Armen Shirikyan
email: armen.shirikyan@u-cergy.fr

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

Keywords: AMS subject classifications: 35Q30, 60H15, 60J05, 93B05, 93C20 Navier-Stokes equations, stationary measures, exponential mixing Contents measures under finite-dimensional transformations . . . 27

Control and mixing for 2D Navier-Stokes equations with space-time localised noise

Armen Shirikyan

To cite this version:

Armen Shirikyan. Control and mixing for 2D Navier-Stokes equations with space

Introduction

The main results of this paper can be summarised as follows: first, suitable controllability properties of a non-linear PDE imply the uniqueness and exponential mixing for the associated stochastic dynamics and, second, these properties are satisfied for 2D Navier-Stokes equations with space-time localised noise. To be precise, let us consider from the very beginning the 2D Navier-Stokes system in a bounded domain D ⊂ R 2 with smooth boundary ∂D:

u + u, ∇ u -ν∆u + ∇p = f (t, x), div u = 0, x ∈ D, (1.1)
u ∂D = 0.

(1.2)

Here u = (u 1 , u 2 ) and p are unknown velocity and pressure of the fluid, ν > 0 is the viscosity, and f is an external force. Let us assume that f is represented as the sum of two functions h and η, the first of which is a given function that is H 1 smooth in space and time and has a locally bounded norm, while the second is either a control or a random force: f (t, x) = h(t, x) + η(t, x).

(1.3)

In both cases, we assume that η is sufficiently smooth and bounded, and its restriction to any cylinder of the form J k × D with J k = [k -1, k] is localised in both space and time (see below for a more precise description of this hypothesis).

Let us denote by n the outward unit normal to the boundary ∂D and introduce the space H = {u ∈ L 2 (D, R 2 ) : div u = 0 in D, u, n = 0 on ∂D}, (1.4) which will be endowed with the usual L 2 norm • . It is well known that for any u 0 ∈ H problem (1.1), (1.2) supplemented with the initial condition u(0, x) = u 0 (x) (1.5) has a unique solution u = u(t; u 0 , f ), which is a continuous function of time valued in H.

Our main result concerns the property of exponential mixing for the discretetime Markov process in H associated with (1.1)-(1.3), and we now present a simplified version of the hypotheses under which it is valid. We assume that the deterministic force h is a 1-periodic function of time whose restriction to any bounded subset of R × D is H 1 -regular. As for the random force η, we assume that it satisfies the four conditions below. Let Q ⊂ J 1 × D be an open set and let Q k = {(t, x) : (t -k + 1, x) ∈ Q}.

Localisation. For any integer k ≥ 1, the restriction of η to the cylinder

J k × D is supported by Q k .
Let us denote by η k (t, x) the restriction of η(t + k -1, x) to the domain J 1 × D.

Independence. The functions η k form a sequence of i.i.d. random variables in H 1 (J 1 × D, R 2 ) with a law λ.

Non-degeneracy. The measure λ is decomposable in the following sense: there is an orthonormal basis {e j } in the space L 2 (Q, R 2 ) such that e j ∈ H 1 0 (Q, R 2 ) for all j ≥ 1, and

η k (t, x) = ∞ j=1 b j ξ jk e j (t, x),
where ξ jk are independent random variables valued in [-1, 1] and {b j } are positive numbers such that j b j e j H 1 < ∞. Moreover, the laws of ξ jk possess a C 1 -smooth densities with respect to the Lebesgue measure on R.

Approximate controllability. There is û ∈ H such that problem (1.1)-(1.3) is approximately controllable to û with a control function η such that, for any k ≥ 1, the restriction of η(t + k -1, x) to J 1 × D belongs to supp λ, and the time of control can be chosen the same for the initial functions u 0 from a given bounded subset of H.

The 1-periodicity of h and the second of the above hypotheses imply that the restrictions of solutions for (1.1)-(1.3) to integer times form a family of Markov chains in H. The following theorem, which is the main result of the paper, describes the long-time asymptotics of this chain.

Main Theorem. Under the above hypotheses, the Markov chain associated with problem (1.1)-(1.3) has a unique stationary measure µ. Moreover, there are positive constants C and γ such that, for any 1-Lipschitz function F : H → R and any u 0 ∈ H, we have

E F u(k; u 0 , h + η) - H F (v)µ(dv) ≤ C 1 + u 0 e -γk , k ≥ 0. (1.6)
We refer the reader to Section 2.1 for a more general result on uniqueness of a stationary measure and exponential mixing. The proof of the above theorem is based on a detailed study of controllability properties1 of problem (1.1)-(1.3) (in which η plays the role of a control), a general criterion for mixing of Markov chains, and a result on the image of measures on a Hilbert space under finitedimensional transformations; see Section 2.2 for more details.

Let us mention that the problem of ergodicity for the 2D Navier-Stokes system was studied intensively in the last twenty years. First results in this direction were established in [FM95, KS00, EMS01, BKL02], and we refer the reader to the book [START_REF]Mathematics of Two-Dimensional Turbulence[END_REF] for further references and description of the methods used in various works. Most of the results established so far concern the situation in which the random force is non-degenerate in a set of determining modes of the problem. In the case when the equation is studied on the torus and the deterministic force is zero, it was proved in [START_REF] Hairer | Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing[END_REF][START_REF]Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations[END_REF] that the Navier-Stokes dynamics is exponentially mixing for any ν > 0, provided that the noise is white in time and has a few non-zero Fourier modes as a function of x (thus, it is finite-dimensional in x, infinite-dimensional in time, and localised in the Fourier space). This result was extended to the case of 2D sphere in the paper [START_REF]A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs[END_REF], which also fixes an error in [START_REF] Hairer | Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing[END_REF]. The main theorem stated above is valid for all ν > 0 and, to the best of our knowledge, provides a first result on mixing properties for Navier-Stokes equations with a space-time localised noise.

In conclusion, let us mention that the results of this paper remain valid in the case when the noises act through the boundary of the domain. This situation will be addressed in a subsequent publication.

The paper is organised as follows. In Section 2, we formulate the main result of this paper on exponential mixing for the Navier-Stokes system with spacetime localised noise, outline its proof, and discuss some examples. Section 3 is devoted to studying a control problem associated with the stochastic system in question. The details of proof of the main result are given in Section 4. The appendix gathers some auxiliary results used in the main text.

Acknowledgment. The author is grateful to Lihu Xu for stimulating discussion. This research was developed within the MME-DII Center of Excellence (ANR-11-LABX-0023-01) and supported by the ANR grant STOSYMAP (ANR 2011 BS01 015 01).

Notation

For an open set Q of a Euclidean space, a closed interval J ⊂ R, and Banach spaces X ⊂ Y , we introduce the following function spaces.

L p = L p (Q) is the Lebesgue space of measurable scalar or vector functions on Q whose p th power is integrable. We shall sometimes write L p (Q, R d ) to emphasise the range of functions. In the case p = 2, the corresponding norm will be denoted by • . H s = H s (Q) is the Sobolev space of order s with the usual norm • s . As in the previous case, we use the same notation for spaces of scalar and vector functions.

H s 0 = H s 0 (Q) is the closure in H s of
the space of infinitely smooth functions with compact support. B X (R) stands for the ball in X of radius R centred at zero. L p (J, X) is the space of Borel-measurable functions u : J → X such that

u L p (J,X) = J u(t) p X dt 1/p < ∞ ;
in the case p = ∞, this norm should be replaced by u ∞ = ess sup J u(t) X . W 1,p (J, X) is the space of functions u ∈ L p (J, X) whose derivative belongs to L p (J, X). It is endowed with a natural norm.

W (J, X, Y ) is the space of functions u ∈ L 2 (J, X) such that ∂ t u ∈ L 2 (J, Y ). L(X, Y
) is the space of continuous linear operators from X to Y with the natural norm. In the case X = Y , we write L(X). C b (X) stands for the space of bounded continuous functions F : X → R; it is endowed with the norm

F ∞ = sup u∈X |F (u)|.
L b (X) denotes the space of functions F ∈ C b (X) such that

F L := F ∞ + sup u =v |F (u) -F (v)| u -v X < ∞.
P(X) is the set of probability Borel measures on X. The space P(X) is endowed with the topology of weak convergence, which is generated by the dual-Lipschitz metric

µ 1 -µ 2 * L := sup F L ≤1 |(F, µ 1 ) -(F, µ 2 )|,
where (F, µ) stands for the integral of F over X with respect to µ.

We denote by D a bounded domain with C 2 boundary ∂D. For T > 0, we set

J T = [0, T ] and D T = J T × D.
The following functional spaces arise in the theory of Navier-Stokes equations:

V = H ∩ H 1 0 (D), X T = W (J T , V, V * ), Y δ,T = u ∈ X T : u| (δ,T ) ∈ W ((δ, T ), V ∩ H 3 , V ) ,
where δ ∈ (0, T ), H is defined by (1.4), and V * denotes the dual space of V (identified with a quotient space in H -1 (D, R 2 ) with the help of the scalar product in L 2 ). These spaces are endowed with natural norms.

2 Main result and scheme of its proof

Exponential mixing

Let D ⊂ R2 be a bounded domain with a C 2 -smooth boundary ∂D and let D 1 = J 1 × D. Consider the Navier-Stotes system (1.1) with the Dirichlet boundary condition (1.2) and an external force of the form (1.3). We assume that h ∈ H 1 loc (R + × D, R 2 ) is a given function which is 1-periodic in time and η is a stochastic process of the form

η(t, x) = ∞ k=1 I k (t)η k (t -k + 1, x), t ≥ 0, ( 2
u ∈ C(R + , H) ∩ L 2 loc (R + , V ), • 0 p dt ∈ L ∞ loc (R + , L 2 ),
Equations (1.1), and the initial condition

u(0, x) = u 0 (x), x ∈ D. (2.2)
In what follows, we shall drop the p component of solutions and write simply u(t). Let us denote by S : H × L 2 (D 1 , R 2 ) → H the operator that takes a pair of functions (u 0 , f ) to u(1), where u(t) is the solution of (1.1), (1.2), (2.2).

Well-known properties of 2D Navier-Stokes equations imply that S is a continuous mapping. Moreover, the range of S is contained in V , and the mapping S : H × L 2 (D 1 , R 2 ) → V is uniformly Lipschitz continuous on bounded subsets; e.g., see Chapter III in [START_REF] Temam | Navier-Stokes Equations[END_REF].

Let us consider a solution u(t) of (1.1)-(1.3) and denote u k = u(k). What has been said implies that

u k = S(u k-1 , h + η k ), k ≥ 1. (2.3) Since η k are i.i.d. random variables in L 2 (D 1 , R 2 ),
} ⊂ H 1 (Q, R 2 ) an or- thonormal basis in L 2 (Q, R 2 ). Let χ ∈ C ∞ 0 (
Q) be a non-zero function and let ψ j = χϕ j . In what follows, we shall assume that {ψ j } are linearly independent 2 , and the function h and random process η satisfy hypotheses (H1) and (H2) formulated below. Note that if {ϕ j } is a complete set of eigenfunctions of the Dirichlet Laplacian in Q, then the functions ψ j are linearly independent for any choice of χ. This is an immediate consequence of the unique continuation property of solutions for elliptic equations; see Theorem 8.9.1 in [START_REF] Hörmander | Linear Partial Differential Operators[END_REF].

(H1) Structure of the noise. The random variables η k can be represented in the form

η k (t, x) = ∞ j=1 b j ξ jk ψ j (t, x), (2.4)
where ξ jk are independent scalar random variables such that |ξ jk | ≤ 1 with probability 1, and {b j } ⊂ R is a non-negative sequence such that

B := ∞ j=1 b j ψ j 1 < ∞. (2.5)
Moreover, the law of ξ jk possesses a C 1 -smooth density ρ j with respect to the Lebesgue measure on the real line.

Let us denote by K ⊂ L 2 (Q, R 2 ) the support of the law of η k . The hypotheses imposed on η k imply that K is a compact subset in H 1 0 (Q, R 2 ). Continuing the elements of K by zero outside Q, we may regard K as a compact subset of H 1 0 (D 1 , R 2 ).

(H2) Approximate controllability. There is û ∈ H such that for any positive constants R and ε one can find an integer l ≥ 1 with the following property: given v ∈ B H (R), there are ζ 1 , . . . , ζ l ∈ K such that

S l (v, ζ 1 , . . . , ζ l ) -û ≤ ε, (2.6)
where S l (v, ζ 1 , . . . , ζ l ) stands for the vector u l defined by (2.3) with η k = ζ k and u 0 = v.

The following theorem, which is the main result of this paper, establishes the uniqueness and exponential mixing of a stationary distribution for the Markov family generated by (2.3).

