L Desvillettes
email: desville@cmla.ens-cachan.fr

About Entropy Methods for Reaction-Diffusion Equations

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About Entropy Methods for Reaction-Diffusion Equations

Introduction

This paper summarizes a series of lectures given by the author at the University of Milano in the winter of 2007. It is devoted to a self-contained and elementary approach to the mathematical properties of reversible reactiondiffusion equations, almost entirely based on the entropy estimate.

The only mathematical prerequisites for this paper are some familiarity with PDEs methods (like, for example, Sobolev inequalities), together with a few basic facts about the solutions to linear parabolic equations.

Our goal is to show how the entropy estimate (which, together with the conservation of atoms, is the only physically relevant estimate) can be used in a systematic way in order to get bounds leading to existence of strong or weak solutions, as well as bounds for the rate of convergence to the equilibrium.

We have decided to present the estimates for three typical systems corresponding to basic chemical reactions involving a few species, rather than introducing general systems with an arbitrary number of species and reactions. We hope that this simplified presentation, which retains many of the difficulties of the general case, will help the readers.

Reversible Reactions

Our aim is to study reversible reaction processes for a set of chemical species A i , i = 1, 2, . . . , q, of the type

α 1 A 1 + • • • + α q A q ⇋ β 1 A 1 + • • • + β q A q , α i , β i ∈ N . (1)
We define by a i := a i (t) the concentration at time t of species A i . Then, thanks to the mass action law, the functions a i , i = 1, 2, . . . , q, satisfy (Cf. for example [START_REF] Giovangigli | Multicomponent flow modeling[END_REF]) the following system of ordinary differential equations (ODEs) :

a ′ i (t) = (β i -α i )   l q j=1 a α j j (t) -k q j=1 a β j j (t)   . (2)
Here, α i , β i are the stoichiometric coefficients, and k > 0, l > 0 are strictly positive reaction rates corresponding to a reversible reaction.

Convection; Diffusion

We now assume that the chemical reaction happens in a reactor (that is, from the point of view of mathematics, a bounded box Ω ⊂ IR N (N ≥ 1), assumed to be smooth (C 2 ), bounded and connected), where the molecules of the various species are moving.

We shall concentrate on the case when the molecules of the various species diffuse (each species has its own velocity of diffusion) within a nonreactive background supposed to be at rest. It means that the unknown is a i := a i (t, x) ≥ 0, i = 1, . . . , q (where x ∈ Ω). It satisfies (Cf. for example [START_REF] De Masi | Mathematical methods for hydrodynamic limits[END_REF]) the following system of partial differential equations (PDE) :

∂ t a i -d i ∆ x a i = (β i -α i )   l q j=1 a α j j -k q j=1 a β j j   , (3)
where d i is the diffusion velocity (in the non-reacting background) of species A i .

Note that many important problems involve some form of convection. The simplest modeling involving such a phenomenon concerns the case when the (non-reactive) fluid is not at rest, but has a velocity u := u(t, x) ∈ IR N . In this case, equation (3) becomes

∂ t a i + ∇ x • (u a i ) -d i ∆ x a i = (β i -α i )   l q j=1 a α j j -k q j=1 a β j j   . (4)
However, the most interesting situation concerns the case of a mixture of reactive gases satisfying reactive Euler equations or reactive Navier-Stokes equations. We refer to [START_REF] Giovangigli | Multicomponent flow modeling[END_REF] for a precise description of such a situation.

We shall assume that the molecules are confined in the reactor, so that the flux of concentration of each species at the boundary ∂Ω is 0. This gives for equation [START_REF] De Masi | Mathematical methods for hydrodynamic limits[END_REF] the homogeneous Neumann boundary condition :

n(x) • ∇ x a i = 0, (5)
where n(x) is (and always will be in the sequel) the outer normal unit vector at point x of ∂Ω.

Three typical chemical reactions

We shall from now on concentrate on the three typical chemical reactions

A + A ⇋ B, (6)
A + B ⇋ C, (7)
and

A + B ⇋ C + D. (8)
Without loss of generality, it is convenient to assume that (denoting by |Ω| the measure of Ω)

l = k = 1 , |Ω| = 1 . (9)
This can be obtained (for example in the case of eq. ( 8)) thanks to the rescaling t → 1 k t, x → |Ω| Then, we first describe the system associated to A + A ⇋ B. It can be written as :

∂ t a -d a ∆ x a = -2 ( a 2 -b), ∂ t b -d b ∆ x b = a 2 -b, (10)
together with the homogeneous Neumann conditions

n(x) • ∇ x a = 0, n(x) • ∇ x b = 0 x ∈ ∂Ω, (11)
and the nonnegative initial data

a(0, x) = a in (x) ≥ 0, b(0, x) = b in (x) ≥ 0. ( 12
)
Then, we turn to the system associated to A + B ⇋ C. It writes

   ∂ t a -d a ∆ x a = -a b + c, ∂ t b -d b ∆ x b = -a b + c, ∂ t c -d c ∆ x c = a b -c, (13)
with a, b, c satisfying homogeneous Neumann conditions

n(x) • ∇ x a = 0, n(x) • ∇ x b = 0, n(x) • ∇ x c = 0 x ∈ ∂Ω, (14)
and the nonnegative initial data

a(0, x) = a in (x) ≥ 0, b(0, x) = b in (x) ≥ 0, c(0, x) = c in (x) ≥ 0 . (15)
Finally, we write down the system associated to A + B ⇋ C + D :

       ∂ t a -d a ∆ x a = -a b + c d, ∂ t b -d b ∆ x b = -a b + c d, ∂ t c -d c ∆ x c = a b -c d, ∂ t d -d d ∆ x d = a b -c d, (16)
with a, b, c, d satisfying homogeneous Neumann conditions

n(x) • ∇ x a = 0, n(x) • ∇ x b = 0, x ∈ ∂Ω, n(x) • ∇ x c = 0, n(x) • ∇ x d = 0 x ∈ ∂Ω, (17)
and the nonnegative initial data

a(0, x) = a in (x) ≥ 0, b(0, x) = b in (x) ≥ 0, c(0, x) = c in (x) ≥ 0, d(0, x) = d in (x) ≥ 0 . ( 18
)

Plan of the paper

The rest of the paper is devoted to the mathematical study of the systems (10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], and ( 16) -(18).

In section 2 are presented the basic a priori estimates for our equations: it is proven there that strong solutions (with strictly positive initial data) are strictly positive, satisfy a set of conservation laws (that is, the conservation of the number of atoms of each kind), and (this will be systematically used in the sequel) an entropy estimate. By exploiting these estimates, a priori L p bounds are proven (with p > 1 depending on the dimension).

Section 3 is devoted to the treatment of the Cauchy problem for our equations. For systems ( 10) -( 12) and ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], it is possible to obtain an L ∞ a priori estimate, either thanks to a maximum principle, or to the use of the properties of the heat kernel. This enables to prove the existence (and uniqueness) of a strong solution for those systems. The situation is completely different for system ( 16) -(18). No L ∞ a priori estimate is known in this case, except in dimension 1 (and, thanks to a very recent work, Cf. [START_REF] Goudon | Regularity analysis for systems of reactiondiffusion equations[END_REF], in dimension 2). It is however possible to prove the existence of weak solutions to this system in L 2 (for any dimension), thanks to a method based on duality arguments.

Finally, we present in Section 4 an estimate of exponential decay toward equilibrium for the solution of system (10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF]. This estimate is a direct consequence of the entropy inequality, together with a Cziszar-Kullback type inequality.

A priori estimates; Formal computations

We now prove the standard a priori estimates for systems (10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], and ( 16) -(18). For this, we assume that we have strong (for example C 2 ([0, T [×Ω) for some T > 0) solutions of one of these systems.

Strict positivity

We begin with the question of strict positivity. The following proposition gives a classical general sufficient condition on the coefficients of a parabolic equation for propagation of strict positivity :

Proposition 2.1 Let Ω be a smooth (C 2 ) bounded open set of IR N , T > 0, and d > 0. We also consider coefficients a ∈ C 2 ([0, T ] × Ω) and b ∈ C 2 ([0, T ] × Ω, IR + ). We suppose that u := u(t, x) is a solution in C 2 ([0, T ] × Ω) to equation ∂ t u(t, x) -d ∆ x u(t, x) = a(t, x) u(t, x) + b(t, x), (19)
together with Neumann boundary condition and initial datum

∇ x u(t, x) • n(x) = 0, u(0, x) = u in (x) > 0,
where

u in ∈ C 2 (Ω).
Then, for all t ∈ [0, T ] and x ∈ Ω, u(t, x) > 0.

Proof of proposition 2.1 : We define t 0 > 0 as the infimum of times t > 0 such that there exists x 0 ∈ Ω satisfying u(t 0 , x 0 ) = 0. For t ∈ [0, t 0 [, we can consider f (t, x) = ln u(t, x). Then, f satisfies the inequality

∂ t f (t, x) -d ∆ x f (t, x) = a(t, x) + b(t, x) u(t, x) + d |∇ x u(t, x)| 2 u(t, x) 2 ≥ -||a|| ∞ , ∇ x f (t, x) • n(x) = 0, f (0, x) = ln u in (x) ≥ -|| ln u in || ∞ .
We consider (for any ε > 0) the auxiliary quantity

f ε (t, x) = f (t, x) + ||a|| ∞ t + || ln u in || ∞ + ε (t + 1
). This quantity satisfies the inequality

∂ t f ε -d ∆ x f ε ≥ ε.
together with Neumann boundary condition and initial datum

∇ x f ε • n(x) = 0, f ε (0, x) ≥ ε.
We define t 1 ≤ t 0 as the infimum of times t > 0 such that there exists x 0 ∈ Ω satisfying f ε (t 1 , x 0 ) = 0. Note that (according to the definition of t 1 ), the function x → f ε (t 1 , x) reaches its infimum at point x 0 , and that t 1 > 0.

If

x 0 ∈ Ω, then ∆ x f ε (t 1 , x 0 ) ≥ 0, so that ∂ t f ε (t 1 , x 0 ) > 0. Since t 1 > 0, this is impossible.
If x 0 ∈ ∂Ω, then, we know that ∇ x u(t 1 , x 0 ) • τ (x 0 ) = 0 (where τ (x 0 ) is any vector tangeant to ∂Ω at point x 0 ), so that thanks to the Neumann boundary condition, ∇ x f ε (t 1 , x 0 ) = 0. Therefore, ∆ x f ε (t 1 , x 0 ) ≥ 0, and we conclude as in the case when x 0 ∈ Ω.

Finally, t 1 = t 0 , so that f ε (t, x) > 0 when t ∈ [0, t 0 [, and (letting

ε go to 0), f (t, x) ≥ -||a|| ∞ t -|| ln u in || ∞ (still when t ∈ [0, t 0 [).
This finally ensures that t 0 = T , so that proposition 2.1 is proven. This proposition can be used directly for proving the strict positivity of the concentrations in systems ( 10) -( 12), ( 13) -( 15) and ( 16) -(18). Proposition 2.2 Let T > 0, Ω be a bounded regular C 2 domain of IR N , d a , d b > 0 (together with d c , d d > 0, depending on the system considered) be diffusivity constants, a in := a in (x) > 0, b in := b in (x) > 0 (together with c in := c in (x) > 0, d in := d in (x) > 0 depending on the system considered) be initial data in C 2 (Ω). Let also a := a(t, x), b := b(t, x) (together with c := c(t, x), d := d(t, x), depending on the system considered) be solutions in C 2 ([0, T [×Ω) of one of the systems (10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], or ( 16) -(18).

Then, for all t ∈ [0, T [ and x ∈ Ω, one has a(t, x) > 0, b(t, x) > 0 (and c(t, x) > 0, d(t, x) > 0 depending on the system considered).

Proof of proposition 2.2 : We consider only the system (10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF], since the other ones can be treated in the same way.

It is enough to consider the time t 0 defined as the infimum of the times t where there exists x 0 ∈ Ω such that a(t, x 0 ) = 0 or b(t, x 0 ) = 0. On the interval [0, t 0 ], it is possible to put the two equations of system [START_REF] Kullback | A lower bound for discrimination information in terms of variation[END_REF] under the form (19). Then, one can use proposition 2.1 in order to show that a(t 0 , •) > 0 and b(t 0 , •) > 0, which leads to a contradiction.

Conservation of the number of molecules

We begin by noticing that the flow of equations ( 10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], or ( 16) -(18), conserves a quantity equivalent to the total L 1 -norm : Proposition 2.3 Let T > 0, Ω be a bounded regular (C 2 ) open set of IR N , d a , d b > 0 (together with d c , d d > 0, depending on the system considered) be diffusivity constants, a in := a in (x) > 0, b in := b in (x) > 0 (together with c in := c in (x) > 0, d in := d in (x) > 0 depending on the system considered) be initial data in C 2 (Ω). Let also a := a(t, x), b := b(t, x) (together with c := c(t, x), d := d(t, x), depending on the system considered) be solutions in C 2 ([0, T ] × Ω) of one of the systems (10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], or ( 16) -(18).

