, then for any time

d
t > t 0 1 t -t 0 t t 0 ∞ 0 x 2 n(s, x)dxds ≤ 1 3α(t -t 0 ) ∞ 0 x 3 n(t 0 , x)dx - 3α 4 ∞ 0 x 2 n(t 0 , x)dx 2 + 2 α 2
Proof. We start by recalling a simple consequence of Holder inequality (31) Recalling (31), n(t, 0) can be rewritten as .

n(t, 0) = α - α 2
d
Taking the inverse, this achieves the proof of Theorem 5.5.

Proposition 1 . 2 ( 0 n 1 0 n 2

 120102 Concentration-comparison principle).Let n 1 , n 2 ∈ C 1 (0, T ; R + ) be two solutions of equation (1) satisfying dy, ∀x ≥ 0, then, we have∀t > 0, ∀x ≥ 0, N 1 (t, x) = x (t, y)dy ≥ N 2 (t, x) =x (t, y)dy.

0 x 3 0 x 2 0 x 2

 030202 with R + xn(t, x)dx = 1 α then we differentiate and this leads tod dt ∞ 0 x 3 n(t, x)dx = ∞ ∂ x (∂ x n(t, x) + n(t, 0)n(t, x)) dx = 6 α -3n(t, 0)∞ n(t, x)dx.(30) In the same way, we differentiate the second momentumd dt ∞ 0 x 2 n(t, x)dx = ∞ ∂ x (∂ x n(t, x) + n(t, 0)n(t, x)) dx = 2-2 n(t, 0) α .

  Thus, if the second momentum is enough controlled, we obtain a better speed of convergence than in the Lyapunov functional approach. , .)|ν α ) + C (t -t 0 ).

	therefore combining (26) and (27),
	1 H(n|ν α )	≥	1 H(n(t 0 , .)|ν α )	+	K + 2 t-t 0	t t 0	t -t 0 ∞ 0 x 2 n(s, x)dxds	.	(28)
	Remark 6. If (28) gives	∞ 0 x 2 n(t, x)dx is uniformly bounded in time, then equation 1 H(n|ν α ) 1 ≥ H(n(t 0
			dt	H(n|ν α ) = -I(n|ν 0 ) ≤ -	H 2 (n|ν 0 ) W 2 2 (n, ν 0 )	.
	This inequality becomes very powerful once it is associated with the following
	inequalities.

  dt

	Finally, we have obtained
	d dt	0	∞	x 3 n(t, x)dx -	3α 4	0	∞	x 2 n(t, x)dx	2	=	6 α	-3α	0	∞	x 2 n(t, x)dx,
	hence, after time integration,
					0	∞	x 3 n(t, x)dx -	3α 4		0	∞	x 2 n(t, x)dx	2	+ 3α	t 0 t	0	∞	x 2 n(s, x)dxds
	=				0	∞	x 3 n(t 0 , x)dx -	3α 4				0	∞	x 2 n(t 0 , x)dx	2	+	6(t -t 0 ) α	,
	therefore, recalling (29), we deduce that
	1 t -t 0		t t 0		0	∞	x 2 n(s, x)dxds =	1 3α(t -t 0 )	0	∞	x 3 n(t 0 , x)dx -	3α 4	0	∞	x 2 n(t 0 , x)dx	2	+	2 α 2 .
	Consequently, using Lemma 5.8 with (28), it follows that
									1 H(n|ν α )	≥	1 H(n(t 0 , .)|ν α )	+	t -t 0 K + C 0 t-t 0 + 2 α 2
																	∞	x 2 n(t, x)dx ,
																	0
	allowing us to rewrite equation (30) as
	d dt	0	∞	x 3 n(t, x)dx =		6 α	-3α	0	∞	x 2 n(t, x)dx+	3α 4	d dt	0	∞	x 2 n(t, x)dx	2	.

dx.

Acknowledgement: The authors want to warmly thank Vincent Calvez, without whom this work would not have existed.

 Lemma 5.7. With the previous definition of n, ν 0 and ν α , the following inequalities hold true 1. H(n|ν 0 ) ≥ H(n|ν α ),

n(t,0) 2 . Proof. We compare the two terms of the first inequality

For all X > 0, we recall that X -1 ≥ log X then

Using the definition of the Wasserstein distance ( 5), the second inequality is just a consequence of the triangle inequality

The Wasserstein distance is then controlled by the second momentum and n(t, 0). Recalling Proposition 3.5, for t ≥ log 2 (απ) 2 = t 0 , we have a lower bound on n(t, 0): 4 n(t, 0) 2 ≤ K. Furthermore using Lemma 5.7, for any time t ≥ t 0 , we see that

Using that X → 1 X is convex on R * + together with Jensen inequality, for the probability measure