Mahdi Boukrouche
email: mahdi.boukrouche@univ-st-etienne.fr

Imane Boussetouan
email: imane.boussetouan@univ-st-etienne.fr

Laetitia Paoli
email: laetitia.paoli@univ-st-etienne.fr

Unsteady 3D-Navier-Stokes System with Tresca's Friction Law

Keywords: Navier-Stokes system, Tresca's friction law, variational inequality, penalty method

Motivated by extrusion problems, we consider a non-stationary incompressible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part. We construct a sequence of approximate solutions by using a regularization of the free boundary condition due to friction combined with a particular penalty method, reminiscent of the "incompressibility limit" of compressible fluids, allowing to get better insights into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem.

Introduction

Fluid flow problems are involved in several physical phenomena and play an important role in many industrial applications. The fundamental model in fluid mechanics is the well-known Navier-Stokes system for incompressible viscous fluids which has been intensively studied during the last 78 years. Since the pioneering work of J. Leray [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] in 1934, the mathematical analysis of this problem has performed significant progresses: we can mention here only few selected references [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF][START_REF] Girault | Finite element approximation of the Navier-Stokes equations[END_REF][START_REF] Temam | Navier-Stokes equations, theory and numerical analysis[END_REF][START_REF] Boyer | Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF][START_REF] Foias | Navier-Stokes equations and turbulence[END_REF][START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems[END_REF]. Nevertheless it is still a very active research field, from both the theoretical point of view and the numerical point of view (see for instance the very recent research articles [START_REF] Bemelmans | Fluid flows around floating bodies, I: the hydrostatic case[END_REF][START_REF] Biryuk | Bounds on Kolmogorov spectra for Navier-Stokes equations[END_REF][START_REF] Goulart | Global stability analysis of fluid flows using sum-of-squares[END_REF][START_REF] Zhang | An augmented Lagrangian approach to simulating yield stress fluid flows around a spherical gas bubble[END_REF]).

Motivated by extrusion problems we consider in this paper a non-stationary incompressible 3D fluid flow with a temperature dependent viscosity. As usual for this kind of problems the extrusion device is composed of an upper fixed part and a lower moving part. Several experiments have shown that the classical adhesion condition between the fluid and the lower moving part of the boundary of its domain is not satisfied and the real behavior seems to be governed by some friction condition of Tresca's type [START_REF] Hervet | Flow with slip at the wall: from simple to complex fluids[END_REF] [START_REF] Baffico | Existence result for a fluid structure interaction problem with friction type slip boundary condition[END_REF].

More precisely, let ω be a non empty open bounded subset with a Lipschitz continuous boundary, of R d-1 for d = 2, 3. We denote by Ω ⊂ R d the domain of the flow given by

Ω = (x ′ , x d ) ∈ R d : x ′ ∈ ω, 0 < x d < h(x ′ ) , where x ′ = (x 1 , . . . , x d-1 ) ∈ R d-1 , x = (x ′ , x d ) ∈ R d . The boundary of Ω is ∂Ω = Γ 0 ∪ Γ L ∪ Γ 1 , where Γ 0 = {(x ′ , x d ) ∈ Ω : x d = 0}, Γ 1 = {(x ′ , x d ) ∈ Ω : x d = h(x ′ )}
and Γ L is the lateral boundary. We assume that h is a Lipschitz continuous function and there exist two real numbers h min and h max such that 0 < h min < h(x ′ ) < h max for all x ′ ∈ R d- 1 .

Let us emphasize that we do not introduce any restrictive assumption on the thickness of the domain. On the contrary to previous papers where only thin films where studied [START_REF] Rao | The effect of the slip boundary condition on the flow of fluids in a channel[END_REF][START_REF] Boukrouche | On the Navier-Stokes system in a thin film flow with Tresca free boundary condition and its asymptotic behavior[END_REF][START_REF] Boukrouche | Non isothermal lubrication problem with Tresca fluid-solid interface law, part I[END_REF], we can consider here general 3D geometries.

The fluid flow is described by the conservation of momentum

∂v ∂t + (v.∇)v = div(σ) + f in Ω × (0, τ ),
and the incompressibility condition

div(v) = 0 in Ω × (0, τ ),
where v is the velocity field of the fluid flow, f represents the density of body forces and σ is the stress tensor. We assume that the fluid is Newtonian, so

σ = -pI + 2µ(T )D(v),
where T depending on (x, t) ∈ Ω × (0, τ ), is the temperature field. Note that T stands for temperature but it will not appear as a variable of the problem, the time interval on which the equations are considered is : [0, τ ]. We do this for the main reason that we have generalised our work to a coupled problem (velocity-pressuretemperature) which is in final version, so here we give the regularity of T suitable for our coupled problem [START_REF] Boukrouche | Existence for non-isothermal fluid flows with Tresca's friction and Cattano's heat law[END_REF]. µ(T ) is the temperature dependent viscosity of the fluid, p is the pressure and D(v) is the strain rate tensor given by

D(v) = d ij (v) 1≤i,j≤d , d ij (v) = 1 2
∂v i ∂x j + ∂v j ∂x i 1 ≤ i, j ≤ d.
Hence v and p satisfy the Navier-Stokes system

∂v ∂t + (v.∇)v -2div µ(T )D(v) + ∇p = f in Ω × (0, τ ), (1)
div(v) = 0 in Ω × (0, τ ), (2)
with the initial condition

v(0, •) = v 0 in Ω. (3)
Let us now describe the boundary conditions. We denote by s : Γ 0 → R d-1 the shear velocity of the lower surface of the extrusion device at t = 0 and by sζ(t), with ζ : [0, τ ] → R such that ζ(0) = 1, its velocity at any instant t ∈ [0, τ ]. We introduce a function g : ∂Ω → R d such that

Γ L g.n dσ = 0, g = 0 on Γ 1 , g n = g • n = 0 and g τ = g -g n n = s on Γ 0 ,
where n = (n 1 , . . . , n d ) is the unit outward normal vector to ∂Ω. We denote here by u • w the Euclidean inner product of two vectors u and w and by |.| the Euclidian norm. We define respectively the normal and the tangential velocities on Γ 0 by

v n = v • n = v i n i , v T = v T i 1≤i≤d with v T i = v i -v n n i 1 ≤ i ≤ d
and the normal and the tangential components of the stress tensor on Γ 0 by

σ n = (σ • n) • n = σ ij n i n j , σ T = σ T i 1≤i≤d with σ T i = σ ij n j -σ n n i 1 ≤ i ≤ d.
Note that we will use the Einstein's summation convention throughout this paper. We assume that the upper surface of the extrusion device is fixed i.e.

v = 0 on Γ 1 × (0, τ ), (4)
the given velocity on the lateral boundary is the product

g(x)ζ(t) i.e. v = gζ on Γ L × (0, τ ), (5)
and the normal component of the velocity on the lower part of boundary is given by

v n = v • n = 0 on Γ 0 × (0, τ ). ( 6
)
The tangential velocity on Γ 0 × (0, τ ) is unknown and satisfies Tresca's friction law [START_REF] Duvaut | Les inéquations en mécanique et physique[END_REF] |σ

T | < ℓ ⇒ v T = (sζ, 0) |σ T | = ℓ ⇒ ∃λ ≥ 0 such that v T = (sζ, 0) -λσ T (7)
where ℓ : [0, τ ] × Γ 0 → R is the upper limit for the shear stress (i.e. ℓ is the Tresca's friction threshold).

The paper is organized as follows. In Section 2 we introduce the functional framework and the formulation of the problem as a variational inequality for the fluid velocity and pressure fields. In Section 3 we use a regularization of Tresca's functional to obtain a sequence of approximate problems (P ε ) of Navier-Stokes type.

A classical technique to study such problems is to choose divergence free testfunctions in order to "kill" the pressure terms then to solve the derived variational problem for the fluid velocity and to get finally the pressure by applying abstract results of functional analysis (see [START_REF] Temam | Navier-Stokes equations, theory and numerical analysis[END_REF][START_REF] Simon | On the existence of the pressure for solutions of the variational Navier-Stokes equations[END_REF][START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems[END_REF] for instance). The major drawback of this technique is that the pressure appears as a by product. In order to get better insights into the links between the velocity and pressure fields, we adopt in this paper another approach, reminiscent of the "incompressiblity limit" of compressible fluids.

More precisely, following an idea of J.L. Lions [START_REF] Lions | Some problems connected with Navier-Stokes equations[END_REF], we relax the divergence free condition and we propose a sequence of penalized problems (P δ ε ). In Section 4 we establish the existence of solutions to this family of problems (P δ ε ) ε>0,δ>0 and we obtain some a priori estimates. Next in Section 5 we define a sequence of approximate pressures (p δ ε ) ε>0,δ>0 and we study its properties. By using functional spaces that are weaker in time than in space, we succeed in obtaining good enough uniform estimates with respect to the the parameters δ and ε. Then in Section 6 we use compactness arguments to pass to the incompressible limit as δ tends to zero and we show that the limit velocity and pressure fields are solutions to the problems (P ε ). Finally we pass to the limit as ε tends to zero and we get a solution to our original variational inequality.

Variational formulation of the problem

We denote by

H 1 (Ω) = H 1 (Ω) d , L 2 (Ω) = (L 2 (Ω)) d , H 1 2 (∂Ω) = (H 1 2 (∂Ω)) d , H 2 (Ω) = H 2 (Ω) d .
We assume that

f ∈ L 2 0, τ ; L 2 (Ω) , ℓ ∈ L 2 0, τ ; L 2 + (Γ 0 ) ∩ L ∞ 0, τ ; L ∞ + (Γ 0 ) , ζ ∈ C ∞ [0, τ ] with ζ(0) = 1, (8)
with

L 2 + (Γ 0 ) = L 2 (Γ 0 ; R + ) (respectively L ∞ + (Γ 0 ) = L ∞ (Γ 0 ; R + ) and τ > 0. The viscosity µ(T ) is a function of L ∞ 0, τ ; L ∞ (Ω)
depending on the temperature T , and there exist two real numbers µ * , µ * such that

0 < µ * ≤ 2µ(X) ≤ µ * ∀X ∈ R, (9)
and also there exists an extension of g to Ω, denoted by G 0 , such that

G 0 ∈ H 2 (Ω), div(G 0 ) = 0 in Ω, G 0 = g on Γ L , G 0 = 0 on Γ 1 , G 0n = 0 and G 0τ = s on Γ 0 . (10)
We introduce now the following functional framework

V 0 = ϕ ∈ H 1 (Ω) : ϕ = 0 on Γ L ∪ Γ 1 , ϕ n = 0 on Γ 0 ,
endowed with the norm of H 1 (Ω) and

V 0div = ϕ ∈ V 0 : div(ϕ) = 0 in Ω .
Moreover let

L 2 0 (Ω) = q ∈ L 2 (Ω) : Ω q dx = 0 .
We define the following applications

a(T ; •, •) : L 2 (0, τ ; V 0 ) × L 2 (0, τ ; V 0 ) → R (u, v) → a(T ; u, v) = τ 0 Ω 2µ(T )D(u) : D(v) dxdt = τ 0 Ω 2µ(T )d ij (u)d ij (v) dxdt, and
Ψ : L 2 0, τ ; L 2 (Γ 0 ) → R u → Ψ(u) = τ 0 Γ 0 ℓ|u| dx ′ dt.
We may observe that Ψ is convex continuous but not differentiable. Let b be the usual trilinear form given by b :

V 0 × V 0 × V 0 → R (u, v, w) → b(u, v, w) = Ω u i ∂v j ∂x i w j dx.
By definition of V 0 we have the identity

b(u, v, w) = -b(u, w, v) - Ω div(u)v • w dx ∀(u, v, w) ∈ V 0 × V 0 × V 0 . (11)
Moreover, using Korn's inequality [START_REF] Kondrat'ev | Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequailities[END_REF] and assumption [START_REF] Foias | Navier-Stokes equations and turbulence[END_REF], there exists α > 0 such that, for almost every t ∈ (0, τ ), we have

α u 2 H 1 (Ω) ≤ Ω 2µ(T )D(u) : D(u) dx ≤ µ * u 2 H 1 (Ω) ∀u ∈ V 0 . (12)
In order to deal with homogeneous boundary conditions on Γ L ∪ Γ 1 , we set v = v -G 0 ζ. The variational formulation of the problem (1)-( 7) is given by (see for example [START_REF] Duvaut | Les inéquations en mécanique et physique[END_REF] and [START_REF] Boukrouche | On the Navier-Stokes system in a thin film flow with Tresca free boundary condition and its asymptotic behavior[END_REF][START_REF] Boukrouche | Non isothermal lubrication problem with Tresca fluid-solid interface law, part I[END_REF])

Problem (P ) Find v ∈ L 2 0, τ ; V 0div ∩ L ∞ 0, τ ; L 2 (Ω) , ∂ v ∂t ∈ L 4 3 0, τ ; (V 0div ) ′ , p ∈ H -1 0, τ ; L 2 0 (Ω)
such that, for all ϕ ∈ V 0 and for all χ ∈ D(0, τ ), we have

d dt ( v, ϕ) , χ D ′ (0,τ ),D(0,τ ) + b( v, v, ϕ), χ D ′ (0,τ ),D(0,τ ) -p, div(ϕ) , χ D ′ (0,τ ),D(0,τ ) +a(T ; v, ϕχ) + Ψ( v + ϕχ) -Ψ( v) ≥ (f, ϕ), χ D ′ (0,τ ),D(0,τ ) -a(T ; G 0 ζ, ϕχ) - G 0 ∂ζ ∂t , ϕ , χ D ′ (0,τ ),D(0,τ ) -b(G 0 ζ, v + G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) -b( v, G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) (13) with the initial condition v(0, •) = v 0 ∈ H, ( 14
)
where H is the well known closure in L 2 (Ω) of the space {ϕ ∈ C ∞ (Ω) : div ϕ = 0 in Ω}, (•, •) denotes the inner product in L 2 (Ω) and •, • D ′ (0,τ ),D(0,τ ) the duality product between D ′ (0, τ ) and D(0, τ ). Let us emphasize that we identify v + ϕχ and v with their trace on Γ 0 in the definition of Ψ( v + ϕχ) and Ψ( v).