Then, for all t ∈ [0, T ] and x ∈ Ω, one has the following properties of conservation : 1. For system (10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF] :

M := Ω a(t, x) + 2 b(t, x) dx = Ω a in (x) + 2 b in (x) dx . (20) 2.
For system (13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF] :

       M 1 := Ω a(t, x) + c(t, x) dx = Ω a in (x) + c in (x) dx , M 2 := Ω b(t, x) + c(t, x) dx = Ω b in (x) + c in (x) dx . (21)
3. Finally, for system ( 16) -(18) :

               M 1 := Ω a(t, x) + c(t, x) dx = Ω a in (x) + c in (x) dx , M 2 := Ω b(t, x) + c(t, x) dx = Ω b in (x) + c in (x) dx , M 3 := Ω b(t, x) + d(t, x) dx = Ω b in (x) + d in (x) dx . ( 22
)
Proof of proposition 2.3 : Those conservations are simply obtained by taking linear combinations of the equations, and by integrating on the domain Ω, taking into account the Neumann boundary condition.

This ends the proof of proposition 2.3.

Taking into account the properties of strict positivity and the previous properties of conservation, we see that there is a natural L ∞ ([0, T ]; L 1 (Ω)) estimate for all the concentrations appearing in systems (10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], and ( 16) -(18).

Entropy estimate

Our main tool in the sequel will be the entropy/entropy dissipation conservation property. In order to state it precisely, we introduce the following definitions of entropy E i and entropy dissipation D i (i = 1, .., 3 corresponds to each system), defined as functions of the concentrations, seen as functions of x. For a general definition of the entropy of a mixture of reactive species, we refer to [START_REF] Giovangigli | Multicomponent flow modeling[END_REF]. We start with system (10) -( 12) :

E 1 (a, b) := Ω a (ln a -1) + b (ln b -1) dx , (23)
D 1 (a, b) = d a Ω |∇ x a| 2 a dx + d b Ω |∇ x b| 2 b dx + Ω (a 2 -b) (ln(a 2 ) -ln b) dx . (24)
Then, we turn to system (13) -( 15) :

E 2 (a, b, c) := Ω a (ln(a) -1) + b (ln(b) -1) + c (ln(c) -1) dx , (25)
D 2 (a, b, c) = d a Ω |∇ x a| 2 a dx + d b Ω |∇ x b| 2 b dx + d c Ω |∇ x c| 2 c dx + Ω (ab -c) (ln(ab) -ln c) dx . (26)
Finally, we introduce the quantities corresponding to system ( 16) -( 18) :

E 3 (a, b, c, d) := Ω a (ln(a) -1) + b (ln(b) -1) +c (ln(c) -1) + d (ln(d) -1) dx , (27)
D 3 (a, b, c, d) := d a Ω |∇ x a| 2 a dx + d b Ω |∇ x b| 2 b dx + d c Ω |∇ x c| 2 c dx +d d Ω |∇ x d| 2 d dx + Ω (a b -c d) (ln(a b) -ln(c d)) dx . ( 28
)
We are able to prove the following a priori estimates : 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], or ( 16) -(18).

Then, for all t ∈ [0, T ] and x ∈ Ω, one has the following identity : (for the i ∈ {1, .., 3} corresponding to the system under consideration)

∂ t E i = -D i . (29)
Moreover, this identity entails the following a priori estimate (for all T ∈ ]0, +∞]) :

sup t∈[0,T ] E i (t) + T 0 D i (s) ds ≤ E i (0). ( 30
)
Proof of proposition 2.4 : Note first that we already know that our solutions are strictly positive, so that we can compute quantities like the logarithms of concentrations. Then, we detail the computation only in the case of system ( 10) -( 12) :

∂ t E 1 = Ω (ln a ∂ t a + ln b ∂ t b) dx = - Ω d a |∇ x a| 2 a + d b |∇ x b| 2 b dx + ∂Ω (ln a ∇ x a • n(x) + ln b ∇ x b • n(x)) dσ(x) + Ω -2 ln a (a 2 -b) + ln b (a 2 -b) dx.
We conclude by using the Neumann boundary condition.

The a priori estimate is obtained by integrating between times 0 and t (0 < t < T ) identity (29). This concludes the proof of proposition 2.4.

Consequences of the entropy estimate

The a priori estimate obtained above can be exploited in order to obtain L p type bounds for our solutions. More precisely, it is possible to obtain the following proposition (Cf. [START_REF] Desvillettes | About global existence for quadratic systems of reaction-diffusion[END_REF]) :

Proposition 2.5 Let T > 0, Ω be a bounded regular (C 2 ) and connected open set of IR N , d a , d b > 0 (together with d c , d d > 0, depending on the system considered) be diffusivity constants, a in := a in (x) > 0, b in := b in (x) > 0 (together with c in := c in (x) > 0, d in := d in (x) > 0 depending on the system considered) be initial data in C 2 (Ω). Let also a := a(t, x), b := b(t, x) (together with c := c(t, x), d := d(t, x
), depending on the system considered) be solutions in C 2 ([0, T ] × Ω) of one of the systems ( 10) -( 12), ( 13) -( 15), or ( 16) -(18).

Then, for all t ∈ [0, T ] and x ∈ Ω, and for α denoting any concentration appearing in one of the three systems, there exists C > 0 depending only on the initial data and the diffusivity constants such that :

||α|| L inf(1+2/N,2) ([0,T ]×Ω) ≤ C (1 + T ) sup(2,N/(N +2)) . ( 31
)
Proof of proposition 2.5 : Thanks to the a priori estimate (30), we see that for any concentration α appearing in any of the three systems considered (that is, α = a, b, c or d), there exists a constant C depending only on the initial data (in fact, the initial entropy of the mixture) such that

T 0 Ω |∇ x √ α| 2 dxdt ≤ C. ( 32
)
We use the Sobolev-Wirtinger estimate (valid for u ∈ H 1 (Ω), where Ω is a bounded, regular, and connected open set of IR N of volume 1, Cf. [START_REF] Brézis | Analyse Fonctionnelle[END_REF]) :

Ω |u(x) -ū| p * dx 2/p * ≤ K Ω |∇ x u| 2 dx, (33)
where ū = Ω u(y) dy, p * = 2N N -2 , and K := K(Ω). When N = 2, this estimate is true for all p * ∈ [1, +∞[, and when N = 1, it is replaced by the stronger inequality

sup x∈Ω |u(x) -ū| 2 ≤ K Ω |∇ x u| 2 dx. ( 34
)
Thanks to the Poincaré-Wirtinger inequality, estimate (32) becomes

T 0 Ω α(t, x) - Ω α(t, y) dy p * dx 2/p * dt ≤ C. Then, T 0 Ω | α(t, x)| p * dx 2/p * dt ≤ 2 2(p * -1)/p * T 0 Ω α(t, x) - Ω α(t, y) dy p * dx 2/p * dt +2 2(p * -1)/p * T 0 Ω α(t, x) dx 2 dt ≤ 2 2(p * -1)/p * C + T 0 Ω α(t, x) dxdt .
Using the conservation of number of atoms (20), (21), or ( 22), and the definition of p * , we see that that for some constant C depending only on the initial data,

T 0 Ω |α(t, x)| N/(N -2) dx (N -2)/N dt ≤ C (1 + T ). ( 35
)
When N = 2, this estimate is replaced by

T 0 Ω |α(t, x)| p dx 1/p dt ≤ C (1 + T ), for all p ∈ [1, +∞[. Finally, when N = 1, it becomes T 0 sup x∈Ω α(t, x) dt ≤ C (1 + T ).
We shall now interpolate estimate (35) with the estimate coming out of the conservation of number of atoms :

sup t∈[0,T ] Ω α(t, x) dx ≤ C. We compute T 0 Ω (α(t, x)) 1+2/N dxdt ≤ T 0 Ω α(t, x) (α(t, x)) 2/N dxdt ≤ T 0 Ω (α(t, x)) N/(N -2) dx (N -2)/N Ω α(t, x) dx 2/N dt ≤ T 0 Ω (α(t, x)) N/(N -2) dx (N -2)/N dt × sup t∈[0,T ] Ω α(t, x) dx 2/N ≤ C (1 + T ) C 2/N .
Finally, we see that for some constant C depending only on the initial data (and when N ≥ 3),

||α|| L 1+2/N ([0,T ]×Ω) ≤ C (1 + T ) N/(N +2) .
This estimate is replaced by

||α|| L p ([0,T ]×Ω) ≤ C (1 + T ) 1/p (36)
for all p ∈ [1, 2[ when N = 2, and by

||α|| L 2 ([0,T ]×Ω) ≤ C (1 + T ) 1/2 (37)
when N = 1.

In order to treat the case N = 2, we use a refinement of Sobolev-Wirtinger's inequality.

Trudinger's inequality

In order to obtain estimate (36) in the case when N = 2, we first recall Trudinger's inequality (Cf. [START_REF] Brézis | Analyse Fonctionnelle[END_REF]), which is a limit case of Poincaré inequality, saying that (when Ω is a bounded and regular (C 2 ) open set of IR 2 ), there are two absolute strictly positive constants s 0 and C 0 such that, for all u ∈ H 1 (Ω),

Ω exp s 0 |u(x)| 2 u 2 H 1 (Ω) ≤ C 0 . ( 38
)
As a consequence, we can also find two strictly positive absolute constants s and C such that (for all functions u ∈ H 1 (Ω)),

Ω |u(x)| 2 u 2 H 1 (Ω) exp s |u(x)| 2 u 2 H 1 (Ω) ≤ C . (39)
Using this inequality for u = √ α(t, •), and integrating in time between 0 and T , we get the estimate

T 0 Ω α(t, x) exp s α(t, x) √ α(t, •) 2 H 1 (Ω) dxdt ≤ C T 0 √ α(t, •) 2 H 1 (Ω) dt ≤ C (1 + T ). ( 40
)
We note that thanks to Young's inequality (valid for x, y, γ > 0)

x y ≤ e γx + y γ (log( y γ ) -1),
applied to γ = log a a + s q and x = y = a, we have for all a > e and s, q > 0,

a 2 ≤ a e sa q + a log a a + s q log a log a a + s q -1 ≤ a e sa q + aq s log(a 2 ).
Using this last inequality with s being the constant in inequality (39), q = √ α(t, •) 2 H 1 (Ω) and a = max(e, α(t, x)), and using also estimates (40) and (the first part of) (30), we end up with

α 2 L 2 ([0,T ]×Ω) ≤ min{α, e} 2 L 2 ([0,T ]×Ω) + max{α, e} 2 L 2 ([0,T ]×Ω) ≤ e 2 T + T 0 Ω α(t, x) exp s α(t, x) √ α(t, •) 2 H 1 (Ω) dxdt + 2 s T 0 Ω α(t, x) log(α(t, x)) dx √ α(t, •) 2 H 1 (Ω) dt ≤ e 2 T + C (1 + T ) + 2 s C (1 + T ),
which is exactly estimate (37), but in dimension 2.

Existence, uniqueness and smoothness

In this section, we study the traditional problems associated to the systems we are interested in (that is, systems ( 10) -( 12), ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF], and ( 16) -( 18)), that is, existence, uniqueness and smoothness of the solutions. More precisely, we shall show that for the two first systems, existence ans uniqueness of a smooth solution holds, while we only can prove existence of a weak solutions to the third system (except in small dimensions).

Linear Parabolic Equations

We begin by recalling a few classical facts about linear parabolic equations.

Existence, uniqueness and smoothness for linear parabolic equations

We first state the following theorem (Cf. for example [START_REF] Ladyzhenskaya | Linear and quasilinear equations of parabolic type[END_REF]) for the heat equation with Neumann boundary condition : Then, there exists a unique weak solution u ∈ L p ([0, T ] × Ω) to the linear parabolic initial-boundary value problem :

   ∂ t u -d ∆ x u = φ, ∇ x u • n = 0 on ∂Ω, u(0, x) = u in (x). (41) Moreover, u ∈ W 1,p (]0, T [×Ω).
Finally, if we know that for some

α ∈]0, 1[, φ ∈ C 0,α ([0, T ] × Ω), then u ∈ C 1,α ([0, T ] × Ω), and if φ ∈ C 1,α ([0, T ] × Ω), then u ∈ C 2 ([0, T ] × Ω).
Note that in this last case, the solution is in fact strong.

The case of the 1D heat equation

In the one-dimensional case, it is easy to give a complete explicit formula for problem (41), using Fourier series.