Theorem 2.1. Assume that h ∈ H 1 loc (R + × D, R 2
) is 1-periodic in time, and Conditions (H1) and (H2) are satisfied. In this case, there is an integer N ≥ 1, depending on h H 1 (D1) , B, and ν, such that if b j = 0 for j = 1, . . . , N ,

(2.7)

then the following assertions hold.

Existence and uniqueness. The Markov family (u k , P u ) has a unique stationary distribution µ ∈ P(H).

Exponential mixing. There are positive constants C and γ such that

P k (u, •) -µ * L ≤ C(1 + u )e -γk for all u ∈ H, k ≥ 0. (2.8)
Note that condition (2.7) expresses the space-time non-degeneracy of the noise. Thus, the property of exponential mixing holds true even for noises whose space-time dimension is finite.

The general scheme of the proof of Theorem 2.1 is outlined in Section 2.2, and the details are given in Sections 3 and 4. Here we consider two examples for which Condition (H2) is fulfilled and discuss a counterexample showing that, in general, the approximate controllability of (1.1)-(1.3) is not likely to hold for all ν > 0 with the same control set K. Example 2.2. We claim that there is δ > 0 such that if the 1-periodic function h satisfies the inequality h L 2 (D1) ≤ δ, then Condition (H2) is fulfilled, provided that K contains the zero element.

Indeed, if h L 2 (D1) is sufficiently small, then problem (1.1), (1.2) with f = h has a unique solution ũ(t, x) defined throughout the real line and 1-periodic in time. To see this, it suffices to take a sequence {u n } of solutions for the problem in question such that u n (-n) = 0 and to prove that it converges as n → ∞ in the space W ([-N, N ], V, V * ) for any N > 0. The limiting function ũ is the required 1-periodic solution. Using standard estimates for Navier-Stokes equations, it is easy to prove that

ũ W (J1,V,V * ) ≤ c(δ) → 0 as δ → 0.
This implies that ũ is globally exponentially stable as t → +∞. Therefore, for any positive constants R and ε one can find an integer l ≥ 1 such that (2.6) holds with ζ 1 = • • • = ζ l = 0 and û = ũ(0). Since K contains the zero element, we see that Condition (H2) is satisfied.

Example 2.3. Suppose that h is represented in the form h(t, x) = h 0 (λt, x), where h 0 is a continuous 1-periodic function of time with range in V such that 1 0 h 0 (s) ds = 0. We claim that, for sufficiently large integers λ > 0, Condition (H2) is satisfied, provided that K contains the zero element.

Indeed, let us represent a solution of Eq. (1.1) with f = h in the form u = w + g, where g(t) = t 0 h(s) ds. Then the function w must satisfy the equations ẇ + w + g, ∇ (w + g) -ν∆w + ∇p = -∆g, div w = 0.

(2.9)

The condition imposed on h imply that g is a V -valued 1-periodic function such that sup

t∈R g(t) V → 0 as λ → ∞.
Combining this with an argument similar to that used in Example 2.2, one can prove that Eq. (2.9) has a unique 1-periodic solution w, which is globally exponentially stable as t → ∞. It follows that ũ = w + g is a globally exponentially stable solution for problem (1.1), (1.2) with f = h. As in Example 2.2, we conclude that the function û = ũ(0) satisfies the required property.

In both examples considered above, Condition (H2) was satisfied due to the fact that the unperturbed problem had a globally stable stationary point. The following simple example of an ordinary differential equation suggests that, in the general case, it is unreasonable to expect the property of approximate controllability in infinite time for the Navier-Stokes system with an arbitrary ν > 0 and a fixed control set.

Counterexample 2.4. Let g : R → R be an arbitrary smooth function vanishing at zero such that

|u| -a g(u) sgn(u) ≥ c > 0 for |u| ≥ 1, (2.10)
where a < 1 and c are positive numbers and sgn(u) denotes the sign of u. Let us consider the equation

u = -νu + g(u) + η(t), (2.11)
where ν > 0 is a parameter and η(t) is a control taking values in an interval [-K, K] ⊂ R. We claim that there is ν 0 > 0 depending on K, a, and c such that the following property holds for 0 < ν ≤ ν 0 : for any û ∈ R there is u 0 ∈ R such that, for any measurable function η(t) defined on the positive half-line and taking values in [-K, K], we have

inf t≥0 |u(t) -û| > 0, (2.12)
where u(t) stands for the solution of (2.11) issued from u 0 . Indeed, let us fix any û ∈ R and any function η(t) with range in [-K, K]. We shall assume that 4K ≥ c and û ≤ 0 (the other case can be treated by a similar argument). Let us denote by V (t, u) the right-hand side of (2.11). Setting ū = (4Kc -1 ) 1/a and ν 0 = c/(2ū 1-a ), we see from (2.10) that

V (t, ū) > 0 for 0 < ν ≤ ν 0 .
It follows that if u 0 ≥ ū, then u(t) ≥ ū for all t ≥ 0, whence we conclude that (2.12) holds.

General criterion for mixing and application

In this section, we outline the proof of Theorem 2.1, which is based on two key ingredients: a coupling approach developed in [KS01, Mat02, KPS02, MY02, Hai02, Shi04] in the context of stochastic PDE's and a property of stabilisation to a non-stationary solution of Navier-Stokes equations [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF]. We first recall an abstract result established in [START_REF]Exponential mixing for randomly forced partial differential equations: method of coupling, Instability in models connected with fluid flows[END_REF].

Let X be a compact metric space with a metric d X and let (u k , P u ), u ∈ X, be a family of Markov chains in X. We denote by P k (u, Γ) its transition function. Let (u k , P u ) be another family of Markov chains in the extended phase space X = X × X such that

Π * P k (u , •) = P k (u, •), Π ′ * P k (u , •) = P k (u ′ , •) for u = (u, u ′ ) ∈ X , k ≥ 0, (2.13)
where P k (u, Γ) denotes the transition function for (u k , P u ) and Π, Π ′ : X → X stand for the natural projections to the components of a vector u = (u, u ′ ).

In other words, relations (2.13) mean that, for any integer k ≥ 1, the random variable u k considered under the law P u with u = (u, u ′ ) is a coupling for the pair of measures (P k (u, •), P k (u ′ , •)). We shall say that (u k , P u ) satisfies the mixing hypothesis if there is a closed subset B ⊂ X and positive constants C and β such that the following properties hold.

Recurrence. Let τ (B ) be the first hitting time of the set B:

τ (B) = min{k ≥ 0 : u k ∈ B}.
Then τ (B ) is P u -almost surely finite for any u ∈ X , and there are positive constants C 1 and δ 1 such that

E u exp δ 1 τ (B ) ≤ C 1 for u ∈ X . (2.14) Exponential squeezing. Let us set σ = min{k ≥ 0 : d X (u k , u ′ k ) > C e -βk }.
(2.15)

Then there are positive constants C 2 , δ 2 , and δ 3 such that, for any u ∈ B, we have

P u {σ = ∞} ≥ δ 3 ,
(2.16)

E u I {σ<∞} exp(δ 2 σ) ≤ C 2 .
(2.17)

The following proposition is a particular case of a more general result established3 in [START_REF]Exponential mixing for randomly forced partial differential equations: method of coupling, Instability in models connected with fluid flows[END_REF] (see Theorem 2.3).

Proposition 2.5. Let (u k , P u ) be a family of Markov chains for which there exists another Markov family (u k , P u ) in the extended space X that satisfies relation (2.13) and the mixing hypothesis. Then (u k , P u ) has a unique stationary distribution µ ∈ P(X), and there are positive constants C and γ such that

P k (u, •) -µ * L ≤ C e -γk for u ∈ X, k ≥ 0. (2.18)
To prove Theorem 2.1, we first observe that the Markov family (u k , P u ) possesses a compact absorbing invariant set X ⊂ H, and it suffices to study its restriction to X, for which we retain the same notation. We shall prove that (u k , P u ) satisfies the hypotheses of Proposition 2.5. A crucial point of our construction is the following result, which says, roughly speaking, that if two points u, u ′ ∈ X are sufficiently close, then the pair (P 1 (u, •), P 1 (u ′ , •)) admits a coupling whose components are close with high probability; cf. Lemma 3.3 in [START_REF]A coupling approach to randomly forced nonlinear PDE's. I[END_REF].

Proposition 2.6. Under the hypotheses of Theorem 2.1, there exists a constant d > 0 such that for any points u, u ′ ∈ X satisfying the inequality u -u ′ ≤ d the pair

(P 1 (u, •), P 1 (u ′ , •)) admits a coupling (V (u, u ′ ), V ′ (u, u ′ )) such that P V (u, u ′ ) -V ′ (u, u ′ ) > 1 2 u -u ′ ≤ C u -u ′ , (2.19)
where C > 0 is a constant not depending on u, u ′ ∈ X.

The proof of this proposition is based on a controllability property for the Navier-Stokes system and application of a concept of optimal coupling; see Sections 3 and 5.1. We now define a coupling operator R = (R, R ′ ) by the relation

R(u, u ′ ) = V (u, u ′ ), V ′ (u, u ′ ) for u -u ′ ≤ d, S(u, ζ), S(u ′ , ζ ′ ) for u -u ′ > d, (2.20)
where ζ and ζ ′ are independent random variables whose law coincides with that of η 1 . Without loss of generality, we can assume that ζ and ζ ′ are defined on the same probability space as V and V ′ , and to emphasise the dependence on ω, we shall sometimes write R(u,

u ′ ; ω) instead of R(u, u ′ ). The required Markov family (u k , P u ) is constructed by iterations of R. Namely, let (Ω k , F k , P k ),
k ≥ 1, be countably many copies of the probability space on which R is defined and let (Ω, F , P) be the direct product of these spaces. We set

u 0 = (u, u ′ ), u k = R(u k-1 , ω k ), k ≥ 1. (2.21)
The recurrence property will follow from approximate controllability (see Hypothesis (H2) in Section 2.1), while the exponential squeezing will be implied by Proposition 2.6.

3 Control problem

Squeezing

In this section, we consider the controlled Navier-Stokes system (1.1)-(1.3) on the time interval J 1 = [0, 1]. We assume that the function h belongs to the space H 1 (D 1 ) (where D 1 = J 1 × D) and denote by {ψ j } the sequence of functions entering Hypothesis (H1) of Section 2.1. Extending the functions ψ j by zero outside Q, we may regard them as elements of H 1 0 (D 1 ). We denote by E m the vector span of ψ 1 , . . . , ψ m endowed with the L 2 norm and by B R the ball in H 1 (D 1 ) of radius R centred at origin. The following theorem whose weaker version was established in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] is a key step in the proof of Theorem 2.1. Theorem 3.1. Under the above hypotheses, for any R > 0 and q ∈ (0, 1) there is an integer m ≥ 1, positive constants d and C, and a continuous mapping

Φ : B R × B H (R) → L(H, E m ), (h, û0 ) → η,
such that the following properties hold.

Contraction. For any functions

h ∈ B R and û0 , u 0 ∈ B H (R) satisfying the inequality u 0 -û0 ≤ d, (3.1)
we have

S û0 , h -S u 0 , h + Φ(h, û0 )(u 0 -û0 ) ≤ q u 0 -û0 . (3.2)
Regularity. The mapping Φ is infinitely smooth in the Fréchet sense.

Lipschitz continuity. The mapping Φ is Lipschitz continuous with the constant C. That is,

Φ(h 1 , û1 ) -Φ(h 2 , û2 ) L ≤ C h 1 -h 2 H 1 + û1 -û2 , (3.3)
where • L stands for the norm in the space L(H, E m ).

An immediate consequence of this theorem is a refinement of a result established in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] on stabilisation of a non-stationary for the Navier-Stokes system. Since that result is not necessary for proving Theorem 2.1, we postpone its formulation and proof until Section 3.4.

The proof of Theorem 3.1 repeats essentially the argument used in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF]. However, since the finite-dimensionality in time for the control and the regularity and Lipschitz properties of Φ are important for the stochastic part of this work, we present a rather complete proof of Theorem 3.1. We begin with a description of the main steps and give the details in the next two subsections.

Step 1: Reduction to the linearised problem. Denote by û(t, x) the solution of (1.1), (1.3) issued from û0 and corresponding to η ≡ 0. In view of the regularising property of the Navier-Stokes system, for any interval J = (δ, 1) with δ > 0 we have

û ∈ L 2 (J, H 3 ∩ V ), ∂ t û ∈ L 2 (J, V ), (3.4)
and the corresponding norms are bounded by a constant depending only on δ and R. In particular, the truncated observability inequality holds for the adjoint of the Navier-Stokes system linearised around û; see Section 5.2.

