Colin Guillarmou
email: colin.guillarmou@math.u-psud.fr

Thibault

Thibault Lefeuvre
email: thibault.lefeuvre@u-psud.fr

THE MARKED LENGTH SPECTRUM OF ANOSOV MANIFOLDS

come

Introduction

Let (M, g) be a smooth closed Riemannian manifold. If the metric g admits an Anosov geodesic flow, the set of lengths of closed geodesics is discrete and is called the length spectrum of g. It is an old problem in Riemannian geometry to understand if the length spectrum determines the metric g up to isometry. Vigneras [Vi] found counterexamples in constant negative curvature. On the other hand we know that the closed geodesics are parametrised by the set C of free-homotopy classes, or equivalently the set of conjugacy classes of the fundamental group π 1 (M ). Indeed, for each c ∈ C, there is a unique closed geodesic γ c of g in the class c. Particular examples of manifolds with Anosov geodesic flow are negatively curved compact manifolds. We can thus define a map, called the marked length spectrum

L g : C → R + , L g (c) := g (γ c ) (1.1)
where, if γ is a C 1 -curve, g (γ) denotes its length with respect to g. We recall the following long-standing conjecture stated in Burns-Katok [BuKa] (and probably considered even before):

Conjecture 1. [START_REF] Burns | Manifolds with non-positive curvature[END_REF]Problem 3.1] If (M, g) and (M, g 0 ) are two closed manifolds with negative sectional curvature and same marked length spectrum, i.e L g = L g 0 , then they are isometric, i.e. there exists a smooth diffeomorphism φ : M → M such that φ * g = g 0 .

Note that if φ : M → M is a diffeomorphism isotopic to the identity, then L φ * g 0 = L g 0 . The analysis of the linearised operator at a given metric g 0 is now well-understood, starting from the pionnering work of Guillemin-Kazhdan [GuKa], and pursued by the works of Croke-Sharafutdinov [CrSh], Dairbekov-Sharafutdinov [DaSh] and more recently by Paternain-Salo-Uhlmann [START_REF] Paternain | Spectral rigidity and invariant distributions on Anosov surfaces[END_REF][START_REF] Paternain | Invariant distributions, Beurling transforms and tensor tomography in higher dimensions[END_REF] and the first author [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF]. It is known that the linearised operator, the so called X-ray transform, is injective for non-positively curved manifolds with Anosov geodesic flows in all dimensions, and for all Anosov geodesic flows in dimension 2. These works imply the deformation rigidity result: there is no 1-parameter family of such metrics with the same marked length spectrum.

Concerning the non-linear problem (Conjecture 1), there are only very few results: in dimension 2 and non-positive curvature, the breakthrough was due to Otal [Ot] and Croke [START_REF] Croke | Rigidity for surfaces of nonpositive curvature[END_REF] who solved that problem 1 . It was extended by Croke-Fathi-Feldman [CFF] to surfaces when one of the metrics has non positive curvature and the other has no conjugate points. Katok [Ka] previously had a short proof for metrics in a fixed conformal class, in dimension 2 but that can be easily extended to higher dimensions. Beside the conformal case, for higher dimension the only known result is that of Hamenstädt [Ha] by applying the famous entropy rigidity work of Besson-Courtois-Gallot [BCG]: if two negatively curved metrics (M, g) and (M, g 0 ) have the same marked length spectrum and their Anosov foliation is C 1 , then Vol g (M ) = Vol g 0 (M ), and since L g determines the topological entropy, the result of [BCG] solves Conjecture 1 when g 0 is a locally symmetric space. For general metrics the problem is largely open. The main difficulty is that the linearised operator takes values on functions on a discrete set and is not a tractable operator to obtain non-linear results. We refer to the surveys/lectures of Croke and Wilkinson [Cr1,[START_REF] Wilkinson | Lectures on marked length spectrum rigidity[END_REF] for an overview of the subject.

The Conjecture 1 actually also makes sense for Anosov geodesic flows, without the negative curvature assumption, but it might be more reasonable to conjecture that only finitely many non isometric Anosov metrics have same marked length spectrum.

Our first result is a local rigidity statement that says that the marked length spectrum parametrizes locally the isometry classes of metrics. This is the first (non-linear) progress towards Conjecture 1 in dimension n ≥ 3 for general metrics.

Theorem 1. Let (M, g 0 ) be:

• either a closed smooth Riemannian surface with Anosov geodesic flow, • or a closed smooth Riemannian manifold of dimension n ≥ 3 with Anosov geodesic flow and non-positive sectional curvature, and let N > n 2 + 8. There exists > 0 such that for any smooth metric g with same marked length spectrum as g 0 and such that g -g 0 C N (M ) < , there exists a diffeomorphism φ : M → M such that φ * g = g 0 .

We actually prove a slightly stronger result in the sense that g can be chosen to be in the Hölder space C N,α with (N, α) ∈ N × (0, 1) satisfying N + α > n/2 + 8. Note also 1 Otal's work was in negative curvature and Croke in non-positive curvature that > 0 is chosen small enough so that the metrics g have Anosov geodesic flow too. This result is new even if dim(M ) = 2, as we make no assumption on the curvature. If dim(M ) > 2 and g is Anosov, the same result holds outside a finite dimensional manifold of metrics, see Remark 4.1. This implies a general result supporting Conjecture 1:

Corollary 1.1. Let (M, g 0 ) be an n-dimensional compact Riemannian manifold with negative curvature and let N > n 2 + 8. Then there exists > 0 such that for any smooth metric g with same marked length spectrum as g 0 and such that g -g 0 C N (M ) < , there exists a diffeomorphism φ : M → M such that φ * g = g 0 .

Since two C 0 -conjugate Anosov geodesic flows that are close enough have the same marked length spectrum, we also deduce that for g 0 fixed as above, each metric g which is close enough to g 0 and has geodesic flow conjugate to that of g 0 is isometric to g.

