Matthieu Alfaro

Elisabeth Logak

Convergence to a propagating front in a degenerate Fisher-KPP equation with advection

Keywords: density-dependent diffusion, Fisher-KPP equation, chemotaxis, drift effect, singular perturbation. 2

We consider a Fisher-KPP equation with density-dependent diffusion and advection, arising from a chemotaxis-growth model. We study its behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We analyze, for small times, the emergence of transition layers induced by a balance between reaction and drift effects. Then we investigate the propagation of the layers. Convergence to a free-boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided.

Introduction

In this paper we consider a Fisher-KPP equation with density-dependent diffusion and advection, namely

(P ε )            u t = ε∆(u m ) -∇ • (u∇v ε ) + 1 ε u(1 -u) in (0, ∞) × Ω ∂(u m ) ∂ν = 0 on (0, ∞) × ∂Ω u(0, x) = u 0 (x) in Ω ,
with ε > 0 a small parameter and v ε (t, x) a smooth given function.

Here Ω is a smooth bounded domain in R N (N ≥ 2), ν is the Euclidian unit normal vector exterior to ∂Ω and m ≥ 2. We are concerned with the behavior of the solutions u ε (t, x) as ε → 0.

Assumption 1.1 (Initial data). Throughout this paper, we make the following assumptions on the initial data.

(i) Let Ω 0 be a nonempty open bounded set with a smooth boundary and such that Ω 0 ⊂ Ω. Let u 0 : Ω 0 → R be C 0 in Ω 0 and C 2 in Ω 0 , strictly positive on Ω 0 and such that u 0 (x) = 0 for all x ∈ ∂Ω 0 . Define the map u 0 : Ω → R by

u 0 (x) := u 0 (x) if x ∈ Ω 0 0 if x ∈ Ω \ Ω 0 .
(ii) Ω 0 is convex.

(iii) there exists δ > 0 such that, if n denotes the Euclidian unit normal vector exterior to the "initial interface" Γ 0 := ∂Ω 0 , then ∂ u 0 ∂n (y) ≥ δ for all y ∈ Γ 0 .

(1.1)

Remark 1.2. Note that the comparison principle allows to relax the regularity assumption on u 0 . See [START_REF] Alfaro | Sharp interface limit of the Fisher-KPP equation[END_REF] for details.

Assumption 1.3 (Structure of v ε ). We assume that

v ε (t, x) = v(t, x) + εv ε 1 (t, x) , (1.2)
with v and v ε 1 smooth functions on [0, ∞) × Ω. We assume that, for all T > 0, there exists C > 0 such that, for all ε > 0 small enough, it holds that v ε 1 C 1,2 ([0,T ]×Ω) ≤ C. Finally we assume ∂v ε ∂ν = 0 on (0, ∞) × ∂Ω .

(1.3)

Remark 1.4. In the sequel we smoothly extend v(t, x) in time and space on the whole of R × R N , as well as v ε 1 (t, x) in space on [0, ∞) × R N . Moreover since we are investigating local in time phenomena we will assume in the sequel, without loss of generality, that the extensions v(t, x) and v ε 1 (t, x) vanish outside of a large time-space ball.

Problem (P ε ) is a simpler version of a chemotaxis-growth system with logistic nonlinearity, where v ε (t, x) is not a given function but is coupled to u either through the parabolic equation εv t = ∆v + uγv or through the elliptic equation 0 = ∆v + uγv, supplemented with the Neumann boundary condition (1.3) (see e.g. [START_REF] Mimura | Aggregating pattern dynamics in a chemotaxis model including growth[END_REF]). Note that, in the case of linear diffusion (corresponding to m = 1) and a bistable nonlinearity, the asymptotic behavior of the corresponding system as ε → 0 has been studied using the Green's function associated to the homogeneous Neumann boundary value problem on Ω for the operator -∆ + γ (see [START_REF] Bonami | A free boundary problem arising in a chemotaxis model[END_REF] and [START_REF] Alfaro | The singular limit of a chemotaxis-growth system with general initial data[END_REF]) .

Motivation and biological background. Before describing our results, let us briefly comment about the relevance of (P ε ) in population dynamics models. The evolution equation in Problem (P ε ) combines logistic growth, chemotaxis and degenerate diffusion. We recall below how these terms appear in mathematical models that attempt to capture remarkable biological features.

Reaction diffusion equations with a logistic nonlinearity were introduced in the pioneering works [START_REF] Fisher | The wave of advance of advantageous genes[END_REF], [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]. The simplest equation reads

u t = ∆u + u(1 -u) ,
and has been widely used to model phenomena arising in population genetics [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] or in biological invasions [START_REF] Shigesada | Biological Invasion: Theory and Practise[END_REF]. Its main mathematical property is to sustain travelling wave solutions with a semi-infinite interval of admissible wave speeds, with the minimal one having a crucial biological interpretation.

Chemotaxis, i.e. the tendency of biological individuals to direct their movements according to certain chemicals in their environment, is induced in (P ε ) by the advection term -∇ • (u∇v ε ): the population, whose density is u(t, x), has an oriented motion in the direction of a positive gradient of the chemotactic substance, whose concentration is v ε (t, x). The first PDE model to describe such movements was proposed in [START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF] and involves linear diffusion for u and a parabolic equation coupling v to u. The Keller-Segel model has received considerable attention in mathematical literature, particularly focusing on the finite-time blow-up of solutions (see [START_REF] Hillen | A user's guide to PDE models for chemotaxis[END_REF] for a recent review). This provides a mathematical tool to analyze aggregation phenomena as observed in bacteria colonies. Chemotaxis systems involving linear diffusion and a growth term, either logistic or bistable, have later been considered in, e.g., [START_REF] Mimura | Aggregating pattern dynamics in a chemotaxis model including growth[END_REF], [START_REF] Bonami | A free boundary problem arising in a chemotaxis model[END_REF], [START_REF] Alfaro | The singular limit of a chemotaxis-growth system with general initial data[END_REF] and [START_REF] Winkler | Boundeness in the higher-dimensional parabolicparabolic chemotaxis system with logistic source[END_REF] .

Variants of the Fisher-KPP equation involving a degenerate diffusion have been proposed in order to take into account population density pressure. Actually one can introduce density-dependent birth or death rates as an attempt to control the size of a population. Nevertheless as shown in [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF], the introduction of a nonlinearity into the dispersal behavior of a species, which behaves in an otherwise linear way, may lead, in an inhomogeneous environment, to a similar regulatory effect. Moreover this assumption is consistent with ecological observations as reported for instance in [START_REF] Carl | Population control in arctic ground squirrels[END_REF], where it is shown that arctic ground squirrels migrate from densely populated areas into sparsely populated areas, even when the latter is less favorable (due to reduced availability of burrow sites or exposure to intensive predation). For such species, migration to avoid crowding, rather than random motion, is the primary cause of dispersal. To describe such movements, the authors in [START_REF] Shigesada | Biological Invasion: Theory and Practise[END_REF] and [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF] use the directed motion model where individuals can only stay put or move down the population gradient; this model yields the degenerate equation

u t = ∆(u 2 ) + G(x)u , (1.4)
in which the population regulates its size below the carrying capacity set by the supply of nutrients. Later in [START_REF] Gurtin | On the diffusion of biological populations[END_REF] a larger class of equations with degenerate diffusion and nonlinear reaction was considered, namely

u t = ∆(u m ) + f (u) , m ≥ 2 . (1.5)
Note that in the absence of f (u), equation (1.5) reduces to the so-called porous medium equation

u t = ∆(u m ) , (1.6)
which has been extensively investigated in the literature. We refer to the book [START_REF] Vásquez | The Porous Medium Equation. Mathematical theory[END_REF] and the references therein. The main feature of this equation is that it is degenerate at the points where u = 0. As a consequence, a loss of regularity of solutions occurs and disturbances propagate with finite speed, a property which has a relevant interpretation in a biological context (see for instance [START_REF] Simpson | Models of collective cell spreading with variable cell aspect ratio: A motivation for degenerate diffusion models[END_REF]).

Formal asymptotic analysis. Problem (P ε ) possesses a unique solution u ε (t, x) in a sense that is explained in Section 3. As ε → 0, the qualitative behavior of this solution is the following. In the very early stage, the nonlinear diffusion term ε∆(u m ) is negligible compared with the drift term -∇u • ∇v ε and the reaction term ε -1 u(1u). Hence, in some sense, the equation is well approximated by a coupling between the transport equation u t + ∇u • ∇v ε = 0 and the ordinary differential equation

u t = ε -1 u(1 -u).
Therefore, as suggested by the analysis in [START_REF] Alfaro | Sharp interface limit of the Fisher-KPP equation[END_REF], u ε quickly approaches the values 0 or 1, and an interface is formed between the regions {u ε ≈ 0} and {u ε ≈ 1} (emergence of the layers). Note that, in this very early stage, the balance of the transport equation and the ordinary differential equation will generate an interface not exactly around Γ 0 but in a slightly drifted place. Once such an interface is developed, the diffusion term becomes large near the interface, and comes to balance with the drift and the reaction terms so that the interface starts to propagate, on a much slower time scale (propagation of the front).

Our goal in this paper is to provide a rigorous analysis that supports this formal approach and makes it more precise. To study the interfacial behavior, we consider the asymptotic limit of (P ε ) as ε → 0. Then the limit solution will be a step function ũ(t, x) taking the value 1 on one side of a moving interface, and 0 on the other side. We show that this sharp interface, which we will denote by Γ t , obeys the law of motion

(P 0 )    V n = c * + ∂v ∂n on Γ t Γ t t=0 = Γ 0 ,
where V n is the normal velocity of Γ t in the exterior direction, c * the minimal speed of travelling waves solutions of a related degenerate one-dimensional problem (see Section 5 for details) and n the outward normal vector on Γ t .