A solution with a non-zero control η is sought in the form u = û + v. Then v must be a solution of the problem

v + v, ∇ v + û, ∇ v + v, ∇ û -ν∆v + ∇p = η(t, x), div v = 0, (3.5) v ∂D = 0, v(0) = v 0 , (3.6)
where v 0 = u 0 -û0 . Together with Eq. (3.5), consider its linearisation around zero:

v + û, ∇ v + v, ∇ û -ν∆v + ∇p = η(t, x), div v = 0. (3.7)
Suppose that we have constructed η ∈ E m such that the solution w(t, x) of (3.7), (3.6) satisfies the inequalities

w(1) ≤ q 2 v 0 , w X1 ≤ C 1 v 0 . (3.8)
A standard perturbative argument shows that if v 0 is sufficiently small, then the solution of (3.5), (3.6) satisfies the inequality v(1) ≤ q v 0 , whence it follows that (3.2) holds. Thus, it suffices to construct a continuous linear operator Φ(h, û0 ) : H → E m such that the solution w ∈ X 1 of problem (3.7), (3.6) with η = Φ(h, û0 )v 0 satisfies inequalities (3.8).

Step 2: Application of the Foiaş-Prodi property. Let {e j } be an orthonormal basis in H formed of the eigenfunctions of the Stokes operator L = -Π∆, where Π stands for the Leray projection in L 2 (D, R 2 ) (onto the closed subspace H), let {α j } be the corresponding (non-decreasing) sequence of eigenvalues for L, and let Π N be the orthogonal projection in H on the vector space H N spanned by e 1 , . . . , e N . Denote by R û : H × L 2 (D 1 ) → X 1 a linear operator that takes (v 0 , η) to the solution w of (3.7), (3.6) and by R û t its restriction to the time t.

Suppose that for any integer N ≥ 1 and any δ > 0 we have constructed an integer m ≥ 1 and a family of linear operators Φ = Φ(h, û0 ) : H → E m which is a Lipschitz function of its arguments and is such that

Π N R û 1 (v 0 , Φ(h, û0 )v 0 ) ≤ C 2 δ v 0 , Φ(h, û0 ) L ≤ C 2 , (3.9)
where C 2 > 0 is a constant not depending on N and δ. In this case, the Poincaré inequality and the regularising property of (3.7) imply that

R û 1 (v 0 , Φ(h, û0 )v 0 ) = (I -Π N )R û 1 (v 0 , Φ(h, û0 )v 0 ) + C 2 δ v 0 ≤ α -1/2 N +1 R û 1 (v 0 , Φ(h, û0 )v 0 ) 1 + C 2 δ v 0 ≤ C 3 α -1/2 N +1 v 0 + Φ(h, û0 )v 0 L 2 (D1) + C 2 δ v 0 ≤ C 3 (C 2 + 1)α -1/2 N +1 + C 2 δ v 0 .
Choosing N sufficiently large and δ sufficiently small, we obtain the first inequality in (3.8). The second is an immediate consequence of the continuity of R û and the boundedness of Φ.

Step 3: Minimisation problem. The construction of Φ is based on a study of a minimisation problem for solutions of (3.7) with a cost functional penalising the term Π N w(1) . Namely, let us consider the following problem. Problem 3.2. Given a constant δ > 0, an integer N ≥ 1, and functions v 0 ∈ H and û ∈ X 1 satisfying (3.4), minimise the functional

J(w, ζ) = 1 2 1 0 ζ(t) 2 dt + 1 δ Π N w(1) 2 over the set of functions (w, ζ) ∈ X 1 × L 2 (D 1 , R 2 ) satisfying the equations ẇ + û, ∇ w + w, ∇ û -ν∆w +∇p = χ(P m ζ), div w = 0, w(0) = v 0 , (3.10)
where p = p(t, x) is a distribution in D 1 and P m stands for the orthogonal projection in L 2 (D 1 ) onto E m .

We shall show that Problem 3.2 has a unique solution (w, ζ), which satisfies the inequality

1 δ Π N w(1) 2 + ζ 2 L 2 (D1) ≤ C 3 v 0 2 , (3.11)
where C 3 > 0 is a constant not depending on N and δ. This will imply the required inequalities (3.9), in which Φv 0 = χ(P m ζ). Further analysis shows that the mapping û → ζ is smooth from X 1 to L 2 (D 1 ) and uniformly Lipschitz continuous on bounded balls. Since û is an analytic function of (h, û0 ), this will complete the proof of the theorem.

Minimisation problem

Step 1: Existence, uniqueness, and linearity. Let us prove that Problem 3.2 has a unique optimal solution (w, ζ) in the space X := X 1 × L 2 (D 1 ). Indeed, the function J : X → R is non-negative and therefore has an infimum on any affine subspace of X. Denote by X v0 the affine subspace of X defined by (3.10) and by J * the infimum of J on X v0 . Let (w n , ζ n ) ∈ X v0 be an arbitrary minimising sequence. Then {ζ n } is a bounded sequence in L 2 (D 1 ), and without loss of generality we can assume that it converges weakly to a limit ζ. It follows that the sequence of solutions w n ∈ X 1 of problem (3.10) with ζ = ζ n converges in the space X 1 to a limit w, which satisfies (3.10). The lower semi-continuity of the norm of a Hilbert space now implies that

J(w, ζ) ≤ lim inf n→∞ J(w n , ζ n ) = J * .
Recalling the definition of J * , we conclude that J(w, ζ) = J * .

To prove the uniqueness, note that any affine subspace is a convex set in X. Combining this property with the strict convexity of the norm of a Hilbert space, we see that if (w i , ζ i ), i = 1, 2, are two optimal solutions, then

ζ 1 = ζ 2 . Since the solution of problem (3.10) with a given ζ ∈ L 2 (D 1 ) is unique, we conclude that w 1 = w 2 .
Finally, it is a standard fact of the optimisation theory that the unique minimum of a quadratic functional under a linear constraint can be expressed as a linear function of the problem data (e.g., see Section III.1 in [START_REF] Zabczyk | Mathematical Control Theory[END_REF] for a general theory and Section A.2 of [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] for a simple proof of the result we need). In the case under study, the corresponding operator depends on the reference solution û. We shall denote by Ψ (û) the linear operator that takes v 0 to ζ, where (w, ζ) is the optimal solution for Problem 3.2.

Step 2: Regularity and Lipschitz continuity. We now prove that the mapping Ψ regarded as an application from H to L(H, L 2 (D 1 )) is infinitely differentiable and uniformly Lipschitz continuous on balls. Let us denote by w 0 = w 0 (û, v 0 ) ∈ X 1 the solution of problem (3.10) with ζ ≡ 0. The linear constraint of Problem 3.2 is equivalent to the relation

w = w 0 + R û(0, χ(P m ζ)).
(3.12)

Setting A(û)ζ = R û 1 (0, χ(P m ζ)), we see that (w, ζ) ∈ X is a solution of Problem 3.2 if and only if w is the global minimum of the function F : L 2 (D 1 ) → R defined by

F (ζ; û, v 0 ) = 1 2 1 0 ζ(t) 2 dt + 1 δ Π N w 0 + A(û)ζ 2 .
Using standard methods of the theory of 2D Navier-Stokes equations, we can prove that A(û) is an analytic function from X 1 to L(H) which is uniformly Lipschitz continuous on bounded sets; e.g., see [START_REF] Kuksin | Diffeomorphisms of function spaces that correspond to quasilinear parabolic equations[END_REF] or Chapter 1 in [START_REF] Vishik | Mathematical Problems in Statistical Hydromechanics[END_REF].

It follows that F satisfies the hypotheses of Proposition 5.5 in which U = L 2 (D 1 ) and Y = X 1 × H. Thus, the unique minimum ζ of F is a smooth function of (û, v 0 ) valued in L 2 (D 1 ), and it is Lipschitz continuous on bounded subsets. Recalling relation (3.12), we conclude that the unique solution (w, ζ) ∈ X of Problem 3.2 is a smooth function on X 1 × H which is Lipschitz continuous on bounded subsets. Since (w, ζ) linearly depends on v 0 , it is straightforward to derive the required properties of Ψ .

Step 3: A priori estimate. From now on, we fix R, δ > 0 and assume that û ∈ B Y δ,1 (R). We claim that there is an integer m ≥ 1 depending on R and N , and a constant C > 0 depending only on R such that, if û ∈ B Y δ,1 (R), then inequality (3.11) holds. Indeed, note that the constraint given by (3.10) is equivalent to the equations

ẇ + νLw + B(û, w) + B(w, û) = Π χ(P m ζ) , w(0) = v 0 , (3.13)
where we set B(u 1 , u 2 ) = Π( u 1 , ∇ u 2 ). Thus, the pair (w, ζ) ∈ X constructed in Step 1 is the minimiser of J under constraint (3.11). Applying the Kuhn-Tucker theorem (see Chapter I in [START_REF] Ioffe | Theory of Extremal Problems[END_REF]), we can find functions λ ∈ H and θ ∈ L 2 (J 1 , V ) such that, for any (r, ξ) ∈ X, we have where we used relations (3.15) and the initial condition in (3.13). Now note that (3.14) is equivalent to the backward Navier-Stokes equations (5.12). Therefore, by Proposition 5.4, the function θ must satisfy the truncated observability inequality (5.13). Combining it with (3.16), we obtain (3.11). We have thus shown that (cf. (3.9))

1 0 (ζ, ξ) dt + 2 δ Π N w(1), r(1) + λ, r (0)
Π N R û 1 v 0 , χP m (Ψ (û)v 0 ) ≤ Cδ v 0 , Ψ (û)v 0 L 2 (D1) ≤ C v 0 .
(3.17)

Proof of Theorem 3.1

Let us define Φ(h, û0 ) as the linear operator taking v 0 to χ(P m Ψ (û)v 0 ), where û ∈ X 1 is the solution of problem (1.1), (3.20) with f = h and u 0 = û0 . Standard regularity results for 2D Navier-Stokes equations imply that û ∈ Y δ,1 for any δ ∈ (0, 1) (see Theorem 3.5 in Chapter III of [START_REF] Temam | Navier-Stokes Equations[END_REF]), so that Ψ (û) is well defined. Moreover, the norm û Y δ,1 is bounded by a constant depending only on R and δ. We claim that Φ(h, û0 ) satisfies the contraction property stated in the theorem. If this assertion is proved, then the regularity and Lipschitz continuity of Ψ combined with similar properties of the resolving operator for the 2D Navier-Stokes system will imply the remaining assertions on Φ. Inequalities (3.17) imply that the solution (w, ζ) of Problem 3.2 satisfies (3.11). Therefore, choosing N and δ -1 sufficiently large, we ensure that the function w = R û(v 0 , Φ(h, û0 )v 0 ) satisfies (3.8). Let us represent a solution of the non-linear problem (3.5), (3.6) with the right-hand side η = Φ(h, û0 )(u 0 -û0 ) in the form v = w + z. Then z ∈ X 1 must be a solution of the problem

ż + z, ∇ z + û + w, ∇ z + z, ∇ (û + w) -ν∆z + ∇p = -w, ∇ w, (3.18) div z = 0, z ∂D = 0, z(0) = 0. (3.19)
Taking the scalar product of (3.18) with 2z in H and carrying out some standard transformations, we derive

∂ t z 2 + 2ν ∇z 2 ≤ C 1 û + w 1 z + w 1 w ∇z .
It follows that

∂ t z 2 + ν ∇z 2 ≤ C 2 û + w 2 1 z 2 + w 2 1 w 2 .
Applying the Gronwall inequality and using the initial condition in (3.19), we obtain

z(t) 2 ≤ t 0 exp C 2 t s û + w 2 1 dr w 2 1 w 2 ds.
Recalling that û X1 is bounded by a constant depending only on R and using the second inequality in (3.8), we derive

sup 0≤t≤1 z(1) ≤ C 3 (R) w 2 X1 ≤ C 4 (R) u 0 -û0 2 .
Since u 0 -û0 ≤ d, choosing d > 0 sufficiently small ensures that

z(1) ≤ q 2 u 0 -û0 .
Combining this with the first inequality of (3.8) in which v 0 = u 0 -û0 , we get (3.2). This completes the proof of the theorem.

Stabilisation of a non-stationary solution

In this section, we prove a simple corollary of Theorem 3.1 on stabilisation of a non-stationary solution for the Navier-Stokes system by a finite-dimensional localised control. This result is not necessary for the proof of Theorem 2.1, and the reader not interested in the control problem may safely skip this section.

Recall that we denote by u(t; u 0 , f ) the solution of (1.1), (1.2), (1.5).