To prove these results, a natural strategy would be to apply an implicit function theorem. The linearised operator I 2 , called the X-ray transform, consists in integrating 2tensors along closed geodesics of g 0 (see Section 2.5). It is known to be injective under the assumptions of Theorem 1 by [START_REF] Croke | Spectral rigidity of a compact negatively curved manifold[END_REF][START_REF] Paternain | Spectral rigidity and invariant distributions on Anosov surfaces[END_REF][START_REF] Paternain | Invariant distributions, Beurling transforms and tensor tomography in higher dimensions[END_REF][START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF], but as mentioned before, the main difficulty to apply this to the non-linear problem is that I 2 maps to functions on the discrete set C and it seems unlikely that its range is closed. To circumvent this problem, we use some completely new approach from [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF] that replaces the operator I 2 by a more tractable Fredholm one, that is constructed using microlocal methods in Faure-Sjöstrand [FaSj] and Dyatlov-Zworski [DyZw]. This new operator, denoted by Π 2 , plays the same role as the normal operator I * 2 I 2 that is strongly used in the context of manifolds with boundary but Π 2 is not constructed from I 2 . The additional crucial ingredient that allows us to relate the operators I 2 and Π 2 is a "positive Livsic theorem" due to Pollicott-Sharp [PoSh] and Lopes-Thieullen [LoTh]. We manage to obtain a sort of stability estimate for the X-ray operator with some loss of derivatives, but that is sufficient for our purpose. A corollary of this method is a completely new stability estimate for the X-ray transform on divergence-free tensors, that quantifies the smallness of a divergence-free symmetric m-tensor f ∈ C α (M ; S m T * M ) (for m ∈ N,α > 0) in terms of the supremum of its integrals 1 (γ) γ f over all closed geodesics γ of g 0 , see Theorem 5.

Combining these methods with some ideas developed by Croke-Dairbekov-Sharafutdinov [CDS] and the second author [Le] in the case with boundary, we are able to prove a new rigidity result which has similarities with the minimal filling volume problem appearing for manifolds with boundary and is a problem asked by Croke in [START_REF] Croke | Rigidity theorems in Riemannian geometry[END_REF]Question 6.8].

Theorem 2. Let (M, g 0 ) be as in Theorem 1 and let N > n 2 + 2. There exists > 0 such that for any smooth metric g satisfying g -g 0 C N < , the following holds true: if L g (c) ≥ L g 0 (c) for all conjugacy class c ∈ C of π 1 (M ), then Vol g (M ) ≥ Vol g 0 (M ). If in addition Vol g (M ) = Vol g 0 (M ), then there exists a diffeomorphism φ : M → M such that φ * g = g 0 .

Again, in the proof, we actually just need g ∈ C N,α with (N, α) ∈ N × (0, 1) satisfying N + α > n/2 + 2. This result (but without the assumption that g is close to g 0 ) was proved by Croke-Dairbekov [CrDa] for negatively curved surfaces and for metrics in a conformal class in higher dimension (by applying the method of [Ka]). Our result is the first general one in dimension n > 2 and is new even when n = 2 as we do not assume negative curvature.

Next, we get Hölder stability estimates quantifying how close are metrics with close marked length spectrum. In that aim we fix a metric g 0 with Anosov geodesic flow and define for g close to g 0 in some C N (M ) norm

L(g) ∈ ∞ (C), L(g) = L g L g 0 .
We are able to show (here and below, H s (M ) is the usual L 2 -based Sobolev space of order s ∈ R on M ):

Theorem 3. Let (M, g 0 ) satisfy the assumptions of Theorem 1 and let N > 3n/2 + 9. For all s > 0 small there is a positive ν = O(s) and a constant C > 0 such that the following holds: for all δ > 0 small, there exists > 0 small such that for any C N metric g satisfying g -g 0 C N < , there is a diffeomorphism φ such that (here 1(c) = 1 for each c ∈ C):

φ * g -g 0 H -1-s (M ) ≤ Cδ L(g) -1 (1-ν)/2 ∞ (C) + C L(g) -1 ∞ (C)
We note that this is the first quantitative estimate on the marked length rigidity problem. It is even new for negatively curved surfaces where the injectivity of g → L g (modulo isometry) is known by [START_REF] Croke | Rigidity for surfaces of nonpositive curvature[END_REF][START_REF] Otal | Le spectre marqué des longueurs des surfaces à courbure négative[END_REF].

We conclude by some finiteness results. On a closed manifold M , we consider for ν 1 ≥ ν 0 > 0, θ 0 > 0 and C 0 > 0 the set of smooth metrics g with Anosov geodesic flow satisfying the estimates (2.2) where the constants C, ν verify

C ≤ C 0 , ν ∈ [ν 0 , ν 1 ] and d G (E s , E u ) ≥ θ 0 if d G denotes
the distance in the Grassmanian of the unit tangent bundle SM induced by the Sasaki metric. We write A(ν 0 , ν 1 , C 0 , θ 0 ) for the set of such metrics. This is a closed set that consists of uniform Anosov geodesic flows. For example, metrics with curvatures contained in [-a 2 , -b 2 ] with a > b > 0 satisfy such property [START_REF] Klingenberg | Riemannian geometry[END_REF]Theorem 3.2.17]. In what follows, we denote by R g the curvature tensor of g. Theorem 4. Let M be a smooth closed manifold and let ν 1 ≥ ν 0 > 0, C 0 > 0 and θ 0 > 0. If dim M = 2, for each sequence of positive numbers B := (B k ) k∈N , there is at most finitely many isometry classes of metrics g in A(ν 0 , ν 1 , C 0 , θ 0 ) satisfying the curvature bounds |∇ k g R g | g ≤ B k and with the same marked length spectrum. If dim M > 2 the same holds true if in addition g have non-positive curvature.

Restricting to negatively curved metrics we get the finiteness results (new if dim M > 2):

Corollary 1.2. Let M be a compact manifold. Then, for each a > 0 and each sequence B = (B k ) k∈N of positive numbers, there is at most finitely many smooth isometry classes of metrics with sectional curvature bounded above by -a 2 < 0, curvature tensor bounded by B (in the sense of Theorem 4) and same marked length spectrum.

We remark that the C ∞ assumptions on the background metric g 0 in all our results and the boundedness assumptions on the C ∞ norms of the curvatures in Theorem 4 can be relaxed to C k for some fixed k depending on the dimension. 2 2. The marked length spectrum and its linearisation 2.1. Marked length spectrum. We consider a smooth manifold M equipped with a smooth Riemannian metric g. We let π 1 (M ) be the fundamental group of M and C be the set of conjugacy classes in π 1 (M ). It is well-known that C corresponds to the set of free-homotopy classes of M . Assume now that the geodesic flow ϕ t of g on the unit tangent bundle SM is Anosov. We will call Anosov manifolds such Riemannian manifolds and let

A := {g ∈ C ∞ (M ; S 2
+ T * M ); g has Anosov geodesic flow}. We recall that ϕ t with generating vector field X is called Anosov if there exists some constants C > 0 and ν > 0 such that for all z = (x, v) ∈ SM , there is a continuous flow-invariant splitting

T z (SM ) = RX(z) ⊕ E u (z) ⊕ E s (z), (2.1)
where E s (z) (resp. E u (z)) is the stable (resp. unstable) vector space in z, which satisfy

|dϕ t (z).ξ| ϕt(z) ≤ Ce -νt |ξ| z , ∀t ≥ 0, ξ ∈ E s (z) |dϕ t (z).ξ| ϕt(z) ≤ Ce -ν|t| |ξ| z , ∀t ≤ 0, ξ ∈ E u (z) (2.2)
The norm, here, is given in terms of the Sasaki metric of g. By Anosov structural stability [An, DMM],

A is an open set. In particular, a metric g ∈ A has no conjugate points (see [Kl]) and there is a unique geodesic γ c in each free-homotopy class c ∈ C. We can thus define the marked length spectrum of g by (1.1).