Plan. The organization of this paper is as follows. We present our results in Section 2. In Section 3, we briefly recall known results concerning the wellposedness of Problem (P ε ); in particular, it admits a comparison principle so that the sub-and super-solutions method can be used to investigate the behavior of the solutions u ε . In Section 4, we prove a generation of interface property for Problem (P ε ). In Section 5 we investigate the motion of interface. Finally, we prove our main result in Section 6.

Results and comments

The question of the convergence of Problem (P ε ) to (P 0 ) has been addressed in [START_REF] Dkhil | Singular limit of a degenerate chemotaxis-Fisher equation[END_REF]. However, the author considers only a very restricted class of initial data, namely those having a specific profile with well-developed transition layers. In other words the generation of interface from arbitrary initial data is not studied. In the present paper we study both the emergence and the propagation of interface. Moreover we prove a sharp O(ε) estimate of the thickness of the transition layers of the solutions u ε . The authors in [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF] prove the convergence of the solutions of (P ε ) with arbitrary initial data with convex compact support to solutions of (P 0 ), when there is no advection (i.e. v ε ≡ 0). They provide an O(ε| ln ε|) estimate of the thickness of the transition layers. Therefore, even in the particular case v ε ≡ 0, our O(ε) estimate was not known.

As mentioned in the introduction, the drift term and the reaction term in (P ε ) are of the same magnitude for small times. Therefore the emergence of the layers, initiated by the ODE u t = ε -1 u(1u), will occur in the neighborhood of a slightly drifted initial interface Γ ε,drift 0

. To analyze such a phenomenon we shall use the Lagrangian coordinates. Recall that we have smoothly extended v(t, x) in time-space on the whole of R × R N , with v ≡ 0 outside of a large time-space ball. Then, for (t 0 , x 0 ) ∈ R × R N , we denote by ϕ (t 0 ,x 0 ) the solution, defined on R, of the Cauchy problem

     dX dt (t) = ∇v (t, X(t)) , X(t 0 ) = x 0 . (2.1)
We denote by Φ the associated flow defined on

R × R × R N , that is Φ(t 1 , t 2 , x 3 ) := ϕ (t 2 ,x 3 ) (t 1 ) . (2.2)
Recall that Γ 0 = ∂Ω 0 = ∂(Supp u 0 ) is the initial interface. From t = 0 to

t ε := ε| ln ε| (generation time) , (2.3)
we let each point on Γ 0 evolve with the law (2.1) and then define a drifted initial interface Γ ε,drift

0 by Γ ε,drift 0 := {Φ(t ε , 0, x) : x ∈ Γ 0 } . (2.4)
Next we consider the free boundary problem

(P 0 ε,drift )    V n = c * + ∂v ∂n on Γ ε,drift t Γ ε,drift t t=0 = Γ ε,drift 0 .
Well-posedness of (P 0 ) and of (P 0 ε,drift ). Using the level set formulation (see, e.g., [START_REF] Barles | Front propagation and phase field theory[END_REF]), the motion law in Problem (P 0 ) can be rewritten as a first order Hamilton-Jacobi equation with a convex Hamiltonian. This approach, combined with the results in [START_REF] Ley | Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts[END_REF], has been used in [START_REF] Dkhil | Singular limit of a degenerate chemotaxis-Fisher equation[END_REF] in order to prove the following.

Theorem 2.1 ( [START_REF] Dkhil | Singular limit of a degenerate chemotaxis-Fisher equation[END_REF], Well-posedness of (P 0 )). Let Ω 0 ⊂⊂ Ω be a smooth subdomain of Ω and let Γ 0 = ∂Ω 0 be the given smooth initial interface. Then there exists T max (Γ 0 ) > 0 such that Problem (P 0 ) has a unique smooth solution on [0, T ] for any 0 < T < T max (Γ 0 ). More precisely, there exists a family of smooth subdomains (Ω t ) t∈(0,T ] with Ω t ⊂⊂ Ω such that, denoting

Γ t = ∂Ω t , Γ := 0≤t≤T ({t} × Γ t ) is the unique solution to Problem (P 0 ) on [0, T ].
Moreover, T max (Γ 0 ) depends smoothly on Γ 0 . Therefore we can choose ε 0 > 0 small enough and T > 0 such that

0 < T < inf 0≤ε≤ε 0 T max (Γ ε,drift 0 ) , (2.5)
which guarantees the existence of a unique smooth solution on [0, T ] to both (P 0 ) and (P 0 ε,drift ) for any 0 < ε ≤ ε 0 . We denote by Γ ε,drift = 0≤t≤T ({t} × Γ ε,drift t ) the smooth solution to (P 0 ε,drift ) and by Ω ε,drift t the region enclosed by Γ ε,drift t . In the sequel we work on [0, T ], with T satisfying (2.5), and define

Q T := (0, T ) × Ω.
Our main result, Theorem 2.2, contains generation, motion and thickness of the transition layers properties. It asserts that: given an initial data u 0 , the solution u ε quickly (at time t ε = ε| ln ε|) becomes close to 1 or 0, except in a small neighborhood of the drifted interface Γ ε,drift t ε , creating a steep transition layer around Γ ε,drift t ε (generation of interface). The theorem then states that the solution u ε remains close to the step function associated with (P 0 ε,drift ) on the time interval [t ε , T ] (motion of interface); in other words, the motion of the transition layer is well approximated by the limit interface equation (P 0 ε,drift ). Moreover, the estimate (2.6) in Theorem 2.2 implies that, once a transition layer is formed, its thickness remains within order O(ε) for the rest of time.

Theorem 2.2 (Generation, motion and thickness of the layers). Let η ∈ (0, 1/2) be arbitrary. Then, there exists C > 0 such that, for all ε > 0 small enough and all

t ε = ε| ln ε| ≤ t ≤ T ,
we have

u ε (t, x) ∈      [0, 1 + η] if x ∈ N Cε (Γ ε,drift t ) [1 -η, 1 + η] if x ∈ Ω ε,drift t \ N Cε (Γ ε,drift t ) {0} if x ∈ (Ω \ Ω ε,drift t ) \ N Cε (Γ ε,drift t ) , (2.6
)

with N r (Γ ε,drift t ) := {x : dist(x, Γ ε,drift t ) < r} the tubular r-neighborhood of Γ ε,drift t .
Note that (2.6) shows that, for any 0 < a < 1, for all t ε ≤ t ≤ T , the a-level set L ε t (a) := {x : u ε (t, x) = a} lives in a tubular O(ε) neighborhood of the interface Γ ε,drift t . In other words, we provide a new O(ε) estimate of the thickness of the transition layers of the solutions u ε . Concerning the localization of the level sets L ε t (a), it is made with respect to a slightly drifted free boundary problem (P 0 ε,drift ). Nevertheless, since the solution of (P 0 ε,drift ) on [0, T ] is continuous w.r.t. the initial hypersurface Γ ε,drift 0 , we recover, as ε → 0, the original free boundary problem (P 0 ) and obtain the expected result. More precisely, let us define the step function ũ(t, x) by

ũ(t, x) := 1 in Ω t 0 in Ω \ Ω t for t ∈ (0, T ] . (2.7)
As a consequence of Theorem 2.2, we obtain the following convergence result which shows that ũ is the sharp interface limit of u ε as ε → 0.

Corollary 2.3 (Convergence).

As ε → 0, u ε converges to ũ, defined in (2.7), everywhere in 0<t≤T ({t} × Ω t ) and 0<t≤T {t} × (Ω \ Ω t ) .

3 Comparison principle, well-posedness for (P ε )

Since the diffusion term degenerates when u = 0 a loss of regularity of solutions occurs. We define below a notion of weak solution for Problem (P ε ), which is very similar to the one proposed in [START_REF] Aronson | Stabilization of solutions of a degenerate nonlinear diffusion problem[END_REF] for the one dimensional problem with homogeneous Dirichlet boundary conditions. Concerning the initial data, we suppose here that u 0 ∈ L ∞ (Ω) and u 0 ≥ 0 a.e. Note that in this section, and only in this section, we assume, for ease of notation, that ε = 1 and that v ε ≡ v; we then denote the associated Problem (P ε ) by (P ).

In the sequel

f (u) = u(1 -u). Definition 3.1. A function u : [0, ∞) → L 1 (Ω) is a solution of Problem (P ) if, for all T > 0, (i) u ∈ C [0, ∞); L 1 (Ω) ∩ L ∞ (Q T ) (ii) for all ϕ ∈ C 2 (Q T ) such that ϕ ≥ 0 and ∂ϕ ∂ν = 0 on ∂Ω, it holds that Ω u(T )ϕ(T ) - Q T (uϕ t + u m ∆ϕ + u∇v • ∇ϕ) = Ω u 0 ϕ(0) + Q T f (u)ϕ . (3.1)
A sub-solution (a super-solution) of Problem (P ) is a function satisfying (i) and (ii) with equality replaced by ≤ (respectively ≥).

Theorem 3.2 (Existence and comparison principle). Let T > 0 be arbitrary.

The following properties hold.

(i) Let u -(u + ) be a sub-solution (respectively a super-solution) with initial data u - 0 (respectively u + 0 ).