Proposition 3.3. For any ρ > 0 and α > 0 there is a finite-dimensional subspace E ⊂ H 1 0 (Q, R 2 ) and positive constants C and d such that the following assertions hold for any functions û0 ∈ H and h ∈ H

1 loc (R + × D, R 2 ) satisfying the inequalities û0 ≤ ρ, h H 1 (J k ×D) ≤ ρ for all k ≥ 1.
(i) For any u 0 ∈ H satisfying the condition u 0 -û0 ≤ d there is a control η such that the restriction of η(t + k -1, x) to J 1 × D belongs to E for any integer k ≥ 1 and

u(t; u 0 , h + η) -u(t; û0 , h) ≤ Ce -αt u 0 -û0 , t ≥ 0. (3.20) (ii)
The mapping (û 0 , u 0 , h) → η is Lipschitz continuous in the sense specified below. Moreover,

η L 2 (Q k ,R 2 ) ≤ Ce -αk u 0 -û0 , k ≥ 1. (3.21)
As was mentioned Section 3.1, a similar result was proved earlier in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] for the 3D Navier-Stokes system (see also [START_REF] Fursikov | Stabilization for the 3D Navier-Stokes system by feedback boundary control[END_REF][START_REF] Barbu | Feedback stabilization of Navier-Stokes equations[END_REF][START_REF] Barbu | Internal stabilization of Navier-Stokes equations with finite-dimensional controllers[END_REF] for some results on stabilisation to a stationary solution). Proposition 3.3 establishes some additional properties of the control. Namely, we show that it is finite-dimensional in both space and time and can be chosen to be a smooth function with respect to the force and the initial state corresponding to the reference solution.

Proof of Proposition 3.3. Let us fix positive constants ρ and α and take an initial function û0 ∈ B H (ρ). The boundedness of the resolving operator for the Navier-Stokes system implies that the corresponding solution û satisfies the inequality û(t) ≤ R for all t ≥ 0, (3.22

)
where R is a constant depending only on ρ. Let q ∈ (0, 1) be such that e -α = q. Denote by d, C, and m the parameters constructed in Theorem 3.1. Let us take any u 0 ∈ H satisfying the inequality u 0 -û0 ≤ d and construct a control function η consecutively on the intervals J k , k ≥ 1. To this end, we denote by h k the restriction of the function h(t -k + 1) to D 1 and set

η(t) = Φ(û 0 , h 1 )(u 0 -û0 ) for 0 ≤ t ≤ 1.
Then, by Theorem 3.1, we have

u(1) -û(1) ≤ q u 0 -û0 = e -α u 0 -û0 ,
where u(t) denotes the solution of (1.1)-(1.3), (1.5) on the interval J 1 . Assume we have constructed a control η on the interval (0, k) with k ≥ 1, and the corresponding solution u satisfies the inequality

u(l) -û(l) ≤ e -αl u 0 -û0 (3.23)
for l = 1, . . . , k. In this case, setting

η(k + t) = Φ(û(k), h k+1 ) u(k) -û(k) for 0 ≤ t ≤ 1, (3.24)
we see that (3.23) remains valid for l = k + 1. Thus, we can construct η on the half-line R + , and the corresponding solution will satisfy (3.23) for all integers l ≥ 1. Combining this with the Lipschitz continuity of the resolving operator of the Navier-Stokes system, we see that (3.20) holds. Inequality (3.21) is a straightforward consequence of (3.24) and (3.23). Finally, it is not difficult to check that if η 1 and η 2 are two controls corresponding to (u i 0 , ûi 0 , h i ), i = 1, 2, then

η 1 k -η 2 k L 2 (D k ) ≤ C k 1 u 1 0 -u 2 0 + û1 0 -û2 0 + max 1≤l≤k h 1 l -h 2 l H 1 (D l ) ,
where C 1 > 0 depends only on ρ and α, η i l stands for the restriction of η i to D l , and h i l are defined in a similar way. This completes the proof.

Proof of Theorem 2.1

We first outline the main steps. A well-known dissipativity argument shows that the random dynamical system defined by (2.1) has a compact invariant absorbing set X ⊂ H. Therefore it suffices to prove the uniqueness of an invariant measure and the property of exponential mixing for the restriction of (u k , P u ) to X. This will be done with the help of Proposition 2.5. Namely, we shall prove Proposition 2.6 and use relations (2.19) and (2.20) to define a Markov chain (u k , P u ) in the extended phase space X = X × X. This Markov chain is an extension of (u k , P u ) and possesses the recurrence and exponential squeezing properties of Section 2.2, and therefore the hypotheses of Proposition 2.5 are satisfied. This will complete the proof of Theorem 2.1.

Reduction to a compact phase space

Well-known properties of the resolving operator for the Navier-Stokes system imply that S satisfies the inequality

S(u, f ) ≤ κ u + C 1 f L 2 (D1) for u ∈ B H (R), f ∈ L 2 (D 1 )
where κ < 1 and C 1 > 0 are some universal constants. Let r > 0 be so large that h + η k L 2 (D1) ≤ r almost surely. Then, with probability 1, we have

S(u, h + η k ) ≤ κρ + C 1 r for u ∈ B H (ρ). (4.1) It follows that if R ≥ C1r 1-κ , then the ball B H (R) is invariant for the Markov chain (u k , P u ). Let us take R = 2C1r
1-κ and denote by X the image of the set B H (R) × B L 2 (D1) (r) under the mapping S. Then X is an invariant subset for (u k , P u ), and the regularising property of the Navier-Stokes dynamics implies that X is compact in H. Iterating (4.1), we see that

P u {u k ∈ X for k ≥ k 0 (ρ)} = 1 for any u ∈ B H (ρ),
where k 0 (ρ) = (ln ρ + C 2 )/ ln κ -1 with a large constant C 2 > 0. It follows that (u k , P u ) has at least one stationary measure µ, and any such measure is supported by X. It is easy to see that to prove (2.8), it suffices to establish inequality (2.18). The latter is proved in the next three subsections.

Proof of Proposition 2.6

We shall apply Proposition 5.3 to construct a measurable coupling (V, V ′ ) and Proposition 5.2 to prove (2.19). Namely, fix R > 0 so large that X ⊂ B H (R -1) and max{ η k H 1 (D1) , h+η k H 1 (D1) } ≤ R-1 almost surely. Denote by C, d > 0 and m ≥ 1 the parameters constructed in Theorem 3.1 with q = 1/4, define the Polish space

Z = {(u, u ′ ) ∈ X × X : u -u ′ ≤ d},
and introduce the function ε(u, u ′ ) = 1 2 u -u ′ on the space Z. Let us consider the pair of measures (P 1 (u, •), P 1 (u ′ , •)). By Proposition 5.3 with θ = 1 2 , there is a probability space (Ω, F , P) and measurable functions

V, V ′ : Ω × Z → H such that for any (u, u ′ ) ∈ Z the pair (V (u, u ′ ), V ′ (u, u ′ )) is a coupling for (P 1 (u, •), P 1 (u ′ , •)) and P V (u, u ′ ) -V ′ (u, u ′ ) > 1 2 u -u ′ ≤ C u-u ′ /4 P 1 (u, •), P 1 (u ′ , •) , (4.2)
where the function C ε (µ 1 , µ 2 ) is defined in Section 5.1. In view of (5.6), to estimate the right-hand side of this inequality, it suffices to bound the function

K u-u ′ /4 (P 1 (u, •), P 1 (u ′ , •)).
To this end, we shall apply Propositions 5.2, 5.6 and Theorem 3.1. Let us endow the ball B R ⊂ H 1 (D 1 ) with the law λ of the random variables η k . Then where {F k } stands for the filtration generated by the Markov family (u k , P u ). By iteration, the above inequality implies that P u {τ (B ) > mℓ} ≤ p m for all m ≥ 1.

(4.6)

A simple application of the Borel-Cantelli lemma now implies τ (B) is P u -almost surely finite. Furthermore, inequality (4.6) immediately implies that τ (B) satisfies (2.14). We now prove (4.5). Let η and η ′ be the random variables entering the definition of the coupling operator R (see (2.20)) and let

ζ k (ω) = η(ω k ), ζ ′ k (ω) = η ′ (ω k ), k ≥ 1. Then {ζ k , ζ ′ k , k ≥ 1} is a family of i.i.d. random variables in L 2 (D 1
) whose law coincides with that of η k . Let û ∈ H be the point defined in Hypothesis (H2). Since X is a closed absorbing subset, we have û ∈ X. Using the definitions of R and τ , we can write

P u {τ (B ) > ℓ} = P u {τ (B) > ℓ, u ℓ -u ′ ℓ > d} = P u {τ (B) > ℓ, S ℓ (u, ζ 1 , . . . , ζ ℓ ) -S ℓ (u, ζ ′ 1 , . . . , ζ ′ ℓ ) > d} ≤ P{ S ℓ (u, ζ 1 , . . . , ζ ℓ ) -S ℓ (u ′ , ζ ′ 1 , . . . , ζ ′ ℓ ) > d} = 1 -P{ S ℓ (u, ζ 1 , . . . , ζ ℓ ) -S ℓ (u ′ , ζ ′ 1 , . . . , ζ ′ ℓ ) ≤ d} ≤ 1 -P (ℓ, d) 2 ,
where the operator S l is defined in Hypothesis (H2), and we set

P (ℓ, d) = inf v∈X P{ S ℓ (v, ζ 1 , . . . , ζ ℓ ) -û ≤ d/2}.
Thus, inequality (4.5) will be established if we show that P (ℓ, d) > 0 for a sufficiently large integer ℓ.

To this end, let us denote by ℓ the integer defined in Hypothesis (H2) with ε = d/4 and R > 0 so large that X ⊂ B H (R). We fix v ∈ X and denote by ζ k = ζ v k ∈ K some vectors for which (2.6) holds. Let δ > 0 be so small that for any ζk ∈ K, k = 1, . . . , ℓ, satisfying the inequalities ζk -ζ v k < δ we have

S ℓ (v, ζ1 , . . . , ζℓ ) -û ≤ d/2. Define the event Γ(v, ℓ) = { ζ k -ζ v k < δ, k = 1, . . . , ℓ} ⊂ Ω. What has been said implies that S ℓ (v, ζ 1 , . . . , ζ ℓ ) -û ≤ d/2 for ω ∈ Γ(v, ℓ).
We see that

P (ℓ, d) ≥ inf v∈X P Γ(v, ℓ) ≥ inf v∈X ℓ k=1 P{ ζ k -ζ v k < δ} ≥ inf ξ∈K P{ ζ 1 -ξ < δ} ℓ , (4.7)
Using the Markov property, inequality (4.8), and the fact that u k -u ′ k ≤ 2 -k d on the set {σ > k}, we derive

P u {σ = n} = E u I {σ>n-1} P u { u n -u ′ n > 2 -n d | F n-1 } = E u I {σ>n-1} P u n-1 { u 1 -u ′ 1 > 2 -n d} ≤ E u I {σ>n-1} P un-1 { u 1 -u ′ 1 > 1 2 u 0 -u ′ 0 } ≤ 2 1-n Cd P u {σ > n -1}.
We thus obtain inequality (4.10) with C 2 = Cd. The proof of Theorem 2.1 is complete.

Appendix

Optimal coupling

Let X be a Polish space with a metric d and let µ 1 , µ 2 ∈ P(X). Recall that a pair of X-valued random variables (ξ 1 , ξ 2 ) is called a coupling for (µ 1 , µ 2 ) if D(ξ i ) = µ i , i = 1, 2. We denote by Π(µ 1 , µ 2 ) the set of all couplings for (µ 1 , µ 2 ). Let us fix ε ≥ 0 and define a symmetric function d ε : X × X → R by the relation

d ε (u 1 , u 2 ) = 1 if d(u 1 , u 2 ) > ε, 0 if d(u 1 , u 2 ) ≤ ε.
(5.1) Definition 5.1. We shall say that a coupling (ξ

1 , ξ 2 ) ∈ Π(µ 1 , µ 2 ) is ε-optimal if it minimizes the function (ζ 1 , ζ 2 ) → E d ε (ζ 1 , ζ
2 ) defined on the set of all couplings for (µ 1 , µ 2 ). That is,

E d ε (ξ 1 , ξ 2 ) = inf (ζ1,ζ2)∈Π(µ1,µ2) E d ε (ζ 1 , ζ 2 ).
(5.2)

In particular, for ε = 0 we obtain the usual concept of maximal coupling of measures; e.g., see [START_REF] Thorisson | Coupling, Stationarity, and Regeneration[END_REF].