Approximate problems

The variational formulation of the problem (1)-( 7) leads to an inequality involving Tresca's functional Ψ, which is convex continuous but not differentiable. To overcome this difficulty we use a regularization of Ψ. More precisley, for any ε > 0, we introduce Ψ ε defined by

Ψ ε (u) = τ 0 Γ 0 ℓ ε 2 + |u| 2 dx ′ dt ∀u ∈ L 2 0, τ ; L 2 (Γ 0 ) which is Gâteaux-differentiable in L 2 0, τ ; L 2 (Γ 0 ) , with Ψ ′ ε (u) ∈ L 2 0, τ ; L 2 (Γ 0 ) ′ = L 2 0, τ ; L 2 (Γ 0 ) for all u ∈ L 2 0, τ ; L 2 (Γ 0 ) given by Ψ ′ ε (u), w = τ 0 Γ 0 ℓ u • w ε 2 + |u| 2 dx ′ dt ∀w ∈ L 2 0, τ ; L 2 (Γ 0 )
where •, • denotes the inner product in L 2 0, τ ; L 2 (Γ 0 ) . Then we consider a sequence of initial data ( v ε0 ) ε>0 such that

v ε0 -→ ε→0 v 0 strongly in H (15)
and we approximate problem (P ) by the following problems (P ε ), ε > 0:

Problem (P ε ) Find v ε ∈ L 2 0, τ ; V 0div ∩L ∞ 0, τ ; L 2 (Ω) , ∂ v ε ∂t ∈ L 4 3 0, τ ; (V 0div ) ′ , p ε ∈ H -1 0, τ ; L 2 0 (Ω)
such that, for all ϕ ∈ V 0 and for all χ ∈ D(0, τ ), we have

d dt ( v ε , ϕ) , χ D ′ (0,τ ),D(0,τ ) + b( v ε , v ε , ϕ), χ D ′ (0,τ ),D(0,τ ) -p ε , div(ϕ) , χ D ′ (0,τ ),D(0,τ ) +a(T ; v ε , ϕχ) + Ψ ′ ε ( v ε ), ϕχ = (f, ϕ), χ D ′ (0,τ ),D(0,τ ) -a(T ; G 0 ζ, ϕχ) - G 0 ∂ζ ∂t , ϕ , χ D ′ (0,τ ),D(0,τ ) -b(G 0 ζ, v ε + G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) -b( v ε , G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) (16) with the initial condition v ε (0, •) = v ε0 ∈ H. ( 17
)
As it has been explained in Section 1, a classical technique to solve such problems consists in choosing divergence free test-functions. Indeed if ϕ ∈ V 0div , the term p ε , div(ϕ) , χ D ′ (0,τ ),D(0,τ ) vanishes and we simply get a variational problem for the fluid velocity v ε . Then the existence of p ε ∈ H -1 0, τ ; L 2 0 (Ω) is derived via abstract results of functional analysis (see [START_REF] Temam | Navier-Stokes equations, theory and numerical analysis[END_REF][START_REF] Simon | On the existence of the pressure for solutions of the variational Navier-Stokes equations[END_REF][START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems[END_REF] for instance).

With this technique the pressure appears as a by product of the study. In order to get better insights into the links between the velocity and pressure fields, we will follow an idea proposed by J.L. Lions in [START_REF] Lions | Some problems connected with Navier-Stokes equations[END_REF] and recently used in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF], which consists in relaxing the divergence free condition. More precisely, we consider the following penalized problems (P δ ε ), δ > 0, ε > 0:

Problem (P δ ε ) Find v δ ε ∈ L 2 0, τ ; V 0 ∩ L ∞ 0, τ ; L 2 (Ω) , ∂ v δ ε ∂t ∈ L 4 3 0, τ ; (V 0 ) ′
such that, for all ϕ ∈ V 0 and for all χ ∈ D(0, τ ), we have

d dt v δ ε , ϕ , χ D ′ (0,τ ),D(0,τ ) + b( v δ ε , v δ ε , ϕ), χ D ′ (0,τ ),D(0,τ ) + 1 2 Ω v δ ε div( v δ ε )ϕ dx, χ D ′ (0,τ ),D(0,τ ) + 1 δ div( v δ ε ), div(ϕ) , χ D ′ (0,τ ),D(0,τ ) + a(T ; v δ ε , ϕχ) + Ψ ′ ε ( v δ ε ), ϕχ = (f, ϕ), χ D ′ (0,τ ),D(0,τ ) -a(T ; G 0 ζ, ϕχ) - G 0 ∂ζ ∂t , ϕ , χ D ′ (0,τ ),D(0,τ ) -b(G 0 ζ, v δ ε + G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) -b( v δ ε , G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) (18) with the initial condition v δ ε (0, •) = v δ ε0 ∈ L 2 (Ω) (19)
and we assume that the sequence of initial data ( v δ ε0 ) δ>0 satisfies

v δ ε0 -→ δ→0 v ε0 strongly in L 2 (Ω). (20)
Let us emphasize that the last term of the first line of ( 18) is added for technical reasons (see (28)) while the first term of the second line is the penalty term:

-1 δ div( v δ ε ) will play the role of an approximate pressure (see Section 5). Furthermore, the approximate initial velocities ( v δ ε0 ) ε>0,δ>0 and ( v ε0 ) ε>0 are not assumed to be more regular that v 0 .

4 Existence result for the penalized problems (P δ ε )

We prove the existence of solutions for the system ( 18)-( 19), for any ε > 0 and δ > 0, by using the Galerkin method. Since V 0 is a closed subspace of H 1 (Ω), it admits an Hilbertian basis (w i ) i≥1 , which is orthogonal for the inner product of H 1 (Ω) and orthonormal for the inner product of L 2 (Ω). Then, for all m ≥ 1, we look for a function v δ εm given by

v δ εm (t, x) = m j=1 g δ εj (t)w j (x), ∀t ∈ (0, τ ), ∀x ∈ Ω, (21)
such that, for all k ∈ {1, . . . , m}, we have

∂ v δ εm ∂t , w k + b( v δ εm , v δ εm , w k ) + 1 2 Ω v δ εm div( v δ εm )w k dx + 1 δ div( v δ εm ), div(w k ) + Ω 2µ(T )D( v δ εm ) : D(w k ) dx + Γ 0 ℓ v δ εm • w k ε 2 + | v δ εm | 2 dx ′ = (f, w k ) - Ω 2µ(T )D(G 0 ζ) : D(w k ) dx -G 0 ∂ζ ∂t , w k -b(G 0 ζ, v δ εm + G 0 ζ, w k ) -b( v δ εm , G 0 ζ, w k ) a.e. in (0, τ ) (22) with the initial condition v δ εm (0, •) = v δ εm0 (23)
where v δ εm0 is defined as the orthogonal projection of v δ ε0 in L 2 (Ω) on Span w 1 . . . w m . For all i, j, k ∈ {1, . . . , m} we denote

F k = (f, w k ) - Ω 2µ(T )D(G 0 ζ) : D(w k ) dx -G 0 ∂ζ ∂t , w k -b(G 0 ζ, G 0 ζ, w k ) ∈ L 2 (0, τ ) and A j,k (T ) = Ω 2µ(T )D(w j ) : D(w k ) dx ∈ L ∞ (0, τ ), B i,j,k = b(w i , w j , w k ) ∈ R.
By replacing v δ εm by its expression [START_REF] Simon | On the existence of the pressure for solutions of the variational Navier-Stokes equations[END_REF] in equation ( 22) and using the orthonormality of (w i ) i≥1 in L 2 (Ω), we obtain

(g δ εk ) ′ + m i,j=1 g δ εj g δ εi B i,j,k + 1 2 m i,j=1 g δ εi g δ εj Ω w i div(w j )w k dx + 1 δ m j=1 g δ εj div(w j ), div(w k ) + m j=1 g δ εj A j,k (T ) + Γ 0 ℓ m j=1 g δ εj w j • w k ε 2 + m j=1 g δ εj w j 2 dx ′ = F k - m j=1 g δ εj b(G 0 ζ, w j , w k ) - m j=1 g δ εj b(w j , G 0 ζ, w k ) ∀k ∈ {1, . . . , m}. (24)
We can rewrite this differential system as

(g δ ε ) ′ = G(t, g δ ε ), g δ ε = g δ εj 1≤j≤m
where G satisfies the assumptions of the Caratheodory theorem (see [START_REF] Coddington | Theory of ordinary differential equations[END_REF]). Moreover, the function G is locally Lipschitz continuous with respect its the second argument. It follows that, for any given initial data, the differential system (24) admits an unique maximal solution g δ εj in H 1 (0, τ m ), 1 ≤ j ≤ m, with 0 < τ m ≤ τ , which implies the existence of a maximal solution v δ εm ∈ H 1 (0, τ m ; V 0 ) to ( 22)- [START_REF] Zhang | An augmented Lagrangian approach to simulating yield stress fluid flows around a spherical gas bubble[END_REF]. In the following lemma, some a priori estimates independent of m, δ and ε will be established, which allow us to extend this solution to the whole interval [0, τ ]. Lemma 4.1. Assume that (8), ( 9) and [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems[END_REF] hold and that ( v δ ε0 ) ε>0,δ>0 is a bounded sequence of L 2 (Ω). The problem ( 22)-( 23) admits a unique solution v δ εm ∈ H 1 0, τ ; V 0 which satisfies the following estimates

v δ εm L ∞ (0,τ ;L 2 (Ω)) ≤ C (25) v δ εm L 2 (0,τ ;H 1 (Ω)) ≤ C (26) div( v δ εm ) L 2 (0,τ ;L 2 (Ω)) ≤ C √ δ ( 27
)
where C is a constant independent of m, δ and ε.

Proof. By multiplying equation ( 22) by g δ εk (t) and adding from k = 1 to m, we obtain

∂ v δ εm ∂t , v δ εm + b( v δ εm , v δ εm , v δ εm ) + 1 2 Ω v δ εm div( v δ εm ) v δ εm dx + 1 δ div( v δ εm ), div( v δ εm ) + Ω 2µ(T )D( v δ εm ) : D( v δ εm ) dx + Γ 0 ℓ | v δ εm | 2 ε 2 + | v δ εm | 2 dx ′ = (f, v δ εm ) - Ω 2µ(T )D(G 0 ζ) : D( v δ εm ) dx -G 0 ∂ζ ∂t , v δ εm -b(G 0 ζ, v δ εm + G 0 ζ, v δ εm ) -b( v δ εm , G 0 ζ, v δ εm ) a.e. in (0, τ m ).
With [START_REF] Girault | Finite element approximation of the Navier-Stokes equations[END_REF] we get

b( v δ εm , v δ εm , v δ εm ) + 1 2 Ω v δ εm div( v δ εm ) v δ εm dx = 0 ( 28
)
and since div(G 0 ) = 0 in Ω, we have also b(G 0 , v δ εm , v δ εm ) = 0. Furthermore since ℓ ∈ L 2 (0, τ ; L 2 + (Γ 0 )), we obtain

∂ v δ εm ∂t , v δ εm + 1 δ div( v δ εm ), div( v δ εm ) + Ω 2µ(T )D( v δ εm ) : D( v δ εm ) dx ≤ (f, v δ εm ) - Ω 2µ(T )D(G 0 ζ) : D( v δ εm ) dx -G 0 ∂ζ ∂t , v δ εm -b(G 0 ζ, G 0 ζ, v δ εm ) -b( v δ εm , G 0 ζ, v δ εm )
a.e. in (0, τ m ). Let us estimate now the terms in the right-hand side of the previous inequality. We denote hereinafter by K the constant of the continuous injection of H 1 (Ω) into L 4 (Ω). By using Cauchy-Schwarz's and Young's inequalities, we obtain

(f, v δ εm ) ≤ f L 2 (Ω) v δ εm L 2 (Ω) ≤ 1 2 f 2 L 2 (Ω) + 1 2 v δ εm 2 L 2 (Ω) , Ω 2µ(T )D(G 0 ζ) : D( v δ εm ) dx ≤ µ * |ζ| v δ εm H 1 (Ω) G 0 H 1 (Ω) ≤ α 4 v δ εm 2 H 1 (Ω) + µ 2 * α |ζ| 2 G 0 2 H 1 (Ω) , G 0 ∂ζ ∂t , v δ εm ≤ ∂ζ ∂t G 0 L 2 (Ω) v δ εm L 2 (Ω) ≤ 1 2 ∂ζ ∂t 2 G 0 2 L 2 (Ω) + 1 2 v δ εm 2 L 2 (Ω) , b(G 0 ζ, G 0 ζ, v δ εm ) ≤ |ζ| 2 G 0 L 4 (Ω) ∇G 0 L 4 (Ω) v δ εm L 2 (Ω) ≤ 1 2 v δ εm 2 L 2 (Ω) + K 4 2 |ζ| 4 G 0 2 H 1 (Ω) ∇G 0 2 H 1 (Ω) , and
b( v δ εm , G 0 ζ, v δ εm ) ≤ |ζ| v δ εm L 4 (Ω) ∇G 0 L 4 (Ω) v δ εm L 2 (Ω) ≤ α 4 v δ εm 2 H 1 (Ω) + K 4 α |ζ| 2 ∇G 0 2 H 1 (Ω) v δ εm 2 L 2 (Ω) .
With [START_REF] Hervet | Flow with slip at the wall: from simple to complex fluids[END_REF] and an integration from 0 to s, with 0 < s < τ m , we get

1 2 v δ εm (s) 2 L 2 (Ω) + 1 δ s 0 div( v δ εm ) 2 L 2 (Ω) dt + α 2 s 0 v δ εm 2 H 1 (Ω) dt ≤ 1 2 v δ εm (0) 2 L 2 (Ω) + 1 2 s 0 f 2 L 2 (Ω) dt + µ 2 * α G 0 2 H 1 (Ω) s 0 |ζ| 2 dt + 1 2 G 0 2 L 2 (Ω) s 0 ∂ζ ∂t 2 dt + 3 2 0 v δ εm 2 L 2 (Ω) dt + K 4 α ∇G 0 2 H 1 (Ω) s 0 |ζ| 2 v δ εm 2 L 2 (Ω) dt + K 4 2 G 0 2 H 1 (Ω) ∇G 0 2 H 1 (Ω) s 0 |ζ| 4 dt.
Reminding that v δ εm0 is defined as the orthogonal projection of v δ ε0 in L 2 (Ω) on Span w 1 . . . w m and that the sequence ( v δ ε0 ) ε>0,δ>0 is bounded in L 2 (Ω), we infer that there exists a constant C 0 , independent of δ and ε such that

v δ εm (0) L 2 (Ω) = v δ εm0 L 2 (Ω) ≤ v δ ε0 L 2 (Ω) ≤ C 0 ∀m ≥ 1, ∀δ > 0, ∀ε > 0.
It follows that

1 2 v δ εm (s) 2 L 2 (Ω) + 1 δ s 0 div( v δ εm ) 2 L 2 (Ω) dt + α 2 s 0 v δ εm 2 H 1 (Ω) dt ≤ C 1 +C 2 s 0 v δ εm 2 L 2 (Ω) dt, (29)
where C 1 and C 2 are two constants independent of m, δ and ε, namely

C 1 = 1 2 C 2 0 + 1 2 τ 0 f 2 L 2 (Ω) dt + µ 2 * α G 0 2 H 1 (Ω) τ 0 |ζ| 2 dt + 1 2 G 0 2 L 2 (Ω) τ 0 ∂ζ ∂t 2 dt + K 4 2 G 0 2 H 1 (Ω) ∇G 0 2 H 1 (Ω) τ 0 |ζ| 4 dt
and

C 2 = 3 2 + K 4 α ∇G 0 2 H 1 (Ω) ζ 2 L ∞ (0,τ ) .
With Grönwall's lemma, we obtain

v δ εm (s) 2 L 2 (Ω) ≤ 2C 1 exp (2sC 2 ) ≤ 2C 1 exp (2τ C 2 ) ∀s ∈ [0, τ m ). (30)
With ( 21) and ( 24) we infer that the functions g δ εj , 1 ≤ j ≤ m, admit a limit at τ m and, by definition of the maximal solution, we may conclude that τ m = τ . Now, (25) follows from (30). By inserting (30) in (29) with s = τ , we obtain (26) and (27).