If u - 0 ≤ u + 0 a.e. then u -≤ u + in Q T ;
(ii) Problem (P ) has a unique solution u on [0, ∞) and

0 ≤ u ≤ max(1, u 0 L ∞ (Ω) ) in Q T ; (3.2) (iii) u ∈ C(Q T ).
Since (1.3) holds, the proof of Theorem 3.2 is standard and follows the same steps of that of [START_REF] Aronson | Stabilization of solutions of a degenerate nonlinear diffusion problem[END_REF]Theorem 5]. The continuity of u follows from [START_REF] Dibenedetto | Continuity of weak solutions to a general porous medium equation[END_REF].

The following lemma proved in [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF], will be very useful when constructing smooth sub-and super-solutions in later sections. ) is sufficiently smooth and let ν supp t be the outward normal vector on Γ supp t . Suppose moreover that

(i) ∇(u m ) is continuous in Q T (ii) L ε [u] := u t -∆(u m )+∇•(u∇v)-f (u) = 0 in {(t, x) ∈ Q T : u(t, x) > 0} (iii) ∂(u m ) ∂ν supp t = 0 on ∂Ω supp t , for all t ∈ [0, T ] .
Then u is a solution of Problem (P ). Similarly a function satisfying (i) and (ii)-(iii) with equality replaced by ≤ (≥) is a sub-solution (respectively a super-solution) of Problem (P ).

Emergence of the transition layers

In this section, we investigate the generation of interface which occurs very quickly around Γ ε,drift t ε

. We prove that, given a virtually arbitrary initial datum u 0 , the solution u ε of (P ε ) quickly becomes close to 1 or 0 in most part of Ω. More precisely -recalling that Φ(t 1 , t 2 , x 3 ), defined in (2.2), denotes the flow associated with the Cauchy problem (2.1)-the following holds.

Theorem 4.1 (Emergence of the layers). Let η ∈ (0, 1/2) be arbitrary. Then there exists M 0 > 0 such that, for all ε > 0 small enough, the following holds with t ε = ε| ln ε|.

(i) for all x ∈ Ω, we have that

0 ≤ u ε (t ε , x) ≤ 1 + η ; (4.1)
(ii) for all x ∈ Ω, we have that

if u 0 (Φ(0, t ε , x)) ≥ M 0 ε then u ε (t ε , x) ≥ 1 -η ; (4.2) (iii) for all x ∈ Ω, we have if dist(Φ(0, t ε , x), Ω 0 ) ≥ M 0 ε then u ε (t ε , x) = 0 , (4.3)
where we recall that Ω 0 = {x : u 0 (x) > 0} (see Assumption 1.1).

In order to prove the above theorem, we shall construct sub-and supersolutions. As mentioned before, in this very early stage, we have to take into account both the reaction and the drift terms. We start with some preparations.

A related ODE and the flow Φ

An ODE. The solution of the problem without diffusion nor advection, namely ūt = ε -1 f (ū) supplemented with the condition ū(0, x) = u 0 (x), is written in the form ū(t, x) = Y t ε , u 0 (x) , where Y (τ, ξ) denotes the solution of the ordinary differential equation Y τ (τ, ξ) = f (Y (τ, ξ)) supplemented with the initial condition Y (0, ξ) = ξ. Nevertheless, in order to take care of the term -u∆v ε , we need a slight modification of f .

Let f be the smooth odd function that coincides with f (u) = u(1u) on [0, ∞): f has exactly three zeros -1 < 0 < 1 and

f ′ (-1) = -1 < 0 , f ′ (0) = 1 > 0 , f ′ (1) = -1 < 0 , (4.4)
i.e. f is of the bistable type. Next, we define fδ (u) := f (u) + δ .

For |δ| small enough, this function is still of the bistable type: if δ 0 is small enough, then for any δ ∈ (-δ 0 , δ 0 ), fδ has exactly three zeros α -(δ) < a(δ) < α + (δ) and there exists a positive constant C such that

|α -(δ) + 1| + |a(δ)| + |α + (δ) -1| ≤ C|δ| , (4.5
)

|µ(δ) -1| ≤ C|δ| , (4.6)
where µ(δ) is the slope of fδ at the unstable zero, namely

µ(δ) := f ′ δ (a(δ)) = f ′ (a(δ)) . (4.7)
Now for each δ ∈ (-δ 0 , δ 0 ), we define Y (τ, ξ ; δ) as the solution of the ordinary differential equation

Y τ (τ, ξ ; δ) = fδ (Y (τ, ξ ; δ)) for τ > 0 Y (0, ξ ; δ) = ξ , (4.8)
where ξ varies in (-C 0 , C 0 ), with

C 0 := u 0 L ∞ (Ω) + 1 . (4.9)
We claim that Y (τ, ξ ; δ) has the following properties.

Lemma 4.2 (Behavior of Y ).

There exist positive constants δ 0 and C such that the following holds for all (τ, ξ ; δ) ∈ (0, ∞) × (-C 0 , C 0 ) × (-δ 0 , δ 0 ).

(i) If ξ > a(δ) then Y (τ, ξ ; δ) > a(δ) If ξ < a(δ) then Y (τ, ξ ; δ) < a(δ) (ii) |Y (τ, ξ ; δ)| ≤ C 0 (iii) 0 < Y ξ (τ, ξ ; δ) ≤ Ce µ(δ)τ (iv) Y ξξ Y ξ (τ, ξ ; δ) ≤ C(e µ(δ)τ -1).
Properties (i) and (ii) are direct consequences of the bistable profile of fδ . For proofs of (iii) and (iv) we refer to [START_REF] Alfaro | The singular limit of a chemotaxis-growth system with general initial data[END_REF].

The flow Φ. Let us briefly recall well known facts concerning the flow Φ(t 1 , t 2 , x 3 ). By definition we have

∂Φ ∂t 1 (t, t 0 , x 0 ) = ∇v(t, Φ(t, t 0 , x 0 )) . (4.10)
Next, note that, by uniqueness,

Φ(t, t 0 , Φ(t 0 , t, x 0 )) = x 0 , for all (t, t 0 , x 0 ) ∈ R × R × R N .
Differentiating this identity with respect to t 0 , we get

∂Φ ∂t 2 (t, t 0 , x) + D 3 Φ(t, t 0 , x) ∂Φ ∂t 1 (t 0 , t, x 0 ) = 0 R N ,
where x := Φ(t 0 , t, x 0 ) and where D 3 Φ(t 1 , t 2 , x 3 ) denotes the Jacobian matrix of Φ w.r.t. the third variable. Hence, using (4.10) we infer that

∂Φ ∂t 2 (t, t 0 , x) + D 3 Φ(t, t 0 , x)∇v(t 0 , x) = 0 R N , (4.11)
which is of crucial importance for our analysis.

4.2 Proof of (4.1) and (4.2)

We use the notation z + = max(z, 0). Our sub-and super-solutions are given by

w ± ε (t, x) := Y t ε , u 0 (Φ(0, t, x)) ± ε 2 C ⋆ (e µ(±εM )t/ε -1) ; ±εM + , (4.12
) or equivalently by

w ± ε (t, Φ(t, 0, x)) := Y t ε , u 0 (x) ± ε 2 C ⋆ (e µ(±εM )t/ε -1) ; ±εM + .
(4.13) Here Y (τ, ξ ; δ) is the solution of (4.8), µ(δ) the slope defined in (4.7), Φ(t 1 , t 2 , x 3 ) the flow defined in (2.2) and M is chosen such that, for all ε > 0 small enough, M ≥ C 0 ∆v ε L ∞ (Q T ) , with C 0 defined by (4.9).

Lemma 4.3 (Sub-and super-solutions for small times).

There exists C ⋆ > 0 such that, for all ε > 0 small enough, (w - ε , w + ε ) is a pair of sub-and supersolutions for Problem (P ε ), in the domain [0, t ε ] × Ω.

Before proving the lemma, we remark that w - ε (0, x) = w + ε (0, x) = u 0 (x). Consequently, by the comparison principle, we have

w - ε (t, x) ≤ u ε (t, x) ≤ w + ε (t, x) for all (t, x) ∈ [0, t ε ] × Ω . (4.14)
Proof. In order to prove that (w - ε , w + ε ) is a pair of sub-and super-solutions for Problem (P ε ) -if C ⋆ is appropriately chosen-we check the sufficient conditions stated in Lemma 3.3.

On the one hand, concerning the sub-solution w - ε , for (t, x) such that

x ∈ Ω supp t [w - ε ] := {x : w - ε (t, x) > 0} we have, at point (t, x), ∇(w - ε ) m = (mY m-1 Y ξ ) t ε , u 0 (Φ(0, t, x)) -ε 2 C ⋆ (e µ(-εM )t/ε -1) ; -εM ∇ x (u 0 (Φ(0, t, x)) . If (t, x) → (t 0 , x 0 ) such that x 0 ∈ Γ supp t 0 [w - ε ] := ∂Ω supp t 0 [w - ε ] then the equality above implies lim (t,x)→(t 0 ,x 0 ) ∇(w - ε ) m (t, x) = 0 R N ,
and conditions (i) and (iii) of Lemma 3.3 are checked for the sub-solution.

On the other hand, concerning the super-solution w + ε , note that ξ := u 0 (Φ(0, t, x)) + ε 2 C ⋆ (e µ(εM )t/ε -1) is positive. Therefore the cubic profile of fδ shows that, for t > 0,

Ω supp t [w + ε ] = Ω Γ supp t [w + ε ] := ∂Ω supp t [w + ε ] = ∂Ω .
Recall that u 0 = 0 in a neighborhood V of ∂Ω; if x is sufficiently close to ∂Ω, Φ(0, t, x) lives in V for all t ∈ [0, t ε ] (with ε > 0 sufficiently small). Therefore (4.12) shows that w + ε is independent on x near ∂Ω and condition (iii) of Lemma 3.3 for the super-solution is checked (and condition (i) is obviously checked).