Proposition 3.1 Let u in ∈ C 2 ([0, 1]) be an initial datum (compatible with the Neumann boudary condition) and φ ∈ C 2 ([0, T ] × [0, 1]) be a second member.
Then, the unique solution u := u(t, x) to eq. ( 41) is given by the formula

u(t, x) = 1 2 √ π 1 -1 ũin (y) ∞ k=-∞ 1 √ d t e -(2k+x-y) 2 4 d t dy + 1 2 √ π t 0 1 -1 φ(s, y) ∞ k=-∞ 1 √ d (t-s) e - (2k+x-y) 2 4d (t-s) dy ds ,
where

ũin (x) = u in (x) x ∈ [0, 1], u in (-x) x ∈ [-1, 0], (42)
and φ(t, x) = φ(t, x) x ∈ [0, 1], φ(t, -x) x ∈ [-1, 0]. ( 43
)
Proof of proposition 3.1 : The proof uses Fourier series, which simplify when (41) is mirrored evenly around x = 0, i.e. when the functions are extended like

ũ(t, x) = u(t, x) x ∈ [0, 1], u(t, -x) x ∈ [-1, 0], (44)
and when φ and ũin are defined analogously by (42), (43). Then, the eigenvalue-problem ψxx = λ ψ on [-1, 1] with homogeneous Neumann boundary and periodicity conditions is satisfied by the eigenvalue-eigenfunction pairs

(λ k , ψk (x)) = (-(kπ) 2 , cos(kπx)) for k = 0, 1, 2, . . . ,
and yields the Fourier representation

ũ(t, x) = 1 -1 ũin (y) dy + 2 ∞ k=1 e λ k dt 1 -1 ũin (y) ψk (y) dy ψk (x) + t 0 1 -1 φ(s, y) dyds (45) +2 ∞ k=1 t 0 e λ k d (t-s) 1 -1 φ(s, y) ψk (y)dy ds ψk (x) . ( 46
)
Thanks to Poisson's summation formula (Cf. for example [START_REF] Cl | Eléments d'Analyse pour l'Agrégation[END_REF] in this context), we can write down the final form of the explicit solution to the 1-D heat equation with Neumann boundary condition :

ũ(t, x) = 1 2 √ π 1 -1 ũin (y) ∞ k=-∞ 1 √ d t e -(2k+x-y) 2 4 d t dy + 1 2 √ π t 0 1 -1 φ(s, y) ∞ k=-∞ 1 √ d (t-s) e -(2k+x-y) 2 4d (t-s) dy ds .
This ends the proof of proposition 3.1.

Nonlinear reaction-diffusion equations with bounded r.h.s.

We now turn to the problem of building solutions to nonlinear reactiondiffusion systems. Thanks to a simple fixed point argument in L 2 , it is classical to prove the following theorem, which holds when the nonlinearity lies in the space W 1,∞ of bounded and Lipschitz-continuous functions (we refer for example to [START_REF] Rothe | Global solutions of reaction-diffusion systems[END_REF] for many results on reaction-diffusion systems including variants of the result below) :

Proposition 3.2
Let Ω be a bounded and regular (C 2 ) open set of IR N , and let D be a diagonal matrix with entries

d i > 0, i = 1, .., k. We consider an initial datum U in ∈ C 2 (Ω, IR k ) compatible with the Neumann boundary con- dition, and f ∈ C 2 (IR k ; IR k ). We suppose moreover that f ∈ W 1,∞ (IR k ; IR k )
(space of bounded and globally Lipschitz-continous functions), which means that there exists K > 0, such that for all x, y ∈ IR k ,

|f (x)| ≤ K, |f (x) -f (y)| ≤ K |x -y|.
Then, there exists a unique strong solution

U ∈ C 2 (IR + × Ω; IR k ) to equation    ∂ t U -D∆ x U = f (U ), n • ∇ x U = 0 for x ∈ ∂Ω, U (0, x) = U in (x). ( 47
)
Proof of proposition 3.2 : We introduce U 0 (t, x) = U in (x). According to theorem 3.1 for linear parabolic equations, we know that for any integer n ≥ 0, there exists a (unique) function

U n+1 ∈ C 2 (IR + × Ω; IR k ) such that    ∂ t U n+1 -D∆ x U n+1 = f (U n ), n(x) • ∇ x U n+1 (t, x) = 0 for x ∈ ∂Ω, U n+1 (0, x) = U in (x). (48) We denote r n (t) = ||U n -U n-1 || 2 L 2 (Ω) (t).
We multiply the difference of (48) for n + 1 and (48) for n by U n+1 -U n , and perform an integration by parts. Denoting by U j n the components of U n , we get

∂ t Ω 1 2 |U n+1 -U n | 2 dx + k j=1 d j Ω |∇ x (U j n+1 -U j n )| 2 dx = Ω (U n+1 -U n ) (f (U n ) -f (U n-1 )) dx, so that Ω |U n+1 -U n | 2 dx(t) ≤ t 0 Ω |U n+1 -U n | 2 dx(s) + K Ω |U n -U n-1 | 2 dx(s) ds,
which we can rewrite as

r n+1 (t) ≤ t 0 r n+1 (s) + K r n (s) ds.
Then, we observe that

r 1 (t) ≤ R T := 2 ||U in || 2 L 2 (Ω) + 2 (||U in || L 2 (Ω) + K T ) 2
for all t ∈ [0, T ]. Using the auxiliary quantity

w n (t) = e -t t 0 r n (s) ds, it is possible to prove that w n+1 (t) ≤ K t 0 w n (s) ds, so that w n (t) ≤ R t (K t) n-1
(n-1)! . This in turn implies that for all t ∈ [0, T ],

r n (t) ≤ (2 R T K (1 + T ) e T ) (K t) n-2 (n -2)! .
Since this sequence is the general term of a series which converges uniformy on [0, T ], and recalling that L ∞ ([0, T ]; L 2 (Ω)) is complete, we see that U n converges to some function U in this space.

Passing to the limit in eq. ( 48), we obtain that U is a weak solution of eq. ( 47) on [0, T ].

We now use the "smoothness" part of theorem 3.1. Since U is a (weak) solution (on [0, T ]) of the heat equation with a r.h.s. in L ∞ , it lies in W 1,p (]0, T [×Ω) for all p > 1, and therefore in C 0,α ([0, T ]×Ω) for all α ∈]0, 1[. Thanks to the hypothesis on f , this is also true of f (U ). As a consequence, still using theorem 3.1, U ∈ C 1,α ([0, T ]×Ω) for all α ∈]0, 1[, and (still thanks to the hypothesis on f ), so does f (U ). We use a last time theorem 3.1, and get that U ∈ C 2 ([0, T ] × Ω), so that U is a strong solution to the system (47) on [0, T ].

Uniqueness of strong solutions to system (47) on [0, T ] is easily obtained by noticing that if U, V are two such strong solutions, then taking the difference of the equations for U and V , multiplying by U -V , and integrating by parts, we obtain :

∂ t ||U -V || L 2 (Ω) (t) ≤ K ||U -V || L 2 (Ω) (t). ( 49
)
Finally, it is possible to build a unique strong solution U to system (47) on IR + by sticking together all the solutions to system (47) on [0, T ], with T > 0. This concludes the proof of proposition 3.2.

Nonlinear reaction-diffusion equations with unbounded r.h.s : use of L ∞ bounds

Proposition 3.2 can be used in order to prove existence (together with uniqueness and smoothness) of solutions to reaction-diffusion systems with general r.h.s, once some L ∞ bound has been proven for the unknowns. Such bounds are not available in general, but they can be obtained for particular systems (among which ( 10) -( 12) and ( 13) -( 15) in any dimension, and ( 16) -( 18) in small dimension). They can be based either on "geometrical properties" of the r.h.s. (here we shall follow the method described in [START_REF] Kirane | On stabilization of solutions of the system of parabolic differential equations describing the kinetics of an autocatalytic reversible chemical reaction[END_REF] for systems of two equations), or on the fact that (the nonnegative part of) the r.h.s. is a function of the unknowns that does not increase too much (we refer to [START_REF] Rothe | Global solutions of reaction-diffusion systems[END_REF] for examples of such a situation).

In general, L ∞ bounds are obtained all the more easily than the system has few equations, and the dimensionality of the space is low.

Maximum principle

We begin with a maximum (and minimum) principle property of our simplest system (namely system (10) -( 12)). It writes like this : 10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF]. We suppose that the initial conditions satisfy :

∀x ∈ Ω, 0 < A 0 < a in (x) < A 1 , 0 < B 0 < b in (x) < B 1 , (50)
for some constants A 0 , A 1 , B 0 , B 1 > 0.

Then, a and b satisfy the bounds

∀t ∈ [0, T ], x ∈ Ω, inf(A 0 , B 0 ) ≤ a(t, x) ≤ sup(A 1 , B 1 ), ( 51
) ∀t ∈ [0, T ], x ∈ Ω, inf(A 2 0 , B 0 ) ≤ b(t, x) ≤ sup(A 2 1 , B 1 ). ( 52
)
Proof of proposition 3.3 : The proof shall be carried out following the ideas of [START_REF] Kirane | On stabilization of solutions of the system of parabolic differential equations describing the kinetics of an autocatalytic reversible chemical reaction[END_REF]. Let us consider the functions

a ε (t, x) = a(t, x) e ε t , b ε (t, x) = b(t, x) e ε t . (53)
From equations [START_REF] Kullback | A lower bound for discrimination information in terms of variation[END_REF], it follows that the evolution of a ε , b ε is governed by the system

   ∂ t a ε -d a ∆ x a ε = -2 ((a ε ) 2 e -εt -b ε ) + ε a ε , ∂ t b ε -d b ∆ x b ε = ((a ε ) 2 e -εt -b ε ) + ε b ε . ( 54
)
We define the set B ε as

B ε = τ > 0 : a ε (t, x) > inf (A 0 , B 0 ), b ε (t, x) > inf(A 2 0 , B 0 ) e -εt , ∀ (t, x) ∈ [0, τ [× Ω . (55)
Then, we consider t = sup B ε . Note first that t > 0. Then, there must exist x ∈ Ω such that at least one of the following equalities holds :

a ε ( t, x) = inf (A 0 , B 0 ) or b ε ( t, x) = inf(A 2 0 , B 0 ) e -εt .
Case 1:

a ε ( t, x) = inf (A 0 , √ B 0 )
. By definition of t and x, the function x → a ε ( t, x) takes it miminum at point x, so that ∆ x a ε ( t, x) ≥ 0 . Evaluating the chemical contributions in the first line of (54) at ( t, x), we get

-2 (a ε ) 2 ( t, x) e -ε t + 2 b ε ( t, x) + 2 ε a ε ( t, x) = -2 inf (A 0 , √ B 0 ) 2 e -ε t + 2 b ε ( t, x) + 2 ε inf (A 0 , √ B 0 ) ≥ -2 inf (A 0 , √ B 0 ) 2 e -ε t + 2 inf(A 2 0 , B 0 ) e -εt + 2 ε inf (A 0 , √ B 0 ) > 0.
Consequently, eq. ( 54) for a ε implies that ∂ t a ε ( t, x) > 0, hence a ε (t, x) < a ε ( t, x) for some t < t, contradicting the definition of t.

Case 2: b ε ( t, x) = inf(A 2 0 , B 0 ) e -εt . In this case, we have b ε ( t, x) ≤ b ε ( t, x) ∀x ∈ Ω, so that b ε ( t, x) takes its minimum at point x = x. Therefore we get, as above, ∆ x b ε ( t, x) ≥ 0. As concerns the r.h.s. of the second line of (54), we obtain

(a ε ) 2 ( t, x) e -ε t -b ε ( t, x) + ε b ε ( t, x) = (a ε ) 2 ( t, x) e -ε t -inf(A 2 0 , B 0 ) e -ε t + ε inf(A 2 0 , B 0 ) e -ε t , ≥ (inf A 0 , √ B 0 ) 2 e -ε t -inf(A 2 0 , B 0 ) e -ε t + ε inf(A 2 0 , B 0 ) e -ε t > 0.
It follows that ∂ t b ε ( t, x) > 0, which leads to a contradiction as in case 1. Consequently, the set B ε is unbounded, and

∀t ≥ 0, x ∈ Ω, a ε (t, x) > inf (A 0 , B 0 ), b ε (t, x) > inf(A 2 0 , B 0 ) e -ε t .
This means that a(t, x) > inf (A 0 , √ B 0 ) e -ε t and b(t, x) > inf(A 2 0 , B 0 ) e -2 ε t . Thus, passing to the limit ε → 0, we have a(t, x) ≥ inf (A 0 , √ B 0 ) and b(t, x) ≥ inf(A 2 0 , B 0 ).