In the following lemma, we establish an estimate of the time derivative for the approximate velocity. Lemma 4.2. Under the same assumptions as in Lemma 4.1, we have

∂ v δ εm ∂t L 4 3 (0,τ ;V ′ 0 ) ≤ C δ (31)
where C δ is a constant independent of m and ε.

Proof. Let ϕ ∈ V 0 . For all m ≥ 1, we define ϕ m as the othogonal projection with respect to the inner product of H 1 (Ω) of ϕ on Span w 1 , . . . , w m . With [START_REF] Temam | Navier-Stokes equations, theory and numerical analysis[END_REF] we get

∂ v δ εm ∂t , ϕ m = -b( v δ εm , v δ εm , ϕ m ) - 1 2 Ω v δ εm div( v δ εm )ϕ m dx - 1 δ div( v δ εm ), div(ϕ m ) - Ω 2µ(T )D( v δ εm ) : D(ϕ m ) dx - Γ 0 ℓ v δ εm • ϕ m ε 2 + | v δ εm | 2 dx ′ + (f, ϕ m ) - Ω 2µ(T )D(G 0 ζ) : D(ϕ m ) dx -G 0 ∂ζ ∂t , ϕ m -b(G 0 ζ, v δ εm + G 0 ζ, ϕ m ) -b( v δ εm , G 0 ζ, ϕ m ) a.e. in (0, τ ).
We estimate all the terms in the right hand side of the previous equality, we obtain

∂ v δ εm ∂t , ϕ m ≤ v δ εm L 3 (Ω) ∇ v δ εm L 2 (Ω) + 1 2 v δ εm L 3 (Ω) div( v δ εm ) L 2 (Ω) ϕ m L 6 (Ω) + 1 δ div( v δ εm ) L 2 (Ω) div(ϕ m ) L 2 (Ω) + µ * v δ εm H 1 (Ω) ϕ m H 1 (Ω) + ℓ L 2 (Γ 0 ) ϕ m L 2 (Γ 0 ) + f L 2 (Ω) ϕ m L 2 (Ω) + µ * |ζ| G 0 H 1 (Ω) ϕ m H 1 (Ω) + ∂ζ ∂t G 0 L 2 (Ω) ϕ m L 2 (Ω) + |ζ| G 0 L 4 (Ω) ∇ v δ εm + ∇G 0 L 2 (Ω) ϕ m L 4 (Ω) +|ζ| v δ εm L 4 (Ω) ∇G 0 L 2 (Ω) ϕ m L 4 (Ω)
a.e. in (0, τ ).

By using the classical inequality

u L 3 (Ω) ≤ u 1 2 L 2 (Ω) u 1 2 L 6 (Ω) ∀u ∈ L 6 (Ω) ∩ L 2 (Ω)
and the injection of H 1 (Ω) in L 6 (Ω), we infer that there exists a constant c, independent of m, δ and ε, such that

v δ εm L 3 (Ω) ∇ v δ εm L 2 (Ω) ϕ m L 6 (Ω) ≤ c v δ εm 1 2 L 2 (Ω) v δ εm 3 2 H 1 (Ω) ϕ m H 1 (Ω) .
As (w j ) j≥1 is an orthogonal family of L 2 (Ω) and ϕ m is the orthogonal projection with respect to the inner product of H 1 (Ω) of ϕ on Span w 1 , . . . , w m , we have

ϕ m H 1 (Ω) ≤ ϕ H 1 (Ω) and v δ εm ∂t , ϕ m = ∂ v δ εm ∂t , ϕ k ∀k ≥ m.
Since (w j ) j≥1 is an Hilbertian basis of V 0 , the sequence (ϕ k ) k≥1 converges strongly to ϕ in H 1 (Ω) and we get

v δ εm ∂t , ϕ m = ∂ v δ εm ∂t , ϕ .
Then, we obtain

∂ v δ εm ∂t , ϕ ≤ √ 3 2 + 1 c v δ εm 1 2 L 2 (Ω) v δ εm 3 2 H 1 (Ω) ϕ H 1 (Ω) + √ 3 δ div( v δ εm ) L 2 (Ω) ϕ H 1 (Ω) + µ * v δ εm H 1 (Ω) ϕ H 1 (Ω) + c ℓ L 2 (Γ 0 ) ϕ H 1 (Ω) + f L 2 (Ω) + µ * |ζ| G 0 H 1 (Ω) + ∂ζ ∂t G 0 L 2 (Ω) ϕ H 1 (Ω) + K 2 |ζ| G 0 H 1 (Ω) v δ εm + G 0 ζ H 1 (Ω) + K 2 |ζ| v δ εm H 1 (Ω) G 0 H 1 (Ω) ϕ H 1 (Ω) a.e. in (0, τ ),
where c is the norm of the trace operator γ 0 :

H 1 (Ω) → L 2 (Γ 0 ). Hence ∂ v δ εm ∂t V ′ 0 ≤ √ 3 2 + 1 c v δ εm 1 2 L 2 (Ω) v δ εm 3 2 H 1 (Ω) + √ 3 δ div( v δ εm ) L 2 (Ω) + µ * v δ εm H 1 (Ω) +c ℓ L 2 (Γ 0 ) + f L 2 (Ω) + µ * |ζ| G 0 H 1 (Ω) + ∂ζ ∂t G 0 L 2 (Ω) +K 2 |ζ| G 0 H 1 (Ω) v δ εm + G 0 ζ H 1 (Ω) + K 2 |ζ| v δ εm H 1 (Ω) G 0 H 1 (Ω) a.e. in (0, τ ).
Observing that

τ 0 v δ εm 1 2 L 2 (Ω) v δ εm 3 2 H 1 (Ω) 4 3 dt = τ 0 v δ εm 2 3 L 2 (Ω) v δ εm 2 H 1 (Ω) dt ≤ v δ εm 2 3 L ∞ (0,τ ;L 2 (Ω)) v δ εm 2 L 2 (0,τ ;H 1 (Ω)) ,
we infer from the estimates of Lemma 4.1 that there exists a constant C δ > 0, independent of m and ε, such that

τ 0 ∂ v δ εm ∂t 4 3 V ′ 0 dt ≤ C δ
which concludes the proof.

In order to pass to the limit as m tends to +∞, we will use also the following Lemma.

Lemma 4.3. Let ε > 0 and ℓ ∈ L 2 0, τ ; L 2 + (Γ 0 ) ∩ L ∞ 0, τ ; L ∞ + (Γ 0 ) . Then the mapping Ψ ′ ε is Lipschitz continuous from L 2 0, τ ; L 2 (Γ 0 ) to L 2 0, τ ; L 2 (Γ 0 ) . Proof. Let us recall that, for all u ∈ L 2 0, τ ; L 2 (Γ 0 ) , Ψ ′ ε (u) ∈ L 2 0, τ ; L 2 (Γ 0 ) ′ = L 2 0, τ ; L 2 (Γ 0 ) is defined by Ψ ′ ε (u), w = τ 0 Γ 0 ℓ u • w ε 2 + |u| 2 dx ′ dt ∀w ∈ L 2 0, τ ; L 2 (Γ 0 )
where •, • denotes the inner product in L 2 0, τ ; L 2 (Γ 0 ) , i.e.

Ψ ′ ε (u) = ℓ u ε 2 + |u| 2 ∀u ∈ L 2 0, τ ; L 2 (Γ 0 ) .
But the mapping

h ε :    R d → R d u → u ε 2 + |u| 2 is Fréchet differentiable on R d and Jac(h ε )(u) = ∂h εi ∂x j (u) 1≤i,j≤d =   δ i,j ε 2 + |u| 2 - u i u j ε 2 + |u| 2 3 2   1≤i,j≤d
where δ i,j = 1 if i = j and δ i,j = 0 if i = j. It follows that

∂h εi ∂x j (u) ≤ 2 ε ∀i, j ∈ {1, . . . , d}, ∀u ∈ R d and h ε is Lipschitz continuous on R d . Since ℓ ∈ L ∞ 0, τ ; L ∞ + (Γ 0 ) , we infer that Ψ ′ ε is Lipschitz continuous from L 2 0, τ ; L 2 + (Γ 0 ) into L 2 0, τ ; L 2 + (Γ 0 ) .
Now, by using the estimates obtained in Lemma 4.1 and Lemma 4.2 combined with compactness arguments, we can prove the following existence result for the penalized problems (P δ ε ).

Theorem 4.4. Let ε > 0 and δ > 0. Assume that ( 8), ( 9) and ( 10) hold and that ( v δ ε0 ) ε>0,δ>0 is a bounded sequence of L 2 (Ω). Then, there exists a subsequence of

( v δ εm ) m≥1 , still denoted ( v δ εm ) m≥1 , such that v δ εm ⇀ v δ ε weakly star in L ∞ 0, τ ; L 2 (Ω) (32) v δ εm ⇀ v δ ε weakly in L 2 (0, τ ; V 0 ) (33)
and v δ ε is solution of (P δ ε ). Furthermore

∂ v δ ε ∂t belongs to L 4 3 (0, τ ; V ′ 0 ).
Proof. The convergences (32)-(33) follow immediately from the a priori estimates (25)-( 26) obtained in Lemma 4.1. From the estimate (31) obtained in Lemma 4.2, we infer that, possibly extracting another subsequence still denoted denoted ( v δ εm ) m≥1 , we have

∂ v δ εm ∂t ⇀ ∂ v δ ε ∂t weakly in L 4 3 (0, τ ; V ′ 0 ). ( 34
)
By using Aubin's lemma [START_REF] Temam | Navier-Stokes equations, theory and numerical analysis[END_REF] and the convergences (33) and (34), with X 0 = V 0 , X = L 4 (Ω) and

X 1 = V ′ 0 we obtain v δ εm → v δ ε strongly in L 2 (0, τ ; L 4 (Ω)).
We may use again Aubin's lemma with X 0 = V 0 , X = H s (Ω) and X 1 = V ′ 0 with 1 2 < s < 1: the embedding of X 0 into X is compact, so we obtain

v δ εm → v δ ε strongly in L 2 (0, τ ; H s (Ω)).
Then, with trace theorem [START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF], we infer that

v δ εm → v δ ε strongly in L 2 (0, τ ; L 2 (Γ 0 ))
where we identify here the functions v δ εm and v δ ε with their trace on Γ 0 . Now, using (32)-(34) and Simon's lemma [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] and possibly extracting another subsequence, still denoted ( v δ εm ) m≥1 , we obtain

v δ εm → v δ ε strongly in C 0 (0, τ ; H), (35)
for any Banach space H such that L 2 (Ω) ⊂ H ⊂ V ′ 0 with continuous injections and compact embedding of L 2 (Ω) into H.

Let χ ∈ D(0, τ ) and ϕ ∈ V 0 . For all m ≥ 1 we define again ϕ m as the othogonal projection with respect to the inner product of H 1 (Ω) of ϕ on Span w 1 , . . . , w m . With [START_REF] Temam | Navier-Stokes equations, theory and numerical analysis[END_REF] we have

τ 0 ∂ v δ εm ∂t , ϕ m + b( v δ εm , v δ εm , ϕ m ) + 1 2 Ω v δ εm div( v δ εm )ϕ m dx χ dt + 1 δ τ 0 div( v δ εm ), div(ϕ m )χ dt + a(T ; v δ εm , ϕ m χ) + Ψ ′ ε ( v δ εm ), ϕ m χ = τ 0 (f, ϕ m )χ dt -a(T ; G 0 ζ, ϕ m χ) - τ 0 G 0 ∂ζ ∂t , ϕ m + b(G 0 ζ, v δ εm + G 0 ζ, ϕ m ) + b( v δ εm , G 0 ζ, ϕ m ) χ dt.
With an integration by parts of the first term we get

τ 0 v δ εm , ϕ m ∂χ ∂t dt + τ 0 b( v δ εm , v δ εm , ϕ m ) + 1 2 Ω v δ εm div( v δ εm )ϕ m dx χ dt + 1 δ τ 0 div( v δ εm ), div(ϕ m )χ dt + a(T ; v δ εm , ϕ m χ) + Ψ ′ ε ( v δ εm ), ϕ m χ = τ 0 (f, ϕ m )χ dt -a(T ; G 0 ζ, ϕ m χ) - τ 0 G 0 ∂ζ ∂t , ϕ m + b(G 0 ζ, v δ εm + G 0 ζ, ϕ m ) + b( v δ εm , G 0 ζ, ϕ m ) χ dt.
Reminding that (ϕ m ) m≥1 converges strongly to ϕ in H 1 (Ω) and using Lemma 4.3, we can pass to the limit in all the terms and we obtain

τ 0 v δ ε , ϕ ∂χ ∂t dt + τ 0 b( v δ ε , v δ ε , ϕ) + 1 2 Ω v δ ε div( v δ ε )ϕ dx + 1 δ div( v δ ε ), div(ϕ) χ dt +a(T ; v δ ε , ϕχ) + Ψ ′ ε ( v δ ε ), ϕχ = τ 0 (f, ϕ)χ dt -a(T ; G 0 ζ, ϕχ) - τ 0 G 0 ∂ζ ∂t , ϕ + b(G 0 ζ, v δ ε + G 0 ζ, ϕ) + b( v δ ε , G 0 ζ, ϕ) χ dt.
which gives [START_REF] Goulart | Global stability analysis of fluid flows using sum-of-squares[END_REF]. It remains to chek that the initial condition ( 19) is satisfied. Indeed, with (35), we have

v δ εm (0) → v δ ε (0) strongly in H with L 2 (Ω) ⊂ H ⊂ V ′
0 and we have also

v δ εm (0) = v δ εm0 → v δ ε0 strongly in L 2 (Ω). Hence v δ ε (0) = v δ ε0 .