Then it remains to prove that

L ε [w - ε ] := (w - ε ) t -ε∆(w - ε ) m + ∇ • (w - ε ∇v ε ) - 1 ε f (w - ε ) ≤ 0 , in {(t, x) ∈ [0, t ε ] × Ω : w - ε (t, x) > 0} and that L ε [w + ε ] ≥ 0 in {(t, x) ∈ [0, t ε ] × Ω}.
We will only prove the latter inequality since the proof of the former is similar.

We compute

∂ t w + ε = 1 ε Y τ + ∂ ∂t [u 0 (Φ(0, t, x))] Y ξ + εµ(εM )C ⋆ e µ(εM )t/ε Y ξ ∇w + ε = ∇ x [u 0 (Φ(0, t, x))] Y ξ ∇ [(w + ε ) m ] = ∇ x [u 0 (Φ(0, t, x))] (Y m ) ξ ∆ [(w + ε ) m ] = |∇ x [u 0 (Φ(0, t, x))] | 2 (Y m ) ξξ + ∆ x [u 0 (Φ(0, t, x))] (Y m ) ξ
, where the function Y and its derivatives are taken at the point

(τ, ξ ; δ) := (t/ε, u 0 (Φ(0, t, x)) + ε 2 C ⋆ (e µ(εM )t/ε -1) ; εM ) . Note that ∂ ∂t [u 0 (Φ(0, t, x))] = ∇u 0 (Φ(0, t, x)) • ∂Φ ∂t 2 (0, t, x) ,
and that

∇ x [u 0 (Φ(0, t, x))] = (D 3 Φ(0, t, x)) T ∇u 0 (Φ(0, t, x)) ,
with (D 3 Φ(t 1 , t 2 , x 3 )) T the transpose of the Jacobian matrix of Φ w.r.t. the third variable. Therefore, using f (w

+ ε ) = f (w + ε ) = fεM (Y ) -εM and the equation Y τ = fεM (Y ), we infer that L ε [w + ε ] = E 1 + E 2 + εY ξ E 3
, where

E 1 = M + Y ∆v ε E 2 = ∇u 0 (Φ(0, t, x)) • ∂Φ ∂t 2 (0, t, x) + D 3 Φ(0, t, x)∇v(t, x) Y ξ E 3 = C ⋆ µ(εM ) e µ(εM )t/ε + (D 3 Φ(0, t, x)) T ∇u 0 (Φ(0, t, x)) • ∇v ε 1 (t, x) -∆ x [u 0 (Φ(0, t, x))] (Y m ) ξ Y ξ -|∇ x [u 0 (Φ(0, t, x))] | 2 (Y m ) ξξ Y ξ .
We note that, for ε > 0 sufficiently small, δ = εM ∈ (-δ 0 , δ 0 ) and that, in the range 0

≤ t ≤ t ε = ε| ln ε|, ξ = u 0 (Φ(0, t, x)) + ε 2 C ⋆ (e µ(εM )t/ε -1) ∈ (-C 0 , C 0 ) ,
so that estimates of Lemma 4.2 on Y (τ, ξ; δ) will apply.

Since we have chosen

M ≥ C 0 ∆v ε L ∞ (Q T )
, E 1 ≥ 0 holds. Moreover, (4.11) implies E 2 = 0. In the sequel we denote by C various positive constants which may change from place to place but do not depend on ε. From Lemma 4.2 (ii)-(iv) we see that

(Y m ) ξ Y ξ = |mY m-1 | ≤ C and that (Y m ) ξξ Y ξ ≤ m(m -1)|Y m-2 Y ξ | + mY m-1 Y ξξ Y ξ ≤ C + C(e µ(εM )t/ε -1) , since m ≥ 2. Hence E 3 ≥ (C ⋆ µ(εM ) -C)e µ(εM )t/ε -C .
Since µ(εM ) → 1 as ε → 0, by choosing C ⋆ ≫ C we see that E 3 ≥ 0 for all ε > 0 small enough.

Recalling that Y ξ > 0, we get L ε [w + ε ] ≥ 0 and the lemma is proved.

We are now in the position to prove (4.1) and (4.2).

Proof. Let η ∈ (0, 1/2) be arbitrary. Then [START_REF] Alfaro | The singular limit of a chemotaxis-growth system with general initial data[END_REF]Lemma 3.11] provides a constant C Y > 0 such that, for all ε > 0 small enough, for all ξ ∈ (-C 0 , C 0 ),

Y (| ln ε|, ξ ; ±εM ) ≤ 1 + η ; (4.15) if ξ ≥ C Y ε then Y (| ln ε|, ξ ; ±εM ) ≥ 1 -η . (4.16)
By setting t = t ε = ε| ln ε| in (4.14), we obtain

Y | ln ε|, u 0 (Φ(0, t ε , x)) -C ⋆ ε 2 e µ(-εM )| ln ε| + C ⋆ ε 2 ; -εM + ≤ u ε (t ε , x) ≤ Y | ln ε|, u 0 (Φ(0, t ε , x)) + C ⋆ ε 2 e µ(εM )| ln ε| -C ⋆ ε 2 ; εM + . (4.17)
Therefore, the assertion (4.1) of Theorem 4.1 is a direct consequence of (4.17) and (4.15). Next we prove (4.2). Note that in view of (4.6), we have εe µ(-εM )| ln ε| → 1 as ε → 0. Therefore, for ε > 0 small enough (since

Y ξ > 0), u ε (t ε , x) ≥ Y | ln ε|, u 0 (Φ(0, t ε , x)) - 3 2 C ⋆ ε + C ⋆ ε 2 ; -εM + . (4.18) Choose M 0 ≫ 0 so that M 0 ε -3 2 C ⋆ ε + C ⋆ ε 2 ≥ max(C Y ε, a(-εM ))
, with C Y as in (4.16). Then, for any x ∈ Ω such that u 0 (Φ(0, t ε , x)) ≥ M 0 ε, we have

u 0 (Φ(0, t ε , x)) - 3 2 C ⋆ ε + C ⋆ ε 2 ≥ C Y ε .
Combining this, (4.18) and (4.16), we see that

u ε (t ε , x) ≥ 1 -η .
This completes the proof of (4.2).

Proof of (4.3)

Let us recall that a finite speed of propagation property, as is (4.3), is proved in [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF]: the authors construct a super-solution using a related travelling wave U of minimal speed, and they obtain an O(ε| ln ε|) estimate of the thickness of the transition layers. We borrow some ideas from this paper but, in order to obtain the improved O(ε) estimate, we again use the solution Y of the ordinary differential equation (4.8).

Let z ε be the solution of the Cauchy problem (recall that v ε (t, x) has been extended on [0, ∞) × R N in Remark 1.4)

(Q ε )    z t = ε∆(z m ) -∇ • (z∇v ε ) + 1 ε f (z) in (0, ∞) × R N z(0, x) = u 0 (x) in R N .
Lemma 4.4 (Super-solutions for (Q ε ) for small times). Choose K ≥ 1 and C ⋆ > 0 appropriately. For all x 0 ∈ ∂Ω 0 = ∂Supp u 0 , denote by n 0 the unit outward normal vector to ∂Ω 0 at x 0 . For t ≥ 0, x ∈ R n , define the function

z + ε (t, x) := K Y t ε , -(Φ(0, t, x) -x 0 ) • n 0 + ε 2 C ⋆ (e µ(εM )t/ε -1) ; εM + .
Here Y (τ, ξ ; δ) is the solution of (4.8), µ(δ) the slope defined in (4.7), Φ(t 1 , t 2 , x 3 ) the flow defined in (2.2) and M is chosen such that, for ε > 0 small enough, M ≥ C 0 ∆v ε L ∞ (Q T ) . Then, for all ε > 0 small enough,

u 0 (x) ≤ z + ε (0, x) for all x ∈ R N , (4.19)
and

L ε [z + ε ] := (z + ε ) t -ε∆(z + ε ) m + ∇ • (z + ε ∇v ε ) - 1 ε f (z + ε ) ≥ 0 , (4.20)
in the domain [0,

t ε ] × R N .
Proof. Recall that Ω 0 is convex. Therefore, in view of (1.1), we can choose K ≥ 1 sufficiently large so that, for all x 0 ∈ ∂Ω 0 and all x ∈ Ω 0 ,

u 0 (x) ≤ -K(x -x 0 ) • n 0 . (4.21)
We prove (4.19). If Φ(0, 0, x) = x / ∈ Ω 0 this is obvious since u 0 (x) = 0. Let us now assume that Φ(0, 0, x) = x ∈ Ω 0 . Since Ω 0 is convex, it lies on one side of the tangent hyperplane at x 0 so that (

x -x 0 ) • n 0 < 0. Recall that Y (0, ξ ; δ) = ξ so that z + ε (0, x) = -K(x -x 0 ) •
ε [z + ε ] = Kf (Y ) -f (KY ) + εK(M + Y ∆v ε ) +ε 2 KY ξ C ⋆ µ(εM ) e µ(εM )t/ε -D 3 Φ(0, t, x)n 0 • ∇v ε 1 (t, x) +∆ x [Φ(0, t, x) • n 0 ]K m-1 (Y m ) ξ Y ξ -|∇ x [Φ(0, t, x) • n 0 ]| 2 K m-1 (Y m ) ξξ Y ξ . Note that Kf (Y ) -f (KY ) = K(K -1)Y 2 ≥ 0.
Then, by using similar arguments to those in the proof of Lemma 4.3, we see that

L ε [z + ε ] ≥ 0, if C ⋆ > 0 is sufficiently large.
We now prove (4.3).