This ends the proof of the minimum principle for system (10) -( 12). The maximum principle can be proven in exactly the same way. We have therefore finished the proof of proposition 3.3

We are now in position to state a theorem of existence (and uniqueness, smoothness) for system (10) -( 12

∀x ∈ Ω, 0 < A 0 < a in (x) < A 1 , 0 < B 0 < b in (x) < B 1 . (56)
Then, there exists a unique (strong) solution a, b in C 2 (IR + × Ω) to system ( 10) -( 12) such that :

∀t ∈ IR + , x ∈ Ω, inf(A 0 , B 0 ) ≤ a(t, x) ≤ sup(A 1 , B 1 ), inf(A 2 0 , B 0 ) ≤ b(t, x) ≤ sup(A 2 1 , B 1 ). ( 57
lies in C 2 (IR 2 , IR 2 ) ∩ W 1,∞ (IR 2 , IR 2 ). According to proposition 3.2, there exists a solution U = (a, b) ∈ C 2 (IR + × Ω) to the system    ∂ t U -D∆ x U = f (U ), n • ∇ x U = 0 for x ∈ ∂Ω, U (0, x) = (a in (x), b in (x)), ( 58
)
where D is a (2 × 2) diagonal matrix with entries d a , d b .

On any interval of time [0, T ] for which (for all

x ∈ Ω) |a(t, x)|, |b(t, x)| ≤ 2 sup(A 1 , A 2 1 , √ B 1 , B 1 ), (a, b
) is also a (strong) solution of system ( 10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF], and according to proposition 3.3, it satisfies

|a(t, x)|, |b(t, x)| ≤ sup(A 1 , A 2 1 , B 1 , B 1 ).
Proof of lemma 3.1: We recall that according to proposition 3.1, the solution of the heat equation ( 59) is given by formula (42) :

u(t, x) = 1 2 √ π 1 -1 ũin (y) ∞ k=-∞ 1 √ dt e -(2k+x-y) 2 4 d t dy (62) + 1 2 √ π t 0 1 -1 φ(s, y) ∞ k=-∞ 1 √ d (t-s) e - (2k+x-y) 2 4 d (t-s) dy ds .
This yields the estimate

u L r ([0,T ]×[-1,1]) ≤ 1 2 √ π ũin * x S L r ([0,T ]×[-1,1]) + 1 2 √ π φ * t,x S L r ([0,T ]×[-1,1]) , (63)
where

S(t, x) := ∞ k=-∞ 1 √ d t e -(2k+x) 2 4 d t satisfies (for q ∈ [1, 3[) S(t, •) L 1 ([-1,1]) = 2 √ π , S L q ([0,T ]×[-1,1]) ≤ C 1 + T 1 q + 1 2 . (64)
The second formula of (64) can be obtained by using (when n = 0) (2n + x) 2 ≥ |2n + x| ≥ 2n -1 in order to estimate

S L q ([0,T ]×[-1,1]) ≤ (d t) -1 2 e -x 2 4 d t + 2 ∞ n=1 e -2n-1 4 d t L q ([0,T ]×[-1,1]) ≤ (d t) -1 2 e -x 2 4 d t L q ([0,T ]×[-1,1]) + 2 (d t) -1 2 e 1/4 d t -e -1/4 d t -1 L q ([0,T ]×[-1,1]) ≤ T 0 (d t) -q/2 2 √ π d t q erf √ q 2 √ d t dt 1/q + 4 T 0 |2(d t) 1/2 | q dt 1/q .
Returning to (63), we can estimate each term in the right-hand side in order to obtain lemma 3.1, the last term being the most difficult. In order to treat it, we apply Young's inequality φ * S L r ≤ φ L p S L q for 1 r +1 = 1 p + 1 q (remember that 1 r + 2 3 > 1 p , which is compatible with q ∈ [1, 3[), and estimate (64) for S L q . The same is true when r = ∞ and p > 3 2 . This concludes the proof of lemma 3.1.

Remark : The lemma above extends to dimension N > 1, with a certain number of changes in the assumption and the conclusion. The main one is that 1 r + 2 3 > 1 p is replaced by 1 r + 2 N +2 > 1 p , so that the solution of the heat equation lies in a L r space which is not as good as in dimension 1.

We use this lemma in order to prove an a priori L ∞ estimate for the concentrations solving system ( 13) -( 15) in dimension 1. 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF].

Then, there exists a constant C T (which grows at most polynomially with T , and depend only on the initial data) such that

||a, b, c|| L ∞ ([0,T ]×[0,1]) ≤ C T . ( 65
)
Proof of proposition 3.4 : According to proposition 2.5, we already know that ||a, b, c|| L 2 ([0,T ]×[0,1]) ≤ C T , where C T is (and will systematically be in the sequel of this proof) a constant which grows at most polynomially with T . Then,

||a -b c|| L 1 ([0,T ]×[0,1]) ≤ C T . We see thanks to lemma 3.1 that ||a, b, c|| L r ([0,T ]×[0,1]) ≤ C T for all r < 3. As a conse- quence, ||a -b c|| L 3/2 ([0,T ]×[0,1]) ≤ C T . A second use of lemma 3.1 yields ||a, b, c|| L r ([0,T ]×[0,1]) ≤ C T for all r < +∞. Then, ||a -b c|| L r/2 ([0,T ]×[0,1]) ≤
C T for all r < +∞, and a third use of lemma 3.

1 yields ||a, b, c|| L ∞ ([0,T ]×[0,1]) ≤ C T .
This ends the proof of proposition 3.4.

Remark : We note that the same proof holds for system ( 16) -( 18) in dimension 1.

A variant of this proof (in which the nonnegativity of the concentrations is used : the nonpositive part of the r.h.s of the equations is not estimated) shows that the L ∞ estimate also remains valid for system (13) -( 15) in any dimension.

This a priori estimate enables to prove the existence (and uniqueness, smoothness) of a (strong) solution to system ( 13) -( 15 Then, the function

f (a, b, c) = ((b c -a) χ K (a, b, c), (a -b c) χ K (a, b, c), (a -b c) χ K (a, b, c)) lies in C 2 (IR 3 , IR 3 ) ∩ W 1,∞ (IR 3 , IR 3
). According to proposition 3.2, there exists a (strong

) solution U = (a, b, c) ∈ C 2 (IR + × [0, 1]) to the system    ∂ t U -D∆ x U = f (U ), n • ∇ x U = 0 for x ∈ ∂Ω, U (0, x) = (a in (x), b in (x), c in (x)), ( 66
)
where D is the diagonal matrix with entries d a , d b , d c .

On any interval of time [0, T ] for which (for all x ∈ Ω) |a(t, x)|, |b(t, x)|, |c(t, x)| ≤ K, the functions (a, b, c) are also a (strong) solution of system ( 13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF]. According to proposition 3.4, we can thus build a (strong) solution to system ( 13) -( 15) on any interval [0, T ] such that C T ≤ K.

Letting K → +∞ and using the "uniqueness" part of proposition 3.2 (on bounded time intervals), we obtain a (strong) solution of system ( 13) -( 15) on IR + by sticking together solutions on [0, T ] (remember that C T < +∞ for all T > 0 since C T grows at most polynomially with T ).

Finally, if we now have two solutions of system ( 13) -( 15) on IR + which satisfy estimate (65), they are also solutions to system (66) for a certain K on a time interval [0, T ] depending on K but arbitrarily large (that is, T is going to +∞ when K → +∞). Thanks to the "uniqueness" part of proposition 3.2, they are equal on this time interval. This concludes the proof of theorem 3.3.

Unbounded Source : The duality method

Duality for parabolic equations with non smooth coefficients

We begin with the following proposition, which shows that the solutions of a parabolic differential inequality with bounded coefficients satisfy a natural L 2 bound. We refer to [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF] for this type of arguments of duality : Proposition 3.5 Let T > 0 and Ω be a bounded, connected and regular

(C 2 ) open set of IR N . We suppose that A : [0, T ] × Ω → IR is a smooth (belonging to C 2 ([0, T ] × Ω)) function such that A 0 ≤ A(t, x) ≤ A 1 , for some strictly positive constants A 0 , A 1 .
We assume that z : [0, T ] × Ω → IR is a strong (belonging to C 2 ([0, T ] × Ω)) nonnegative solution to the parabolic inequation :

   ∂ t z -∆ x (A z) ≤ 0, ∇ x (A z) • n = 0 on ∂Ω, z(0, •) = z in . (67)
Then, there exists a constant C (depending on T, A 0 , A 1 ) such that

||z|| L 2 ([0,T ]×Ω) ≤ C ||z in || L 2 (Ω) .
Proof of proposition 3.5 : For any smooth (C 2 ) nonnegative function H := H(t, x), let us consider the nonnegative (smooth) solution of the dual problem:

   -∂ t w -A∆ x w = H, ∇ x w • n = 0, on ∂Ω, w(T, •) = 0. (68)
Integrating by parts, we see that

T 0 Ω z ∂ t w + A∆ x w dxdt = - T 0 Ω w ∂ t z -∆ x (A z) dxdt - Ω z in (x) w(0, x) dx + T 0 ∂Ω z A ∇ x w • n dσ(x) dt - T 0 ∂Ω w ∇ x (A z) • ndσ(x) dt.
Using the Neumann boundary conditions, we end up with

- T 0 Ω z(t, x) H(t, x) dxdt ≥ - Ω z in (x) w(0, x) dx. (69)
Let us estimate w(0, •) in L 2 (Ω). Multiplying eq. ( 68) by -∆ x w and integrating by parts in the variable x gives (for any time t ∈ [0, T ])

- 1 2
d dt Ω |∇ x w(t, x)| 2 dx + Ω A(t, x) (∆ x w(t, x)) 2 dx = - Ω H(t, x) ∆ x w(t, x) dx ≤ Ω A 0 2 (∆ x w(t, x)) 2 + 1 2 A 0 H 2 (t, x) dx.
It follows, after integration in time, that

T 0 Ω (∆ x w(t, x)) 2 dxdt ≤ 1 A 2 0 T 0 Ω H 2 (t, x) dxdt.
Integrating eq. ( 68) between times 0 and T , we obtain

w(0, x) + T 0 A(s, x) ∆ x w(s, x) ds = T 0 H(s, x) ds, so that Ω w(0, x) 2 dxdt ≤ 2 T + 2 T A 2 1 A 2 0 T 0 Ω H 2 (t, x) dxdt.
Therefore, thanks to estimate (69), we see that

T 0 Ω z(t, x) H(t, x) dxdt ≤ Ω z in (x) w(0, x) dx ≤ z in L 2 (Ω) w(0) L 2 (Ω) ≤ 2 T + 2 T A 2 1 A 2 0 1/2 z in L 2 (Ω) H L 2 ([0,T ]×Ω) .
By density, this estimate is true for any nonnegative function H in L 2 , and, using H = H + -H -, we see that for any function H in L 2 , one has (for

C = (2 T + 2 T A 2 1 A 2 0 ) 1/2 ) T 0 Ω z(t, x) H(t, x) dxdt ≤ C z in L 2 (Ω) H L 2 ([0,T ]×Ω) .
By duality (L 2 is the dual of itself), this gives a bound for z in L 2 (0, T ] × Ω) in terms of z in L 2 (Ω) , and the proof of proposition 3.5 is ended.

The L 2 LogL estimate

We shall now follow [START_REF] Desvillettes | About global existence for quadratic systems of reaction-diffusion[END_REF] and use the duality argument of proposition 3.5 in order to build a weak (L 2 ) solution to problem ( 16) -(18) : Then, there exists a weak solution a, b, c, d ∈ L 2 loc (IR + ; L 2 (Ω)) to system ( 16) -(18).