Let us denote by C ε (µ 1 , µ 2 ) the quantity defined by the right-hand side of (5.2) and call it the ε-optimal cost for the pair (µ 1 , µ 2 ). Note that, in view of (5.1) and (5.2), for any ε-optimal coupling (ξ 1 , ξ 2 ) we have

P{d(ξ 1 , ξ 2 ) > ε} = C ε (µ 1 , µ 2 ).
(5.3) Thus, to estimate the probability of the event that the distance between the components of an ε-optimal coupling for (µ 1 , µ 2 ) is larger than ε it suffices to estimate the corresponding ε-optimal cost. We now establish a simple result that enables one to do it. Let us introduce the following function on the space P(X) × P(X):

K ε (µ 1 , µ 2 ) = sup f,g (f, µ 1 ) -(g, µ 2 ) , (5.4)
where the supremum is taken over all functions f, g ∈ C b (X) satisfying the inequality f (u 1 ) -g(u 2 ) ≤ d ε (u 1 , u 2 ) for u 1 , u 2 ∈ X.

(5.5)

Then, by the Kantorovich duality (see Theorem 5.10 in [START_REF] Villani | Optimal Transport. Old and New[END_REF]), we have

K ε (µ 1 , µ 2 ) = C ε (µ 1 , µ 2 ) (5.6)
Thus, to estimate the ε-optimal cost it suffices to estimate the function K ε . The following proposition reduces this question to a "control" problem.

Proposition 5.2. Let X be a compact metric space with metric d, let U 1 , U 2 be two X-valued random variables defined on a probability space (Ω, F , P), and let µ 1 , µ 2 be their laws. Suppose there is a measurable mapping

Ψ : Ω → Ω such that d U 1 (ω), U 2 (Ψ (ω)) ≤ ε for almost every ω ∈ Ω, (5.7)
where ε > 0 is a constant. Then

K ε (µ 1 , µ 2 ) ≤ 2 P -Ψ * (P) var .
(5.8)

In particular, any ε-optimal coupling (ξ 1 , ξ 2 ) for the pair (µ 1 , µ 2 ) satisfies the inequality P{d(ξ 1 , ξ 2 ) > ε} ≤ 2 P -Ψ * (P) var .

(5.9)

Proof. Inequality (5.9) is a straightforward consequence of (5.8), (5.6), and (5.3).

To prove (5.8), we use an argument applied in the proof of Lemma 11.8.6 in [START_REF] Dudley | Real Analysis and Probability[END_REF]. Namely, note that if f, g ∈ C b (X) are such that (5.5) holds, then the function h(u) = sup v∈X (f (v) -d ε (u, v)) satisfies the inequalities h ≤ g and

f (u 1 ) -h(u 2 ) ≤ d ε (u 1 , u 2 ), |h(u 1 ) -h(u 2 )| ≤ 1 for u 1 , u 2 ∈ X. It follows that (f, µ 1 ) -(g, µ 2 ) = E f (U 1 ) -g(U 2 ) ≤ E f (U 1 ) -h(U 2 • Ψ ) + E h(U 2 • Ψ ) -h(U 2 ) ≤ E d ε (U 1 , U 2 • Ψ ) + (h • U 2 , Ψ * (P)) -(h • U 2 , P) ≤ (h • U 2 , P) -(h • U 2 , Ψ * (P)) .
It remains to take the supremum over f, g and to note that right-hand side of this inequality does not exceed that of (5.8).

We now study the question of existence and measurability of an ε-optimal coupling. Let (Z, B) be a measurable space and let {µ z i , z ∈ Z}, i = 1, 2, be two families of probability measures on X such that the mapping z → µ z i is measurable from Z to the space P(X) endowed with the topology of weak convergence and the corresponding σ-algebra. The following proposition establishes the existence of an "almost" ε-optimal coupling that is a measurable function of z, provided that X is a subset of a Banach space.

Proposition 5.3. In addition to the above hypotheses, assume that X is a compact subset of a Banach space Y . Then for any positive measurable function ε(z), z ∈ Z, and any θ ∈ (0, 1) there is a probability space (Ω, F , P) and measurable functions ξ z i (ω) : Ω × Z → Y , i = 1, 2, such that, for any z ∈ Z, the pair (ξ z 1 , ξ z 2 ) is a coupling for (µ z 1 , µ z 2 ) and

E d ε(z) (ξ z 1 , ξ z 2 ) ≤ C θε(z) (µ z 1 , µ z 2 ).
(5.10)

Proof. The existence of a measurable optimal ε(z)-coupling would be an immediate consequence of Corollary 5.22 in [START_REF] Villani | Optimal Transport. Old and New[END_REF] if the function d ε(z) (u 1 , u 2 ) was continuous and independent of z. Since this is not the case, we now outline the modifications that are needed to prove the existence of a measurable coupling satisfying (5.10).

Let us write Z as the union of countably many disjoint measurable subset Z k such that the image of the restriction of ε(z) to any of them is contained in a bounded interval separated from zero. For instance, we can take Z k = {z ∈ Z : (k + 1) -1 < ε(z) ≤ k -1 } with k ≥ 1 and use a similar partition for (1, +∞). If we construct measurable couplings on each set Z k for which (5.10) holds with Z = Z k , then by gluing them together, we get the required random variables ξ z i , i = 1, 2.

To construct a measurable coupling on Z k , we first reduce the problem to the case when ε ≡ 1. Let us consider stretched measures μz i defined by the formula μz

i (Γ) = µ z i (ε(z)Γ) for Γ ∈ B Y . The measures μz i are supported by a compact set X k ⊂ Y , and if ( ξz 1 , ξz 2 ) is a measurable coupling such that E d 1 ( ξz 1 , ξz 2 ) ≤ C θ (μ z 1 , μz 2 ) for any z ∈ Z k , then ε(z)( ξz 1 , ξz 2
) is a measurable coupling for the original measures that satisfies (5.10). Thus, we can assume from the very beginning that ε ≡ 1.

Let d : X × X → R be an arbitrary continuous symmetric function such that

d 1 (u 1 , u 2 ) ≤ d(u 1 , u 2 ) ≤ d θ (u 1 , u 2 ) for all u 1 , u 2 ∈ X.
(5.11) By Corollary 5.22 in [START_REF] Villani | Optimal Transport. Old and New[END_REF], there is a probability space (Ω, F , P) and measurable functions ξ z

i : Z × Ω → X such that (ξ z 1 , ξ z 2 ) is a coupling for (µ z 1 , µ z 2 ) for any z ∈ Z, and E d(ξ z 1 , ξ z 2 ) = inf (ζ1,ζ2)∈Π(µ z 1 ,µ z 2 ) E d(ζ 1 , ζ 2 ).
Combining this relation with (5.11), we arrive at (5.10) with ε ≡ 1.

Truncated observability inequality

Let δ ∈ (0, 1) and let û ∈ Y δ,1 be an arbitrary function. Consider the adjoint equation for the Navier-Stokes system linearised around û: ġ + û, ∇ g + ∇g, û + ν∆g + ∇π = 0, div g = 0.

(5.12)

Let us fix an open set Q ⊂ (δ, 1) × D, an orthonormal basis {ϕ j } ⊂ L 2 (Q, R 2 ), and a function χ ∈ C ∞ 0 (Q) and denote by P m the orthogonal projection in

Minima of second-order polynomials on Hilbert spaces

Let U and Y be real separable Hilbert spaces and let F : U × Y → R be a function of the form

F (u, y) = (Q y u, u) U + (a y , u) U + b y ,
where Q y ∈ L(U ) is a self-adjoint operator, a y , b y ∈ U for all y ∈ Y , and (•, •) U stands for the scalar product in U . We assume that

(Q y u, u) U ≥ c u 2 U for all u ∈ U , y ∈ Y .
(5.17)

In this case, it is easy to see that for any y ∈ Y the function u → F (u, y) has a unique global minimum u * = u * (y). The following simple proposition establishes some properties of u * . Therefore, the required smoothness of u * will be established if we show that the implicit function theorem can be applied to (5.19). This is a straightforward consequence of the fact that ∂ u F is an infinitely smooth function of its arguments, and its derivative in u coincides with 2Q y . We now prove that if a y , b y , and Q y are Lipschitz continuous on bounded balls, then so is u * (y). Indeed, fix R > 0 and take two points y 1 , y 2 ∈ B Y (R). It follows from (5.19) and the explicit form of the derivative ∂ u F that

Q y1 u * (y 1 ) -u * (y 2 ) = Q y2 -Q y1 u * (y 2 ) + 1 2 a y2 -a y1 .
Taking the scalar product of this equation with u * (y 1 ) -u * (y 2 ) and using inequality (5.17) and the Lipschitz continuity Q y and a y , we obtain the required result.

Image of measures under finite-dimensional transformations

Let X be a separable Banach space endowed with a norm • and represented as the direct sum X = E ∔ F, (5.20)

where E ⊂ X is a finite-dimensional subspace. Denote by P E and P F the projections corresponding to decomposition (5.20). Let λ ∈ P(X) be a measure that can be written as the tensor product of its projections λ E = (P E ) * λ and λ F = (P F ) * λ. Assume also that λ has a bounded support and that λ E possesses a C 1 -smooth density ρ with respect to the Lebesgue measure on E. The following simple result gives an estimate for the total variation distance between the measure λ and its image under a diffeomorphism of X acting only along E.

Proposition 5.6. Under the above hypotheses, let Ψ : X → X be a transformation that can be written in the form

Ψ (u) = u + Φ(u), u ∈ X,
where Φ is a C 1 -smooth map such that the image of Φ is contained in E and Φ(u 1 ) ≤ κ, Φ(u 1 ) -Φ(u 2 ) ≤ κ u 1 -u 2 for all u 1 , u 2 ∈ X. (5.21)

Then the total variation distance between λ and its image under Ψ admits the estimate λ -Ψ * (λ) var ≤ Cκ, (

where C > 0 is a constant not depending on κ.

This proposition is a simple particular case of more general results presented in Chapter 10 of [START_REF] Bogachev | Differentiable Measures and the Malliavin Calculus[END_REF]. However, for the reader's convenience, we give a complete proof of Proposition 5.6.

Proof. Inequality (5.22) needs to be proved for κ ≪ 1, because it is trivial if κ is separated from zero. Furthermore, since λ has a bounded support, there is no loss of generality in assuming that Φ vanishes outside a large ball.

We first note that if f ∈ C b (X), then where we used the first inequality in (5.21). It is straightforward to see that the right-hand side of the above inequality can be estimated by Cκ.

2

  Main result and scheme of its proof 2.1 Exponential mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 General criterion for mixing and application . . . . . . . . . . . . 3 Control problem 3.1 Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Minimisation problem . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stabilisation of a non-stationary solution . . . . . . . . . . . . . .

  + νLr + B(û, r) + B(r, û) -χ(P m ξ) dt = 0. It follows that θ -νLθ -B * (û)θ = 0, (3.14) θ(1) = -2 δ Π N w(1), ζ = P m (χθ), (3.15) where B * (û) denotes the (formal) adjoint of the operator B(û, •) + B(•, û) in H, and Eq. (3.14) holds in the sense of distributions. Multiplying Eq. (3.14) by w and the first equation in (3.13) by θ, adding together the resulting relations, and integrating over D 1 , we derive 2 δ Π N w(1) 2 + 1 0 ζ(t) 2 dt = -(θ(0), v 0 ), (3.16)

Proposition 5. 5 .