2

The smoothness assumptions come from the fact we are using certain results based on microlocal analysis; it is a standard fact that only finitely many derivatives are sufficient for microlocal methods. It is likely that with some technical works one could improve the result to C 3 or C 4 regularity.

It will also be important for us to consider the mapping g → L g from the space of metrics to the set of sequences. In order to be in a good functional setting and since we shall work locally, we fix a smooth metric g 0 ∈ A and consider the metrics g in a neighborhood U g 0 of g 0 in C N (M ; S 2 + T * M ) for some N large enough and which will be chosen later. We can consider the map

L : U g 0 → ∞ (C), L(g)(c) := L g (c) L g 0 (c) . (2.3)
which we call the g 0 -normalized marked length spectrum. We notice from the definition of the length that

L(g) ∈ [0, 2] if g ≤ 2g 0 , justifying that L maps to ∞ (C). Proposition 2.1. The functional (2.3) is C 2 near g 0 if we choose the C 3 (M ; S 2 + T * M ) topology. In particular, there is a neighborhood U g 0 ⊂ C 3 (M ; S 2 + T * M ) of g 0 and C = C(g 0 ) > 0 such that for all g ∈ U g 0 L(g) -1 -DL g 0 (g -g 0 ) ∞ (C) ≤ C g -g 0 2 C 3 (M ) .
(2.4)

Proof. Let M := S g 0 M be the unit tangent bundle for g 0 and X 0 the geodesic vector field.

We use the stability result in the work of De la Llave-Marco-Moryion [DMM, Appendix A] (see also the proof of [START_REF] Dang | Fried conjecture in small dimensions[END_REF]Lemma 4.1]) which says that there is a neighborhood

V X 0 in C 2 (M; T M) of X 0 and a C 2 map X ∈ V X 0 → θ X ∈ C 0 (M) such that for each X ∈ V X 0
and each fixed periodic orbit γ X 0 of X 0 , there is a closed orbit γ X freely-homotopic to γ X 0 and the period (γ

X ) is C 2 as a map X ∈ V X 0 → (γ X ) ∈ R + given by (γ X ) = γ X 0 θ X .
In particular, we see that X ∈ V X 0 → (γ X )/ (γ X 0 ) is C 2 and its derivatives of order j = 1, 2 are bounded:

D j (γ X )/ (γ X 0 ) C 2 →R ≤ sup X∈V X 0 D j θ X C 2 →C 0 ≤ C
for some C depending on V X 0 . Now we fix c ∈ C and choose the geodesic γ c (g 0 ) for g 0 as being the element γ X 0 above, and we take U g 0 a small neighborhood of g 0 in the C 3 topology. The map X : g ∈ U g 0 → X g ∈ C 2 (M; T M) is defined so that X g is the geodesic vector field of g, where we used the natural diffeomorphism between M = S g 0 M and

S g M := {(x, v) ∈ T M ; g x (v, v) = 1}
obtained by scaling the fibers to pull-back the field on M. It is a C ∞ map between the Banach space C 3 (M ; S 2 + T * M ) and C 2 (M; T M). Thus the composition g → (γ Xg ), which is simply the map g → L g (c), is C 2 on U g 0 and the second derivative is uniformly bounded in U g 0 . The inequality (2.4) follows directly.

2.2. The X-ray transform. The central object on which stands our proof is the X-ray transform over symmetric 2-tensors, which is nothing more than the linearization DL that appeared in Proposition 2.1. It is a direct computation, which appeared already in [GuKa], that for

h ∈ C 3 (M ; S 2 T * M ) (DL(g 0 ).h)(c) = 1 2L g 0 (c) Lg 0 (c) 0 h γc(t) ( γc (t), γc (t))dt
where γ c (t) is the geodesic for g 0 homotopic to c and γc (t) its time derivative. This leads us to define the so-called X-ray transform on 2-tensors for g 0 as the operator

I g 0 2 : C 3 (M ; S 2 T * M ) → ∞ (C), I g 0 2 h(c) := 1 L g 0 (c) Lg 0 (c) 0 h γc(t) ( γc (t), γc (t))dt (2.5)
Note that if ϕ t is the geodesic flow for g 0 (the flow of X g 0 ), this can be rewritten as

I g 0 2 h(c) = 1 L g 0 (c) Lg 0 (c) 0 π * 2 h(ϕ t (z))dt,
where z ∈ γ c is any point on the closed orbit and here, for m ∈ N, we denote by π * m is the natural continuous maps (for all k ∈ N ∪ {∞})

π * m : C k (M, S m T * M ) → C k (SM ), f → (π * m f )(x, v) = f (x)(⊗ m v)
, where we now use SM as a notation for the unit tangent bundle for g 0 . More generally, if we define the X-ray transform on SM by

I g 0 : C 0 (SM ) → ∞ (C), I g 0 h(c) := 1 L g 0 (c) Lg 0 (c) 0 h(ϕ t (z))dt (2.6)
with z ∈ γ c , we will also define the X-ray transform on m-tensors as the operator (for m ∈ N) defined on C 0 (M ; S m T * M ) by

I g 0 m := I g 0 π * m .
(2.7)

When the background metric is fixed, we will remove the g 0 index and just write I m , I instead of I g 0 m , I g 0 . There is also a dual operator acting on distributions

π m * : C -∞ (SM ) → C -∞ (M, S m T * M ), π m * u, f := u, π * m f where •, • denotes the distributional pairing. Let H s (SM ) (resp. H s (M ; S m T * M )) denote the L 2 -based
Sobolev space of order s ∈ R on SM (resp. on m-tensors on M ). We note that for all s ∈ R, the following map are bounded

π * m : H s (M ; S m T * M ) → H s (SM ), π m * : H s (SM ) → H s (M ; S m T * M ).
(2.8)