Proof of theorem 3.4 : We introduce the approximated system :

             ∂ t a n -d a ∆ x a n = - an bn-cn dn 1+ 1 n (a 2 n +b 2 n +c 2 n +d 2 n ) , ∂ t b n -d b ∆ x b n = - an bn-cn dn 1+ 1 n (a 2 n +b 2 n +c 2 n +d 2 n ) , ∂ t c n -d c ∆ x c n = an bn-cn dn 1+ 1 n (a 2 n +b 2 n +c 2 n +d 2 n ) , ∂ t d n -d d ∆ x d n = an bn-cn dn 1+ 1 n (a 2 n +b 2 n +c 2 n +d 2 n ) , (70)
together with the homogeneous Neumann conditions

n(x) • ∇ x a n = 0, n(x) • ∇ x b n = 0, n(x) • ∇ x c n = 0, n(x) • ∇ x d n = 0, x ∈ ∂Ω, (71)
and the strictly positive initial data

a n (0, x) = a in (x), b n (0, x) = b in (x), c n (0, x) = c in (x), d n (0, x) = d in (x) . (72)
We know thanks to proposition 3.2 that there exists a unique strong solution to this system, since the right-hand side is C 2 , bounded and globally Lipschitz continuous (W 1,∞ ). We introduce the quantity

z n = φ(a n ) + φ(b n ) + φ(c n ) + φ(d n )
, where φ(x) = x ln x -x + 1. Note that φ is a nonnegative function whose second derivative is x → 1

x . The computation gives

∂ t z n -d a ∆ x φ(a n ) -d b ∆ x φ(b n ) -d c ∆ x φ(c n ) -d d ∆ x φ(d n ) = -ln a n a n b n -c n d n 1 + 1 n (a 2 n + b 2 n + c 2 n + d 2 n ) -ln b n a n b n -c n d n 1 + 1 n (a 2 n + b 2 n + c 2 n + d 2 n ) + ln c n a n b n -c n d n 1 + 1 n (a 2 n + b 2 n + c 2 n + d 2 n ) + ln d n a n b n -c n d n 1 + 1 n (a 2 n + b 2 n + c 2 n + d 2 n ) -d a |∇ x a n | 2 a n -d b |∇ x b n | 2 b n -d c |∇ x c n | 2 c n -d d |∇ x d n | 2 d n .
Therefore,

∂ t z n -d a ∆ x φ(a n ) -d b ∆ x φ(b n ) -d c ∆ x φ(c n ) -d d ∆ x φ(d n ) + d a |∇ x a n | 2 a n + d b |∇ x b n | 2 b n + d c |∇ x c n | 2 c n + d d |∇ x d n | 2 d n ≤ 0, (73)
and finally

∂ t z n -∆ x (A n z n ) ≤ 0,
where

A n = d a φ(a n ) + d b φ(b n ) + d c φ(c n ) + d d φ(d n ) z n . Note that ∀t > 0, x ∈ Ω, inf(d a , d b , d c , d d ) ≤ A n (t, x) ≤ sup(d a , d b , d c , d d ),
and

∇ x (A n z n ). • n(x) = 0 for x ∈ ∂Ω.
According to proposition 3.5, we obtain the estimate

||z n || L 2 ([0,T ]×Ω) ≤ C ||φ(a in )|| L 2 (Ω) + ||φ(b in )|| L 2 (Ω) +||φ(c in )|| L 2 (Ω) + ||φ(d in )|| L 2 (Ω) ,
where C does not depend on n (but depends on T and the diffusivity constants). This ensures that the sequences a n , b n , c n and d n are bounded in the space L 2 (ln L) 2 ([0, T ] × Ω). In particular, the quantities a n b n and c n d n are equiintegrable.

We conclude the proof ot theorem 3. We start again from estimate (73). Integrating between times 0 and T and for x ∈ Ω this estimate, we see that

Ω z n (T, x) dx + d a T 0 Ω |∇ x a n | 2 a n (t) dxdt + d b T 0 Ω |∇ x b n | 2 b n (t) dxdt + d c T 0 Ω |∇ x c n | 2 c n dxdt + d d T 0 Ω |∇ x d n | 2 d n dxdt ≤ Ω z(0, x) dx.
As a consequence, we obtain that

∇ x ( √ 1 + a n ) is bounded in L 2 ([0, T ] × Ω), and therefore in L 1 ([0, T ] × Ω).
Then, we compute

(∂ t -d a ∆ x ) √ 1 + a n = d a 4 |∇ x a n | 2 (1 + a n ) 3/2 + 1 2 c n d n -a n b n √ 1 + a n (1 + 1 n (a 2 n + b 2 n + c 2 n + d 2 n )) , so that (∂ t -d a ∆ x ) √ 1 + a n is bounded in L 1 ([0, T ] × Ω).
We consider now g n = √ 1 + a n , and we prove that this quantity is strongly compact (in L 1 loc (]0, T [×Ω)), using the fact that ∇ x g n is bounded in L 1 ([0, T ] × Ω) and that (∂ t -d a ∆ x ) g n is also bounded in L 1 ([0, T ] × Ω). This is a variant of Aubin's lemma (Cf. [START_REF] Temam | Studies in Mathematics and its Applications[END_REF]).

We denote by ω ν the set x ∈ Ω : d(x, ∂Ω) > ν . Let x ∈ ω ν (with ν positive and such that ω ν is not an empty set), and

k ∈ IR N , with | k| ≤ ν. Since ∇ x g n is bounded in L 1 ([0, T ] × Ω), we know that T 0 ων | g n (t, x + k) -g n (t, x)| dxdt ≤ C T | k| , (74)
where C T is some constant depending on T . Then, introducing any smooth function φ := φ(x), with compact support in Ω, we have

∂ t Ω g n φ dx = d a Ω g n ∆ x φ dx + Ω β n φ dx ,
where β n is bounded in L 1 ([0, T ] × Ω). Therefore T 0 ∂ t Ω g n φ dx dt ≤ d a g n L 1 ∆ x φ L ∞ + β n L 1 φ L ∞ (75) ≤ C T φ W 2,∞ .
We introduce ν > 0 and a mollifying sequence φ δ (x) = δ -N φ(x/δ) of smooth functions with compact support B(0, δ), so that B(0, δ) + ω ν ⊂ Ω when δ < ν. Then, for any t ∈ [µ, T -µ] ⊂ [0, T ] (0 < µ < T /2), and for any h ∈ IR, | h| ≤ µ, we get

T -µ µ ων | g n (t + h, x) -g n (t, x)| dxdt ≤ T -µ µ ων | (g n * x φ δ )(t + h, x) -(g n * x φ δ )(t, x)| dxdt + 2 T 0 ων | (g n * x φ δ )(t, x) -g n (t, x)| dxdt ≤ T -µ µ ων | (g n * x φ δ )(t + h, x) -(g n * x φ δ )(t, x)| dxdt + 2 C T δ.
Using (75), we see that

T -µ µ ων | g n (t + h, x) -g n (t, x)| dxdt ≤ T -µ t=µ x∈ων |h| 1 u=0 y∈Ω g n (•, y) φ δ (x -y) dy ′ (t + uh)du dxdt + 2 C T δ ≤ | h| x∈ων 1 u=0 T t=0 ∂ t y∈Ω
g n (t, y) φ δ (x -y) dy dtdudx

+ 2 C T δ ≤ C T | h| x∈ων ||φ δ (x -•)|| W 2,∞ dx + 2 C T δ .
Finally, optimizing in δ, we get

T -µ µ ων | g n (t + h, x) -g n (t, x)| dxdt ≤ C T | h| δ -N -2 + 2 C T δ ≤ C T |h| 1/(N +3) .
Recalling estimate (74), we obtain

T -µ µ ων | g n (t + h, x + k) -g n (t, x)| dxdt ≤ C T |h| 1/(N +3) + |k| .
This ensures that g n is a strongly compact sequence in L 1 (]µ, T -µ[×ω ν ) for all (sufficiently small) µ > 0, ν > 0, or, more simply, that g n is a strongly compact sequence in L 1 loc (]0, T [×Ω).

Then, up to extraction of a subsequence, g n converges a.e., and so does a n . The same proof applied to the other concentrations shows that b n , c n and d n also converge (up to extraction) a.e..

It is now possible to pass to the limit in the weak formulation of system (70) -(72).

We recall that this formulation (written here only for the first equation) is the following : for any smooth (C 2 ) test function ϕ := ϕ(t, x) with compact support in [0, +∞[×Ω and such that n(x)

• ∇ x ϕ(t, x) = 0 for x ∈ ∂Ω , +∞ 0 Ω a n ∂ t ϕ dxdt + Ω a in (x) ϕ(0, x) dx + +∞ 0 Ω d a a n ∆ x ϕ dxdt = +∞ 0 Ω cn d -an bn 1+ 1 n (a 2 n +b 2 n +c 2 n +d 2 n ) ϕ dxdt . (76)
At this point, let us recall that since a n converges a.e. (up to a subsequence), and is bounded in L 2 (ln L) 2 ([0, T ] × Ω), it converges strongly in L 2 ([0, T ] × Ω) to a function a lying in this space (and so do the other concentrations), so that the quantity a n b n and c n d n , which are equiintegrable, converge respectively to a b and c d in L 1 .

We can pass finally to the limit in the left-hand side of (76), obtaining

+∞ 0 Ω a ∂ t ϕ dxdt + Ω a in ϕ(0, •) dx + +∞ 0 Ω d a a ∆ x ϕ dxdt. (77)
As concerns the right-hand side of (76), we see that

+∞ 0 Ω c n d -a n b n 1 + 1 n (a 2 n + b 2 n + c 2 n + d 2 n ) ϕ dxdt converges to +∞ 0 Ω (c d -a b) ϕ dxdt .
We end up in this way with the weak form of the (first equation) of system ( 16) -(18).

4 Large Time Behaviour

Entropy dissipation estimate : general theory

We first describe the traditional strategy (sometimes called entropy/entropy dissipation method in the context of kinetic equations) used to get explicit estimates of convergence toward equilibrium for equations where dissipative effects are predominant.

Lyapounov functionals; De La Salle Principle

The principles of the approach described below and many applications can be found in [START_REF] Desvillettes | Entropic methods for the study of the long time behavior of kinetic equations[END_REF].

We consider an abstract equation

∂ t f = A f, (78)
where A is an operator which can be linear or nonlinear, and can involve derivatives or integrals. We suppose that there exists a (bounded below) Lyapounov functional H := H(f ) (usually called entropy (or opposite of the entropy)) and a functional D := D(f ) (usually called entropy dissipation) such that (what is called in kinetic theory the first and second part of Boltzmann's H-theorem) holds :

∂ t H(f ) = -D(f ) ≤ 0, (79)
D(f ) = 0 ⇐⇒ A f = 0 ⇐⇒ f = f eq , (80)
where f eq is a given function.

Then, one can often prove that the decreasing function t → H(f (t)) converges toward its minimum, that t → D(f (t)) converges toward 0, and that as a consequence lim t→+∞ f (t) = f eq (81)

in a convenient topology (a precise theorem could for example be written down, in the case when H is coercive). Unfortunately, the situation described above is often slightly more intricate, because one has to take into account quantities which are conserved in the evolution of eq. (78). In particular, in eq. ( 80), f eq is in general not a given function, but a set of functions depending on a number of parameters equal to the number of conserved quantities.

Use of the Gronwall Lemma

In order to obtain an explicit rate of convergence toward equilibrium, one looks for functional inequalities of the form

D(f ) ≥ Φ(H(f ) -H(f eq )), (82)
where Φ : IR + → IR + is a function such that Φ(x) = 0 ⇐⇒ x = 0. One tries to find a function Φ which increases as much as possible near 0.

Note that we look for such an inequality for functions f which are not necessarily solutions of eq. (78) (they do not even depend upon t !). This is a way of transforming a problem on PDEs in a problem of functional inequalities.

In the case when there are conserved quantities in the evolution of eq. ( 78), it is enough to prove estimate (82) when the corresponding quantities are fixed.

Assuming that estimate (82) holds, we apply it to the function f := f (t) solution of eq. ( 78) and get thanks to estimate (79) the differential inequality

∂ t (H(f )(t) -H(f eq )) ≤ -Φ(H(f )(t) -H(f eq )).
(83)

Then, Gronwall's lemma ensures that

H(f )(t) -H(f eq ) ≤ R(t), (84)
where R is the reciprocal of a primitive of -1/Φ. When one can take, for some constant C 0 > 0, Φ(x) = C 0 x, one gets R(t) ≤ Cst e -C 0 t . Sometimes, it is however only possible to take Φ(x) = Cst ε x 1+ε for some (or all) ε > 0, and consequently R(t) = Cst ε t -1/ε .

Cziszar-Kullback inequality

It is often more interesting to estimate ||f (t) -f eq || for some norm || ||, than to estimate the more abstract quantity H(f (t)) -H(f eq ). Therefore, we are interested in inequalities of the form

H(f ) -H(f eq ) ≥ ψ ||f (t) -f eq || , (85)
where we look for a function ψ ≥ 0 which increases as much as possible near 0. Note that since H reaches its minimum at f = f eq , we expect at best that ψ behaves quadratically at point 0.

Next proposition, sometimes called Cziszar-Kullback or Pinsker inequality (Cf. [START_REF] Csiszar | Information-type measures of difference of probability distributions and indirect observations[END_REF] and [START_REF] Kullback | A lower bound for discrimination information in terms of variation[END_REF]), enables to prove such an estimate in the case when

H(f ) = Ω (f ln f -f ), where Ω ⊂ IR N . Proposition 4.1 Let Ω be a measurable subset of IR N and f, g : Ω → R + . Then Ω f (x) ln f (x) g(x) -f (x) + g(x) dx ≥ 3 2 Ω f (x) dx + 4 Ω g(x) dx × Ω |f (x) -g(x)| dx 2 .
Proof of proposition 4.1 : We recall the following elementary inequality, due to Pinsker : for all u ∈ IR,

3 |u -1| 2 ≤ (2 u + 4) (u log u -u + 1). Then, Ω |f (x) -g(x)| dx = Ω f (x) g(x) -1 g(x) dx ≤ Ω 2 f (x) g(x) + 4 f (x) g(x) ln f (x) g(x) - f (x) g(x) + 1 g(x) √ 3 dx ≤ 1 √ 3 Ω (2 f (x) + 4 g(x)) dx Ω f (x) ln f (x) g(x) -f (x) + g(x) dx.
This concludes the proof of proposition 4.1.

In the sequel, since we shall use Lyapounov functionals which are sums of quantities like Ω (α ln α -α), for various concentrations α, we shall have to use Cziszar-Kullback inequality in order to obtain estimates like (85).