 5 Let us assume that the functions y → Q y , y → (a y , b y ) (5.18) are infinitely smooth from Y to the spaces L(U ) and U × U , respectively. Then the unique minimum u * (y) is an infinitely smooth function of y ∈ Y . Moreover, this implication remains true if the property of infinite smoothness is replaced by Lipschitz continuity on bounded balls. Proof. Inequality (5.17) implies that u * (y) is the only solution of the linear equation (∂ u F )(u, y) = 0. (5.19)

FλF

  (f, λ) = F λ F (dw) E f (v + w)ρ(v) dv. Since v → v + Φ(v + w) is a C 1 -diffeomorphism of E for |κ| ≪ 1, we see that (f, Ψ * (λ)) = F λ F (dw) E f v + w + Φ(v + w) ρ(v) dv = (dw) E f (v ′ + w)ρ(Θ w (v ′ )) det I + (DΦ)(Θ w (v ′ ) + w) dv ′ ,where Θ w (v ′ ) denotes the solution of the equation v + Φ(v + w) = v ′ and DΦ stands for the Fréchet derivative of Φ. Combining the above formulas, we getδ(f, λ) := (f, Ψ * (λ)) -(f, λ) = F λ F (dw) E f (v + w)∆(v, w) dv,(5.23)where we set∆(v, w) = ρ(Θ w (v)) det I + (DΦ)(Θ w (v) + w) -ρ(v).The κ-Lipschitz continuity of Φ implies thatdet I + (DΦ)(Θ w (v) + w) -1 -1 = O(κ) for all v ∈ E, w ∈ F ,where O(κ) denotes any function whose absolute value can be estimated by Cκ (uniformly with respect to all other variables). It follows that∆(v, w) = ρ(Θ w (v)) -ρ(v) det I + (DΦ)(Θ w (v) + w) + ρ(v)O(κ).(5.24) Furthermore, using inequalities (5.21), we deriveΘ w (v) ≤ 1 1-κ v + κ w , Θ w (v) -v ≤ κ 1-κ v + w .(5.25) Substituting (5.24) into (5.23) and using inequalities (5.25), we obtain|δ(f, λ)| ≤ F λ F (dw) E |f (v + w)| |ρ(Θ w (v)) -ρ(v)| det I + (DΦ)(Θ w (v) + w) dv + O(κ) ≤ f ∞ F λ F (dw) E ρ(v) -ρ(v + Φ(v + w)) dv + O(κ).Taking the supremum over f ∈ C b (X) satisfying f ∞ ≤ 1 and noting that the integrals on the right-hand side can be taken over sufficiently large balls, we obtain λ -Ψ * (λ) var ≤ 1 2 BF (R) λ F (dw) BF (R) ρ(v) -ρ(v + Φ(v + w)) dv + O(κ) + θΦ(v + w)) dθ dv + O(κ),

  Eq. (2.3) defines a homogeneous family of Markov chains in H, which is denoted by (u k , P u ), u ∈ H. Let P k (u, Γ) be the transition function for (u k , P u ).Let us fix an open set Q ⊂ D 1 and denote by {ϕ j

Note, however, that we do not deal at all with the Gramian of the control problem in question, and the property we use may be called squeezing by a finite-dimensional modification.

The assumption on the linear independence of {ψ j } is not really needed, and Theorem 2.1 below remains true without it. We make, however, this assumption to simplify the proof of Proposition 2.6.

In[START_REF]Exponential mixing for randomly forced partial differential equations: method of coupling, Instability in models connected with fluid flows[END_REF], the proof is carried out in the particular case when B = B × B; however, the same argument applies in the general situation.

P 1 (u, •) is the image of λ under the mapping η → S(u, h + η) acting from B R to H. Let E m be the subspace entering Theorem 3.1 and let

for any η ∈ B R and any u, u ′ ∈ B H (R) satisfying the inequality u -u ′ ≤ d.

The existence of such mapping was established in Theorem 3.1. Let us define a transformation Ψ = Ψ u,u ′ of the space H 1 (D 1 ) by the relation

where χ : R + → R + is a smooth function such that χ(r) = 1 for r ≤ R -1 and χ(r) = 0 for r ≥ R. The choice of the constant R and inequality (4.3) imply that

Therefore, by Proposition 5.2, we have

Now note that the mapping Ψ satisfies the hypotheses of Proposition 5.6 with a constant κ proportional to u -u ′ . Combining (4.2), (4.4), and (5.22), we arrive at the required inequality (2.19).

Recurrence

Let us recall that the probability space (Ω, F , P) and the X -valued Markov chain {u k = (u k , u ′ k )} on it were defined in Section 2.2. We shall denote by P u the probability measure associated with the initial condition u . Let us set

where d > 0 is a small constant to be chosen later. In this and next subsections, we shall prove that the recurrence and exponential squeezing properties are satisfied for (u k , P u ) with the above choice of B.

To prove that τ (B) is P u -almost surely finite and satisfies (2.14), it suffices to show that p := sup

where ℓ ≥ 1 is an integer. Indeed, if this inequality is proved, then using the Markov property, for any integer m ≥ 1 we can write

where we used the fact that ζ k are i.i.d. random variables. It remains to note that the function ξ → P{ ζ 1 -ξ < δ} is lower semicontinuous and strictly positive on K, and therefore the right-hand side of (4.7) is a positive constant. This completes the verification of the recurrence property.

Exponential squeezing

Let us take arbitrary initial points u, u ′ ∈ X such that (u, u ′ ) ∈ B. Then, by (2.19) and (2.20), we have

For any integer n ≥ 1, let us set

Combining this observation with inequality (4.8) and the Markov property, we derive

Iterating this inequality, for any n ≥ 1 we obtain

whence it follows that

Choosing d > 0 sufficiently small, we arrive at (2.16).

To prove inequality (2.17), it suffices to show that

where the positive constant C does not depend on u . To this end, note that

L 2 (Q, R 2 ) onto the m-dimensional subspace spanned by ϕ j , j = 1, . . . , m. Recall that H N stands for the vector span of the first N eigenfunctions of the Stokes operator. The following result is a simple consequence of the observability inequality for the linearised Navier-Stokes system (see [START_REF] Fursikov | Exact controllability of the Navier-Stokes and Boussinesq equations[END_REF][START_REF] Yu | Remarks on exact controllability for the Navier-Stokes equations[END_REF][START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]).

Proposition 5.4. For any δ, ρ > 0 and any integer N ≥ 1 there is an integer m ≥ 1 such that if û ∈ B Y δ,1 (ρ), then any solution g ∈ X 1 of Eq. (5.12) with g(1) ∈ H N satisfies the inequality

where C > 0 is a constant depending only on δ and ρ.

Proof. We essentially repeat the argument used in the paper [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF] in which inequality (5.13) is proved for a time-independent function χ and a projection P m acting in the space variables. We claim that if g ∈ X 1 is a solution of (5.12) with g(1) ∈ H N , then

where C 1 > 0 is a constant depending on δ, ρ, and N . Once this inequality is established, the required result can be derived by using the fact that

, where {δ m } is a sequence going to zero as m → ∞; cf. proof of Proposition 5.3 in [START_REF] Barbu | Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations[END_REF].

Suppose that (5.14) is false. Then there are functions ûn ∈ B Y δ,1 (ρ) and solutions g n ∈ X 1 of (5.12) with û = ûn such that g n (1) ∈ H N , g n (1) = 1, (5.15)

χg n H 1 (D1) ≥ n χg n L 2 (D1) .

(5.16)

Now note that g n (1) H 2 ≤ C 2 for all n ≥ 1, whence it follows, by standard estimates for the 2D Navier-Stokes system, that

Passing to a subsequence, we can assume that {û n } and {g n } converge weakly (in appropriate functional spaces) to some functions û and g, respectively, such that û ∈ B Y δ,1 (ρ), g ∈ X 1 is a solution of (5.12), and g(1) = 1. Moreover, it follows from (5.16) that χg ≡ 0. Let us show that the latter is impossible. Indeed, let an interval (a, b) ⊂ J and a ball B ⊂ D be such that χ(t, x) ≥ α > 0 for (t, x) ∈ (a, b) × B.

Then, by the observability inequality (see Lemma 1 in [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF]), we obtain g(a) ≤ C 4 g L 2 ((a,b)×D) ≤ C 4 α -1 χg L 2 (D1) = 0.

The backward uniqueness for the linearised Navier-Stokes system (see Section II.8 in [START_REF] Babin | Attractors of Evolution Equations[END_REF]) implies that g(t) = 0 for a ≤ t ≤ 1. This contradicts the fact that g(1) = 1. The proof is complete.


Ngally Sabouang

Jean Mbey
email: mbey25@yahoo.fr

Aimé

TALC AS RAW MATERIAL FOR CEMENTITIOUS PRODUCTS FORMULATION

Keywords: Talc, Magnesia, Cement, compressive strength 1-Introduction

This study reports the characterisation of a talc from Cameroon as a possible source material for cement formulation. To that end, the talc sample was characterized and mixed with a solution of sodium polyphosphaste to formulate the cementitious products.

Addition of magnesia (MgO) was done to analyse the effect of available MgO on the products. Fourier transform infra red, X-rays diffraction, linear shrinkage, compressive strength and Scanning electron microscopy were used to analysed the products. The compressive strength increased with MgO addition and the linear shrinkage decreased. All the analyses indicate that talc is a raw material of interest in cementitious products formulation however, the availability of the MgO is a key factor for a better performance of the products.

warming and other detrimental ecological changes has spurred the research and development of new kinds of cementitious materials produced from non conventional resources. The reason is that the production of 1ton of hydraulic cement will generate about 900 kg of carbon dioxide and consume about 5 billion joules of electrical power and fuel energy [1,2]. The geopolymers for instance are a family of cementitious materials developed in late 70's and appear as an alternative of interest [3]. Kaolinite is nowadays a material largely used in geopolymers making as shown by the increasing number of publications that deal with metakaolin based geopolymers [4][5][6][7][8]. Some new classes of raw materials are now being introduced in this domain. For instance, the use of volcanic ashes as raw material in geopolymer making is now established [9][10].

Talc is a hydrated layered magnesium silicate with chemical formula Mg 3 S i4 O 10 (OH) 2 . It is a hydrophobic material that easily blends and disperses within organic media including polymers. Talc is widely used as reinforcing filler in several industrial products such as paper, paints, rubbers, polymers, ceramics, refractory materials [11,12]. The composition of talc make it a potential raw material in cement formulation since magnesium can beneficially interact with phosphate to give cementitious material, as in phosphate-magnesia cements [13][14][15][16]. Talc-based cementitious material are already used as prosthesis in dental surgery [17].

Recently, Pusch et al., (2013) [18], have used talc as a super plasticizer for the formulation of cement poor concrete to be used for boreholes sealing in rocks.

Given that some talc deposits are found along the pan-African Yaoundé fold belt at the border of the northwestern edge of the Congo craton [19,20], it is obvious that research seeking for talc valorization is to be undertaken. In the present work, talc from Cameroon is used as raw material for the formulation of cementitious products. Because the mobilization of magnesia is expected, a study of the effect brought by the addition of magnesia is also carried out. Analysis of some properties of cement paste made from talc is presented here as a first step towards valorization of talc as raw material in cement formulation. Powder X-ray diffraction (XRD), Fourier Infra red spectroscopy (FTIR), scanning electron microscopy (SEM), mechanical testing (compressive strength) and linear shrinkage measurements are used to characterize the formulations.

2-Materials and methods

A talc from Lamal Pougue in the Boumyébèl area at about 93 Km from Yaoundé (Center region , Cameroon) was used. The area description is presented in Nkoumbou et al., (2008) [19]. The raw material was manually ground and sieved at 100 µm prior to characterization.

Sodium polyphosphate ((NaPO 3 ) n ) analytical grade containing 60% P 2 O 5 purchase at Riedel de Haen AG, was used as phosphate source.

The magnesia used is a analytical grade product from Rhône-Poulenc. The IR spectrum (Figure 1) shows traces of sulphur compounds through the stretching band of sulphate at 1423 cm -1 and S-O stretching at 847 cm -1 . The bands at 1459 cm -1 indicates the presence of carbonate. Some silicates impurities are detected from the band at 1077 cm -1 due to forsterite (Mg 2 SiO 4 ) as also observed from the XRD pattern (Figure 2). The X ray pattern also indicates that the magnesite is almost completely converted into magnesia.

The cement pastes were prepared as follows:

In an acidic solution of sodium polyphosphate, known amounts of talc and calcined MgO were added. The mixture is thoroughly mixed during 5 minutes using a M&O mixer. This mixture is used to prepare cylindrical samples using cylindrical mould having a diameter of 30 mm and a height of 60 mm. The paste in the mould are vibrated for 10 minutes on an electrical vibrating table (M & O, type 202, No 106) to remove entrapped air bubbles. The samples are cover with a polyethylene bag and left for consolidation. The consolidated samples are kept at the ambient temperature (24-27°C) and the mechanical analysis are carried out on the 28 days aged samples. The sample indexed CITA 0 is prepared without any added magnesia. For this sample, 50 g of talc with 50 g of 40% (in weight) solution of sodium polyphosphate are used. For samples in which magnesia is added, the amounts of magnesia, expressed in percentage mass with respect to the initial mass of talc (50 g), were: 5%, 10%, 15% and 20%. The obtained products are indexed CITA 1 to CITA 4 in the increasing order of magnesia content.

The cement sample are analyzed using powder X-ray diffraction (XRD), Fourier transform infra red (FTIR) spectroscopy, Scanning electron microscopy (SEM) coupled to energy dispersive X-ray spectroscopy (EDS), linear shrinkage and compressive strength measurements.

Chemical analysis of major elements in the raw talc was obtained by inductive coupled plasma by atomic emission spectrometry (ICP-AES) at the Service d'Analyse des Roches et des Minéraux (SARM, Nancy).

Powder X-ray diffraction patterns were recorded using a D 8 Advance Bruker diffractometer equipped with a Co Kα radiation ( = 1.7890 Å) operating at 35 kV and 45 mA. The diffraction patterns were obtained from 1.5° to 32° at a scanning rate of 1° min -1 .