Let us now explain the notion of solenoidal injectivity of the X-ray transform. If ∇ denotes the Levi-Civita connection of g 0 and σ : ⊗ m+1 T * M → S m+1 T * M the symmetrisation operation, we define the symmetric derivative

D := σ • ∇ : C ∞ (M ; S m T * M ) → C ∞ (M ; S m+1 T * M ).
The divergence operator is its formal adjoint given by D * f := -Tr(∇f ),

where Tr :

C ∞ (M ; S m T * M ) → C ∞ (M ; S m-2 T * M ) denotes the trace map defined by Tr(q)(v 1 , ..., v m-2 ) = n i=1 q(e i , e i , v 1 , ..., v m-2 ), if (e 1 , ...e n ) is a local orthonormal basis of T M for g 0 . If f ∈ C k,α (M ; S m T * M ) with (k, α) ∈ N × (0, 1)
, there exists a unique decomposition of the tensor f such that

f = f s + Dp, D * f s = 0,
(2.9)

where f s ∈ C k,α (M ; S m T * M ) and p ∈ C k+1,α (M ; S m-1 T * M ) (see [START_REF] Sharafutdinov | Integral geometry of tensor fields[END_REF]Theorem 3.3.2]). The tensor f s is called the divergence-free part (or solenoidal part) of f . It is direct to see that for each f ∈ C k (M ; S m T * M ), we have π * m+1 Df = Xπ * m f and that for u ∈ C k (SM ) with k ≥ 1 we have I(Xu) = 0 if X = X g 0 is the geodesic vector field for g 0 on SM . This implies that for k ≥ 1

∀f ∈ C k (M ; S m T * M ), I m+1 (Df ) = 0.
Thus in general it is impossible to recover the exact part Dp of a tensor f from I m f . We know recall some results about solenoidal injectivity of I m , defined as the property

ker I m ∩ C ∞ (M ; S m T * M ) ∩ ker D * = 0.
(2.10) Proposition 2.2. Let (M, g 0 ) be a smooth Riemannian manifold and assume that the geodesic flow of g 0 is Anosov. Then I m is solenoidal injective in the sense (2.10) when:

(1) m = 0 or m = 1, see [DaSh, Theorem 1.1 and 1.3],

(2) m ∈ N and dim(M ) = 2, see [Gu1, Theorem 1.4],

(3) m ∈ N and g 0 has non-positive curvature, see [START_REF] Croke | Spectral rigidity of a compact negatively curved manifold[END_REF]Theorem 1.3].

The case (2) with m = 2 was first proved in [START_REF] Paternain | Spectral rigidity and invariant distributions on Anosov surfaces[END_REF]Theorem 1.1].

The operator Π and stability estimates

In this section, we briefly review the results of the paper [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF] and in particular the operator Π defined there. As before we assume that (M, g 0 ) has Anosov geodesic flow and let X = X g 0 be its geodesic vector field.

3.1. The operator Π. Since X preserves the Liouville measure µ, the operator -iX is an unbounded self-adjoint operator on L 2 (SM ) := L 2 (SM, dµ). The L 2 -spectrum is then contained in R and the resolvents R ± (λ) := (-X ± λ) -1 are well-defined and bounded on L 2 (SM ) for Re(λ) > 0, they are actually given by

R + (λ)f (z) = +∞ 0 e -λt f (ϕ t (z))dt, R -(λ)f (z) = - 0 -∞ e λt f (ϕ t (z))dt
In [FaSj], Faure-Sjöstrand (see also [BuLi, DyZw]) proved that for Anosov flows, there exists a constant c > 0 such that for any s > 0, r < 0, one can construct a Hilbert space H r,s such that H s (SM ) ⊂ H r,s ⊂ H r (SM ) and -X -λ : Dom H r,s (X) → H r,s (with Dom H r,s (X) := {u ∈ H r,s ; Xu ∈ H r,s }) is an unbounded Fredholm operator with index 0 on Re(λ) > -c min(|r|, s); for -X + λ the same holds with a Sobolev space H s,r satisfying the same properties as above. Moreover, -X ± λ is invertible on these spaces for Re(λ) large enough, the inverses coincide with R ± (λ) when acting on H s (SM ) and extend meromorphically to the half-plane Re(λ) > -c min(|r|, s), with poles of finite multiplicity.

An Anosov geodesic flow is mixing [An], and R ± (λ) has a simple pole at λ = 0 with rank 1 residue operator ([Gu1, Lemma 2.5]): one can then write the Laurent expansion 3 :

R + (λ) = 1 ⊗ 1 λ + R 0 + O(λ), R -(λ) = - 1 ⊗ 1 λ -R * 0 + O(λ), (3.1)
where R 0 , R * 0 : H s (SM ) → H r (SM ) are bounded. The operator Π is then defined by:

Π := R 0 + R * 0 (3.2)
The following Theorem was obtained by the first author in [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF]:

Proposition 3.1. [Gu1, Theorem 1.1] The operator Π : H s (SM ) → H r (SM ) is bounded, for any s > 0, r < 0, with infinite dimensional range, dense in the space of invariant distributions

C -∞ inv (SM ) := {w ∈ C -∞ (SM ); Xw = 0}.
It is a self-adjoint map H s (SM ) → H -s (SM ), for any s > 0, and satisfies:

(1) ∀f ∈ H s (SM ), XΠf = 0, (2) ∀f ∈ H s (SM ) such that Xf ∈ H s (SM ), ΠXf = 0. 4
If f ∈ H s (SM ) with f, 1 L 2 = 0, then f ∈ ker Π if and only if there exists a solution u ∈ H s (SM ) to the cohomological equation Xu = f , and u is unique modulo constants.

The link between the X-ray transform I and the operator Π is rather unexplicit and given by the Livsic theorem [Li]. For instance if f ∈ C ∞ (SM ) is in the kernel of the X-ray transform, i.e. If = 0, then we know by the smooth Livsic theorem that there exists u ∈ C ∞ (SM ) such that f = Xu and thus Πf = ΠXu = 0 by Theorem 3.1, (ii).

Remark 3.1. In the study of the X-ray transform on a manifold with boundary, it is natural to introduce the normal operator I * I. It satisfies XI * Iu = 0, for any u ∈ C ∞ and I * IXu = 0 if u vanishes on the boundary. For closed manifolds, the operator Π is the replacement of the operator I * I used for manifolds with boundary (e.g. in [START_REF] Pestov | Two dimensional compact simple Riemannian manifolds are boundary distance rigid[END_REF][START_REF] Guillarmou | Lens rigidity for manifolds with hyperbolic trapped set[END_REF][START_REF] Stefanov | Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge[END_REF][START_REF] Lefeuvre | Local marked boundary rigidity under hyperbolic trapping assumptions[END_REF]).