Properties of the approximate pressure

For any ε > 0 and δ > 0 we define and approximate pressure p δ ε ∈ L 2 0, τ ; L 2 (Ω) by

p δ ε = - 1 δ div( v δ ε ) (36)
where v δ ε is the solution of the penalized problem (P δ ε ) obtained in the previous Section. From [START_REF] Goulart | Global stability analysis of fluid flows using sum-of-squares[END_REF] we get

d dt v δ ε , ϕ , χ D ′ (0,τ ),D(0,τ ) + b( v δ ε , v δ ε , ϕ), χ D ′ (0,τ ),D(0,τ ) + 1 2 Ω v δ ε div( v δ ε )ϕ dx, χ D ′ (0,τ ),D(0,τ ) -p δ ε , div(ϕ) , χ D ′ (0,τ ),D(0,τ ) + a(T ; v δ ε , ϕχ) + Ψ ′ ε ( v δ ε ), ϕχ = (f, ϕ), χ D ′ (0,τ ),D(0,τ ) -a(T ; G 0 ζ, ϕχ) - G 0 ∂ζ ∂t , ϕ , χ D ′ (0,τ ),D(0,τ ) -b(G 0 ζ, v δ ε + G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) -b( v δ ε , G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) , ∀ϕ ∈ V 0 , ∀χ ∈ D(0, τ ).
(37) Furthermore, with Green's formula, we obtain

Ω p δ ε dx = - 1 δ ∂Ω v δ ε • n dx = 0 a.e. in (0, τ ) (38)
and, with (27) and (33), we have

p δ ε L 2 (0,τ ;L 2 (Ω)) ≤ C √ δ
where C is a constant independent of δ and ε. Unfortunately this last estimate does not allow us to pass to the limit in the term p δ ε , div(ϕ) , χ D ′ (0,τ ),D(0,τ ) as δ tends to zero. So we will establish an estimate independent of ε and δ by using the same kind of technique as in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF]).

Lemma 5.1. Under the same assumptions as in Lemma 4.1, there exists a constant C, independent of δ and ε, such that

p δ ε H -1 (0,τ ;L 2 (Ω)) ≤ C. (39)
Proof. Let χ ∈ D(0, τ ) and w ∈ L 2 0 (Ω). Then there exists ϕ ∈ H 1 0 (Ω) such that div(ϕ) = w in Ω and ϕ = P (w) where P is a linear continuous operator from L 2 0 (Ω) into H 1 0 (Ω) (see [START_REF] Lions | Some problems connected with Navier-Stokes equations[END_REF]). With an integration by parts of the first term of (37), we get

τ 0 p δ ε , div(ϕ) χ dt = - τ 0 v δ ε , ϕ ∂χ ∂t dt + τ 0 b( v δ ε , v δ ε , ϕ) + 1 2 Ω v δ ε div( v δ ε )ϕ dx χ dt +a(T ; v δ ε , ϕχ) - τ 0 (f, ϕ)χ dt + a(T ; G 0 ζ, ϕχ) + τ 0 G 0 ∂ζ ∂t , ϕ + b(G 0 ζ, v δ ε + G 0 ζ, ϕ) + b( v δ ε , G 0 ζ, ϕ) χ dt
Let us denote ϕχ = η and recall that K is the constant of the continuous injection of H 1 (Ω) into L 4 (Ω). We get

τ 0 b( v δ ε , v δ ε , ϕ) + 1 2 Ω v δ ε div( v δ ε )ϕ dx χ dt ≤ ≤ 1 + √ 3 2 v δ ε L 2 (0,τ ;L 4 (Ω)) v δ ε L 2 (0,τ ;H 1 (Ω)) η L ∞ (0,τ ;L 4 (Ω)) ≤ K 2 1 + √ 3 2 v δ ε 2 L 2 (0,τ ;H 1 (Ω)) η L ∞ (0,τ ;H 1 (Ω)) .
Then we obtain

τ 0 (p δ ε , w)χ dt ≤ v δ ε L 2 (0,τ ;L 2 (Ω)) ∂η ∂t L 2 (0,τ ;L 2 (Ω)) +K 2 1 + √ 3 2 v δ ε 2 L 2 (0,τ ;H 1 (Ω)) η L ∞ (0,τ ;H 1 (Ω)) +µ * √ τ v δ ε L 2 (0,τ ;H 1 (Ω)) η L ∞ (0,τ ;H 1 (Ω)) + √ τ f L 2 (0,τ ;L 2 (Ω)) η L ∞ (0,τ ;L 2 (Ω)) +µ * G 0 H 1 (Ω) ζ L 1 (0,τ ) η L ∞ (0,τ ;H 1 (Ω)) + G 0 L 2 (Ω) ∂ζ ∂t L 1 (0,τ ) η L ∞ (0,τ ;L 2 (Ω)) +K 2 G 0 H 1 (Ω) ζ L 2 (0,τ ) v δ ε + G 0 ζ L 2 (0,τ ;H 1 (Ω)) η L ∞ (0,τ ;H 1 (Ω)) +K 2 G 0 H 1 (Ω) ζ L 2 (0,τ ) v δ ε L 2 (0,τ ;H 1 (Ω)) η L ∞ (0,τ ;H 1 (Ω)) .
By using the continuity of the operator P and the continuous injection of H 1 (0, τ ) into L ∞ (0, τ ), we have

η L ∞ (0,τ ;H 1 (Ω)) = χ L ∞ (0,τ ) P (w) H 1 (Ω) ≤ C χ H 1 (0,τ ) w L 2 (Ω) ≤ C η H 1 (0,τ ;L 2 (Ω)) ,
where C is a constant independent of δ and ε. With the estimates (25) and (26), we infer that v δ ε is bounded in L 2 (0, τ ; H 1 (Ω)) ∩ L ∞ (0, τ ; L 2 (Ω)) independently of δ and ε, and we obtain

τ 0 (p δ ε , w)χ dt ≤ C wχ H 1 (0,τ ;L 2 (Ω)) ∀w ∈ L 2 0 (Ω), ∀χ ∈ D(0, τ ) (40)
where we denote again by C a constant independent of δ and ε.

Let now w ∈ L 2 (Ω). We can apply (40) with

w = w - 1 mesΩ Ω w dx.
Indeed, w ∈ L 2 0 (Ω). Furthermore, with (38), we have

τ 0 p δ ε , w - 1 mesΩ Ω w dx χ dt = = τ 0 (p δ ε , w)χ dt - 1 mesΩ τ 0 Ω p δ ε dx Ω w dx χ dt = τ 0 (p δ ε , w)χ dt .
Observing that w L 2 (Ω) ≤ w L 2 (Ω) , we obtain

τ 0 (p δ ε , w)χ dt = τ 0 (p δ ε , w)χ dt ≤ C wχ H 1 (0,τ ;L 2 (Ω)) , ∀ w ∈ L 2 (Ω), ∀χ ∈ D(0, τ ).( 41
)
Then the density of D(0, τ )⊗L 2 (Ω) into H 1 0 (0, τ ; L 2 (Ω)) allows us to conclude.

6 Existence results for the problems (P ε ) and (P )

Now we can pass to the limit in the penalized problems (P δ ε ) when δ tends to zero.

Theorem 6.1. Let ε > 0 and assume that ( v δ ε0 ) ε>0,δ>0 is a bounded sequence of L 2 (Ω). Assume moreover that (8), ( 9), ( 10) and (20) hold. Then, there exists a subsequence of (

v δ ε , p δ ε ) δ>0 , still denoted ( v δ ε , p δ ε ) δ>0 , such that v δ ε ⇀ v ε weakly star in L ∞ 0, τ ; L 2 (Ω) (42) v δ ε ⇀ v ε weakly in L 2 (0, τ ; V 0 ) (43) p δ ε ⇀ p ε weakly in H -1 0, τ ; L 2 0 (Ω) (44) and ( v ε , p ε ) is solution of (P ε ). Furthermore ∂ v ε ∂t belongs to L 4 3 0, τ ; (V 0div ) ′ .
Proof. Observing that the estimates obtained in Lemma 4.1 are independent of m, δ and ε, we infer that the sequence ( v δ ε , p δ ε ) δ>0 is bounded in L 2 (0, τ ; V 0 ) ∩ L ∞ 0, τ ; L 2 (Ω) . Moreover, with Proposition 5.1, the sequence (p δ ε ) δ>0 is bounded in H -1 0, τ ; L 2 (Ω) and the convergences (43)-( 42)-(44) follow immediately.

From (27) we infer that

div( v δ ε ) L 2 (0,τ ;L 2 (Ω)) ≤ C √ δ
with a constant C independent of δ and ε. Thus

div( v δ ε ) → 0 strongly in L 2 0, τ ; L 2 (Ω) .
Finally we can obtain an estimate of

∂ v δ ε ∂t in L 4 3 0, τ ; (V 0div )
′ by using the same kind of computations as in Lemma 4.2. Indeed, let ϕ ∈ V 0div and χ ∈ D(0, τ ). With (37) we get

d dt v δ ε , ϕ , χ D ′ (0,τ ),D(0,τ ) = τ 0 Ω ∂ v δ ε ∂t ϕχ dxdt = -b( v δ ε , v δ ε , ϕ), χ D ′ (0,τ ),D(0,τ ) - 1 2 Ω v δ ε div( v δ ε )ϕ dx, χ D ′ (0,τ ),D(0,τ ) -a(T ; v δ ε , ϕχ) -Ψ ′ ε ( v δ ε ), ϕχ + (f, ϕ), χ D ′ (0,τ ),D(0,τ ) -a(T ; G 0 ζ, ϕχ) - G 0 ∂ζ ∂t , ϕ , χ D ′ (0,τ ),D(0,τ ) -b(G 0 ζ, v δ ε + G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) -b( v δ ε , G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) .
We can estimate all the terms in the right hand side of the previous equality and we obtain

τ 0 Ω ∂ v δ ε ∂t ϕχ dxdt ≤ √ 3 2 + 1 c τ 0 v δ ε 1 2 L 2 (Ω) v δ ε 3 2 H 1 (Ω) ϕ H 1 (Ω) |χ| dt +µ * τ 0 v δ ε H 1 (Ω) ϕ H 1 (Ω) |χ| dt+c τ 0 ℓ L 2 (Γ 0 ) ϕ H 1 (Ω) |χ| dt + τ 0 f L 2 (Ω) + µ * |ζ| G 0 H 1 (Ω) + ∂ζ ∂t G 0 L 2 (Ω) ϕ H 1 (Ω) |χ| dt + τ 0 K 2 |ζ| G 0 H 1 (Ω) v δ ε + G 0 ζ H 1 (Ω) + K 2 |ζ| v δ ε H 1 (Ω) G 0 H 1 (Ω) ϕ H 1 (Ω) |χ| dt.
Observing that

τ 0 v δ ε 1 2 L 2 (Ω) v δ ε 3 2 H 1 (Ω) ϕ H 1 (Ω) |χ| dt ≤ ≤ v δ ε 1 2 L ∞ (0,τ ;L 2 (Ω)) τ 0 v δ ε 2 H 1 (Ω) dt 3 4 ϕ H 1 (Ω) χ L 4 (0,τ ) ≤ v δ ε 1 2 L ∞ (0,τ ;L 2 (Ω)) v δ ε 3 2 L 2 (0,τ ;H 1 (Ω)) ϕχ L 4 (0,τ ;H 1 (Ω))
and reminding that ( v δ ε ) δ>0 is bounded in L 2 0, τ ; H 1 (Ω) ∩ L ∞ 0, τ ; L 2 (Ω) independently of δ and ε, we infer that

∂ v δ ε ∂t L 4 3 (0,τ ;(V 0div ) ′ ) ≤ C (45)
with a constant C independent of δ and ε.

It follows that, possibly extracting another subsequence still denoted ( v δ ε ) δ>0 , we have

∂ v δ ε ∂t ⇀ ∂ v ε ∂t weakly in L 4 3 0, τ ; (V 0div ) ′ . ( 46
)
By using Aubin's lemma, with X 0 = V 0 , X = L 4 (Ω) and X 1 = (V 0div ) ′ , we obtain

v δ ε → v ε strongly in L 2 0, τ ; L 4 (Ω)
and, with

X 0 = V 0 , X = H s (Ω), 1 2 < s < 1, and X 1 = (V 0div ) ′ , v δ ε → v ε strongly in L 2 0, τ ; H s (Ω) . Hence v δ ε → v ε strongly in L 2 0, τ ; L 2 (Γ 0 ) .
Finally, using (42)-( 46) and Simon's lemma, and possibly extracting another subsequence still denoted ( v δ ε ) δ>0 , we obtain

v δ ε → v ε strongly in C 0 (0, τ ; H)
for any Banach space H such that L 2 (Ω) ⊂ H ⊂ (V 0div ) ′ with continuous injections and compact embedding of L 2 (Ω) into H. With all these convergences and with the assumption [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF], we can pass to the limit in [START_REF] Goulart | Global stability analysis of fluid flows using sum-of-squares[END_REF] and ( 19) by the same techniques as in Theorem 4.4 and we get ( 16) and [START_REF] Biryuk | Bounds on Kolmogorov spectra for Navier-Stokes equations[END_REF]. Now, observing that Ψ ε is convex, we obtain that

Ψ ε ( v ε + ϕχ) -Ψ ε ( v ε ) ≥ Ψ ′ ε ( v ε ),
) for all ϕ ∈ V 0 and for all χ ∈ D(0, τ ), with the initial condition

v ε (0, •) = v ε0 . (48)
In order to pass to the limit as ε tends to zero in the previous inequality, we use the following lemma. Lemma 6.2. Let (w ε ) ε>0 be a sequence of L 2 0, τ ; L 2 (Γ 0 ) and w ∈ L 2 0, τ ; L 2 (Γ 0 ) such that (w ε ) ε>0 converges strongly to w in L 2 0, τ ; L 2 (Γ 0 ) . Then lim Now we can prove that problem (P ) admits a solution.

Theorem 6.3. Assume that ( v δ ε0 ) ε>0,δ>0 is a bounded sequence of L 2 (Ω). Assume moreover that (8), ( 9), ( 10) and ( 15) hold. Then, there exists a subsequence of ( v ε , p ε ) ε>0 , still denoted ( v ε , p ε ) ε>0 such that

v ε ⇀ v weakly star in L ∞ 0, τ ; L 2 (Ω) (49)
v ε ⇀ v weakly in L 2 (0, τ ; V 0 ) (50) p ε ⇀ p weakly in H -1 0, τ ; L 2 0 (Ω) (51)

and ( v, p) is solution of (P ). Furthermore ∂ v ∂t belongs to L 4 3 0, τ ; (V 0div ) ′ .