Infrared spectra were recorded in diffuse reflection mode using a Bruker IFS 55 spectrometer.

The spectra, recorded from 4000 cm -1 to 600 cm -1 with a resolution of 4 cm -1 , are accumulation of 200 scans.

The SEM micrographs and EDS data were obtained on a Hitachi S-4800 using a YAG (Yttrium Aluminium Garnet) backscatter secondary electron detector. The samples were carbon coated prior to analysis. Measurement of linear shrinkage was done using a Calliper on the hardened geopolymer mortars aged of 1, 7, 14 or 28 days respectively. Linear shrinkage was calculated using the following equation:

R L = (L 0 -L)x100/L 0
Where L 0 is the initial length of specimens at first days and L is the length of specimens after a given number of days [10].

Compressive strength was determined on cylinder samples with 60 mm height and aged 28 days using an electro-hydraulic press (M & O, type 11.50, No. 21) with capacity of 60 KN.

The samples are submitted to a compressive force at average rate of 3 mm/min until the sample failed according to ASTM C 109 standard test method. The compressive strength was calculate using the following equation: δ= F/A, where F is the failure force and A is the initial cross section of the sample.

Results and discussion

Characterization of the used talc

The chemical analysis of the used talc is given in table I. For the ideal talc formulation (Mg 3 Si 4 O 10 (OH) 2 ), the theoretical content of equivalent magnesium oxide is 31.9% and silica is 63.4% . Hence, the content in Magnesia (32.08%, MgO) and silica (51.71%, SiO 2 ) is consistent with the magnesium silicate nature of the Talc. However, the material shows a deficiency in silicon, probably substituted by part of iron (6,30%, Fe 2 O 3 ). The IR spectra and the XRD pattern of the raw material are given in figures 1 and 2. On the FTIR spectrum (Figure 1), the O-H stretching vibration band of OH surrounded by 3 Mg (Mg 3 -OH) appears at 3676 cm -1 . The band at 3660 cm -1 is consistent with the stretching of OH surrounded by 2Mg and 1 Fe (Mg 2 Fe-OH) while the 3649 cm -1 band is associated to OH surrounded by 1 Mg and 2 Fe (MgFe 2 -OH). The OH libration band is observed at 667 cm -1 ; the stretching band for Si-O-Si is observed at 1050 cm -1 and the combination bands of Si-O are observed at 779 cm -1 [21][22][23][24]. Trace of calcite and/or dolomite is evidenced by the vibration bands of carbonate group at 2535 cm -1 , 2922 cm -1 , 1818 cm -1 and 1459 cm -1 [START_REF] Huang | Infrared study of the carbonate minerals[END_REF]. The OH bands for chlorite (in the range 3590-3415 cm -1 ) [24,[START_REF] Villiéras | Etude des modifications des propriétés du talc et de la chlorite par traitement thermique[END_REF] was not clearly observed and this indicates that the content in chlorite is rather low.

The XRD pattern (Figure 2) shows the typical talc reflections corresponding to (001), ( 002) and (003) basal planes at 9.37 Å, 4.68 Å and 3.12 Å respectively. The presence of chlorite as associated mineral to talc is shown by the reflections at 14. 38 Å, 7.10 Å and 4.76 Å. The intensities of these peaks are relatively low and confirm the conclusion from FTIR of low content in chlorite. Trace amount of amphibole is suggested by a low intensity diffraction peak at 8.25 Å. The carbonate trace content observed from IR is also indicated by the reflections at 3.05 Å on the XRD pattern of the talc.

X-ray diffraction and FTIR analysis of the cementitious products

In figures 1 and 2, the infra red spectra and XRD patterns of the cements are presented together with the spectra of the raw products (talc and magnesia). The XRD pattern of CITA 0 (formulation based on raw talc only) is given as reference together with CITA 3 and CITA 4 .

Because the addition of MgO is done in order to improve the compressive strength, through an increase of cement phase formation, hence, given that CITA 3 and CITA 4 exhibited the highest values of compressive strength (see compressive strength section), which indicate more cement phase formation, they were chosen for both FTIR and XRD analyses.

It is easily noticeable from both IR spectra and XRD patterns that there is not formation of new crystalline phase. This observation is in accordance with what is reported for phosphomagnesia cement prepared with sodium phosphate [START_REF] Zurz | [END_REF]. The remaining Talc is clearly evidenced from X ray through the reflections at 9.31 Å and 3.11 Å. The disappearance of characteristic diffraction peak of magnesia in CITA 3 and CITA 4 clearly shows that almost all the magnesia has reacted. A part from water characteristic bands (3280 cm -1 ( water OH stretching) and 1649 cm -1 (in plane deformation)), the OH bands between 3700 and 3400 cm -1 are characteristic of OH linked to magnesium (Mg-OH). The OH stretching of OH in a 3 magnesium environment at 3676 cm -1 are characteristic of talc. The changes in the intensity of these bands are indicative of reaction of the magnesium from the talc. The bands at 667 cm -1 show a more marked change. Given that this band was assigned to talc OH libration, then it is suggested to be an indicative of the magnesium from the Mg-OH interactions with the added phosphate. The band at 3699 cm -1 is rather characteristic of magnesium from the added magnesia and the change in intensity of this band compared to the bands in the pure magnesia spectrum is indicative of the reaction of this magnesia. Two new bands at 1129 cm -1 at 918 cm -1 are observed in the FTIR spectra of CITA 3 and CITA 4 . The band at 1129 cm -1 is assigned to asymmetric bending of P-O in PO 3 or PO 4 groups and the bands at 918 cm -1 is assigned to asymmetric bending of P-O-P groups [28]. The presence of these phosphate characteristic bands is an evidence for the presence of phosphate in the matrix of the obtained products. It is then obvious that cement formation occurs and magnesium interaction with the phosphate is responsible of cement phase formation but the added magnesia is more reactive than the magnesia from the talc structure.

Linear shrinkage

The linear shrinkage as a function of the age of the hardened mortars, determined at the ambient temperature of the lab (24 ± 3), are presented on figure 3. The overall trend indicates that linear shrinkage decrease with increasing MgO content in the mixture. Linear shrinkage originates from capillary tension within the paste framework during synthesis [29]. Thus, high shrinkage is an indication of insufficient cement phase formation. Hence, the comparison of linear shrinkage, amongst the studied samples, indicates that the mobilization of MgO for cement phase formation is rather low in raw talc. From figure 5, it is obvious that increase cement formation is observed for 20% added MgO sample (CITA 4 ). The availability of the MgO from the talc should then be improved in other to ameliorate the binder phase formation and hence the performance of the products.

Compressive strength

The compressive strength of 28 day aged samples are presented on figure 4. The value of the strength varied from 3.35 MPa (Talc cement with 0% added MgO) to 4.50 MPa (Talc cement with 20% added MgO). The addition of the calcined magnesia (MgO) lead to an increase of the compressive strength. This increase is associated to increased cement phase formation and corroborates with more cement phase formation consecutive to MgO addition as previously concluded from the linear shrinkage. The mobilization of the MgO from Talc appears then as a key to better mechanical performances of the samples. It is evidenced that talc is a potential raw material for the formulation of cementitious materials.

Microstructure

From SEM images for CITA 3 and CITA 4 are presented of figure 5. It can be observed that the materials have a homogeneous feature in which platy disc particles characteristic of talc could be observed. These remaining platelets are held together by a binder phase due to the reaction of the magnesium with phosphate. The presence of many pores is coherent with the low mechanical performance and relatively high shrinkage. These pores are favorable to fluids capillary diffusion that induces shrinkage of the products.

The EDS analyses on figure 6 evidence unreacted mixed oxide of iron, chrome and Nickel.

These elements are characteristic of identified minerals associated to talc (amphibole, chlorite with iron substitution and probably iron oxide). The sulphur is rather associated to the added calcined magnesia as observed from FTIR.

The cementitious materials could be ameliorated by increasing binder phase formation so that the remaining talc could be better held in the matrix and reduction of pores will be probably observed.

4: Conclusion

The present work was aimed at evaluating the potentiality of talc as a raw material for cementitious products preparation. For this prospect, formulated products were studied by infra-red spectroscopy, X-ray diffraction, linear shrinkage, compressive strength and SEM/EDS. The infra-red and X-ray diffraction indicate that magnesia is the actual reactive phase for the cement phase formation that served as binder for the remaining talc particles to form cement.

The linear shrinkage indicates low binder phase formation which is corroborated by the compressive strength and the SEM micrograph analysis. From compressive strength and linear shrinkage, it is shown that the addition of magnesia improve the performances of the products. It is then concluded that a better mobilization of the MgO from talc could be beneficially used to produce better cementitous products. All the results show that talc is a potentially interesting raw material in cementitious materials formulation. For the purpose of ameliorating the MgO mobilization from talc, a thermal treatment of the talc prior to cementitious product formulation is presently being carried out and will be presented in an upcoming paper.

Acknowledgement:

Professor Charles Nkoumbou, of the university of Yaoundé I, is greatly acknowledged for his assistance in talc collection from the field.


Benoit Cloitre

Broken harmonic functions

come

Introduction

With our tauberian approach 2 we believe that the difficulty of RH relies on the fact that it is not totally an arithmetical problem nor totally an analytical problem. A subtle use of both complex analysis and arithmetic seems needed to settle the problem. Indeed although our functions of good variation are discretely defined (see section 1 for the discrete implicit definition) an analytic conjecture is crucial in order to get information on the good variation index of the Ingham function. In addition this analytic conjecture contains an important discrete condition: the Hardy-Littlewood-Ramanujan criteria (HLR criteria) wich appears to be the corner stone of our strategy. This strategy depends upon 2 conjectures.

In section 2 we indroduce the so called broken harmonic functions and state the comparison conjecture between good variation index of these functions.

In section 3 we consider a special case of broken harmonic functions involving √ 2 and state a conjecture on the value of its good variation index.

The section 4 described how these 2 conjectures allow us to derive RH is true.

Finally in section 5 we generalise the Ingham function and discuss the extension of the method to L-functions. 1 The name comes from a tauberian theorem of Ingham which was the starting point of our study which began 4 years ago. Namely

nan ≥ -C ∧ lim n→∞ n k=1 a k Φ k n = l ⇒ ∞ k=1 a k = l
Cf. [Kor], theorem 18.2.p.110 or [Ten], corollaire II.7.23.

2 Our preliminary study is described in rough unpublished expository papers [START_REF] Cloitre | A tauberian approach to RH[END_REF][START_REF] Cloitre | [END_REF][START_REF] Cloitre | Functions of good variation[END_REF].

This comparison method reveals somewhat a fractal (or multifractal) aspect of RH which is hidden behind the complexity of the Mertens function

M (n) = n k=1 µ(k) or the Liouville function L(n) = n k=1 λ(k).
This fact is well illustrated by the figure 4 in section 4.

1 Functions of good variation (FGV)

Implicit discrete definition

Let (a n ) n≥1 be a real sequence and g : [0, 1] → R be a bounded and measurable function. Defining the sums:

• A(n) := n k=1 a k • A g (n) := n k=1 a k g k n
we say that g is a FGV of index α(g) ∈ R if we have the 2 following tauberian conditions:

1. (β < α(g)) ∧ A g (n) ∼ n -β (n → ∞) ⇒ lim n→∞ n β A(n) exists. 2. (β > α(g)) ∧ A g (n) ∼ n -β (n → ∞) ⇒ n β A(n) is unbounded
It is easy to see that FGV exist. For instance polynomials are FGV. More precisely letting g(x) = m k=0 c k x k with m ≥ 1, c m = 0 and g(0)g(1) = 0 we can say that g is a FGV of index α(g) = min { (ρ i )} where ρ i are the roots of m k=0 c k k-x (cf. [START_REF] Cloitre | Functions of good variation[END_REF] for a sketch of proof).

Analytic conjecture

We propose an analytic method to determine the good variation index α(g) of some FGV g. This method doesn't work for all FGV. For instance the "almost Dirac" function defined by g(x) = 1 on [0, 1] except at x = 1 2 where we have

g 1 2 = r ∈]0, 1[ is a FGV of index α (g) = -log(1-r) log 2
which can be proved only by arithmetical means (cf. [START_REF] Cloitre | A tauberian approach to RH[END_REF] p. 12 and the sketch of proof) .

Before stating this conjecture let us define a function of quasi bounded variation on ]0, 1], the HLR criteria and the little Mellin transform.

Function of quasi bounded variation

We say that a bounded and measurable function g : ]0, 1] → R is of quasi bounded variation if we have:

1. ∀x ∈]0, 1] g is of bounded variation on [x, 1] . 2. ∃x 0 ∈]0, 1] such that g is monotonic by parts on ]0, x 0 ].