The operators Π

m . For m ∈ N, we introduce the operator Π m := π m * Ππ * m mapping C ∞ (M ; S m T * M ) to C -∞ (M ; S m T * M ).
In [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF], the first author studied the microlocal properties of Π m by using in particular the works [FaSj, DyZw].

3 up to assuming that the Liouville measure µ is normalised to have mass 1. 4 In [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF], it is shown that ΠXf = 0 if f ∈ H s+1 (SM ), but this implies the result by a density argument and the approximation result [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF]Lemma E.47].

Proposition 3.2. [Gu1, Theorem 3.5, Lemma 3.6] The operator Π m is a pseudodifferential operator of order -1 which is elliptic on solenoidal tensors in the sense that there exists pseudodifferential operators Q, S, R of respective order 1, -2, -∞ such that:

QΠ m = Id + DSD * + R Moreover for any s > 0, Ππ * m : H -s (M ; S m T * M ) ∩ ker D * → H -s (SM ) is bounded. It is injective if I m is solenoidal injective in the sense of (2.10).
This results implies the following stability estimates.

Lemma 3.3. Assume that I m is solenoidal injective in the sense (2.10). For all s > 0, there exists a constant C > 0 depending on g 0 , s such that for all f ∈ H -s (M ; S m T * M ) ∩ ker D * :

f H -s-1 (M ) ≤ C Ππ * m f H -s (M ) (3.3)
Proof. This is actually a consequence of Proposition 3.2. We know that there exist pseudodifferential operators Q, S, R of respective order 1, -2, -∞ on M such that:

QΠ m = Id + DSD * + R.
For each f ∈ H -s (M ; S m T * M ) with D * f = 0, we have Ππ * 2 f ∈ H -s (SM ) by Theorem 3.2. Then then exists C > 0 (which may change from line to line) such that for all such f

f H -s-1 ≤ C( QΠ m f H -s-1 + Rf H -s-1 ) ≤ C( π m * Ππ * m f H -s + Rf H -s-1 ) ≤ C(||Ππ * m f H -s + Rf H -s-1 ).
where we used (2.8) and the boundedness of pseudodifferential operators on Sobolev spaces.

The proof now boils down to a standard argument of functional analysis. Assume (3.3) does not hold. Then, one can find a sequence of tensors f n ∈ H -s (M ; S m T * M ) ∩ ker D * , such that f n H -s-1 = 1 and thus:

1 = f n H -s-1 ≥ n Ππ * m f n H -s , that is Ππ * m f n → n→∞ 0 in H -s ,

and thus in particular in

H -s-1 . Since R is compact and (f n ) n∈N is bounded in H -s-1 , we can assume (up to extraction) that Rf n → n→∞ v ∈ H -s-1 .
By the previous inequality, we deduce that (f n ) n∈N is a Cauchy sequence in H -s-1 , which thus converges to an element f ∈ H -s-1 (M ; S m T * M )∩ker D * such that f H -s-1 = 1. The operator Ππ * m : H -s-1 → H -s-1 is bounded by Proposition 3.2 so Ππ * m f n → n→∞ 0 = Ππ * m f and it is also injective so f ≡ 0. This is a contradiction.

Remark 3.2. With a bit more work, we can actually get a better estimate with a -(s + 1/2) Sobolev exponent on the left-hand side of (3.3).

Proofs of the main results

As before, we fix a smooth Riemannian manifold (M, g 0 ) with Anosov flow and will shall consider metrics g with regularity C N,α for some N ≥ 3, α > 0 to be determined later and such that g -g 0 C N,α < , for some > 0 small enough so that g also has Anosov flow. 4.1. Reduction of the problem. The metric g 0 is divergence-free with respect to itself: D * g 0 = -Tr(∇g 0 ) = 0, where the Levi-Civita connection ∇ and trace Tr are defined with respect to g 0 . By a standard argument, there is a slice transverse to the diffeomorphism action (φ, g) → φ * g at the metric g 0 ; here φ varies in the group of C N,α -diffeomorphisms on M homotopic to the identity. We shall write Diff N,α 0 (M ) for the group of C N,α (M ) diffeomorphisms homotopic to Id, with N ≥ 2, α ∈ (0, 1). Since L(φ * g 0 ) = L(g 0 ) = 1 for all φ ∈ Diff N,α 0 (M ), it suffices to work on that transverse slice to study the marked length spectrum. This is the content of the following: Lemma 4.1. [CDS, Theorem 2.1] Let N ≥ 2 be an integer, α ∈ (0, 1). For any δ > 0 small enough, there exists > 0 such that for any g satisfying g -g 0 C N,α < , there exists φ ∈ Diff N,α 0 (M ) that is C N,α close to Id such that g := φ * g is divergence-free with respect to the metric g 0 and g -g 0 C N,α < δ.

We introduce f := φ * g -g 0 ∈ C N,α (M ; S 2 T * M ), which is, by construction, divergencefree and satisfies f C N,α < δ. Our goal will be to prove that f ≡ 0, if δ is chosen small enough and L g = L g 0 . 4.2. Geometric estimates. We let g, g 0 be two C 3 -metrics with Anosov geodesic flow.

Lemma 4.2. Assume that L g (c) ≥ L g 0 (c) for each c ∈ C. If γ c denotes the unique geodesic freely homotopic to c for g 0 , then

I g 0 2 f (c) = γc π * 2 f ≥ 0.
Proof. We denote by γ c the g-geodesic in the free-homotopy class c. One has:

γc π * 2 f = γc π * 2 g - γc π * 2 g 0 = E g (γ c ) -L g 0 (c),
where E g (γ c ) = g 0 (γc) 0 g γc(t) ( γc (t), γc (t))dt is the energy functional for g. By using Cauchy-Schwartz, E g (γ c ) ≥ g (γ c ) 2 / g 0 (γ c ) and since γ c is freely-homotopic to c, we get g (γ c ) ≥ g (γ c ). Since g (γ c ) ≥ g 0 (γ c ) by assumption, we obtain the desired inequality.