Proof. We shall first prove that property (4.3) holds for z ε the solution of the Cauchy Problem (Q ε ). Recall that a(δ) is the unstable zero of fδ = f + δ so that a(εM ) < 0. Moreover, in view of (4.5) and (4.6), we can choose M 0 > 0 large enough so that, for ε > 0 small enough,

-M 0 ε + C ⋆ εe (µ(εM )-1)| ln ε| -C ⋆ ε 2 < a(εM ) . For x ∈ Ω such that dist(Φ(0, t ε , x), Ω 0 ) ≥ M 0 ε, we choose x 0 ∈ ∂Ω 0 such that dist(Φ(0, t ε , x), Ω 0 ) = Φ(0, t ε , x) -x 0 and define z +
ε as in Lemma 4.4. It follows from Lemma 4.4 and the comparison principle that, for all ε > 0 small enough, all (t, x) ∈ [0,

t ε ] × R N , 0 ≤ z ε (t, x) ≤ z + ε (t, x) . (4.22)
Since, for t = t ε = ε| ln ε|,

-(Φ(0, t ε , x) -x 0 ) • n 0 + ε 2 C ⋆ (e µ(εM )t ε /ε -1) = -Φ(0, t ε , x) -x 0 + C ⋆ εe (µ(εM )-1)| ln ε| -C ⋆ ε 2 ≤ -M 0 ε + C ⋆ εe (µ(εM )-1)| ln ε| -C ⋆ ε 2 < a(εM ) , it follows from Lemma 4.2 (i) that Y t ε ε , -(Φ(0, t ε , x) -x 0 ) • n 0 + ε 2 C ⋆ (e µ(εM )t ε /ε -1) ; εM < a(εM ) < 0 ,
and therefore z + ε (t ε , x) = 0, which in turn implies z ε (t ε , x) = 0. Hence (4.3) holds for z ε the solution of (Q ε ). Now, a straightforward modification of [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF]Corollary 4.1] shows that there exists T > 0 such that, for all ε > 0 small enough,

u ε (t, x) = z ε (t, x) ,
for all (t, x) ∈ (0, T ) × Ω. This proves (4.3) for u ε the solution of (P ε ). Lemma 5.2 (Sub-and super-solutions for the propagating front). Choose σ > 0 small enough so that

c * (m -1)D 2 (1 + 2σ) m-2 σ ≤ 1 2 , (5.12)
where D is the constant that appears in (5.9). Choose K ≥ 1. Then, if L > 0 is large enough, we have, for ε > 0 small enough,

L ε [u - ε ] ≤ 0 ≤ L ε [u + ε ] in (0, T ) × Ω , (5.13)
∂(u - ε ) m ∂ν supp t [u - ε ] = 0 on ∂Ω supp t [u - ε ] for all t ∈ [0, T ] , (5.14)
∂(u + ε ) m ∂ν supp t [u + ε ] = 0 on ∂Ω supp t [u + ε ] for all t ∈ [0, T ] .
(5.15)

Proof. Properties (5.14) and (5.15) follow from (U m ) ′ (0) = 0 (see (5.2)). We prove below that L ε [u + ε ] ≥ 0, the proof of L ε [u - ε ] ≤ 0 following the same lines. Note that we only need to consider the case d ε (t, x) ≤ εp(t) since, if d ε (t, x) > εp(t) then u + ε (t, x) = 0. Straightforward computations and equation (5.1) yield

ε(u + ε ) t = εq ′ U + (1 + q)(d ε t -εp ′ )U ′ ε∇u + ε = (1 + q)U ′ ∇d ε ε 2 ∆(u + ε ) m = (1 + q) m |∇d ε | 2 (-c * U ′ -U (1 -U )) + ε(1 + q) m ∆d ε (U m ) ′ ,
where arguments are omitted. Thus we get

εL ε [u + ε ] = E 1 + E 2 + E 3 ,
where

E 1 = U ′ (1 + q) d ε t -εp ′ + c * (1 + q) m-1 |∇d ε | 2 + ∇d ε • ∇v ε =: U ′ (1 + q)E ⋆ 1 E 2 = U -(1 + q) + (1 + q) m |∇d ε | 2 + U (1 + q) 2 -(1 + q) m |∇d ε | 2 + εq ′ E 3 = -ε(1 + q) m ∆d ε (U m ) ′ + ε(1 + q)∆v ε U .
In the sequel we define a(t) := 1 + q(t) and denote by C i various positive constants which do not depend on ε.

Since ∆v ε L ∞ (Q T ) is uniformly bounded w.r.t. ε > 0 (see Assumption 1.3), we deduce from (5.9) and (5. (5.17) Indeed, if -d 0 < d ε (t, x) ≤ εp(t), it follows from (5.7) that, for ε > 0 small enough, |∇d ε (t, x)| = 1. Next, if d ε (t, x) ≤ -d 0 , (5.3) implies that

0 < (1 -U ) d ε (t, x) -εp(t) ε ≤ (1 -U )(- d 0 ε ) ≤ Ce -β d 0 ε ,
and (5.17) holds for ε > 0 small enough. Therefore (5.16) and (5.17) imply

E 2 + E 3 ≥ U -a + a m + U a 2 -a m -ε(C 2 + C 3 )a m -εC 3 a + εq ′ .
(5.18)

Next, since

E ⋆ 1 = d ε t + c * |∇d ε | 2 + ∇d ε • ∇v -εp ′ + c * (a m-1 -1)|∇d ε | 2 + ε∇d ε • ∇v ε 1 ,
using (5.10), (5.9) and Assumption 1.3, we see that +c * (m -1)D 2 (1 + 2σ) m-2 q(t) + εCD .

E ⋆ 1 ≤ N |d ε (t, x)| -εp ′ (t) + c * (a m-1 -1)|∇d ε | 2 + ε∇d ε • ∇v ε 1 ≤ N |d ε (t,
The last inequality above comes from the fact that, for ε > 0 small enough, we have 0 ≤ q(t) ≤ σ(1 + εLe LT ) ≤ 2σ, which in turn implies that 0 ≤ a m-1 -1 ≤ (m -1)(1 + 2σ) m-2 q(t) .

(5. [START_REF] Ley | Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts[END_REF] In the following, we distinguish two cases. First, assume that 0 ≤ d ε (t, x) ≤ εp(t) so that, for ε > 0 small enough, In view of (5.12) we get

E ⋆ 1 ≤ ε e Lt (2N -
1 2 L) + 2N K + CD ≤ 0 ,
if L > 0 is sufficiently large. This implies that E 1 = aU ′ E ⋆ 1 ≥ 0. Now, assume that d ε (t, x) ≤ 0 so that d ε (t, x)εp(t) ε ≤ -K ≤ -1 .

(5.20)

Lemma 3 . 3 (

 33 Being sub-and super-solutions). Let u be a continuous nonnegative function in Q T . Define Ω supp t := {x ∈ Ω : u(t, x) > 0} Γ supp t := ∂Ω supp t , for all t ∈ [0, T ]. Suppose the family Γ supp := 0≤t≤T ({t} × Γ supp t

  2) that |E 3 | ≤ εC 3 (a m + a)U so that E 2 + E 3 ≥ Ua + a m + U a 2a m -εC 3 a m -εC 3 a + |∇d ε | 2 -1 a m (1 -U ) + εq ′ . (5.16)We claim that, for ε > 0 small enough,|(|∇d ε | 2 -1)(1 -U )| ≤ εC 2 .

  x)εp(t)| + ε(N p(t)p ′ (t)) + c * (a m-1 -1)D 2 + εCD ≤ N |d ε (t, x)εp(t)| + ε(N p(t)p ′ (t))

E ⋆ 1 ≤

 1 ε(2N p(t)p ′ (t)) + c * (m -1)D 2 (1 + 2σ) m-2 q(t) + εCD ≤ e -t/ε (-ε2N -1 + c * (m -1)D 2 (1 + 2σ) m-2 σ) +e Lt (ε2N -εL + εc * (m -1)D 2 (1 + 2σ) m-2 σL) + ε2N K + εCD .

The first author is supported by the French Agence Nationale de la Recherche within the project IDEE (ANR-2010-0112-01).

AMS Subject Classifications:

35K65, 35B25, 35R35, 92D25.

Acknowledgement. The first author is grateful to Hiroshi Matano for a fruitful discussion in Dresden.

The propagating front

The goal of this section is to construct efficient sub-and super-solutions that control u ε during the latter time range, when the motion of interface occurs. We begin with some preparations.

Materials

In the linear diffusion case (m = 1), it is well-known that the equation u t = ∆u + u(1u) admits travelling wave solutions with some semi-infinite interval of admissible wave speed. The same property holds for the nonlinear diffusion case, namely equation u t = ∆(u m )+u(1-u), m > 1. Nevertheless, it turns out that the travelling wave with minimal speed c * > 0 is both compactly supported from one side and sharp. In the following, U denotes the unique solution of

(5.1)

The travelling wave U is smooth outside 0 and

Moreover, there exist C > 0 and β > 0 such that the following properties hold.

)

for all z ≤ -1 .

(5.4)

For more details and proofs we refer the reader to [START_REF] Atkinson | Travelling wave solutions for some linear diffusion equation[END_REF], [START_REF] Biro | Stability of travelling waves for degenerate reactiondiffusion equations of KPP type[END_REF], [START_REF] Hilhorst | Interface dynamics of the Fisher equation with degenerate diffusion[END_REF], as well as to [START_REF] Malaguti | Travelling wavefronts in reactiondiffusion equations with convection effects and non-regular terms[END_REF] for related results.