Proof. Recalling that the estimates (25)-( 26) are independent of m, δ and ε, we deduce that ( v ε ) ε>0 is bounded in L 2 0, τ ; H 1 (Ω) ∩ L ∞ 0, τ ; L 2 (Ω) . Moreover, since the estimate (39) is independent of δ and ε, the sequence (p ε ) ε>0 is bounded in H -1 0, τ ; L 2 (Ω) and we may infer the convergences (49)-( 50)-(51). Furthermore the estimate (45) implies that that ∂ v ε ∂t ε>0 is bounded in L With all these convergences and the assumption [START_REF] Lions | Some problems connected with Navier-Stokes equations[END_REF], we can pass to the limit in (47) and (48) by the same techniques as in Theorem 4.4 and Theorem 6.1 and we get ( 13) and ( 14).

D

  ϕχ ∀ϕ ∈ V 0 , ∀χ ∈ D(0, τ ) and in[START_REF] Lions | Non-homogeneous boundary value problems and applications[END_REF] we getd dt ( v ε , ϕ) , χ D ′ (0,τ ),D(0,τ ) + b( v ε , v ε , ϕ), χ D ′ (0,τ ),D(0,τ )p ε , div(ϕ) , χ D ′ (0,τ ),D(0,τ ) +a(T ; v ε , ϕχ) + Ψ ε ( v ε + ϕχ) -Ψ ε ( v ε ) ≥ (f, ϕ), χ D ′ (0,τ ),D(0,τ )a(T ; G 0 ζ, ϕχ) ′ (0,τ ),D(0,τ ) b(G 0 ζ, v ε + G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ ) b( v ε , G 0 ζ, ϕ), χ D ′ (0,τ ),D(0,τ )

ε→0Ψℓℓ ε 2 +ℓ

 2 ε (w ε ) = Ψ(w).Proof. Let ε > 0. By definition of Ψ ε and Ψ, we haveΨ ε (w ε ) -Ψ(w) = |w ε | -|w| dx ′ dt + |w ε | 2 -|w ε | dx ′ dt. It follows that Ψ ε (w ε ) -Ψ(w) ≤ |w ε | -|w| dx ′ dt + τ 0 Γ 0 ℓε dx ′ dt ≤ ℓ L 2 (0,τ ;L 2 (Γ 0 )) w εw L 2 (0,τ ;L 2 (Γ 0 )) + ε τ meas(Γ 0 )which allows us to conclude.

4 3 0

 3 , τ ; (V 0div ) ′ . Hence, possibly extracting another subsequence still denoted ( v ε ) ε>0 , we have τ ; (V 0div ) ′ and with the same arguments as in the previous Theorem, we getv ε → v strongly in L 2 0, τ ; L 4 (Ω) , v ε → v strongly in L 2 0, τ ; L 2 (Γ 0 ) , and v δ ε → v ε strongly in C 0 (0, τ ; H),for any Banach space H such that L 2 (Ω) ⊂ H ⊂ (V 0div ) ′ with continuous injections and compact embedding of L 2 (Ω) into H.


Isabelle Tristani

EXPONENTIAL CONVERGENCE TO EQUILIBRIUM FOR THE HOMOGENEOUS BOLTZMANN EQUATION FOR HARD POTENTIALS WITHOUT CUT-OFF

Keywords: Mathematics Subject Classification (2010): 76P05 Rarefied gas flows, Boltzmann equation, 47H20 Semigroups of nonlinear operators, 35B40 Asymptotic behavior of solutions Boltzmann equation without cut-off, hard potentials, spectral gap, dissipativity, exponential rate of convergence, long-time asymptotic

 where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a L 1 space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani et al. in [22]. We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.

1. Introduction 1.1. The model. In the present paper, we investigate the asymptotic behavior of solutions to the spatially homogeneous Boltzmann equation without angular cut-off, that is, for long-range interactions. Previous works have shown that these solutions converge towards the Maxwellian equilibrium with a polynomial rate when time goes to infinity. Here, we are interested in improving the rate of convergence and we show an exponential decay to equilibrium.

We consider particles described by their space homogeneous distribution density f = f (t, v). We hence study the so-called spatially homogeneous Boltzmann equation:

(1.1) ∂ t f (t, v) = Q(f, f )(t, v), v ∈ R 3 , t ≥ 0.
The Boltzmann collision operator is defined as

Q(g, f ) :=
The Boltzmann collision kernel B(vv * , σ) only depends on the relative velocity |vv * | and on the deviation angle θ through cos θ = κ, σ where κ = (vv * )/|vv * | and •, • is the usual scalar product in R 3 . By a symmetry argument, one can always reduce to the case where B(vv * , σ) is supported on κ, σ ≥ 0 i.e 0 ≤ θ ≤ π/2. So, without loss of generality, we make this assumption.

In this paper, we shall be concerned with the case when the kernel B satisfies the following conditions:

• it takes product form in its arguments as

(1.2) B(v -v * , σ) = Φ(|v -v * |) b(cos θ);
• the angular function b is locally smooth, and has a nonintegrable singularity for θ → 0, it satisfies for some c b > 0 and s ∈ (0, 1/2) (moderate angular singularity) Our main physical motivation comes from particles interacting according to a repulsive potential of the form (1.5) φ(r) = r -(p-1) , p ∈ (2, +∞).

The assumptions made on B throughout the paper include the case of potentials of the form (1.5) with p > 5. Indeed, for repulsive potentials of the form (1.5), the collision kernel cannot be computed explicitly but Maxwell [START_REF] Maxwell | On the dynamical theory of gases[END_REF] has shown that the collision kernel can be computed in terms of the interaction potential φ. More precisely, it satisfies the previous conditions (1.2), (1.3) and (1.4) in dimension 3 (see [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Cercignani | The mathematical theory of dilute gases[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF]) with s := 1 p-1 ∈ (0, 1) and γ := p-5 p-1 ∈ (-3, 1). One traditionally calls hard potentials the case p > 5 (for which 0 < γ < 1), Maxwell molecules the case p = 5 (for which γ = 0) and soft potentials the case 2 < p < 5 (for which -3 < γ < 0). We can hence deduce that our assumptions made on B include the case of hard potentials. The equation (1.1) preserves mass, momentum and energy. Indeed, at least formally, we have:

R 3 Q(f, f )(v) ϕ(v) dv = 0 for ϕ(v) = 1, v, |v| 2 ;
from which we deduce that a solution f t to the equation (1.1) is conservative, meaning that

(1.6) ∀ t ≥ 0, R 3 f (t, v) ϕ(v) dv = R 3 f 0 (v) ϕ(v) dv for ϕ(v) = 1, v, |v| 2 .
We introduce the entropy H(f ) = R 3 f log(f ) and the entropy production D(f ). Boltzmann's H theorem asserts that (1.7) d dt H(f ) = -D(f ) ≤ 0 and states that any equilibrium (i.e any distribution which maximizes the entropy) is a Maxwellian distribution µ ρ,u,T for some ρ > 0, u ∈ R 3 and T > 0:

µ f (v) = µ ρ,u,T (v) = ρ e -|v-u| 2 2T (2πT ) 3/2 ,
where ρ, u and T are the mass, momentum and temperature of the gas:

ρ := ρ f = R 3 f (v) dv, u := u f = 1 ρ R 3 f (v) v dv, T := T f = 1 3ρ R 3 f (v) |v -u| 2 dv.
Thanks to the conservation properties of the equation (1.6), the following equalities hold:

ρ f = ρ f 0 , u f = u f 0 , T f = T f 0 where f 0 is the initial datum.
Moreover, a solution f t of the Boltzmann equation is expected to converge towards the Maxwellian distribution µ ρ,u,T when t → +∞.

In this paper, we only consider the case of an initial datum satisfying

(1.8) ρ f 0 = 1, u f 0 = 0, T f 0 = 1,
one can always reduce to this situation (see [START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF]). We then denote µ the Maxwellian with same mass, momentum and energy of f 0 :

µ(v) = (2π) -3/2 e -|v| 2 /2 .
1.2. Function spaces and notations. We introduce some notations about weighted L p spaces. For some given Borel weight function m ≥ 0 on R 3 , we define the Lebesgue weighted space L p (m), 1 ≤ p ≤ +∞, as the Lebesgue space associated to the norm

h L p (m) := h m L p .
We also define the weighted Sobolev space W s,p (m), s ∈ N, 1 ≤ p < +∞, as the Sobolev space associated to the norm

h W s,p (m) :=   |α|≤s ∂ α h p L p (m)   1/p .
Throughout this paper, we will use the same notation C for positive constants that may change from line to line. Moreover, the notation A ≈ B will mean that there exist two constants

C 1 , C 2 > 0 such that C 1 A ≤ B ≤ C 2 A.
1.3. Main results and known results.

1.3.1.

Convergence to equilibrium. We first state our main result on exponential convergence to equilibrium.

Theorem 1.1. Consider a collision kernel B satisfying conditions (1.2), (1.3), (1.4) and f 0 a nonnegative distribution with finite mass, energy and entropy:

f 0 ≥ 0, R 3 f 0 (v) (1 + |v| 2 ) dv < ∞, R 3 f 0 (v) log(f 0 (v)) dv < ∞
and satisfying (1.8). Then, if f t is a smooth solution (see Definition 1.2) to the equation (1.1) with initial datum f 0 , there exists a constant C > 0 such that

∀ t ≥ 0, f t -µ L 1 ≤ C e -λ t
where λ > 0 is defined in Theorem 1.4.

We improve a polynomial result of Villani [START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF] and generalize to our context similar exponential results known for simplified models. Mouhot in [START_REF] Mouhot | Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials[END_REF] proved such a result for the spatially homogeneous Boltzmann equation with hard potentials and Grad's cut-off. Carrapatoso in [START_REF] Carrapatoso | On the trend to equilibrium for the homogeneous landau equation with hard potentials[END_REF] recently proved exponential decay to equilibrium for the homogeneous Landau equation with hard potentials which is the grazing collisions limit of the model we study in the present paper. Let us also mention the paper of Gualdani et al. [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF] where an exponential decay to equilibrium is proved for the inhomogeneous Boltzmann equation for hard spheres (see also [START_REF] Mischler | Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF][START_REF] Mischler | Semigroup factorisation in banach spaces and kinetic hypoelliptic equations[END_REF] for related works).

It is a known fact that our equation (1.1) admits solutions which are conservative and satisfy some suitable properties of regularity, we will call them smooth solutions. We here precise the meaning of this term and give an overview of results on the Cauchy theory of our equation. Definition 1.2. Let f 0 be a nonnegative function defined on R 3 with finite mass, energy and entropy. We shall say that (t, v) → f (t, v) is a smooth solution to the equation (1.1) if the following conditions are fulfilled:

• f ≥ 0, f ∈ C(R + , L 1 ); • for any t ≥ 0, R 3 f (t, v) ϕ(v) dv = R 3 f 0 (v) ϕ(v) dv for ϕ(v) = 1, v, |v| 2 and R 3 f (t, v) log(f (t, v)) dv + t 0 D(f (s, •)) ds ≤ R 3 f 0 (v) log(f 0 (v)) dv ;
where D(f ) is the entropy production defined in (1.7); • for any ϕ ∈ C 1 (R + , D(R 3 )) and for any t ≥ 0,

R 3 f (t, v) ϕ(t, v) dv = R 3 f 0 (v) ϕ(0, v) dv + t 0 R 3 f (τ, v) ∂ t ϕ(τ, v) dv dτ + t 0 R 3 Q(f, f )(τ, v) ϕ(τ, v) dv dτ
where the last integral is define through the following formula

R 3 Q(f, f )(v) ϕ(v) dv = 1 4 R 3 ×R 3 ×S 2 B(v -v * , σ) [f ′ * f ′ -f * f ] (ϕ + ϕ * -ϕ ′ -ϕ ′ * ) dσ dv * dv;
• for any t 0 > 0 and for any ℓ ∈ R + , (1.9) sup

t≥t 0 f (t, •) L 1 ( v ℓ ) < ∞ ;
• for any t 0 > 0 and for any N , ℓ ∈ R + ,

(1.10) sup t≥t 0 f (t, •) H N ( v ℓ ) < ∞.
Such a solution is known to exist. The problem of existence of solutions was first studied by Arkeryd in [START_REF] Arkeryd | Intermolecular forces of infinite range and the Boltzmann equation[END_REF] where existence of solutions is proven for not too soft potentials, that is γ > -1 (Goudon [START_REF] Goudon | On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions[END_REF] and Villani [START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF] then improved this result enlarging the class of γ considered). We mention that uniqueness of solution for hard potentials can be proved under some more restrictive conditions on the initial datum, see the paper of Desvillettes and Mouhot [START_REF] Desvillettes | Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF] where f 0 is supposed to be regular (f 0 ∈ W 1,1 ( v 2 )) and the paper of Fournier and Mouhot [START_REF] Fournier | On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity[END_REF] where f 0 is supposed to be localized ( R 3 f 0 e a|v| γ dv < ∞ for some a > 0) for hard potentials.

Concerning the moment production property, it was discovered by Elmroth [START_REF] Elmroth | Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range[END_REF] and Desvillettes [START_REF] Desvillettes | Some applications of the method of moments for the homogeneous Boltzmann and Kac equations[END_REF] and improved by Wennberg [START_REF] Wennberg | Entropy dissipation and moment production for the Boltzmann equation[END_REF], which justifies our point (1.9) in the definition of a smooth solution. We here point out the fact that this property is not anymore true for Maxwell molecules or soft potentials. As a consequence, our method, which relies partially on this property, works only for hard potentials.

Finally, we mention papers where regularization results are proven for "true" (that is non mollified) physical potentials: [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] by Alexandre et al. and [13] by Chen and He where the initial datum is supposed to have finite energy and entropy, [START_REF] Bally | Regularization properties of the 2D homogeneous Boltzmann equation without cutoff[END_REF] by Bally and Fournier where only the 2D case is treated and [START_REF] Fournier | Finiteness of entropy for the homogeneous boltzmann equation with measure initial condition[END_REF] by Fournier under others conditions on the initial datum. Theorem 1.4 from [START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case[END_REF] explains our point (1.10).