Next, we can use the following result Lemma 4.3. There exists > 0 small such that if g-g 0 C 0 ≤ and Vol g (M ) ≤ Vol g 0 (M ),

then with f = g -g 0 SM π * 2 f dµ ≤ 2 3 f 2 L 2 . Proof. Let g τ := g 0 + τ f with f ∈ C 3 (M ; S 2 T * M ). A direct computation gives that M Tr g 0 (f )dvol g 0 = SM π * 2 f dµ.
Then the argument of [START_REF] Croke | Local boundary rigidity of a compact Riemannian manifold with curvature bounded above[END_REF]Proposition 4.1] by Taylor expanding Vol gτ (M ) in τ directly provides the result.

Finally, we conclude this section with the following: Lemma 4.4. Assume that I g 0 2 f (c) ≥ 0 for all c ∈ C. Then, there exists a constant C = C(g 0 ) > 0, such that:

0 ≤ SM π * 2 f dµ ≤ C L(g) -1 ∞ (C) + f 2 C 3 (M ) (4.1)
where dµ is the Liouville measure of g 0 and g = g 0 + f as above.

Proof. For the Anosov geodesic flow of g 0 , the Liouville measure is the unique equilibrium state associated to the potential given by J u (z) := -∂ t det dϕ t (z)| Eu(z) | t=0 (the unstable Jacobian). By Parry's formula (see [START_REF] Parry | Equilibrium states and weighted uniform distribution of closed orbits[END_REF]Paragraph 3]), we have:

∀F ∈ C 0 (SM ), lim T →∞ 1 N (T ) c∈C,Lg 0 (c)≤T e γc J u L g 0 (c) γc F = 1 Vol(SM ) SM F dµ, (4.2)
where, as before, γ c is the g 0 -geodesic in c and N (T ) is the constant of normalisation corresponding to the sum when F = 1. The first inequality in (4.1) then follows from that formula and the assumption I g 0 2 f ≥ 0. For the second inequality in (4.1) we use Proposition 2.1 with the fact that DL g 0 f = 1 2 I g 0 2 f to deduce that there exists C(g 0 ) > 0 such that

I g 0 2 f ∞ (C) ≤ 2 L(g) -1 ∞ (C) + C(g 0 ) f 2 C 3 . (4.3)
Thus, we get for any T > 0

1 N (T ) c∈C,Lg 0 (c)≤T e γc J u I g 0 2 f (c) ≤ I g 0 2 f ∞ (C) ≤ 2 L(g) -1 ∞ (C) + C(g 0 ) f 2 C 3 (4.4)
and the left-hand side converges to 1 Vol(SM ) SM π * 2 f dµ by Parry's formula (4.2), which is the sought result by letting T → ∞.

We note that in the previous proof, the approximation of SM π * 2 f by I g 0 2 f (c) could also be done using the Birkhoff ergodic theorem and the Anosov closing lemma to approximate SM π * 2 f by some I g 0 2 f (c) for some c ∈ C so that L g 0 (c) is large. The following lemma is another key ingredient in the proof of our main results. It is a positive version of Livsic theorem which was proved independently by Pollicott-Sharp [PoSh] and Lopes-Thieullen [LoTh] (though the stronger version we use is actually that of [LoTh]). Here M 1 denotes the Borel probability measures on SM which are invariant by the geodesic flow of g 0 . Note that, by Sigmund [Si], the Dirac measures u → 1 Lg 0 (c) γc u on closed orbits are dense in M 1 . Proposition 4.5. [LoTh, Theorem 1], [PoSh, Theorem 1] Let α ∈ (0, 1] and let X 0 be the geodesic vector field of g 0 . There exists a constant C = C(g 0 ) > 0 and β ∈ (0, 1) such that for any u ∈ C α (SM ) satisfying

∀c ∈ C, γc u ≥ 0, there exists h ∈ C αβ (SM ) and F ∈ C αβ (SM ) such that F ≥ 0 and u + Xh = F . Moreover F C αβ ≤ C u C α .
4.3. Proof of Theorem 1 and Theorem 2. We fix g 0 with Anosov geodesic flow on M and assume that either M is a surface or that g 0 has non-positive curvature in order to have that I g 0 2 is solenoidal injective by Proposition 2.2. Fix N ≥ 3 to be chosen later and α > 0 small. As explained in Lemma 4.1, we take δ > 0 small and > 0 small so that g -g 0 C N,α < implies that there is φ ∈ Diff N,α 0 (M ) with φ * g -g 0 C N,α < δ and D * (φ * g -g 0 ) = 0.

We write f := φ * g -g 0 and remark that the assumption L g ≥ L g 0 implies L φ * g ≥ L g 0 thus I g 0 2 f (c) ≥ 0 for all c ∈ C by Lemma 4.2. By Proposition 4.5, we know that there exists h ∈ C β (SM ) and F ∈ C β (SM ) for some 0 < β < α (depending on g 0 and linearly on α)

such that π * 2 f + Xh = F ≥ 0, with π * 2 f + Xh C β ≤ C π * 2 f C α ≤ C f C α , (4.5)
where C = C(g 0 ). Take 0 < s β very small (it will be fixed later) and let β < β be very close to β. Thus we obtain (for some constant C = C(g 0 , s, β) that may change from line to line)

f H -1-s ≤ C Ππ * 2 f H -s , by Lemma 3.3 ≤ C Π(π * 2 f + Xh) H -s , since ΠXh = 0 ≤ C π * 2 f + Xh H s , by Theorem 3.1 ≤ C π * 2 f + Xh 1-ν L 2 π * 2 f + Xh ν H β , by interpolation with ν = s/β . (4.6)
Note that by (4.5) we have a control:

π * 2 f + Xh H β ≤ C π * 2 f + Xh C β ≤ C f C α . (4.7)
And we can once more interpolate between Lebesgue spaces so that:

π * 2 f + Xh L 2 ≤ C π * 2 f + Xh 1/2 L 1 π * 2 f + Xh 1/2 L ∞ ≤ C π * 2 f + Xh 1/2 L 1 f 1/2 C α . (4.8)
Next, using that π * 2 f + Xh ≥ 0, we have

π * 2 f + Xh L 1 = SM (π * 2 f + Xh)dµ = SM π * 2 f dµ (4.9)
We will now consider two cases: in case (1) we assume that L g = L g 0 , while in case (2) we assume that Vol g (M ) ≤ Vol g 0 (M ). Combining Lemma 4.2 and Lemma 4.4, we deduce that in case (1), we have 2), we get by Lemma 4.3 that if > 0 is small enough,

π * 2 f + Xh L 1 ≤ C f 2 C 3 , while in case (
π * 2 f + Xh L 1 ≤ C f 2 L 2
These facts combined with (4.8) yield