Another ingredient is a "cut-off signed distance function" d ε (t, x) which is defined as follows. Let d ε = d ε,drift be the signed distance function to Γ ε,drift , the smooth solution of the free boundary problem (P 0 ε,drift ), namely

where dist(x, Γ ε,drift t

) is the distance from x to the hypersurface Γ ε,drift t . We remark that d ε = 0 on Γ ε,drift and that |∇ d ε | = 1 in a neighborhood of Γ ε,drift : there exists d 0 > 0 such that, for all ε > 0 small enough,

Next, let ζ(s) be a smooth increasing function on R such that

We then define the cut-off signed distance function

and that the equation of motion (P 0 ε,drift ) is recast as

(5.8) Moreover, there exists a constant D > 0 such that, for all ε > 0 small enough,

(5.9)

Finally, in view of (5.7) and (5.8), the mean value theorem provides a constant N > 0 such that, for all ε > 0 small enough,

(5.10)

Sub-and super-solutions

We define

where p(t) := -e -t/ε + e Lt + K q(t) := σ(e -t/ε + εLe Lt ) , and where U and d ε were defined in subsection 5.1. In the following lemma,

The above study for the case 0 ≤ d ε (t, x) ≤ εp(t) implies a fortiori that

| , and we deduce from (5.20) and (5.4) that

Summarizing, we obtain that, in any cases,

m ≥ 2, the inequality -a + a m + U a 2a m ≥ -a + a 2 = q + q 2 ≥ q holds. Therefore, using ≤ 1 + 2σ and substituting the expressions for q(t) and q ′ (t), we see that

if L > 0 is sufficiently large. This completes the proof of Lemma 5.2.

Proof of Theorem 2.2

By fitting the sub-and super-solutions for the generation into the ones for the motion, we are now in the position to prove our main result.

Let η ∈ (0, 1/2) be arbitrary. Choose σ that satisfies (5.12) and

By Theorem 4.1, there exists M 0 > 0 such that (4.1), (4.2) and (4.3) hold with the constant η replaced by σ/2. Recall that u ± ε are the sub-and supersolutions constructed in (5.11).

Lemma 6.1 (Ordering initial data).

There exists K > 0 such that for all K ≥ K, all L > 0, all ε > 0 small enough, we have

for all x ∈ Ω.

Proof. We first prove

If x is such that d ε (0, x) ≥ -Kε, this is obvious since the definition of U then implies u - ε (0, x) = 0. Next assume that x is such that d ε (0, x) < -Kε. Let us denote by d(t, x) the signed distance function to Γ t . Note that, in view of hypothesis (1.1), the mean value theorem provides the existence of a constant K0 > 0 such that

Moreover in view of the definition of Γ ε,drift 0 in (2.4) and the compactness of Γ 0 , there exists K 0 > 0 such that, for ε > 0 small enough,

Hence, if we choose K ≥ K 0 , we deduce from (6.5), (

In view of the definition of Γ ε,drift 0 in (2.4) and the compactness of Γ 0 , there exists

If x is such that d ε (0, x) ≥ K 1 ε then it follows from (6.7) and Theorem 4.1 (iii) that u ε (t ε , x) = 0, which proves (6.6). Next assume that x is such that d ε (0, x) < K 1 ε. Since U is non increasing we have

). Then (6.6) follows from (4.1) (with η replaced by σ/2).

We now prove Theorem 2.2.

Proof. We fix K ≥ 1 large enough so that Lemma 6.1 holds, and L > 0 large enough so that Lemma 5.2 holds. Therefore, from the comparison principle, we deduce that

for t > 0, (6.8) is enough to prove the convergence result, namely Corollary 2.3. We now choose C large enough so that In the sequel we prove (2.6).

Obviously, if ε > 0 is small enough, the constant map z + ≡ 1 + η is a super-solution. Therefore we deduce from Theorem 4.1 where we have successively used (6.1) and (6.10). In view of (6.8) this implies that u ε (t + t ε , x) ≥ 1η, for all 0 ≤ t ≤ Tt ε .

Finally we take x ∈ (Ω \ Ω ε,drift t ) \ N Cε (Γ ε,drift t ), i.e. d ε (t, x) ≥ Cε .

(6.13) Using (6.11) we see that, for ε > 0 small enough, d ε (t, x)εp(t) ≥ 0 so that u + ε (t, x) = 0, which, in view of (6.8) implies that u ε (t + t ε , x) = 0, for 0 ≤ t ≤ Tt ε .

This completes the proof of Theorem 2.2.


A Nikitin

J P Champion

Vl G Tyuterev

The MIRS computer package for modeling the rovibrational spectra of polyatomic molecules

The MIRS spectroscopic software for the modeling of ro-vibrational spectra of polyatomic molecules is presented. It is designed for the global treatment of complex band systems of molecules to take full account of symmetry properties. It includes ecient algorithms based on the irreducible tensor formalism. Predictions and simultaneous data tting (positions and intensities) are implemented as well as advanced options related to group theory algebra. Illustrative examples on CH3D, CH4, CH3Cl and PH3 are reported and the present status of data available is given. It is written in C++ for standard PC computer operating under Windows.

Introduction 2

Modeling principles

Spectroscopic investigations of polyatomic molecules play a role of primordial importance for atmospheric applications.

Nowadays, in order to model the dramatic eects of ro-vibrational perturbations on line positions and strengths revealed by modern high-resolution and high-sensitivity techniques, the analysis of complex interacting band systems (so-called polyads) becomes the norm rather than the exception. Decades ago, sophisticated models have been developed

and successfully applied to spherical tops in general and methane in particular [START_REF] Champion | Spherical top spectra[END_REF]. For this type of molecule no simple analytical formulation was available to describe the ro-vibrational energy levels beyond the zero order of approximation. Furthermore, strong resonances between bending and stretching vibrational modes prevent from fully perturbative approaches. These particular features explain why the modeling of spherical tops relied from the beginning on numerical methods based on specic formalisms. A full account for symmetry properties proves to be crucially important for such systems. Besides purely technical advantages, such developments promoted more general concepts suitable for a wider class of molecules. One of the main achievement was the introduction of the socalled vibrational extrapolation scheme allowing the construction of a unied transformed hamiltonian and related transformed transition moments successfully applied to numerous global analyses. Most of the results on CH 4 and CH 3 D are included in the HITRAN database [2].

In the present paper, we describe recent computational developments: the Windows implementation MIRS of the theoretical model published in [START_REF] Nikitin | Improved algorithms for the modeling of vibrational polyads of polyatomic molecules: Application to t-d, o-h, and c-3 upsilon molecules[END_REF] . It is especially designed for the treatment of complex band systems of molecules having symmetry properties (spherical and symmetric tops). The algorithms are based on the eective hamiltonian theory with extensive use of the irreducible tensor formalism. Predictions and global data tting (positions and intensities) are both implemented.

The present paper was primarily motivated by sharing the concern of the HITRAN database to give friendly access to detailed results of advanced spectra analyses of symmetric top molecules. One problem is that the irreducible tensor formalism implies a non-conventional nomenclature of the eective parameters. For a given model and order of approximation, tensorial and classical expansions already available for symmetric tops are equivalent. However, due to the high exibility of the tensorial approach, all symmetry allowed terms, sophisticated or not, are equally easy to implement. The selection of relevant terms is solely governed by physical or statistical considerations whereas in traditional approaches it interfers at least implicitly with intuitive arbitrary criteria. As a result the conversion of tensorial terms into conventional terms is made dicult not only because of the relative algebraic complexity of tensor calculations but also and principally because several tensorial terms have not yet been systematically considered in conventional formalisms. Except for simple systems like isolated vibrational states, this prevents from global conversions of complete sets of tensorial terms into complete sets of conventional terms. The theoretical investigation of the corresponding implicitly reduced hamiltonians remains not obvious in the general case. For searchers not familiar with tensorial calculus, the friendly graphical user interface of the MIRS package includes on-line documentation, and represents a powerful and exible tool to access spectroscopic data from a standard PC computer. For expert researchers, advanced options are included such as group theory algebra and commutator calculations needed to set up reduced forms of eective hamiltonians.

For brevity, in the present paper, only the basic features will be described in some detail. Recent results of the global analysis of the lower polyads of molecules of atmospheric interest (CH 3 D and CH 3 Cl) are reported to illustrate the reliability and originality of the features involved.