We now recall previous results on convergence to equilibrium for solutions to equation (1.1). It was first studied by Carlen and Carvalho [START_REF] Carlen | Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation[END_REF][START_REF] Carlen | Entropy production estimates for Boltzmann equations with physically realistic collision kernels[END_REF] and then by Toscani and Villani [START_REF] Toscani | Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation[END_REF]. Up to now, the best rate of convergence in our case was obtained by Villani in [START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF]: Theorem 1.3. Let us consider f t a smooth solution to (1.1) with an initial datum f 0 satisfying (1.8) with finite entropy. Then f t satisfies the following polynomial decay to equilibrium: for any t 0 > 0 and any ε > 0, there exists C t 0 ,ε > 0 such that

∀ t ≥ t 0 , f t -µ L 1 ≤ C t 0 ,ε t -1 ε .
This result comes from [START_REF] Villani | Cercignani's conjecture is sometimes true and always almost true[END_REF]Theorem 4.1] which states that if f is a function which satisfies the following lowerbound (1.11) ∀ v ∈ R 3 , f (v) ≥ K 0 e -A 0 |v| q 0 with K 0 , A 0 > 0, q 0 ≥ 2 then for any ε > 0, there exists an explicit constant K ε > 0 such that

(1.12) D(f ) ≥ K ε H(f |µ) 1+ε .
It is a result from Mouhot [30, Theorem 1.2] that the lowerbound (1.11) holds for any smooth solution f t of our equation (1.1). Let us mention that lowerbounds of solutions were first studied by Carleman [START_REF] Carleman | Sur la théorie de l'équation intégrodifférentielle de Boltzmann[END_REF] (for hard spheres) and then by Pulvirenti and Wennberg [START_REF] Pulvirenti | A Maxwellian lower bound for solutions to the Boltzmann equation[END_REF] (for hard potentials with cut-off). Finally, Mouhot [START_REF] Mouhot | Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions[END_REF] extended these results to the spatially inhomogeneous case without cut-off. We here state Theorem 1.2 from [START_REF] Mouhot | Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions[END_REF] that we use: for any t 0 > 0 and for any exponent q 0 such that

q 0 > 2 log 2 + 2s 1-s log 2 , a smooth solution f t to (1.1) satisfies ∀ t ≥ t 0 , ∀ v ∈ R 3 , f (t, v) ≥ K 0 e -A 0 |v| q 0 for some K 0 , A 0 > 0.
We can then deduce that the conclusion of Theorem 1.3 holds using the Csiszár-Kullback-Pinsker inequality fµ L 1 ≤ 2H(f |µ) combined with the result of Villani (1.12).

Let us here emphasize that the method of Villani to prove the polynomial convergence towards equilibrium is purely nonlinear. Ours is based on the study of the linearized equation.

1.3.2. The linearized equation. We introduce the linearized operator. Considering the linearization f = µ + h, we obtain at first order the linearized equation around the equilibrium µ

(1.13) ∂ t h = Lh := Q(µ, h) + Q(h, µ), for h = h(t, v), v ∈ R 3 . The null space of the operator L is the 5-dimensional space (1.14) N (L) = Span µ, µ v 1 , µ v 2 , µ v 3 , µ |v| 2 .
Our strategy is to combine the polynomial convergence to equilibrium and a spectral gap estimate on the linearized operator to show that if the solution enters some stability neighborhood of the equilibrium, then the convergence is exponential in time. Previous results on spectral gap estimates hold only in L 2 (µ -1/2 ) and the Cauchy theory for the nonlinear Boltzmann equation is constructed in L 1 -spaces with polynomial weight. In order to link the linear and the nonlinear theories, our approach consists in proving new spectral gap estimates for the linearized operator L in spaces of type L 1 ( v k ). To do that, we exhibit a convenient splitting of the linearized operator in such a way that we may use the abstract theorem from [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF] which allows us to enlarge the space of spectral estimates of a given operator.

Here is the result we obtain on the linearized equation which provides a constructive spectral gap estimate for L in L 1 ( v k ) and which is the cornerstone of the proof of Theorem 1.1.

Theorem 1.4. Let k > 2 and a collision kernel B satisfying (1.2), (1.3) and (1.4). Consider the linearized Boltzmann operator L defined in (1.13). Then for any positive λ < min(λ 0 , λ k ) (where λ 0 is the spectral gap of L in L 2 (µ -1/2 ) defined in Proposition 2.1 and λ k is a constant depending on k defined in Lemma 2.7), there exists an explicit constant C λ > 0, such that for any h ∈ L 1 ( v k ), we have the following estimate

(1.15) ∀ t ≥ 0, S L (t)h -Πh L 1 ( v k ) ≤ C λ e -λt h -Πh L 1 ( v k ) ,
where S L (t) denotes the semigroup of L and Π the projection onto N (L).

Let us briefly review the existing results concerning spectral gap estimates for L. Pao [START_REF] Pao | Boltzmann collision operator with inverse-power intermolecular potentials[END_REF] studied spectral properties of the linearized operator for hard potentials by nonconstructive and very technical means. This article was reviewed by Klaus [START_REF] Klaus | Boltzmann collision operator without cut-off[END_REF]. Then, Baranger and Mouhot gave the first explicit estimate on this spectral gap in [START_REF] Baranger | Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials[END_REF] for hard potentials (γ > 0). If we denote D the Dirichlet form associated to -L:

D(h) := R 3 (-Lh) h µ -1 ,
and N (L) ⊥ the orthogonal of N (L) defined in (1.14) and Π the projection onto N (L), the Dirichlet form D satisfies

(1.16) ∀ h ∈ N (L) ⊥ , D(h) ≥ λ 0 h 2 L 2 (µ -1/2 ) ,
for some constructive constant λ 0 > 0. This result was then improved by Mouhot [START_REF] Mouhot | Explicit coercivity estimates for the linearized Boltzmann and Landau operators[END_REF] and later by Mouhot and Strain [START_REF] Mouhot | Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff[END_REF]. In the last paper, it was conjectured that a spectral gap exists if and only if γ + 2s ≥ 0. This conjecture was finally proven by Gressman and Strain in [START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF].

Another question would be to obtain similar results in other spaces: L p spaces with 1 < p ≤ 2 and a polynomial weight or L p spaces with 1 ≤ p ≤ 2 and a stretched exponential weight. Our computations do not allow to conclude in those cases, more precisely, we are not able to do the computations which allow to obtain the suitable splitting of the linear operator in order to apply the theorem of enlargement of the space of spectral estimates. As a consequence, we can not prove the existence of a spectral gap on those spaces. However, we believe that such results may hold.

We here point out that the knowledge of a spectral gap estimate in L 1 ( v k ) for the fractional Fokker-Planck equation (see [START_REF] Tristani | Fractional fokker-planck equation[END_REF]) is consistent with our result. Indeed, the behavior of the Boltzmann collision operator has been widely conjectured to be that of a fractional diffusion (see [START_REF] Desvillettes | On asymptotics of the boltzmann equation when the collisions become grazing[END_REF][START_REF] Goudon | On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions[END_REF][START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF]).

The linearized equation

Here and below, we denote m(v) := v k with k > 2. The aim of the present section is to prove Theorem 1.4. To do that, we exhibit a splitting of the linearized operator into two parts, one which is bounded and the second one which is dissipative. We can then apply the abstract theorem of enlargement of the functional space of the semigroup decay from Gualdani et al. [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF] (see Subsection 2.4).

2.1. Notations. We now introduce notations about spectral theory of unbounded operators. For a given real number a ∈ R, we define the half complex plane

∆ a := {z ∈ C, ℜe z > a} .
For some given Banach spaces (E, • E ) and (E, • E ), we denote by B(E, E) the space of bounded linear operators from E to E and we denote by • B(E,E) or • E→E the associated norm operator. We write B(E) = B(E, E) when E = E. We denote by C (E, E) the space of closed unbounded linear operators from E to E with dense domain, and

C (E) = C (E, E) in the case E = E.
For a Banach space X and Λ ∈ C (X) we denote by S Λ (t), t ≥ 0, its semigroup, by D(Λ) its domain, by N(Λ) its null space and by R(Λ) its range. We also denote by Σ(Λ) its spectrum, so that for any z belonging to the resolvent set ρ(Λ) := C\Σ(Λ) the operator Λz is invertible and the resolvent operator

R Λ (z) := (Λ -z) -1
is well-defined, belongs to B(X) and has range equal to D(Λ). An eigenvalue ξ ∈ Σ(Λ) is said to be isolated if Σ(Λ) ∩ {z ∈ C, |z -ξ| ≤ r} = {ξ} for some r > 0.

In the case when ξ is an isolated eigenvalue, we may define Π Λ,ξ ∈ B(X) the associated spectral projector by

Π Λ,ξ := - 1 2iπ |z-ξ|=r ′ R Λ (z) dz with 0 < r ′ < r.
Note that this definition is independent of the value of r ′ as the application

C \ Σ(Λ) → B(X), z → R Λ (z) is holomorphic.
For any ξ ∈ Σ(Λ) isolated, it is well-known (see [START_REF] Kato | Perturbation theory for linear operators[END_REF] paragraph III-6.19) that Π 2 Λ,ξ = Π Λ,ξ , so that Π Λ,ξ is indeed a projector. When moreover the so-called "algebraic eigenspace" R(Π Λ,ξ ) is finite dimensional we say that ξ is a discrete eigenvalue, written as ξ ∈ Σ d (Λ).

2.2. Spectral gap in L 2 (µ -1/2 ). We here state a direct consequence of inequality (1.16) from [START_REF] Baranger | Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials[END_REF], which gives us a spectral gap estimate in L 2 (µ -1/2 ). Proposition 2.1. There is a constructive constant λ 0 > 0 such that

∀ t ≥ 0, ∀ h ∈ L 2 (µ -1/2 ), S L (t)h -Πh L 2 (µ -1/2 ) ≤ e -λ 0 t h -Πh L 2 (µ -1/2 ) .
2.3. Splitting of the linearized operator. We first split the linearized operator L defined in (1.13) into two parts, separating the grazing collisions and the cut-off part, we define b δ := 1 θ≤δ b and b c δ := 1 θ≥δ b for some δ ∈ (0, 1) to be chosen later, it induces the following splitting of L:

Lh = L δ h + L c δ h =: R 3 ×S 2 µ ′ * h ′ -µ * h + h ′ * µ ′ -h * µ b δ (cos θ)|v -v * | γ dσ dv * + R 3 ×S 2 µ ′ * h ′ -µ * h + h ′ * µ ′ -h * µ b c δ (cos θ)|v -v * | γ dσ dv * .
In the rest of the paper, we shall use the notations

B δ (v -v * , σ) := b δ (cos θ) |v -v * | γ and B c δ (v -v * , σ) := b c δ (cos θ) |v -v * | γ .
As far as the cut-off part is concerned, our strategy is similar as the one adopted in [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF] for hard-spheres. For any ε ∈ (0, 1), we consider Θ ε ∈ C ∞ bounded by one, which equals one on

|v| ≤ ε -1 and 2ε ≤ |v -v * | ≤ ε -1 and | cos θ| ≤ 1 -2ε
and whose support is included in

|v| ≤ 2ε -1 and ε ≤ |v -v * | ≤ 2ε -1 and | cos θ| ≤ 1 -ε .
We then denote the truncated operator

A δ,ε (h) := R 3 ×S 2 Θ ε µ ′ * h ′ + µ ′ h ′ * -µ h * b c δ (cos θ)|v -v * | γ dσ dv *
and the corresponding remainder operator

B c δ,ε (h) := R 3 ×S 2 (1 -Θ ε ) µ ′ * h ′ + µ ′ h ′ * -µ h * b c δ (cos θ)|v -v * | γ dσ dv * .
We also introduce

ν δ (v) := R 3 ×S 2 µ * b c δ (cos θ)|v -v * | γ dσ dv * ,
so that we have the following splitting:

L c δ = A δ,ε + B c δ,ε -ν δ . Moreover, ν δ satisfies ν δ (v) = K δ (µ * | • | γ )(v)
with

K δ := S 2 b c δ (cos θ) dσ ≈ π/2 δ b(cos θ) sin θ dθ ≈ δ -2s - π 2 -2s
---→ δ→0 +∞ using the spherical coordinates to get the second equality and (1.3) to get the final one; and

(µ * | • | γ )(v) ≈ v γ .
We finally define

B δ,ε := L δ + B c δ,ε -ν δ so that L = A δ,ε + B δ,ε .

Dissipativity properties.

Lemma 2.2. There exists a function ϕ k (δ) depending on k and tending to 0 as δ tends to 0 such that for any h ∈ L 1 ( v γ m), the following estimate holds:

(2.1) R 3 L δ (h) sign(h) m dv ≤ ϕ k (δ) h L 1 ( v γ m) .
Proof. Let us first introduce a notation which is going to be useful in the sequel of the proof:

(2.2)

κ δ := π/2 0 b δ (cos θ) sin 2 (θ) dθ = δ 0 b(cos θ) sin 2 (θ) dθ ≈ δ 1-2s ---→ δ→0 0,
where the last equality comes from (1.3). We here underline the fact that considering a moderate singularity, meaning s ∈ (0, 1/2), is here needed to get the convergence of κ δ to 0 as δ goes to 0.

We split L δ into two parts in the following way:

L δ h = R 3 ×S 2 µ ′ * h ′ -µ * h b δ (cos θ)|v -v * | γ dσ dv * + R 3 ×S 2 h ′ * µ ′ -h * µ b δ (cos θ)|v -v * | γ dσ dv * =: L 1 δ h + L 2 δ h,
this splitting corresponds to the splitting of L δ as Q δ (µ, h) + Q δ (h, µ) if Q δ denotes the collisional operator associated to the kernel B δ .

We first deal with L 1 δ . Let us recall that we have µ µ * = µ ′ µ ′ * . In the following computation, we denote g := h µ -1 :

R 3 L 1 δ (h) sign(h) m dv = R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g ′ -g sign(g) m dσ dv * dv = R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g ′ -g sign(g) -sign(g ′ ) m dσ dv * dv + R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g ′ -g sign(g ′ ) m dσ dv * dv ≤ R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g ′ -g sign(g ′ ) m dσ dv * dv,
where we used that for any a, b ∈ R, (a-b)(sign(a)-sign(b)) ≤ 0 to get the last inequality.

Remark 2.3. We here emphasize that this computation is particularly convenient in the L 1 case since sign(h) = sign(g). In the L p case, it is trickier and for now, we are not able to adapt it to get the wanted estimates.