π * 2 f + Xh L 2 ≤ C f C 3 . f 1/2 C α , case (1) C f L 2 . f 1/2 C α , case (2) Thus we have shown f H -1-s ≤ C f 1-ν C 3 f 1+ν 2 C α , case (1) C f 1-ν L 2 . f 1+ν 2 C α , case (2)
(4.10)

where C = C(g 0 , s, β). We choose α very small and 0 < s β < α, j ∈ {α, 3} and N 0 > n/2 + j + s: by interpolation and Sobolev embedding we have

f C j ≤ f H n/2+j+s ≤ C f 1-θ j H -1-s f θ j H N 0 (4.11) with θ j = n/2+j+1+2s N 0 +s+1 . If N 0 > 3 2 n+8, we see that γ := 1 2 (1-θ α )(1+ν)+(1-θ 3 )(1-ν) > 1 if s >
0 and α are chosen small enough, thus in case (1) we get with γ :

= (1+ν)θ α /2+(1-ν)θ 3 f H -1-s ≤ C f γ H -1-s f γ H N 0 Thus if f = 0 we obtain, if f H N 0 ≤ δ 1 ≤ C f γ-1 H -1-s f γ H N 0 ≤ C f γ-1+γ H N 0 ≤ Cδ γ-1+γ .
Since γ -1 + γ > 0, we see that by taking δ > 0 small enough we obtain a contradiction, thus f = 0. This proves Theorem 1 by choosing N ≥ N 0 . In case (2) (corresponding to Theorem 2), this is the same argument except that we get a slightly better result due to the L 2 norm in (4.10): N 0 can be chosen to be any number N 0 > n/2 + 2. To conclude, we have shown that if g -g 0 C N,α < for N ∈ N with N + α > n/2 + 2, then L g ≥ L g 0 implies that either Vol g (M ) ≤ Vol g 0 (M ) and φ * g = g 0 for some C N,α diffeomorphism, or Vol g (M ) ≥ Vol g 0 (M ). Note that in both cases, if g is smooth then φ is smooth.

4.4. Stability estimates for X-ray transforms. We next give some new stability estimates for the X-ray transform. To the best of our knowledge, these are the first estimates in the closed setting. In the following, we will consider the X-ray transform I m over divergence-free symmetric m-tensors.

Theorem 5. Assume that (M, g 0 ) satisfies the assumptions of Theorem 1. Then for all α > 0, there is β ∈ (0, α) depending linearly on α such that for all s ∈ (0, β) and for all ν ∈ (s/β, 1), there exists a constant C > 0 such that for all f ∈ C α (M ; S m T * M ) ∩ ker D * :

f H -1-s ≤ C I g 0 m f (1-ν)/2 ∞ ( f C α + I g 0 m f ∞ ) (1+ν)/2
Proof. The proof is essentially the same as Theorem 1. By using Lemma 3.3 with π * m f replaced by π * m f + I g 0 m f ∞ and Proposition 4.5, we have, as in (4.6), that for all 0 < α < 1 small, there is 0 < β < α depending on g 0 and linearly on α such that for all 0 < s < β < β, and for all f ∈ C α (M ; S m T * M ) ∩ ker D * :

f H -1-s ≤ C Ππ * m f H -s ≤ C Π(π * m f + Xh) H -s ≤ C π * m f + Xh 1-ν L 2 π * m f + Xh ν H β ,
for some C depending only on (g 0 , s, β, β , α), ν := s/β and where π *

m f +Xh = -I g 0 m f ∞ + F with h, F ∈ C β such that F C β ≤ C( f C α + I g 0 m f ∞ ).
Using (4.7), (4.8), (4.9), (4.4) with I g 0 2 f replaced by I g 0 m f , we get the result.

Remark 4.1. Note that ν and s can be chosen arbitrarily small in the estimate. In the general case of an Anosov manifold (without any assumption on the curvature), the sinjectivity of the X-ray transform is still unknown. However, it was proved in [START_REF] Dairbekov | Some problems of integral geometry on Anosov manifolds[END_REF]Theorem 1.5] and [START_REF] Guillarmou | Invariant distributions and X-ray transform for Anosov flows[END_REF]Lemma 3.6] that its kernel is finite-dimensional and contains only smooth tensors. The same arguments as above then show that Theorem 5 still holds for all f as above with the extra condition f ⊥ ker I m with respect to the L 2 scalar product, and similarly for Theorem 1, if g is not in a finite dimensional manifold.

4.5. Stability estimates for the marked length spectrum. Proof of Theorem 3.

We will apply the same reasoning as before to get a stability estimate for the non-linear problem (the marked length spectrum). We proceed as before and reduce to considering f = φ * g -g 0 where φ ∈ Diff N,α 0 (M ) and f C N,α < δ. By Theorem 5, and using (4.3) we have for 0 < α small, 0 < s α and β, ν as in Theorem 5 (in particular ν, α, s can be made arbitrarily small):

f H -s-1 ≤ C I g 0 2 f (1-ν)/2 ∞ ( f C α + I g 0 2 f ∞ ) (1+ν)/2 ≤ C( L(g) -1 ∞ + f 2 C 3 ) (1-ν)/2 f (1+ν)/2 C α + C( L(g) -1 ∞ + f 2 C 3 ) ≤ C L(g) -1 (1-ν)/2 ∞ f (1+ν)/2 C α + L(g) -1 ∞ + f 1-ν C 3 f (1+ν)/2 C α
We use the interpolation estimate (4.11) and for N 0 > n/2 + 9 we get

f H -s-1 ≤ C L(g) -1 (1-ν)/2 ∞ f (1+ν)/2 C α + L(g) -1 ∞ + f γ H -s-1 f γ C N 0 , (4.12) where γ = 1 2 (1 -θ α )(1 + ν) + (1 -θ 3 )(1 -ν) > 1, γ > 0, θ 3 = n/2+4+2s N 0 +s+1
, if s > 0 is chosen small enough. Assume δ is chosen small enough so that Cδ α/2 ≤ 1/2. Then:

f γ H -s-1 f γ C N 0 ≤ C f H -s-1 f (γ-1)+γ C N 0 ≤ C f H -s-1 δ (γ-1)+γ ≤ 1 2 f H -s-1 ,
if δ > 0 is chosen small enough depending on C = C(g 0 , s, α, β, ν) and N + α > N 0 . The sought result then follows from the previous inequality combined with (4.12).