The case of Two reaction-diffusion equations

Large time behavior : the result

We now present a result which gives an explicit estimate about the speed of convergence of a solution to system (10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF] towards equilibrium. This result is extracted from [START_REF] Desvillettes | Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations[END_REF]. Then, the unique strictly positive smooth (C 2 ) solution t ∈ IR + → (a(t), b(t)) to equations ( 10) -( 12) obeys the following estimate of exponential decay toward equilibrium:

a(t, •) -a ∞ 2 L 1 (Ω) + b(t, •) -b ∞ 2 L 1 (Ω) ≤ C 1 e -C 2 t ,
where C 1 , C 2 > 0.

Here, a ∞ and b ∞ are the unique nonnegative numbers which satisfy the equation

a 2 ∞ = b ∞ , a ∞ + 2b ∞ = M. (86)
4.2.2 The proof : the entropy dissipation estimate Proof of theorem 4.1: We recall that since the initial data are strictly positive and smooth, system (10) -( 12) has indeed a unique smooth C 2 solution, which moreover satisfies (for some constants A 0 , A 1 , B 0 and

B 1 > 0) ∀t ≥ 0, x ∈ Ω, A 0 ≤ a(t, x) ≤ A 1 , B 0 ≤ b(t, x) ≤ B 1 . (87)
This is a direct application of theorem 3.2.

We then prove the following proposition, which corresponds to estimate (82) in the abstract formulation of subsection 4.1 : Proof of proposition 4.2 : We systematically denote the average over Ω of a quantity z by z = Ω z(x) dx (remember that Ω is of measure 1).

We start with the identity |∇xa| 2 a ≥ |∇xa| 2 ||a|| L ∞ , and apply Poincaré's inequality (with Poincaré's constant P ), so that

Ω |∇ x a| 2 a dx ≥ P Ω |a -ā| 2 ||a|| L ∞ dx.
We do the same for b, but we also use the identity

Ω |∇ x b| 2 b dx = 4 Ω |∇ x √ b| 2 dx ≥ P Ω | √ b - √ b| 2 dx.
Using then the inequality (x -y)(ln(x) -ln(y)) ≥ 4 (

√ x - √ y) 2
and the L ∞ bound on a and b, we get (for a strictly positive constant which we denote by C, as any other strictly positive constant in the sequel)

D 1 (a, b) ≥ C a - √ b 2 L 2 + a -a 2 L 2 + b -b 2 L 2 + √ b - √ b 2 L 2 . ( 90
)
We shall prove that the r.h.s. of ( 90) is bounded below by (some constant times) the relative entropy E(a, b) -E(a ∞ , b ∞ ).

Firstly, we use the conservation law (89) to rewrite the relative entropy as

E 1 (a, b) -E 1 (a ∞ , b ∞ ) = Ω a ln a a ∞ -(a -a ∞ ) + b ln b b ∞ -(b -b ∞ ) dx .
We use the elementary inequality ∀a > 0, a ln a -a + 1 ≤ (a -1) 2 , in order to obtain

E 1 (a, b) -E 1 (a ∞ , b ∞ ) ≤ 1 a ∞ a -a ∞ ≤ √ b - √ b 2 L 2 .
Thus, we only have to show that for some constant C > 0,

|a -b| 2 ≥ C |a -a ∞ | 2 + |b -b ∞ | 2 .
Note that this is a statement on numbers and not on functions.

We recall that a + 2 b = a ∞ + 2b ∞ , and that a 2 ∞ = b ∞ , so that we only have to show that for some constant C > 0,

|(a) 2 + 1 2 a -a 2 ∞ - 1 2 a ∞ | 2 ≥ C |a -a ∞ | 2 .
This is easily obtained by noticing that if f (x) = x 2 + 1 2 x, then |(f -1 ) ′ (x)| ≤ 2. This ends the proof of proposition 4.2.

The Cziszar-Kullback type inequality

We now turn to another proposition, which shows that the relative entropy E 1 (a, b) -E 1 (a ∞ , b ∞ ) defined by ( 23) and (86) controls (from above) the squares of the L 1 -distances to the equilibrium : Proposition 4.3 Let Ω be an open subset of IR N of measure 1. For all (measurable) nonnegative functions a, b : Ω → IR such that Ω (a(x) + 2b(x)) dx = M > 0,

E 1 (a, b) -E 1 (a ∞ , b ∞ ) ≥ C a -a ∞ 2 L 1 + b -b ∞ 2 L 1 , (92)
where definitions ( 23) and ( 86) have been used. + (q(a) -q(a ∞ )) + (q(b) -q(b ∞ )) .

We first note that thanks to Cziszar-Kullback inequality (Cf.

Q ′′ (a) = 1 a + 1 2 1 M -a ≥ 3 + 2 √ 2 2M , (93)
and

R ′′ (b) = 1 b + 4 M -b ≥ 6 + 4 √ 2 M , (94)
we combine 2/3 of (93) and 1/3 of (94) to Taylor-expand (q(a) -q(a ∞ )) + (q(b) -q(b ∞ ))

≥ 3 + 2 √ 2 6M |a -a ∞ | 2 + 3 + 2 √ 2 3M |b -b ∞ | 2 .
Finally, we observe that

a -a ∞ 2 L 1 ≤ 6 + 2 √ 2 3 + 2 √ 2 a -a 2 L 1 + 3 + 2 √ 2 3 |a -a ∞ | 2 ,

1 N

 1 x, (a, b, c, d) → k l (a, b, c, d).

Proposition 2 . 4

 24 Let T > 0, Ω be a bounded regular (C 2 ) open set of IR N , d a , d b > 0 (together with d c , d d > 0, depending on the system considered) be diffusivity constants, a in := a in (x) > 0, b in := b in (x) > 0 (together with c in := c in (x) > 0, d in := d in (x) > 0 depending on the system considered) be initial data in C 2 (Ω). Let also a := a(t, x), b := b(t, x) (together with c := c(t, x), d := d(t, x), depending on the system considered) be solutions in C 2 ([0, T ] × Ω) of one of the systems (10) -(

Theorem 3 . 1

 31 We consider a smooth (C 2 ) bounded and connected open subset Ω of IR N . Let u in ∈ C 2 (Ω) be an initial datum (compatible with the Neumann boundary condition) and φ ∈ L p ([0, T ] × Ω) (for some T > 0 and p ∈]1, +∞[) be a right-hand side.

Proposition 3 . 3 (

 33 Minimum and maximum principle) We assume that Ω is a bounded, smooth (C 2 ), and connected open subset of IR N , and that d a , d b are strictly positive diffusivity constants. Let a := a(t, x) and b := b(t, x) be a solution in C 2 ([0, T ] × Ω) to system (

) : Theorem 3 . 2

 32 Let Ω be a bounded, connected and regular (C 2 ) open set of IR N (N ≥ 1), and d a , d b be strictly positive diffusivity constants. We consider initial data a in , b in ∈ C 2 (Ω) (compatible with the Neumann boundary condition) which satisfy the bound :

) Proof of theorem 3 . 2 :

 32 We consider a function χ := χ(a, b) in C 2 (IR 2 , IR) such that χ(a, b) = 1 when |a|, |b| ≤ 2 sup(A 1 , A 2 1 , √ B 1 , B 1 ), and χ(a, b) = 0 when |a| or |b| ≥ 4 sup(A 1 , A 2 1 , √ B 1 , B 1 ). Then, the function f (a, b) = (-2 (a 2 -b) χ(a, b), (a 2 -b) χ(a, b))

Proposition 3 . 4

 34 Let a in , b in and c in be strictly positive initial data lying in C 2 ([0, 1]), and d a , d b , d c be strictly positive diffusivity constants. We consider a, b, c a (strong) solution in C 2 ([0, T ] × [0, 1]) of system (

) in dimension 1 : 3 . 3

 133 Theorem Let a in , b in and c in be strictly positive initial data lying in C 2 ([0, 1]), and d a , d b , d c be strictly positive diffusivity constants. Then, there exists a unique solution a, b, c ∈ C 2 (IR + × [0, 1]) to system (13) -(15) such that ||a, b, c|| L ∞ ([0,T ]×[0,1]) ≤ C T (where C T grows at most polynomially with T ). Proof of theorem 3.3 : We consider a function χ K := χ K (a, b, c) in C 2 (IR 3 , IR) such that χ K (a, b, c) = 1 when |a|, |b|, |c| ≤ K, and χ K (a, b, c) = 0 when |a| or |b| or |c| ≥ 2 K.

Theorem 3 . 4

 34 Let Ω be a bounded, regular (C 2 ) and connected open set of IR N , let d a , d b , d c and d d be strictly positive diffusivity constants, and let a in , b in , c in , d in be strictly positive initial data in C 2 (Ω).

  4 in next paragraph, by showing the strong convergence (i.-e. the convergence a.e.) of the sequences a n , b n , c n and d n .

3. 4 . 3

 43 Existence of weak L 2 solutions In the previous paragraph, we proved that there exists for each n ≥ 1 a smooth solution (a n , b n , c n , d n ) to system (70) -(72) Moreover, we have shown that the sequences a n , b n , c n and d n are bounded in L 2 (ln L) 2 ([0, T ] × Ω), so that the quantities a n b n and c n d n are equiintegrable.

Theorem 4 . 1

 41 Let Ω be a bounded, regular (C 2 ) and connected open set of IR N (N ≥ 1) of measure 1, and d a , d b be two strictly positive diffusivity constants. Let the initial data a in , b in be two strictly positive functions of C 2 (Ω) with mass Ω (a in + 2 b in ) dx = M > 0.

Proposition 4 . 2

 42 Let Ω be a bounded, smooth (C 2 ), and connected subset of IR N of measure 1. We consider d a , d b > 0 two strictly positive diffusion constants.Let a, b : Ω → IR be two functions such that (for some constants A 0 , A 1 , B 0 andB 1 > 0) ∀x ∈ Ω, A 0 ≤ a(x) ≤ A 1 , B 0 ≤ b(x) ≤ B 1 ,(88)andΩ (a(x) + 2 b(x)) dx = M > 0. (89)Then, there exists a constant C > 0 (depending onΩ, M , d a , d b , A 0 , A 1 , B 1 and B 2 ) such that D 1 (a, b) ≥ C (E 1 (a, b) -E 1 (a ∞ , b ∞ )),where a ∞ , b ∞ are defined by (86), and functionals E 1 , D 1 are defined by formulas (23) and (24).

Proof of proposition 4 . 3 :

 43 We still use the notations a = Ω a dx and b = Ω b dx, and define q(x) := x ln x -x. Then,E 1 (a, b) -E 1 (a ∞ , b ∞ ) =

1 ,

 1 and moreover a ≤ M and b ≤ M/2 by (89). Then, we consider Q(a) := q(a) + q( M -a 2 ) for a ∈]0, M [ and R(b) := q(b) + q(M -2b) for b ∈]0, M/2[. Since

Therefore, the sup of all T such that (for all x ∈ Ω) |a(t, x)|, |b(t, x)| ≤ 2 sup(A 1 , A 2 1 , √ B 1 , B 1 ) is infinite, and therefore (a, b) is a (strong, C 2 ) solution of system ( 10) - [START_REF] Martin | Nonlinear reaction-diffusion systems[END_REF] on IR + , which satisfies (thanks to proposition 3.2) estimate (57).

If we now have two solutions of system ( 10) -( 12) on IR + which satisfy estimate (57), they are also solutions of system (58), and thanks to the "uniqueness" part of proposition 3.2, they are equal. This concludes the proof of theorem 3.2.

Using the properties of the heat kernel

We introduce here the idea of using the smoothing properties of the heat kernel in order to obtain L ∞ bounds for system (13) - [START_REF] Temam | Studies in Mathematics and its Applications[END_REF]. Those bounds are only local in time (that is, in L ∞ ([0, T ] × Ω) for a given T > 0), but we take care of controlling what happens when T → +∞.

We give complete proofs for system ( 13) -( 15) in dimension 1, and briefly indicate how these proofs can be extended to the case of system ( 13) -( 15) in any dimension, or to the case of system ( 16) -( 18) in dimension 1.

Those proofs are inspired by the book of Rothe [START_REF] Rothe | Global solutions of reaction-diffusion systems[END_REF].

We begin with a standard estimate of the smoothing effect of the heat equation in dimension 1.

and assume for the initial data u(0, x) = u in (x) and for the source term φ := φ(t, x) that

for some p ≥ 1.

Then, for any exponent r > 1 such that 1 r + 2 3 > 1 p and for all T > 0, the norm u L r ([0,T ]×[0,1]) grows at most polynomially in T like

In the case when p > 3 2 , one can take r = ∞ in estimate (61).

by Young's inequality. This concludes the proof of proposition 4.3.

We come back to the proof of theorem 4.1.