HLR criteria (for Hardy-Littlewood-Ramanujan) 3

We say that a bounded and measurable function g satisfies the HLR criteria if it is a FGV of index 0 ≤ α(g) ≤ 1 such that for any function f (x) converging exponentially fast to zero as x → ∞ one has

A g (n) = f (n) ⇒ ∀ε > 0 lim n→∞ a(n)n 1-ε = 0

Little Mellin transform

We introduce g as the little Mellin transform of a bounded and measurable function g as follows when z < 0

g (z) := ˆ1 0 g(t)t -z-1 dt
We can now state the analytic conjecture.

Analytic conjecture

Let g be a function of quasi bounded variation ]0, 1] → R satisfying g(1) = 0 and lim x→0 g(x) = 0 which has the following properties:

• g is continuous on the left.

• g satisfies the HLR criteria.

• g can be extended analytically to the whole complex plane with possibly some singularities.

• inf { (ρ) | ρ ∈ C ∧ g (ρ) = 0} ∈ [0, 1] . Then g is a FGV of index α(g) = inf { (ρ) | ρ ∈ C ∧ g (ρ) = 0}.

Remark

This conjecture is true for polynomials and for continuous functions of bounded variation which are well approximated by polynomials. Using results from tauberian theory related to the Mellin convolution ([BGT, Bin]), we think the conjecture is also true for any function of bounded variation on [0, 1]. For functions of quasi bounded variation on ]0, 1] the HLR criteria seems to play a central role.

Comparison conjecture

At the early stage of our study on FGV we thought that comparing Φ to a simpler function could shed light on the value of α (Φ). Unfortunately our ideas were too naive. Essentially we were confronted to the same problem than Turan with his approach to RH based on the location of zeros of sections of the zeta function [START_REF] Borwein | The Riemann hypothesis: a ressource for expert and afficionado alike[END_REF]. Indeed approaching the Ingham function by functions with a finite number of discontinuities [START_REF] Cloitre | A tauberian approach to RH[END_REF] led us to the same kind of difficulties.

That's why we are convinced that functions with infinitely many discontinuities are needed and recently we came across a useful family of functions.

Broken harmonic functions

Let a, b be two reals satisfying -1 ≤ a < 1, 0 < a + b ≤ 1 and define x n by x 0 = 1, x 1 = b 1-a and for n ≥ 2

x n+1 = bx n 1 -ax n
Then we define the broken harmonic function g (a,b) for 0 < x ≤ 1 as follows

x n+1 < x ≤ x n ⇒ g (a,b) (x) = x x n

Examples

• g (-1,1) = Φ which has its discontinuities at x = 1 k for k ≥ 2 integer.

• g (-1/2,1) (x) = x 2 2
x .

• For r > 1 we have g (0,1/r) (x) = xr -log x log r .

As we shall see the functions g (0,1/r) will be very interesting for our purpose. In the sequel BHF will stand for broken harmonic function.

Comparison conjecture

BHF deserve attention since they are all built in a same way and their good variation index seem to follow rules. Hence we elaborated a comparison conjecture between them. The strong form of the conjecture relates all BHF whereas the weak form is restricted to the Ingham function and to g (0,1/r) when r > 1.

First of all we conjecture that BHF are FGV which can be devided in 2 sets. The set of BHF which satisfy the HLR crieria and the set of BHF which don't satisfy the HLR criteria.

Strong form of the comparison conjecture

Let a, b, a , b be 4 real values verifying:

• -1 ≤ a < 1, 0 < a + b ≤ 1, -1 ≤ a < 1, 0 < a + b ≤ 1.
Suppose g (a,b) satisfies the HLR criteria and g (a ,b ) doesn't satisfy the HLR crieria. Then we have the following inequality between index of good variation

α g (a,b) ≥ α g (a ,b )
Weak form of the comparison conjecture Let r > 1. If g (0,1/r) doesn't satisfy the HLR criteria then we have the following inequality between its index and the index of Φ which satisfies the HLR criteria

α (Φ) ≥ α g (0,1/r)
Remark There is a rough heuristic argument supporting the conjecture. Let us define 2 sequences a and a as follows

n k=1 a k g (a,b) k n = n k=1 a k g (a ,b ) k n = 1 2 n
then we have according to FGV properties:

• n k=1 a k n -α(g (a,b)) +ε • n k=1 a k n -α(g (a ,b ) )+ε
Now if g (a,b) satisfies the HLR criteria but not g (a ,b ) then we have some chances to get for N large enough

max n k=1 a k 1 ≤ n ≤ N < max n k=1 a k 1 ≤ n ≤ N
because a n is bigger in absolute value than a n . This would mean α g (a,b) ≥ α g (a ,b ) . In section 4 we provide an example using the Liouville function (fig. 4) where this phenomenom seems true.

3 On the good variation index of g (0,1/ √

2)

It is this function which will show the relevance of the comparison conjecture. Firstly the analytic conjecture does'nt allow us to determine the good variation index of g (0,1/ √ 2) . Indeed we have

g (0,1/r) (z) = 1 -r 1-z (1 -z)(1 -r z )
Therefore when g (0,1/r) satisfies the HLR criteria we have simply α g (0,1/r) = 1 which is the case when r ≥ 2 is an integer value (and we believe it is the only case).

When r isn't an integer value however g (0,1/r) clearly doesn't satisfy the HLR criteria and experiments show that α g (0,1/r) = 0 very often and sometime the index is not trivial i.e. 0 < α g (0,1/r) < 1.

The simplest case for which the index seems not trivial is r = √ 2 since we conjecture that we have

α g (0,1/ √ 2) = 1 2

Experimental support

Let a(n) be defined by the recursion

A g (0,1/ √ 2) (n) = 1 2 n
Then we plot A(n) √ n (fig. 1) where a fractal pattern appears which seems to stay bounded along the y axis and just after we plot a(n)

√ n (fig. 2) showing clearly that g (0,1/ √ 2) doesn't satisfy the HLR criteria.

fig. 1 fig. 2 Hereafter we compute a(n) defined by A g (0,1/ √ 2) (n) = 1 n . We can notice that local maxima of A(n) in the halfplane y > 0 occur at values of n of form 3.2 k . Therefore we plot A(3.2 k ) √ 3.2 k for k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

fig.3

The graph could converge or stay bounded by a slowly varying function supporting the conjecture α g (0,1/

√ 2) = 1 2 . Remark α g (0,1/ √ 2) = 1
2 is an interesting conjecture in its own and should be confirmed by more experiments. In general we think that:

• r ≥ 2 integer yields α g (0,1/r) = 1 (see APPENDIX 2 for some examples).

• r = m 1/k for k ≥ 2 fixed integer and m ≥ 2 integer but ot a power of k yields α g (0,m -1/k ) = 1 k (see APPENDIX 3 where we take r = 2 1 3 ).

• If r j is never an integer value for j ≥ 1 integer then α g (0,1/r) = 0 (see APPENDIX 4 for examples).

Right now we have no clue to show these assertions. However the fractal aspect of graphics suggest that a proof could exist. Some diophantin facts like

2 -N -1 2 -k n > C(N )
n 2 should also play a role.

On RH

We have

Φ (z) = ζ(1-z)
1-z and Φ satifies the HLR criteria and the other conditions of the analytic cojecture. Hence we have thanks to the known location of the zeros of the zeta function

0 < α (Φ) ≤ 1 2 (1)
next since g (0,1/ √ 2) doesn't satisfy the HLR criteria from section 3 (fig. 2) the comparison conjecture (strong form or weak form) and the conjecture The plot behaves like a very slowly varying function (of type log(n) ε ) and the fig. 4 shows that the red graphic doesn't come across the minima of the blue graphic. This would mean n k=1 (-1) Ω(k) n 1/2+ε implying RH is true.

α g (0,1/ √ 2) = 1 2 yield α (Φ) ≥ α g (0,1/ √ 2) = 1 2 (

Generalisation

Generalisation of the Ingham function

Let χ be a caracter and let us define the function g χ on ]0, 1]

g χ (x) = x 1≤k≤1/x χ(k) 1 kx
Then we have

• g χ satisfies the HLR criteria

• g χ (z) = ζ(1-z)L(1-z,χ) 1-z
Thus we can extend the comparison conjecture providing a broader definition of BHF.

General definition of broken harmonic functions

This definition is an extension of the construction in section 2. Let u be a real sequence satisfying

• 1 = u 1 > u 2 > u 3 > ... > u ∞ = 0.
Let g u be the left continuous function defined for n ≥ 1 and any x ∈]0, 1] by

u n+1 < x ≤ u n ⇒ g u (x) = p n x
where p n > 0 is an increasing real sequence such that u i p i 1 . Then g u is a BHF.

Generalisation of the comparison conjecture

If g u is a BHF satisfying the HLR criteria and g u is a BHF which doesn't satisfy the HLR criteria then we have α (g u ) ≥ α (g u )

Application to the generalised Riemann hypothesis

RH is true for L(s, χ).

fig. 6

It looks like a(n)n stays of order log n thus we would have lim n→∞ a(n)n 1-ε = 0 for any ε > 0 and g τ would satisfy the HLR criteria.

On the zeta function of Davenport and Heilbronn

It is interesting to see that our method doesn't work for this function. and computing 10000 terms of a(n) given by the recursion A g H (n) = 1 2 n it is clear that g H doesn't satisfy the HLR criteria as shown below (fig. 7) where we have compared na(n) with ± √ n (red). fig. 7 na(n) seems not bounded by a slowly varying function and we wouldn't have lim n→∞ a(n)n 1-ε = 0 for any ε > 0. Therefore g H doesn't satisfy the HLR criteria and we can say nothing about the value of α (g H ).

The fact that h(n) is not multiplicative and that ζ H has no Euler product is certainly the cause of this phenomenon. Indeed to us the HLR criteria has something to do with the quasi-multiplicity of a n which is a concept that will be described in more details later.

A g (0,1/5) (n) = 1 2 n plot of A(n)n A g (0,1/5) (n) = 1 2 n plot of a(n)n

  fig.4

.

  We consider ζ H (s) = n≥1 h(n) n s where h is the 5-periodic sequence [1, ξ, -ξ, -1, 0] where ξ = -2+ √ It is known that ζ H satisfies a functional equation like ζ and has nontrivial zeros off the critical line. Letting g H (x) = x k≥1 h(k) 1 kx

The name HLR comes from the condition a(n) = O 1 n in the first tauberian theorem of Hardy-Littlewood and from the Ramanujan conjecture related to the behaviour of the coefficients of Dirichlet series in the Selberg class (this remark will be relevant in 5.5.1.).

Proof

Without loss of generality we take the caracter modulo 4:

• χ = 1, 0, -1, 0, 1, 0, -1, 0, ... giving L(s, χ) = n≥0 (-1) n (2n+1) s the Dirichlet beta function. Let u i = 1 vi where v is the increasing sequence of integers which can be written as a sum of 2 squares. Then keeping the notations in 5.2 and letting

we get g χ = g u which is a BHF. Then from 5.1 and the analytic conjecture we have from the known location of the zeros of ζ(1 -s)L(1 -s, χ)

= 1 2 and the fact that g (0,1/ √ 2) is a BHF which doesn't satisfy the HLR criteria we get from the conjecture 5.3

Finally (3) and (4) yield α (g χ ) = 1 2 and ζ(1 -s)L(1 -s, χ) has no zero in the half-plane s < 1 2 .

Remarks

Extension to other L functions

The method in section 5 for Dirichlet L functions can be extended naturally to other L functions. For instance let us consider the Ramanujan τ function and the associated automorphic form. It suffices to take the following function

1 kx which is a BHF and we believe that it satisfies the HLR criteria. Indeed computing 2000 terms of a(n) given by the recursion A gτ (n) = 1 2 n it seems that g τ satisfies the HLR criteria as shown by the graphic below (fig. 6) where we compare na(n) to ± log n (red) APPENDIX 1

It could be bounded and so α g (0,1/

APPENDIX 2

It seems bounded and α g (0,2 -1/3 ) = 1 3 is probable. A g (0,2 -1/3 ) (n) = 1 2 n plot of a(n)n 1/3 g (0,2 1/3 ) doesn't satisfy the HLR criteria.

APPENDIX 3

It is clearly bounded with a simple fractal pattern.

The HLR criteria is satisfied.

APPENDIX 4

A g (0,π) (n) = 1 2 n plot of A(n)

It seems bounded and so α g (0,1/π) = 0.

A g (0,2/3) (n) = 1 2 n plot of A(n)

It seems bounded and so α g (0,2/3) = 0.