2 Modeling principles

Unied irreducible tensor method

As described in [START_REF] Champion | Spherical top spectra[END_REF], in most ro-vibrational quantitative studies, the solution of the Schrödinger equation is based on the construction of eective hamiltonians by introducing basis functions in which the hamiltonian matrix has a blockdiagonal form with respect to appropriate vibrational subspaces. There exist many ways (more or less equivalent) of dening basis functions and operators. However, in view of limitations of the standard calculation resources presently available on PC computers, considerations of computational eciency in setting up theoretical models are of importance. As for previously reported softwares [START_REF] Wenger | Spherical top data system (stds) software for the simulation of spherical top spectra[END_REF][START_REF] Wenger | Highly-spherical top data system (htds) software for spectrum simulation of octahedral xy6 molecules[END_REF][START_REF] Wenger | C4(v) top data system (c4(v) tds) software for infrared spectrum simulation of xy(5)z symmetric molecules[END_REF][START_REF] Rotger | Spectroscopy of xy(2)z(2) (c2v) molecules: A tensorial formalism adapted to the o(3) superset of t-d superset of c-2v chain. application to the ground state of so2f2[END_REF] vibrational operators are constructed in tensor form by recursive coupling of creation and annihilation operators associated to the normal modes of the molecule. In MIRS, a specic coupling scheme associated to a binary tree is applied directly to an arbitrary number of interacting vibrational modes and to arbitrarily high polyads. The construction of vibrational basis functions is achieved consistently from the action of creation operators on the vacuum function. This method is not only satisfying conceptually but also quite ecient for the computer calculation of matrix elements and commutators. The ro-vibrational terms are easily generated by further tensor couplings. Once the vibrational polyad scheme is dened for a given molecule belonging to a given point group, the MIRS program determines automatically all symmetry allowed terms up to the desired order of approximation. The standard Amat and Nielsen classication scheme [8] is used in which the order of terms of the type H mn ∼ r n J m is m + n=2 [START_REF] Aliev | Higher-order eects in the vibration-rotation spectra of semirigid molecules[END_REF]. For transformed transition moment, terms of the type μ mn are of order m + n=1. In general, a complete nomenclature of ro-vibrational terms appears rather cumbersome when all tensorial indices are displayed. In many cases, most of the couplings are trivial and the following simplied notation can be used unambiguously

Ω(K, κΓ) {n 1 . . . n v }{m 1 . . . m v }[Γ n , [Γ m ]] . (1)
In the rotational part (rst group of indices), Ω is the power in angular momentum components J a , K is the tensor rank relative to the space rotation group 0(3), κ refers to the projection axis Oz, and Γ designates the common symmetry in the molecular point group of the rotational and vibrational operators. In the vibrational part (last group of indices), the n i s (resp. m j s) are the powers of the creation operators a + i (resp. annihilation operators a j ), and optionally Γ n (resp. Γ m ) species the internal coupling species when non-trivial. For instance, 1(1,0A 2 ) 002000 000011 designates a term from H 14 describing the third-order Coriolis interaction between the v 3 = 2(A 1 ) and v

5 = v 6 = 1(A 2 ) states of an XY Z 3 (C 3v ) molecule.

Model denition and quantum numbers

The polyad structure reects possible vibrational quasi-degeneracies which constitutes the key for dening eective hamiltonians and consequently the theoretical model for the calculation of spectra. For a given molecule, it is essentially governed by the number of the vibrational modes and their fundamental frequencies. All harmonic oscillator energies from zero to the maximum desired value are calculated using the fundamental frequencies. This determines a reference energy level pattern which is not actually used in accurate calculations, but serves for a preliminary polyad denition only. This allows a certain exibility to include or exclude some states according to estimations of coupling matrix elements. First, starting from zero (Ground State = Polyad 0) upwards, MIRS denes automatically the initial polyad structure according to the reference pattern and applying a threshold criterium set by the user: all vibrational states for which the energy distances are closer than a given threshold value have to be gathered in one polyad. Consequently the energy gap between subsequent polyads should be larger than this value. Of course, the strength of a perturbation depends not only on the proximity of the zero order energy levels but also on the corresponding coupling matrix elements. It is possible, at the second step, to take into account such physical considerations indirectly by playing with the reference level pattern: for instance the user can articially increase the distance between reference levels in the case of weak interactions or equivalently decrease the gap between reference levels corresponding to strongly coupled but distant states. These shifts do not introduce an error in nal spectrum calculations, which use true energy values. Illustrations of such exibility are given in subsequent sections. In the frame of this approach the polyad structure of the molecule determines an eective hamiltonian expansion which serves as a theoretical model for ro-vibrational bound state calculations for the considered electronic state. The so-called vibrational extrapolation scheme is automatically implemented through the formal polyad expansion (see [START_REF] Champion | Spherical top spectra[END_REF] for details).

H = H P0 + H P1 + H P2 + . . . . (2)
The polyad structure also determines the construction of the eective transition moments in a similar way. It determines as well the quantum numbers used to label the levels and transitions. Strict quantum numbers are related to the usual invariants. They are: the polyad number P , the rotational quantum number J and the ro-vibrational symmetry species C. Within each P, J, C block the eigenvalues sorted according to increasing energies are numbered by the index n from 1 to the dimension of the block. In addition, approximate quantum numbers are generated on the basis of the eigenvector analysis. For instance, the normal mode assignment and the K value (in the case of symmetric top molecules) are assigned according to the largest contribution of the wavefunction decomposition into the zero-order basis functions. These approximate quantum numbers should correspond to quantum numbers of the zero-order harmonic-oscillator + rigid-rotor approximation under the condition that corresponding physical quantities are kept near-invariant under the eect of perturbations. The latter are usually accounted for by successive contact transformations from the initial molecular hamiltonian to the eective one [START_REF] Brown | Methane line parameters in hitran[END_REF], see [START_REF] Champion | Spherical top spectra[END_REF] and [START_REF] Zhilinskii | Method of irreducible tensorial operators in the theory of molecular spectra[END_REF] for more detail. In strongly perturbed situations, these approximate labels may become physically irrelevant.

Options and limitations

In its present form, the MIRS package contains the basic coupling coecients to work with the C 3v , D 3h , O h and T d point groups as well as with C 1 (no symmetry). Even though, in principle, the recursive algorithms imply no limitations in angular momentum values and coupling of elementary operators, the default coecient tables included in the package are limited to meet standard needs. In particular the angular momentum limit is 80. Note that for the cubic groups the G coecients are calculated numerically and are identical to those included in the STDS package [START_REF] Wenger | Spherical top data system (stds) software for the simulation of spherical top spectra[END_REF][START_REF] Wenger | Highly-spherical top data system (htds) software for spectrum simulation of octahedral xy6 molecules[END_REF]. In contrast with STDS, the number and symmetry species of normal modes are arbitrary. This means that, when physically justied, some normal modes can be omitted for the purpose of simplifying the model. Orders of expansion are also unlimited in principle, but limited in the package to the eighth order to keep the size of the coecient tables within reasonable bounds. As described previously the MIRS program oers a large exibility in dening the polyad structure on the basis of simple key values.

Finally, all symmetry allowed terms in the transformed Hamiltonian, vibrationally diagonal or not, and the transition moments (only dipolar transitions are implemented to date) are automatically generated to the desired order.

Quite large sets of eective parameters can be generated by this procedure which means that limitations arise actually from theoretical rather than from program considerations.

Basic applications

The rst application of the MIRS package is the diusion of the results of spectroscopic analyses in a exible way.

As a matter of fact, from the les containing the model features and the eective parameter values, dierent kinds of prediction can be made by non-expert users.

Prediction of energy levels and transitions

Complete les of energy levels can be calculated using standard menu commands. Each record contains the energy value, the set of strict quantum numbers (see previous section), other approximate quantum numbers, depending on the symmetry point group and optional informations derived from the eigenvector coecients to determine the approximate quantum numbers. Details on the experimental measurements tted are also provided. The number and the rms deviations of assigned transitions are given together with the corresponding averaged observed minus calculated residual. Similar details are provided for transitions (calculated positions and intensities) using standard options. The partition function may be derived from the calculated levels involved in the model or set to an appropriate external estimate. Several output formats including HITRAN [START_REF] Brown | Methane line parameters in hitran[END_REF] and GEISA [START_REF] Jacquinet-Husson | The 1997 spectroscopic geisa databank[END_REF] can be selected. The temperature and intensity threshold can be adjusted to meet specic needs for applications.

Present status of data available

The package presently available on our Web Site 1 includes tutorial examples as well as complete projects related to recent analyses.

1 http://icb.u-bourgogne.fr/OMR/SMA/SHTDS/MIRS.html

The most simple example is the sixth-order model for the ground-state and the bending dyad of 12 CH 4 . In this case, the vibrational coupling scheme coincides exactly with the one used in the STDS program. It means that the eective parameter sets are identical in MIRS and STDS providing a good validation of both programs. Unfortunately, from the next polyad upwards even though the models are perfectly equivalent and include the same number of parameters at a given order of approximation, the two parameter sets are not in one to one correspondence. A similar system is illustrated with the sixth order model for the ground-state and the ν 2 /ν 4 dyad of PH 3 .

The model for the lower two polyads (triad and nonad) of CH 3 D represents the most complex band systems of a symmetric top studied so far. Some 10,000 line positions and 2400 line intensities have been tted to an accuracy close to the experimental precision (about 0.001 cm =1 for positions and 4% for intensities) using a common set of 441 eective hamiltonian parameters to describe 13 vibrational states (ground, triad, nonad) corresponding to 39 bands (6 fundamental, 3 overtone, 3 combination, and 27 hot bands). The intensities of the 9 bands involved in the nonad-ground state system were tted to the second order of approximation using 83 eective dipole moment parameters.

Finally, very recent results of the global analysis of the lower polyads of 12 CH 3 35 Cl and 12 CH 3 37 Cl are included.

A preliminary analysis of the infrared spectrum in the region from 0 to 1800 cm =1 was performed using 125 eective parameters for both isotopomers to model 12 sets of transitions (6 cold bands, 3 hot bands and 3 pure rotational systems). The precision on positions is of the order of 0.000 2 cm =1 .

Advanced applications

The above results have been obtained using the advanced tools included in MIRS for the modeling and the t of experimental data. Various options are available and described in the on-line documentation of the downloadable package. Only the basic ideas will be reported here.

Setting up new models

By default, MIRS is set to built partially transformed hamiltonian matrices and transition moments according to the vibrational extrapolation scheme. This means that for a given molecule a common set of eective hamiltonian parameters is set up to t and predict the subsequent vibrational polyads. The same principle applies for transition moment parameters. For instance a band system from the ground state to a given polyad and all the corresponding hot band systems are described by a common set of eective parameters. Such features are generated automatically by the program and controlled by a few basic modeling parameters entered through the window menu reproduced in Fig. 1.