We consider the unique smooth solution t → (a(t), b(t)) to system (10) -( 12) obtained at the beginning of the proof. We apply proposition 4.2 to a = a(t) and b = b(t). This is possible since condition (88) is satisfied thanks to estimate (87), and condition (89) is satisfied thanks to the conservation property (20). As a consequence, we see that

Thanks to the entropy estimate (29), we obtain

and thanks to Gronwall's lemma :

Using now proposition 4.3 with a = a(t) and b = b(t) (and noticing again that the conservation property (20) enables to fulfill the assumptions of this proposition), we conclude the proof of theorem 4.1.


The City Lightweight and Innovative Cab (CLIC) project was a scientific collaboration gathering public and private organizations. The aim was to propose an innovative lighten truck cab, where a high strength steel was used. As long as it could affect directly the acoustic environment of the cab, it was necessary to be able to simulate the vibroacoustic behavior of the truck cab in the mid frequency range. The dissipative treatments used for noise and vibration control such as viscoelastic patches and acoustic absorbing materials must then be taken into account in the problem. A process based on the SmEdA (Statistical modal Energy distribution Analysis) method was developed and is presented in this paper. SmEdA allows us substructuring the global problem, to study the interaction between the floor and the interior cavity. The process consists in building finite element models (FEM) of each subsystem (floor, internal cavity), including the dissipative material (damping layer, poroelastic material). Standard modal FEM calculations are then performed for each uncoupled subsystem. From the spatial mode shapes, and the modal strain -kinetic energies, the modal loss factors of both subsystems are estimated. Finally, the pressure levels inside the cavity are deduced from the resolution of the SmEdA equations. To validate this process, a truck cabin has been excited mechanically on a rail of the floor and the pressure levels at different positions inside the cabin were measured for different configurations of dissipative treatment. Comparisons between SmEdA and experimental results allows us to assess the accuracy of the proposed method

I-Introduction

With the new security and pollution standards, the industrials have to make more effort on their engines design. The CLIC (City Lightweight and Innovative Cab) project, that is gathering different private and public partners, aims to suggest a new design for lighten cabin truck, where a high strength steel is used. This lightening process decreases the NVH performances, and the mid frequency vibroacoustic behavior should be taken into account to ensure the driver comfort. The finite element method (FEM) is the most common modeling method, despite the drawbacks in time consuming and computer memory requirements. The SEA method (Statistical Energy Analysis) [START_REF] Lyon | Statistical Energy Analysis of Dynamical Systems: Theory and Application[END_REF] is well used for the high frequency studies, but it requires the validation of many hypotheses [START_REF] Bot | Validity diagrams of statistical energy analysis[END_REF]. The SmEdA method (Statistical modal Energy distribution Analysis) [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF][START_REF] Maxit | Extension of SEA model to subsystems with non-uniform modal energy distribution[END_REF][START_REF] Maxit | Statistical modal Energy distribution Analysis (SmEdA), UP2HF summer school[END_REF] may be seen as a good compromise between those two methods. It consists to decompose the global system in different subsystems.

Then, each subsystem (taken separately) is described by its uncoupled (free or blocked) modes. FEM can be used to calculate the subsystem modes when the geometry of the considered subsystem is complex. Then, the modal coupling loss factors describing the coupling between the pair of modes are deduced. Considering the modal injected power by the external source and writing the energy equilibrium of each mode, the energy response of the global system can be deduced. Contrary to the SEA method, it does not assume the equipartition of the modal energies. The method is then well adapted for dealing vibro-acoustic problems in the midfrequency range. In particular, it can describe non-resonant transmission [START_REF] Maxit | Extension of SEA model to subsystems with non-uniform modal energy distribution[END_REF] and take into account the effect of spatially localized excitations when the modal overlap factor is low [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF] or inhomogeneity of the vibratory field within a heterogeneous subsystem [START_REF] Maxit | Statistical modal Energy distribution Analysis (SmEdA), UP2HF summer school[END_REF][START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF]. Even if SmEdA is not implemented in an industrial code, it is easily to build a model with a programming language like MATLAB associated with input data from a commercial FEM code like NASTRAN.

In this work, one focuses on the modelling of the vibroacoustic behavior of a truck cab in the mid-frequency range in order to assess its NVH performances. In this analysis, the effect of the vibration and noise control treatments (trim) should be considered. It may be the visco-elastic layers used for damped the structural vibration or the porous elastic materials used for their sound absorbing ability. An approach based on SmEdA for taking into account these dissipative treatments has been recently developed during the PhD thesis of Ha Dong Hwang [START_REF] Hwang | SmEdA vibro-acoustic modelling in the mid-frequency range including the effect of dissipative treatments[END_REF]. Basically, this method consists to develop an equivalent model of the dissipative materiel in order to be easily implemented in the subsystem FE model, to extract the subsystem modes of the dissipative subsystem and then to estimate the modal damping loss factors using a projection of the complex finite element matrices. These modal loss factors characterize in SmEdA the effect of the dissipative material which depend on its positions, its geometry and its mechanical characteristics. This methodology will be reminded and illustrated on a basic case on section II. Then, one proposes its application for describing the interaction between the floor and the passenger cavity of truck cabin (i.e. the transmission through the lateral faces are supposed negligible). Results are presented in section III. Finally, after resolving the SmEdA equations, one can deduce the pressure level on the driver ears. The numerical results are compared to experimental tests to study its accuracy.

10/19/2016

II-Statistical Modal Energy Distribution Analysis

II.1 Principle

SmEdA [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF][START_REF] Maxit | Extension of SEA model to subsystems with non-uniform modal energy distribution[END_REF][START_REF] Maxit | Statistical modal Energy distribution Analysis (SmEdA), UP2HF summer school[END_REF][START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF][START_REF] Hwang | SmEdA vibro-acoustic modelling in the mid-frequency range including the effect of dissipative treatments[END_REF] is based on a modal description of each subsystems and it predicts the power flow exchanged between these different subsystem modes. Let us considers two subsystems denoted #1 and #2, composed of 𝑁 1 and 𝑁 2 modes respectively in the considered frequency band of excitation. The SmEdA equations expressing the energy conservation for each mode is written as:

  2 2 1 1 1 1 1 12 1 12 2 1 1 1 2 12 1 2 2 12 2 1 1 , 1 , + , N N p inj p p pq p pq q q q N N q inj p q p q q p q q p p EE p ,...,N E E                                          2 1 , q ,...,N             (1)
where :

-

m inj   is the injected power into mode m of subsystem  ( 1, 2   ); - m E  , m   and m 
 are respectively the energy, the angular frequency and the damping loss factor of mode m of subsystem  , and; -12 mn  is the modal coupling loss factor between the mode 𝑚 of subsystem 1 and the mode 𝑛 of the subsystem 2:

                2 12 12 2 2 1 1 2 2 2 1 2 2 2 2 2 1 2 1 1 2 2 1 1 2 2 2 1 W ... , mn mn m m n n n m m n m m n n m m n n n m                                      
(2) where 12 W mn is the intermodal work between the mode 𝑚 of subsystem 1 and the mode 𝑛 of the subsystem 2. This latter term expresses the spatial matching between the two modes and it can be evaluated from the modeshapes on the coupling surface (see [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF]).

In these equations, the angular frequencies m   and the intermodal works 12 W mn can then be deduced from an extraction of the subsystem modes using a standard finite element code. Usually, for the modal damping loss factors, a constant value is considered for all the modes of a given subsystem. This approach is however not adapted to describe the effect of dissipative material. A methodology will be described in the next section to estimate each of these modal loss factors from the characteristics and the position of the dissipative material inside the subsystem.

For illustration, one considers a white noise point source on subsystem 1 in the following. M is the point of excitation and FF S is the density power spectrum of the white noise signal of the excitation. The modal injected power is then given by:

  2 1 1 4 FF p p inj S M       , 2 0 q inj  , (3)
where

  1 p M



is the p th mode shape at point M of subsystem 1.

Knowing the injected power and the modal characteristics of each subsystem, one can compute the modal energies by resolving Eq. ( 1), and then deduce the total energy of each subsystem,

1 2 1 2 1 2 1 1
, ,

N N p q p q E E E E       (4)
and the spatial mean square of the pressure for an acoustic subsystem or of the vibratory velocity for a mechanical subsystem:

2 2 00 cE p V       , 2 E V M      . ( 5
)

II.2 Process for taking the dissipative treatments into account

In this section, we present the outline of the process allowing us to include dissipative material in the SmEdA modelling. The details can be found in Ref. [START_REF] Hwang | SmEdA vibro-acoustic modelling in the mid-frequency range including the effect of dissipative treatments[END_REF].

The process can be decomposed in four steps:

-The first step consists in representing the dissipative materials by using homogenized equivalent material models.

For a mechanical subsystem, an equivalent single layer model may be used for representing the viscoelastic layer patches (as illustrated on Fig. 2). The method proposed in [START_REF] Guyader | Viscoelastic properties of single layer plate material equivalent to multi-layer composites plate[END_REF] based on the zigzag theory and the traveling wave approach gives us the equivalent complex Young modulus and the equivalent Poisson's ratio of the multilayer from the knowledge of the thickness and mechanical parameters of each layer. This approach permit to deal with any number of layers and facilitates the modification of the material properties of each layer. It also facilitates the positioning of the viscoelastic patches on the structure by modifying the element properties of the 2D shell finite element model of the structure.

For an acoustic subsystem, the well-known equivalent fluid model can be used to represent the acoustic behavior of the poroelastic materiel. The complex equivalent fluid density and the complex equivalent fluid compressibility can be estimated from acoustic tube measurements with the Utsuno two cavities method [START_REF] Hwang | SmEdA vibro-acoustic modelling in the mid-frequency range including the effect of dissipative treatments[END_REF]. This type of modelling has two main advantages: (a) it permits increasing the upper frequency bound of the FEM calculation compared to a full detailed model; (b) it is well adapted for 10/19/2016 optimizing dissipative materials. Different positions of the dissipative materials could be considered easily by modifying only the property of the element in the original finite element mesh; -The second step consists in building the finite element models of each subsystem treated. The dissipative treatments are introduced in these models by using the homogenized material models. As the mechanical properties of these materials may vary with frequency, constant mean values are considered for each frequency band of calculation (typically, third-octave band). A finite element model is then created for each subsystem and each frequency band. Boundary conditions at the coupling surface are defined in accordance with DMF [START_REF] Maxit | Estimation of SEA coupling loss factors using a dual formulation and FEM modal information, part I: theory[END_REF]. The normal modes of each subsystem and each frequency band are then extracted. The natural frequencies and mode shapes at the coupling junction (normalized to a unit modal mass) are saved in the database used for the SmEdA calculation; -The modal damping loss factors are estimated in the third step. Modal equations of each subsystem containing a dissipative material are obtained from the dynamic equation of motion of the subsystem considered and the Galerkin method with normal mode shapes is used. Cross-modal terms are then neglected for estimating the modal damping loss factors by analogy between the modal equations obtained and the standard ones. Analytical expressions of the modal damping loss factors are obtained in function of the subsystem modeshapes and the complex matrices of the finite element model of the dissipative subsystem. These theoretical developments are described in details in [START_REF] Hwang | SmEdA vibro-acoustic modelling in the mid-frequency range including the effect of dissipative treatments[END_REF]. These calculations may be achieved in post-processing the modal extraction calculation and they are few time-consuming; -The final step consists in resolving SmEdA equations [START_REF] Lyon | Statistical Energy Analysis of Dynamical Systems: Theory and Application[END_REF] depending on the subsystem modal information estimated numerically in the two previous steps. Finally, the total energy of each subsystem can be estimated by summing the modal energies (4).

II.3 Illustration on a laboratory test case

For illustrating the process, let us considered the steel plate-cavity system excited by a mechanical force as presented in Fig. 2. Three configurations of damped are considered: the bare case (without any viscoelastic treatments); the visco1 plate damped by one viscoelastic patch, and; the visco2 plate damped by two viscoelactic patches. The viscoelastic layer is a 2.6~mm thick prefabricated damping patch currently used in the automotive industry. It can be semi-permanently attached to the plate by heating. The mechanical properties of the equivalent single layer model of the viscoelastic layer patch are obtained using the inhouse code MOVISAND based on the zig-zag method [START_REF] Guyader | Viscoelastic properties of single layer plate material equivalent to multi-layer composites plate[END_REF]. The complex elastic modulus of the viscoelastic material is determined using the timetemperature superposition principle. The frequency-dependent curves of isothermal properties at different temperatures were estimated experimentally by the manufacturer using a dynamic mechanical analyser (DMA+300 from METRAVIB company). For information, the damping loss factors of the viscoelastic layer and the steel are plotted on Fig. 3 for each third octave band.