We now use the classical pre-post collisional change of variables to pursue the computation:

R 3 L 1 δ (h) sign(h) m dv ≤ R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g -g ′ sign(g) m ′ dσ dv * dv = R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g -g ′ sign(g) (m ′ -m) dσ dv * dv + R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g -g ′ sign(g) m dσ dv * dv. = R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g -g ′ sign(g) (m ′ -m) dσ dv * dv - R 3 L 1 δ (h) sign(h) m dv.
We hence deduce that

R 3 L 1 δ (h) sign(h) m dv ≤ 1 2 R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * g -g ′ sign(g) (m ′ -m) dσ dv * dv ≤ R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ µ * |g| |m ′ -m| dσ dv * dv = R 3 ×R 3 ×S 2 B δ (v -v * , σ) µ * |h| |m ′ -m| dσ dv * dv.
We now estimate the difference |m ′ -m|:

|m ′ -m| ≤ sup z∈B(v,|v ′ -v|) |∇m| (z) |v ′ -v|, with |v ′ -v| = |v -v * |/2 sin (θ/2) ≤ 1 2 √ 2 |v -v * | sin θ.
Then, we use the fact

sup z∈B(v,|v ′ -v|) |∇m| (z) ≤ k 2 k-1 v k-2 + v -v ′ k-1 ≤ k 2 2(k-1) v k-2 + v * k-1 , which implies that (2.3) |m ′ -m| ≤ C k |v -v * | sin θ v k-1 + v * k-1 ,
for some constant C k > 0 depending on k.

Remark 2.4. We here point out that this kind of estimate does not hold in the case of a stretched exponential weight. Indeed, taking the gradient of a stretched exponential function, there is not anymore a gain in the degree as in the case of a polynomial function.

We finally obtain (2.4)

R 3 L 1 δ (h) sign(h) m dv ≤ C k R 3 ×R 3 ×S 2 b δ (cos θ) sin θ µ * |v -v * | γ+1 v k-1 + v * k-1 |h| dσ dv * dv ≤ C k π/2 0 b δ (cos θ) sin 2 (θ) dθ 2π 0 dφ R 3 ×R 3 µ * |v -v * | γ+1 v k-1 + v * k-1 |h| dv * dv ≤ C k κ δ R 3 |h| v γ m dv,
where we used spherical coordinates to obtain the second inequality and (2.2) to obtain the last one.

We now deal with L 2 δ . We split it into two parts:

L 2 δ h = R 3 ×S 2 h ′ * µ ′ -h * µ b δ (cos θ)|v -v * | γ dσ dv * = R 3 ×S 2 B δ (v -v * , σ) h ′ * µ ′ -µ dσ dv * + R 3 ×S 2 B δ (v -v * , σ) h ′ * -h * dσ dv * µ =: L 2,1 δ h + L 2,2 δ h. Concerning L 2,2 δ , we use the cancellation lemma [1, Lemma 1]. It implies that L 2,2 δ h = (S δ * h) µ with S δ (z) = 2π π/2 0 sin θ b δ (cos θ) |z| γ cos γ+3 (θ/2) -|z| γ dθ = 2π |z| γ π/2 0 sin θ b δ (cos θ) 1 -cos γ+3 (θ/2) cos γ+3 (θ/2) dθ = 2π |z| γ δ 0 sin θ b(cos θ) 1 -cos γ+3 (θ/2) cos γ+3 (θ/2) dθ ≤ C |z| γ δ 0 sin θ b(cos θ) θ 2 dθ ≤ C δ 2-2s |z| γ ,
where the next-to-last inequality comes from the fact that 1-cos γ+3 (θ/2) cos γ+3 (θ/2) ∼ γ+3 2 θ 2 as θ goes to 0. We hence deduce that for any θ ∈ (0, δ), 1-cos γ+3 (θ/2) cos γ+3 (θ/2) ≤ C θ 2 for some C > 0; and the last inequality comes from (1.3). We deduce that (2.5)

R 3 L 2,2 δ (h) sign(h) m dv ≤ R 3 |S δ * h| m µ dv ≤ C δ 2-2s R 3 (| • | γ * |h|) µ m dv ≤ C δ 2-2s R 3 (| • | γ * µ m) |h| dv ≤ C δ 2-2s R 3 |h| v γ dv.
We now deal with L 2,1 δ . To do that, we introduce the notation M := √ µ and write that µ

′ -µ = (M ′ -M )(M ′ + M ), which implies R 3 L 2,1 δ (h) sign(h) m dv ≤ R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h ′ * | |M ′ -M | (M ′ + M ) m dσ dv * dv ≤ R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h ′ * | |M ′ -M | M ′ |m ′ -m| dσ dv * dv + R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h ′ * | |M ′ -M | M ′ m ′ dσ dv * dv + R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h ′ * | |M ′ -M | M m dσ dv * dv.
We now perform the pre-post collisional change of variables, which gives us:

R 3 L 2,1 δ h sign(h) m dv ≤ R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h * | |M ′ -M | M |m ′ -m| dσ dv * dv + R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h * | |M ′ -M | M m dσ dv * dv + R 3 ×R 3 ×S 2 B δ (v -v * , σ) |h * | |M ′ -M | M ′ m ′ dσ dv * dv =: I 1 + I 2 + I 3 .
For the term I 1 , we use the fact that M is bounded and the estimate (2.3) on |m ′ -m|:

(2.6)

I 1 ≤ C k R 3 ×R 3 ×S 2 b δ (cos θ) sin θ |v -v * | γ+1 |h * | M v k-1 + v * k-1 dσ dv * dv ≤ C k κ δ R 3 |h| v γ m dv.
The term I 2 is treated using that M is Lipschitz continuous, we obtain:

(2.7)

I 2 ≤ C κ δ R 3 |h| v γ+1 dv.
To treat I 3 , we first estimate the integral

R 3 ×S 2 B δ (v -v * , σ) |M ′ -M | M ′ m ′ dσ dv =: J(v * ) = J.
Using the fact that M is Lipschitz continuous, we have

J ≤ C R 3 ×S 2 b δ (cos θ) sin(θ/2) |v -v * | γ+1 M ′ m ′ dσ dv.
Then, for each σ, with v * still fixed, we perform the change of variables v → v ′ . This change of variables is well-defined on the set {cos θ > 0}. Its Jacobian determinant is

dv ′ dv = 1 8 (1 + κ • σ) = (κ ′ • σ) 2 4
,

where κ = (v -v * )/|v -v * | and κ ′ = (v ′ -v * )/|v ′ -v * |. We have κ ′ • σ = cos(θ/2) ≥ 1/ √ 2. The inverse transformation v ′ → ψ σ (v ′ ) = v is then defined accordingly. Using the fact that cos θ = κ • σ = 2(κ ′ • σ) 2 -1 and sin(θ/2) = 1 -cos 2 (θ/2) = 1 -(κ ′ • σ) 2 , we obtain J ≤ C R 3 ×S 2 b δ (2(κ ′ • σ) 2 -1) 1 -(κ ′ • σ) 2 |ψ σ (v ′ ) -v * | γ+1 M (v ′ ) m(v ′ ) dv dσ ≤ C κ ′ •σ≥1/ √ 2 b δ (2(κ ′ • σ) 2 -1) 1 -(κ ′ • σ) 2 |ψ σ (v ′ ) -v * | γ+1 M (v ′ )m(v ′ ) 1 (κ ′ • σ) 2 dv ′ dσ ≤ C κ•σ≥1/ √ 2 b δ (2(κ • σ) 2 -1) 1 -(κ • σ) 2 |ψ σ (v) -v * | γ+1 M (v) m(v) 1 (κ • σ) 2 dv dσ. We now use the fact that |ψ σ (v) -v * | = |v -v * |/(κ • σ). We deduce that J ≤ C κ•σ≥1/ √ 2 b δ (2(κ • σ) 2 -1) 1 -(κ • σ) 2 |v -v * | γ+1 M (v) m(v) 1 (κ • σ) γ+3 dv dσ ≤ C R 3 ×S 2 b δ (2(κ • σ) 2 -1) 1 -(κ • σ) 2 |v -v * | γ+1 M (v) m(v) dv dσ
where we used the fact that κ

• σ ≥ 1/ √ 2 to bound from above 1/(κ • σ) γ+3 . Using the equalities cos(2θ) = 2(κ • σ) 2 -1 and sin θ = 1 -(κ • σ) 2 ,
we obtain

J ≤ C R 3 ×S 2 b δ (cos(2θ)) sin θ |v -v * | γ+1 M m dv dσ ≤ C S 2 b δ (cos(2θ)) sin θ dσ R 3 v γ+1 M m dv v * γ+1 ≤ C κ δ v * γ+1 ,
Using this last estimate, we can conclude that (2.8)

I 3 ≤ C κ δ R 3 |h| v γ+1 dv.
Gathering estimates (2.4), (2.5), (2.6), (2.7) and (2.8), we can conclude that (2.1) holds.

We now want to deal with the part B c δ,εν δ . To do that, we shall review a classical tool in the Boltzmann theory, a version of the Povzner lemma (see [START_REF] Wennberg | The Povzner inequality and moments in the Boltzmann equation[END_REF][START_REF] Bobylev | Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems[END_REF][START_REF] Mischler | On the spatially homogeneous Boltzmann equation[END_REF][START_REF] Bobylev | Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions[END_REF]). The version stated here is a consequence of the proof of Lemma 2.2 from [START_REF] Mischler | On the spatially homogeneous Boltzmann equation[END_REF].

Lemma 2.5. For any k > 2, ∀ v, v * ∈ R 3 , S 2 v ′ * k + v ′ k -v * k -v k b c δ (cos θ) dσ ≤ C k v * k-1 v + v k-1 v * -C ′ k |v| k
for some constants C k , C ′ k > 0 depending on k. Proof. If we adapt the proof of Lemma 2.2 from [START_REF] Mischler | On the spatially homogeneous Boltzmann equation[END_REF] taking ψ = • , we obtain

S 2 v ′ * k + v ′ k -v * k -v k b c δ (cos θ) dσ ≤ C k π/2 0 b c δ (cos θ) sin 2 (θ) dθ v * k-1 v + v k-1 v * -C ′ k,δ |v| k with C ′ k,δ ---→ δ→0
+∞ and C ′ k,δ ≥ C ′ k > 0 for any δ ∈ (0, 1). We then conclude using (1.3) which implies that

π/2 0 b c δ (cos θ) sin 2 (θ) dθ ≈ π 2 1-2s -δ 1-2s ≤ C
for any δ ∈ (0, 1).

We can now prove the following estimate on B c δ,εν δ .

Lemma 2.6. For any k > 2, for any ε ∈ (0, 1) and for δ ∈ (0, 1) small enough, we have the following estimate (2.9)

∀ h ∈ L 1 ( v γ m), R 3 B c δ,ε (h) sign(h) m dv- R 3 ν δ |h| m dv ≤ (Λ k,δ (ε) -λ k ) h L 1 ( v γ m)
where λ k > 0 is a constant depending on k and Λ k,δ (ε) is a constant depending on k and δ which tends to 0 as ε goes to 0 when k and δ are fixed.

Proof. We compute

B c δ,ε h L 1 (m) ≤ R 3 ×R 3 ×S 2 (1 -Θ ε ) B c δ (v -v * , σ) µ ′ * |h ′ | + µ ′ |h ′ * | + µ|h * | m dσ dv * dv.
We first bound from above the truncation function (1 -Θ ε ):

B c δ,ε h L 1 (m) ≤ {| cos θ|∈[1-ε,1]} B c δ (v -v * , σ) µ * |h| (m ′ + m ′ * + m * ) dσ dv * dv + {|v-v * |≤ε} B c δ (v -v * , σ) µ * |h| (m ′ + m ′ * + m * ) dσ dv * dv + {|v|≥ε -1 or |v-v * |≥ε -1 } B c δ (v -v * , σ) µ ′ * |h ′ | + µ ′ |h ′ * | + µ|h * | m dσ dv * dv,
where the pre-post collisional change of variables has been used in the two first terms. We obtain that B c δ,ε h L 1 (m) is bounded from above by (2.10)

C k {| cos θ|∈[1-ε,1]} 1 θ≥δ b(cos θ) dσ + K δ ε γ R 3 ×R 3 µ * v * γ+k |h| v γ+k dv * dv + R 3 ×R 3 ×S 2 χ ε -1 B c δ (v -v * , σ) µ ′ * |h ′ | + µ ′ |h ′ * | + µ|h * | m dσ dv * dv =: J 1 + J 2
where χ ε -1 is the characteristic function of the set

|v| 2 + |v * | 2 ≥ ε -1 or |v -v * | ≥ ε -1 .
The first term of the right hand side of (2.10) is easily controlled as

(2.11) J 1 ≤ C k C δ ε γ h L 1 ( v γ m) .
As far as the second term in (2.10) is concerned, we write

J 2 = R 3 ×R 3 ×S 2 χ ε -1 B c δ (v -v * , σ) µ ′ * |h ′ | + µ ′ |h ′ * | + µ|h * | m dσ dv * dv = R 3 ×R 3 ×S 2 χ ε -1 B c δ (v -v * , σ) µ ′ * |h ′ | + µ ′ |h ′ * | -µ * |h| -µ|h * | m dσ dv * dv + K δ R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ m dv * dv + 2 K δ R 3 ×R 3 χ ε -1 µ|h * | |v -v * | γ m dv * dv =: T 1 + T 2 + T 3 .
We notice that the characteristic function χ ε -1 is invariant under the usual pre-post collisional change of variables as it only depends on the kinetic energy and momentum. We hence bound the term T 1 thanks to Lemma 2.5:

T 1 ≤ R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ S 2 v ′ * k + v ′ k -v * k -v k b c δ (cos θ) dσ dv * dv ≤ C k R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ v k-1 v * + v v * k-1 dv * dv -C ′ k R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ |v| k dv * dv ≤ C k R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ v k-1 v * + v v * k-1 dv * dv + C ′ k R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ dv * dv -C ′ k 2 1-k/2 R 3 ×R 3 χ ε -1 µ * |h| |v -v * | γ v k dv * dv =: T 11 + T 12 + T 13 .
We treat together the terms T 11 , T 12 and T 3 using the following inequality:

χ ε -1 (v, v * ) ≤ 1 {|v|≥ε -1 /2} + 1 {|v * |≥ε -1 /2} ≤ 2 ε(|v| + |v * |).
We obtain:

(2.12)

T 11 + T 12 + T 3 ≤ ε C k R 3 ×R 3 (|v| + |v * |) µ * |h| |v -v * | γ v k-1 v * + v v * k-1 dv * dv + ε C ′ k R 3 ×R 3 (|v| + |v * |) µ * |h| |v -v * | γ dv * dv + ε K δ R 3 ×R 3 (|v| + |v * |) µ|h * | |v -v * | γ m dv * dv ≤ ε C k C δ h L 1 ( v γ m) .
Gathering (2.11) and (2.12), we conclude that (2.13)

J 1 + T 11 + T 12 + T 3 ≤ C k C δ (ε + ε γ ) h L 1 ( v γ m) =: Λ k,δ (ε) h L 1 ( v γ m) .
We now put together the terms T 13 , T 2 and the term coming from ν δ , their sum is bounded from above by

-K δ R 3 ×R 3 (1-χ ε -1 ) µ * |v-v * | γ |h| m dv * dv -C ′ k 2 1-k/2 R 3 ×R 3 χ ε -1 µ * |v-v * | γ |h| m dv * dv.
Since K δ → ∞ as δ → 0, we can take δ small enough so that K δ ≥ C ′ k 2 1-k/2 , we obtain the following bound:

(2.14) -C ′ k 2 1-k/2 R 3 (µ * | • | γ ) |h| m dv ≤ -λ k h L 1 ( v γ m) .
Combining the bounds obtained in (2.13) and (2.14), we can conclude that (2.9) holds, which concludes the proof.