Figure 1. Graphical user interface to enter the modeling parameters: the example of CH3Cl.

The modeling parameters are (i) the symmetry point group of the molecule, (ii) the harmonic frequencies of the desired normal modes, (iii) the corresponding symmetry species, (iv) the frequency gap used for grouping the subsequent vibrational sublevels into relevant vibrational polyads. Note that in the modeling procedure the frequencies can be entered in arbitrary units and do not need to match exactly the physical frequencies in order to provide exibility.

For example, the modeling of the CH 3 D molecule was performed by setting the six fundamental frequencies to 2, 2, 1, 2, 1 and 1 respectively and the frequency gap to 0.1. This yielded the triad and nonad structure described in [12]. For CH 3 Cl the situation was more complex and the fundamental frequencies were set closer to the actual values: 2968, 1354, 733, 3039, 1452 and 1018, respectively with a frequency gap of 200 (Fig. 1) [START_REF] Nikitin | New measurements and global analysis of chloromethane in the region from 0 to 1800 cm(-1)[END_REF]. In all cases, the resulting polyad structure can be displayed for checking and modifying purposes.

The nuclear spin statistical weights involved in intensity calculations are also manually introduced at this stage.

Once the vibrational model is set up, the polyads and the transitions to consider are entered through another window menu examplied in Fig. 2. For each polyad the order of the hamiltonian expansion and the maximum value of J has to be entered. Similarly, for each type of transitions the order of the dipole moment expansion has to be specied. Thanks to its intrinsic exibility MIRS can also be used to create quantum model hamiltonians not necessarily connected to real molecules. For instance, various systems involving coupled harmonic or anharmonic oscillators can be simulated numerically for further theoretical investigations.

Experimental data tting (positions, intensities)

MIRS oers standard options for non-linear least-squares t of spectroscopic data (positions and intensities). In coherence with the vibrational extrapolation scheme well suited for global analyses, the program is set to t simultaneously all transitions involved in a given project. As far as possible the physical quantities tted are directly the measured transition frequencies and intensities. It is also possible to include indirect measurements such as observed energy levels. Note that, due to their high-sensitivity and high-resolution, modern spectroscopic techniques provide wealth of transitions so that many energy levels are involved in several observed transitions. In our approach, the direct simultaneous t of such transitions indirectly accounts for the so-called combination dierences (see Fig. 3 of [START_REF] Tyuterev | Reduced eective-hamiltonians for degenerate excited vibrational-states of tetrahedral molecules -application to 2-nu-2,nu-2+nu-4 and 2-nu-4 of ch4[END_REF]).

The t of intensities is also achieved using a non-linear least-squares procedure with an adequate weighting already described in [START_REF] Wenger | Spherical top data system (stds) software for the simulation of spherical top spectra[END_REF][START_REF] Wenger | Highly-spherical top data system (htds) software for spectrum simulation of octahedral xy6 molecules[END_REF]. Note that in our approach a rigorous coherence is de facto applied between intensity and position ts.

The development of new algorithms and softwares was in fact justied by the increasing complexity of the problems to solve. In particular, the large number of adjustable parameters involved in global analyses required specic features in tting procedures.

Advanced features for spectroscopic non-linear least-squares

In complex situations, it is dicult to select a priori the relevant parameters among a large number of candidates at a given order of approximation. Statistical correlations arise not only from possible deciencies of the coverage by experimental data but also from essential colinearities in eective hamiltonians [START_REF] Tyuterev | Parameters of reduced hamiltonian and invariant parameters of interacting e and f2 fundamentals of tetrahedral molecules -nu-2 and nu-4 bands of (ch-2)-c-4 and(sih-28)-s-4[END_REF][START_REF] Tyuterev | Reduced eective-hamiltonians for degenerate excited vibrational-states of tetrahedral molecules -application to 2-nu-2,nu-2+nu-4 and 2-nu-4 of ch4[END_REF][START_REF] Lobodenko | Reduced eective hamiltonian for coriolis-interacting nn and nt fundamentals of c3v molecules[END_REF]. To help selecting adjustable parameters at any stage of the iterative assignment procedure special options of MIRS allow to run series of iterations with automatic introduction of new parameters or elimination of statistically unsignicant parameters among pre-selected terms specied by the user.

Another diculty is encountered when approximate quantum numbers are not physically meaningful and stable.

As mentioned previously, in MIRS, the assignment of observed to calculated data is based on the strict quantum numbers P , J and C (see previous section) and the numbering index n within each P, J, C block. In some cases close eigenvalues may permute their numbering indices from one iteration to the next causing undesirable assignment jumps. MIRS allows to remove temporarily from the t the assigned levels capable of such behaviour by detecting anomalous variations of the scalar product of the corresponding eigenvectors from one iteration to the next.

The t of intensities is also subject to specic problems. In particular the signs of the transition parameters cannot be completely determined from the t of data. Some of the relative signs may be correlated with those of the interaction hamiltonian parameters. To help in such situations, MIRS can be set to run series of iterations to test the relative signs of pre-selected terms.

However, despite all the above advanced options, it should be emphasized that in any case a purely automatic selection of adjustable terms for positions or intensities is hopeless. All procedures have to rely on physical considerations.

Group theory algebracommutator calculation

MIRS can also be used by expert users to generate various types of coupling coecients as well as commutators of tensor operators in algebraic or numerical formats. Such tools are useful for the theoretical investigation of the fundamental properties of eective hamiltonians and the denition of reduced forms of eective hamiltonians [START_REF] Tyuterev | Parameters of reduced hamiltonian and invariant parameters of interacting e and f2 fundamentals of tetrahedral molecules -nu-2 and nu-4 bands of (ch-2)-c-4 and(sih-28)-s-4[END_REF][START_REF] Nikitin | New measurements and global analysis of chloromethane in the region from 0 to 1800 cm(-1)[END_REF]12]. The present version of the package includes typical sample outputs related to S-generators of contact transformations involved in the eective hamiltonian for the triad of CH 3 D.

Getting and installing

The MIRS package can be downloaded freely from our Web site 2 . It consists of a self-extractive archive containing various tutorial examples and complete projects on CH 3 D and CH 3 Cl. MIRS written in C++ includes all necessary executable les for Windows 95 and higher. The CPU and disk space requirements obviously depend on the complexity of the project. The package itself requires a minimum of 15 MB of free disk space. The tables of coupling coecients occupy up to 150 MB when the highest J value (80) is involved. All examples and projects included in the package can be executed using a computer with 256 MB RAM and 500 MB of free disk space. Typical execution times using a processor at 2 GHz are about 10 min to build the CH 3 D project (all matrix element and prediction les) and less than 15 min to build each of the CH 3 Cl projects.

Standard features and options are directly accessible from the tutorial projects in which all necessary modeling les are included. The Help menu can be accessed at any stage of the project building and subsequent calculations.

Refer to the Web site for complete installation and running instructions.

Conclusion and perspectives

The main features of the MIRS software for modeling the ro-vibrational spectra of polyatomic molecules have been about to be submitted. The MIRS package can be used at dierent levels of expertise. Basically it can be used simply to generate lists of calculated energy levels and transitions from eective parameters. It is especially designed for the modeling and the global analysis of complex vibration rotation spectra (dipolar transition). It may also be used to simulate quantum model hamiltonians. Future developments and results of analyses in progress (higher vibrational polyads of CH 3 Cl) will be made available from the above mentioned Web Site.

Figure 2 .

 2 Figure 2. Graphical user interface to set up model properties. The example of CH3Cl.

Figure 3 .

 3 Figure 3. Graphical user interface for calculation options.

  described. They correspond to the version of the package presently downloadable from the Web site http://icb.ubourgogne.fr/OMR/SMA/SHTDS/MIRS.html. It includes all executable les, tutorial examples as well as complete results of recent analyses of infrared spectra of CH 3 D already transferred into the HITRAN database and CH 3 Cl

http://icb.u-bourgogne.fr/OMR/SMA/SHTDS/MIRS.html

Acknowledgements

Part of the work was supported by the Institut National des Sciences de l'Univers under the frame of the Programme National de Chimie Atmosphérique and by CNRS (France)RFBR (Russia) under the frame of the Programme Intègré de Coopération Scientique No. 1572.

http://icb.u-bourgogne.fr/OMR/SMA/SHTDS/MIRS


The Countenance of Watteau

Hal Opperman

To cite this version:

Hal Opperman. The Countenance of Watteau. 1985. halshs-00587254 Watteau 1684-1721(Washington-Paris-Berlin, 1984-1985)

HAL OPPERMAN THE COUNTENANCE OF WATTEAU

Review of the exhibition and catalogue

AUTHOR'S FOREWORD

The following typescript, prepared in 1985, was conceived as a review article of the exhibition and accompanying catalogue Watteau 1684-1721(Washington-Paris-Berlin, 1984-1985). An edited version, reduced in length, was eventually published in Art Bulletin, LXX (1988), 354-359. The original typescript had entered into circulation by then, however, and has been cited on occasion by scholars of Watteau to the present day. This scan of the typescript is being made available in response to bibliographical enquiries, old photocopies having become increasingly difficult to find.

Hal Opperman Professor Emeritus of Art History University of Washington March 2011

Recommended Citation: Hal Opperman, "The Countenance of Watteau," review of the exhibition and catalogue Watteau 1684-1721(Washington-Paris-Berlin, 1984-1985). Unpublished typescript, 1985.