Once the parameters of the equivalent models for the viscoelastic multilayer, a finite element mesh of each subsystem was created. The plate was represented with 13 776 quadrilateral shell elements whereas the cavity was represented with 4 031 412 tetrahedral solid elements. The mesh sizes were chosen in order to have a minimum of six elements per wavelength at 8 kHz. The modal analysis was performed for each subsystem for each third octave band using the MSC/NASTRAN code and the Lanczos method. A NASTRAN DMAP program allowed estimating the modal damping loss factors in the same run as the modal analysis.

Figure 3 shows the modal damping loss factors for the two damped plates. It can be seen that the damping pads provide substantially high damping compared to the bare plate. Their effects decrease slightly with increasing frequencies, similar to the decreasing values of the damping loss factors of the equivalent viscoelastic model. The damping values can vary substantially from one mode to another for the same configuration. This indicates that some modes exhibiting low strains at the pad positions are less influenced by the damping pad than others. This is illustrated in Fig. 4 where the spatial shapes of two modes of visco1 plate (identified by two red circles on Fig. 3) are plotted: the first has a natural frequency of 1584 Hz and a damping loss factor of 0.007, whereas the second has a natural frequency of 1818 Hz and a damping loss factor of 0.05. It can be clearly seen that the first mode does not exhibit significant spatial deformation around the damping pad, contrary to the second mode. Compared to the visco1 plate, the visco2 plate presents averaged 10/19/2016 values roughly 50 % higher and lower variations from one mode to another. This can be explained by the fact that the two damping pads occupied an area twice as large as a single pad. The influence of the dissipative material on the intermodal coupling loss factors and on the modal energy distribution for the cavity are illustrated in Fig. 5, by comparing these quantities between the bare and visco2 configurations. These results concern the 1 kHz third octave band for which the bare plate and the visco2 plate have 19 and 18 modes, respectively, whereas the empty cavity has 21 modes. It can be seen that for the bare case, only two couples of plate-cavity modes ( i.e. couples (4,3) and (10,10)) have significant intermodal coupling factor values. This shows that the coupling strength between these couples of modes is high compared to the other couples. Although the number of subsystem modes is relatively high in this band, few of them participate significantly in the energy sharing between the two subsystems. This is in particular due to the fact that the frequency coincidences play a significant role when the damping is low. On the contrary, for the visco2 configuration, a large number of couples have significant values for the intermodal coupling factors and participate in the energy sharing between the plate and the cavity. Almost all the cavity modes are significantly coupled to at least one plate mode. As observed in Fig. 4, the result is that the distribution of the modal energies of the cavity is more uniform for the visco2 configuration than for the bare configuration: a variation of 35 dB for the bare configuration compared to 20 dB for the visco2 configuration. Moreover, a considerable decrease of the highest values of the intermodal coupling factors for the visco2 case compared to the bare case. It results more cavity modes are significantly coupled with the plate modes for the visco2 case, but with less intensity.

Subsystem energies are finally obtained by summing the modal energies of each subsystem. For the 1 kHz third octave, the energy level of the cavity is estimated to: 103 dB (ref. 10 -12 J) for the bare configuration, 92 dB for the visco1 configuration and 89 dB for the visco2 configuration.

The energy difference between the bare and the visco configurations may be directly related to the difference observed on Fig. 3 for the modal damping loss factors. The effect of one viscoelastic patch is very significant (i.e. cavity energy reduced of 10 dB) whereas adding a supplementary one permits a supplementary decrease of cavity energy of around 3 dB, only . Results have been presented considering viscoelastic patches on the panel. The same type of results have been obtained considering a fibrous material inside the cavity. These results and a comparison with a laboratory experiment can be found in [START_REF] Hwang | SmEdA vibro-acoustic modelling in the mid-frequency range including the effect of dissipative treatments[END_REF].

Equivalent viscoelastic layer

III-Truck Cabin Modelling

III.1 Presentation of the industrial case

In order to assess the accuracy of the SmEdA method for predicting the vibroacoustic behaviour of the concept lighten cabin of the CLIC project, an experimental validation study was carried out previously on an existing cabin supplied by the industrial partners (see figure [START_REF] Maxit | Non resonant transmission modelling with Statistical modal Energy distribution Analysis[END_REF]). One presents some results of this study in this section. The objective was to predict the pressure level inside the cabin (globally) and on the driver ears when a mechanical force excited a truck rail. The cabin was based on four air suspensions which allowed us to decouple it from the ground for frequencies above 20 Hz. A hammer impact was used to excite the left rail on the back part, close to an air suspension as shown on figure 6. An accelerometer was used for measuring the vibratory acceleration at a point close to the excitation point (on the rail) and different microphones were used for measuring the pressure at different points inside the cavity and at the driver ear position. Two types of dissipative treatments had been considered (see figure 7): viscoelastic patches (called "IFF in French) stuck on the cabin floor and an acoustic panel composed of fibrous materials (felt) positioned on the rear of the cabin (see figure 7). The acoustic panel is composed of two parts, depending of the compression of the felt. Measurements inside the cabin were achieved with and without the acoustic panel.

III.2 SmEdA modelling

For the numerical simulation, we supposed that the acoustic radiation into the cabin is dominated by the floor radiation. Then, we neglected the contributions of the walls and the roof in the modelling. These structures were considered as rigid and simply supported boundary conditions were imposed at the edges of the floor.

The viscoelastic patches are identical to the ones considered in section 2. The mechanical characteristics of the viscoelastic material were measured with the dynamic mechanical analyser and the equivalent single layer parameters of the multilayer were obtained using MOVISAND. For the fibrous materials, an acoustic tube with the two cavities method was used to estimate the equivalent characteristic impedance and the equivalent wavenumber of each felt. The equivalent fluid parameters (i.e. complex density and complex celerity) were deduced in function of frequency and averaged for each third octave band. QUAD elements with a shell property taken the MOVISAND equivalent parameters into account. The FE models of the cavity and of the floor were composed of 3 Kdof and 1 Mdof, respectively. The two meshes were not coincident at the coupling surface between the floor and the cavity. A projection method was used for the calculation of the intermodal works from the modeshapes of each subsystem [START_REF] Gerges | Vibroacoustic modelling of a trimmed truck cab in the mid frequency range[END_REF]. As the parameters of the equivalent models (for the viscoelastic patches and the fibrous material) are frequency-dependant, finite element models were generated for each third octave band. The averaged values of the equivalent parameters were assigned to the finite elements related to the equivalent model. Modal analysis were performed using MSC/NASTRAN code. The process was automated as it was repeated for each third octave band of interest. For the different frequency bands, the MATLAB program allowed: (a), creating the input NASTRAN data file with the appropriate properties for the elements corresponding to the equivalent material; (b), running the NASTRAN calculation, and; (c), exporting the NASTRAN outputs in a MATLAB file. The data saved in the MATLAB file are the resonance frequencies, the mode shapes for the nodes belonging to the coupling surface and the modal damping loss factors. The SmEdA calculations were then performed under the MATLAB environment: first, the intermodal works were calculated from the modeshapes and the projection method (see illustration figure 9); second, the modal coupling loss factors were estimated using Eq. ( 2) with the intermodal works, the modal damping loss factors, and the modal frequencies; third, the injected power were calculated using Eq. ( 3); fourth, the modal energies were deduced from Eq. ( 1) with the quantities calculated previously; finally, the spatial mean square pressure inside the cavity was deduced from Eq. (4-5) and the pressure at the ear drivers was estimated using the process described in [START_REF] Totaro | Extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems[END_REF].

III.3 Results and comparison with the measurements

Before analysing the SmEdA results, the acceleration level at the excitation point was calculated by FEM considering the floor model only. The result is compared to the measurement on Fig. 10. Globally, one observes a relatively fair agreement between the two results: the tendency is well respected and the discrepancies are lower than 2 dB in the lower part of the frequency band. However, one notes significant discrepancies (between 5 and 13) for 4 third octave bands. The reliability of the FEM can be questioned. In particular, the boundary conditions applied on the floor edge can be not appropriated. As the injected power by the force may be directly related to the acceleration at the force point, one can expect to observe discrepancies between the SmEdA and the experiment results for these frequency bands.

Figure 11 shows the pressure level inside the cabin calculated by SmEdA, with and without the damping patches. As it was not possible in practice to remove the patches, these results cannot be compared to experimental results. One notes however that the effect of the patches is very important (a decrease of around 10 dB on the pressure level inside the cavity). Figure 12 present the pressure level at the driver ears. One observes a good accuracy of the SmEdA modelling in general. Some discrepancies can be observed in the band [800 Hz-1200 Hz], that correspondd to the band for which discrepancies have been observed on the acceleration level (figure 10). The pressure difference between the two ear positions is smaller in the SmEdA modelling than observed experimentally. This can be expected as SmEdA is not adapted for describing small variations of the pressure/vibration field. The process described in [START_REF] Gerges | Vibroacoustic modelling of a trimmed truck cab in the mid frequency range[END_REF] is well adapted when a subsystem exhibit a small number of modes (i.e. when the vibratory response is dominated by the response of few modes) or when a subsystem is non-homogeneous (i.e. with significant spatial variations of the vibratory field).

IV-Conclusions

An approach based on the SmEdA method has been proposed to predict the vibroacoustic behavior in the mid frequency range of a truck cabin including the dissipative materials (i.e. visco-elastic layers, acoustic absorbing materials). The calculation process requires to achieve modal analysis from Finite Element models of the uncoupled-subsystems including the dissipative treatments. These latter are represented with simplified equivalent models. This has the advantages to easier their introduction in the FE models in the practical applications, to reduce the computing times compared to a full model and to be well adapted for re-analyzing in an optimization purpose. A key point of this approach relies on the use of the Galerkin procedure for estimating the modal damping loss factors of the treated subsystems. A NASTRAN DMAP program allowed estimating these factors in the same run as the modal analysis. The application on the truck cabin has shown the feasibility of the approach for predicted the sound pressure up to 2 kHz. Comparisons with experiment results were acceptable even if additional investigations are necessary to completely assess the ability of the approach for predicting the vibroacoustic behaviour of an industrial structure like the truck cabin.

Figure 1 .

 1 Figure 1. Schematic representation of equivalent single layer modelling for a plate treated with viscoelastic materials.

Figure 2 .

 2 Figure 2. Plate-cavity system excited by a mechanical point force. Two damped configurations: (a), visco1 case ; (b), visco2 case.

Figure 3 .

 3 Figure 3. Plate modal damping loss factors as a function of the modal frequency: cross, visco1 plate; point, visco2 plate.

Figure 4 .

 4 Figure 4. Example of 2 modeshapes of plate for the visco1 configuration: (a) at 1584 Hz ; (b) at 1828 Hz.

Figure 5 .

 5 Figure 5. SmEdA results for the third-octave band 1kHz band: upper, modal coupling loss factors; lower, modal energy distribution of the receiving cavity. Two configurations: left, bare case ; right, visco2 case.

Steel 10 /

 10 19/2016

Figure 6 .

 6 Figure 6. Upper, experimental set-up on the truck cabin; Lower, air suspension fixed the truck rail and force of excitation.

Figure 7 .

 7 Figure 7. Upper, fibrous materials used to compose the rear panel; Lower, viscoelastic patches on the floor.

Figure 8 .

 8 Figure 8. Upper, FEM of the cavity with the fibrous material. Green, compressed felt; pink, normal felt. Lower, FEM of the floor with the viscoelastic patches (indicated by strong red color).

Figure 9 .

 9 Figure 9. Illustration of the intermodal work calculation between a floor mode and a cavity mode.

Figure 10 .

 10 Figure 10. Acceleration level at the excitation point: comparison between the FEM result and the experimental measurements.

Figure 11 .

 11 Figure 11. Global pressure level inside the cabin with and without the viscoelastic (IFF) patches. SmEdA results. Configuration without the acoustic panel.

Figure 12 .

 12 Figure 12. Pressure level on the driver ears: comparison between the SmEdA results and the measurements. Configuration without the acoustic panel.

Figure 13 Figure 13 .

 1313 Figure13shows the difference of the global pressure level between the cabin with and the cabin without the acoustic panel at the rear of the cabin. The global tendency is respected. Excepted for the third octave band 1 kHz, SmEdA predict correctly or slightly underestimate the effect of the panel. Even if these comparisons are not fully satisfactory, they show that the SmEdA model can take into account the effect of the dissipative material. A particular study should be carry out to define the most appropriate boundary conditions for the floor. Another way would consist to model the entire cabin. It would certainly lead to better results because the issue raised by the boundary conditions of the floor would be relaxed.

Acknowledgements

This work was co-funded by the French government (FUI 12 -Fonds Unique Interministériel) and European Union (FEDER -Fonds européen de développement régional). It was carried out in the framework of the LabEx CeLyA ("Centre Lyonnais d'Acoustique", ANR-10-LABX-60) and the research project CLIC ("City Lightweight Innovative Cab") labelled by LUTB cluster (Lyon Urban Truck and Bus), in partnership with Renault Trucks, Arcelor-Mittal, ACOEM, ALTRAN, FEMTO-ST (Univ. de Franche-Comté) and LVA (INSA de Lyon).