We can now prove the dissipativity properties of B δ,ε = L δ + B c δ,εν δ . Lemma 2.7. Let us consider a ∈ (-λ k , 0) where λ k is defined in Lemma 2.6. For δ > 0 and ε > 0 small enough, B δ,εa is dissipative in L 1 (m), namely

∀ t ≥ 0, S B δ,ε (t) L 1 (m)→L 1 (m) ≤ e at .
Proof. Gathering results coming from lemmas 2.2 and 2.6, we obtain

R 3 B δ,ǫ (h) sign(h) m dv ≤ R 3 (ϕ k (δ) + Λ k,δ (ε) -λ k ) |h| v γ m dv
We first take δ small enough so that ϕ k (δ) ≤ (a + λ k )/2. We then chose ε small enough so that Λ k,δ (ε) ≤ (a + λ k )/2. With this choice of δ and ε, we have the following inequality:

ϕ k (δ) + Λ k,δ (ε) -λ k ≤ a.
It implies that

R 3 B δ,ǫ (h) sign(h) m dv ≤ a h L 1 ( v γ m) ,
which concludes the proof.

Regularization properties.

We first state a regularity estimate on the truncated operator A δ,ε which comes from [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF]Lemma 4.16].

Lemma 2.8. The operator A δ,ε maps L 1 ( v ) into L 2 functions with compact support. In particular, we can deduce that A δ,ε ∈ B L 2 µ -1/2 and A δ,ε ∈ B L 1 (m) .

We now study the regularization properties of T (t) := A δ,ε S B δ,ε (t).

Lemma 2.9. Consider a ∈ (-λ k , 0). For a choice of δ, ε such that the conclusion of Lemma 2.7 holds, there exists a constant C > 0 such that

T (t)h L 2 (µ -1/2 ) ≤ C e at h L 1 (m) .
Proof. We here use Lemma 2.8. We introduce a constant R > 0 such that for any h in L 1 ( v ), supp (Ah) ⊂ B(0, R). We then compute

T (t)h L 2 (µ -1/2 ) ≤ C B(0,R) (T (t)h) 2 dv 1/2 ≤ C S B δ,ε (t)h L 1 ( v ) ≤ C S B δ,ε (t)h L 1 (m) ≤ C e at h L 1 (m) ,
where the last inequality comes from Lemma 2.7.

2.4. Spectral gap in L 1 ( v k ).

The abstract theorem.

Let us now present an enlargement of the functional space of a quantitative spectral mapping theorem (in the sense of semigroup decay estimate).

The aim is to enlarge the space where the decay estimate on the semigroup holds. The version stated here comes from [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF]Theorem 2.13].

Theorem 2.10. Let E, E be two Banach spaces such that E ⊂ E with dense and continuous embedding, and consider L ∈ C (E), L ∈ C (E) with L |E = L and a ∈ R. We assume:

(1) L generates a semigroup S L (t) and

Σ(L) ∩ ∆ a = {ξ} ⊂ Σ d (L)
for some ξ ∈ C and La is dissipative on R (Id -Π L,ξ ). (2) There exist A, B ∈ C (E) such that L = A + B (with corresponding restrictions A and B on E) and a constant C a > 0 so that

(i) B -a is dissipative on E, (ii) A ∈ B(E) and A ∈ B(E), (iii) T (t) := AS B (t) satisfies ∀ t ≥ 0, T (t) B(E,E) ≤ C a e at .
Then the following estimate on the semigroup holds:

∀ a ′ > a, ∀ t ≥ 0, S L (t) -S L (t)Π L,ξ B(E) ≤ C a ′ e a ′ t
where C a ′ > 0 is an explicit constant depending on the constants from the assumptions. 2)-(iii) from Lemma 2.9. We can conclude that estimate (1.15) holds.

The nonlinear equation

We first establish bilinear estimates on the collisional operator and we then prove our main result: Theorem 1.1.

3.1. The bilinear estimates. Proposition 3.1. Let B satisfying (1.2), (1.3) and (1.4). Then

Q(h, h) L 1 (m) ≤ C h L 1 ( v γ m) h L 1 (m) + h L 1 ( v γ+1 ) h W 1,1 ( v γ+1 m)
for some C > 0.

Proof. We split Q(h, h) into two parts and we use the pre-post collisional change of variables for the second one, we obtain

Q(h, h) L 1 (m) = R 3 R 3 ×S 2 B(v -v * , σ) (h ′ * -h * )h + (h ′ -h)h ′ * dσ dv * m dv ≤ R 3 R 3 ×S 2 B(v -v * , σ) (h ′ * -h * ) dσ dv * |h| m dv + R 3 ×R 3 ×S 2 B(v -v * , σ) |h ′ -h| |h ′ * | m dσ dv * dv ≤ R 3 R 3 ×S 2 B(v -v * , σ) (h ′ * -h * ) dσ dv * |h| m dv + R 3 ×R 3 ×S 2 B(v -v * , σ) |h ′ -h| |h * | m ′ dσ dv * dv =: T 1 + T 2 .
We first deal with T 1 using the cancellation lemma [1, Lemma 1]:

T 1 = R 3 |S * h| |h| m dv with S(z) = 2π π/2 0 sin θ b(cos θ) |z| γ cos γ+3 (θ/2) -|z| γ dθ = 2π |z| γ π/2 0 sin θ b(cos θ) 1 -cos γ+3 (θ/2) cos γ+3 (θ/2) dθ ≤ C |z| γ .
We deduce that

(3.1) T 1 ≤ C h L 1 ( v γ ) h L 1 ( v γ m) .
We now treat the term T 2 which is splitted into two parts:

T 2 = R 3 ×R 3 ×S 2 B(v -v * , σ) |h ′ m ′ -hm ′ | |h * | dσ dv * dv ≤ R 3 ×R 3 ×S 2 B(v -v * , σ) |h ′ m ′ -hm| |h * | dσ dv * dv + R 3 ×R 3 ×S 2 B(v -v * , σ) |m ′ -m| |h| |h * | dσ dv * dv =: T 21 + T 22 .
Concerning T 21 , we have to estimate

R 3 ×S 2 b(cos θ) |v -v * | γ |h ′ m ′ -hm| dv dσ =: J(v * ) = J.
To do that, we use Taylor formula denoting v u := (1u)v + uv ′ for any u ∈ [0, 1], which allows us to estimate |h ′ m ′ -hm|:

|h ′ m ′ -hm| = 1 0 ∇(hm)(v u ) • (v -v ′ ) du ≤ 1 0 |∇(hm)(v u )| |v -v * | sin(θ/2) du.
It implies the following inequality on J:

J ≤ C R 3 ×S 2 ×[0,1] b(cos θ) sin(θ) |v -v * | γ+1 |∇(hm)(v u )| du dσ dv.
Moreover, if v = v * , we have the following equality:

|v -v * | = 1 1 -u 2 κ + u 2 σ |v u -v * |.
Using the fact that 0 ≤ κ, σ ≤ 1, one can show that for any u ∈ [0, 1],

1 - u 2 κ + u 2 σ ≥ 1 √ 2 .
We can thus deduce that for any u ∈

[0, 1], we have |v -v * | ≤ C|v u -v * | for some C > 0, which implies J ≤ C R 3 ×S 2 ×[0,1] b(cos θ) sin(θ) |v u -v * | γ+1 |∇(hm)(v u )| du dσ dv.
For u, v * and σ fixed, we now perform the change of variables v → v u . Its Jacobian determinant is

dv u dv = 1 - u 2 2 1 - u 2 + u 2 κ, σ ≥ 1 - u 2 3 ≥ 1 8
since κ, σ ≥ 0. Gathering all the previous estimates, we obtain

J ≤ C S 2 b(cos θ) sin(θ) dσ R 3 |v -v * | γ+1 |∇(hm)(v)| dv.
We thus obtain :

(3.2) T 21 ≤ C h L 1 ( v γ+1 ) h W 1,1 ( v γ+1 m) .
Let us finally deal with T 22 . We here use the inequality (2.3):

(3.3)

T 22 ≤ C R 3 ×R 3 ×S 2 B(v -v * , σ) |h| |h * | v k-1 + v * k-1 |v ′ -v| dσ dv * dv ≤ C S 2
b(cos θ) sin(θ) dσ

R 3 ×R 3 |h| |h * | v k-1 + v * k-1 |v -v * | γ+1 dv * dv ≤ C h L 1 ( v γ m) h L 1 ( v γ+1 ) .
Inequalities (3.1), (3.2) and (3.3) together yields the result.

We now recall a classical result from interpolation theory (see for example Lemma B.1 from [START_REF] Mischler | Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF]). Lemma 3.2. For any s, s * , q, q * ∈ Z with s ≥ s * , q ≥ q * and any θ ∈ (0, 1), there exists C > 0 such that for any h ∈ W s * * ,1 ( v q * * ), we have h W s,1 ( v q ) ≤ C h 1-θ W s * ,1 ( v q * ) h θ W s * * ,1 ( v q * * ) with s * * , q * * ∈ Z such that s = (1θ)s * + θs * * and q = (1θ)q * + θq * * .

It allows us to prove the following corollary which is going to be useful in the proof of our main theorem. Corollary 3.3. Let B satisfying (1.2), (1.3) and (1.4). Then

Q(h, h) L 1 (m) ≤ C h 3/2 L 1 (m) h 1/2 L 1 ( v 2γ m) + h 3/2 L 1 (m) h 1/2 H 4 ( v 4γ+k+6 ) .
Proof. On the one hand, using Lemma 3.2, we obtain:

h L 1 ( v γ m) ≤ h 1/2 L 1 ( v 2γ m) h 1/2 L 1 (m) .
On the other hand, again using twice Lemma 3.2, we obtain

h L 1 ( v γ+1 ) h W 1,1 ( v γ+1 m) ≤ h 2 W 1,1 ( v γ+k+1 ) ≤ C h L 1 (m) h W 2,1 ( v 2γ+k+2 ) ≤ C h 3/2 L 1 (m) h 1/2 W 4,1 ( v 4γ+k+4 ) .
To conclude we use that for any q ∈ N, we can show using Hölder inequality that

h L 1 ( v q ) ≤ C h L 2 ( v q+2 ) .
3.2. Proof of Theorem 1.1. Let f 0 = µ + h 0 and consider the equation (3.4) ∂ t h t = Lh t + Q(h t , h t ), h(t = 0) = h 0 .

Let us notice that for any t ≥ 0, we have Π h t = 0. Indeed, f 0 has same mass, momentum and energy as µ, it implies that Π h 0 = 0 and these quantities are conserved by the equation.

We now state a nonlinear stability theorem which is the third key point (with Theorems 1.3 and 1.4) in the proof of Theorem 1.1. Theorem 3.4. Consider a solution h t to (3.4) for any positive λ < min(λ 0 , λ k ) (see Theorem 1.4).

Proof. We use Duhamel's formula for the solution of (3.4): We denote η ′ := C K 1/2 η 1/4 + η 3/4 . We end up with a similar differential inequality as in [START_REF] Mouhot | Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials[END_REF]Lemma 4.5]. We can then conclude in the same way that

h t =
∀ t ≥ 0, h t L 1 (m) ≤ C ′ e -λt h 0 L 1 (m) ,
for some C ′ > 0.

To conclude the proof of Theorem 1.1, we consider η > 0 defined in Theorem 3.4. Using Theorem 1.3, we can choose t 1 > 0 such that

∀ t ≥ t 1 , h t L 1 (m) = f t -µ L 1 (m) ≤ η.
Thanks to the properties of a smooth solution, we also have

∀ t ≥ t 1 , h t H 4 ( v 4γ+k+6 ) ≤ f t H 4 ( v 4γ+k+6 ) + µ H 4 ( v 4γ+k+6 ) ≤ K
for some K > 0. We can hence apply Theorem 3.4 to h t starting from t 1 . We finally obtain ∀ t ≥ t 1 , f tµ L 1 (m) ≤ C ′ e -λt h t 1 L 1 (m) ≤ C ′′ e -λt , for some C ′′ > 0. The conclusion of Theorem 1.1 is hence established.

( 1 . 3 )

 13 ∀ θ ∈ (0, π/2], c b θ 1+2s ≤ sin θ b(cos θ) ≤ 1 c b θ 1+2s ; • the kinetic factor Φ satisfies (1.4) Φ(|vv * |) = |vv * | γ with γ ∈ (0, 1),this assumption could be relaxed to assuming only that Φ satisfies Φ(•) = C Φ | • | γ for some C Φ > 0.

h s 5 / 4 L 1 0 e 1 / 4 + η 3 /4 h s 5 / 4 L 1

 5410143541 (m) ds ≤ e -λt h 0 L 1 (m) + C t -λ(t-s) K 1/2 η (m) ds.

  such that∀ t ≥ 0, h t H 4 ( v 4γ+k+6 ) ≤ Kfor some K > 0. There exists η > 0 such that if moreover∀ t ≥ 0, h t L 1 ( v 2γ m) ≤ ηthen there exists C > 0 (depending on K and η) such that

∀ t ≥ 0, h t L 1 (m) ≤ C e -λt h 0 L 1 (m)

  S L (t) h 0 + t 0 S L (ts) Q(h s , h s ) ds.We now estimate h t L 1 (m) thanks to Theorem 1.4 and Corollary 3.3:h t L 1 (m) ≤ e -λt h 0 L 1 (m)

	+ C	0	t	e -λ(t-s) h s	1/4 L 1 (m) h s	1/2 H 4 ( v 4γ+k+6 ) + h s	3/4 L 1 ( v 2γ m)

Acknowledgments. We thank Stéphane Mischler for fruitful discussions and his encouragement.