4.6. Compactness theorems and proof of Theorem 4. We let M be a compact smooth manifold equipped with an Anosov geodesic flow. By the proof of [START_REF] Knieper | New results on noncompact harmonic manifolds[END_REF]Theorem 4.8], the universal cover M and the fundamental group π 1 (M ) are hyperbolic in the sense of Gromov [Gr]. We shall denote by R g the curvature tensor associated to the metric g and by inj(g) the injectivity radius of g. We proceed by contradiction: let (g n ) n≥0 be a sequence of smooth metrics on M in the class A(ν 0 , ν 1 , C 0 , θ 0 ) (defined in the Introduction) such that L gn = L g 0 and such that for each k ∈ N there is B k > 0 such that |∇ k gn R gn | gn ≤ B k for all n, and we assume that for each n = n , g n is not isometric to g n . Since the metrics have Anosov flow, they have no conjugate points and thus inj(g n ) = 1 2 min c∈C L gn (c) = 1 2 min c∈C L g 0 (c).

By Hamilton compactness result [START_REF] Hamilton | A compactness property for solutions of the Ricci flow[END_REF]Theorem 2.3], there is a family of smooth diffeomorphisms φ n on M such that g n := φ * n g n converges to g ∈ A(ν 0 , ν 1 , C 0 , θ 0 ) in the C ∞ topology (note that A(ν 0 , ν 1 , C 0 , θ 0 ) is invariant by pull-back through smooth diffeomorphisms). Denote by φ n * ∈ Out(π 1 (M )) the action of φ n on the set of conjugacy classes C. The universal cover M of M is a ball since M has no conjugate points, and π 1 (M ) is a hyperbolic group thus we can apply the result of Gromov [START_REF] Gromov | Hyperbolic Groups, in Essays in Group Theory[END_REF]Theorem 5.4.1] saying that the outer automorphism group Out(π 1 (M )) is finite if dim M ≥ 3. This implies in particular that there is a subsequence (φ n j ) j∈N such that φ n j * (c) = φ n 0 * (c) for all c ∈ C and all j ∈ N where as before C is the set of conjugacy classes of π 1 (M ). But φ * n 0 g n j have the same marked length spectrum as φ * n 0 g 0 for all j, thus L g n j = L φ * n 0 g 0 for all j. Since g n j → g in C ∞ , we have L g = L g n j for all j and by Theorem 1, we deduce that there is j 0 such that for all j ≥ j 0 , g n j is isometric to g. This gives a contradiction. Now, if dim M = 2, Out(π 1 (M )) is a discrete infinite group. We first show that for each c ∈ C, the set of classes (φ -1 n ) * (c) ∈ C is finite as n ranges over N. Assume the contrary, then consider γ n the geodesic for g n in the class c, one has L gn (c) = gn (γ n ) = g 0 (γ 0 ), by assumption. Now φ -1 n (γ n ) is a g n geodesic in the class (φ -1 n ) * (c) with length g n (φ -1 n (γ n )) = gn (γ n ) = g 0 (γ 0 ). We know that there are finitely many g-geodesics with length less than g 0 (γ 0 ), but we also have

L g ((φ -1 n ) * (c)) ≤ g (φ -1 n (γ n )) ≤ g n (φ -1 n (γ n ))(1 + ) ≤ g 0 (γ 0 )(1 + ), if g n -g C 3 ≤ .
Thus we obtain a contradiction for n large. The extended mapping class group5 Mod(M ) is isomorphic to Out(π 1 (M )) (see [START_REF] Farb | A primer on mapping class group[END_REF]Theorem 8.1]). By [START_REF] Farb | A primer on mapping class group[END_REF]Proposition 2.8] 6 , if M has genus at least 3, there is a finite set C 0 ⊂ C such that if φ * ∈ Mod(M ) is the identity on C 0 then φ is homotopic to Id, while if M has genus 2, the same condition implies that φ is either homotopic to Id or to an hyperelliptic involution h.

In both cases, we can extract a subsquence φ n j such that φ n j * = φ n 0 * for all j ≥ 0 and we conclude like in the higher dimensional case.

extended in the sense that it includes orientation reversing elements.

see also the proof of Theorem 3.10 in[FaMa]

Acknowledgements. We would like to thank S. Gouëzel, G. Knieper, G. Massuyeau, M. Mazzucchelli, D. Monclair, F. Naud, F. Paulin, G. Rivière, M. Salo, J-M. Schlenker, R. Tessera for useful discussions. C.G. is supported partially by ERC consolidator grant IPFLOW.


Semyon Dyatlov
email: dyatlov@math.berkeley.edu

Colin Guillarmou

DYNAMICAL ZETA FUNCTIONS FOR AXIOM A FLOWS

We show that the Ruelle zeta function of any smooth Axiom A flow with orientable stable/unstable spaces has a meromorphic continuation to the entire complex plane. The proof uses the meromorphic continuation result of [DyGu16] together with [CoEa71, GMT17] which imply that every basic hyperbolic set can be put into the framework of [DyGu16].

Let M be a compact manifold and X be a C ∞ vector field on M. Denote by ϕ t = exp(tX) the corresponding flow. In this note we prove Theorem. Assume that ϕ t is an Axiom A flow (see Definition 2.3) with orientable stable/unstable spaces. Define the Ruelle zeta function

ζ(λ) = γ 1 -e -λT γ , Re λ 1, (1.1)
where the product is taken over the primitive closed orbits γ of ϕ t , with the exception of fixed points, and T γ are the corresponding periods. Then ζ(λ) extends meromorphically to λ ∈ C.

We refer the reader to §2 for examples of Axiom A flows and for the dynamical systems terminology used here.

The above theorem was first conjectured by Smale [Sm67, bottom of page 802].It was proved in the special case of Anosov flows by Giulietti-Liverani-Pollicott [START_REF] Giulietti | Anosov flows and dynamical zeta functions[END_REF]. Dyatlov-Zworski [START_REF] Dyatlov | Dynamical zeta functions for Anosov flows via microlocal analysis[END_REF] gave a simple microlocal proof in the Anosov case. In our previous work [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF], we generalized the microlocal techniques used in [START_REF] Dyatlov | Dynamical zeta functions for Anosov flows via microlocal analysis[END_REF] to handle the case of open hyperbolic systems, defined as locally maximal hyperbolic sets with a smooth neighborhood on which the flow is strictly convex. We emphasize that the introduction of microlocal tools and the use of escape functions for the study of hyperbolic diffeomorphisms and flows appeared first in the works of Faure-Roy-Sjöstrand [START_REF] Faure | A semiclassical approach for Anosov diffeomorphisms and Ruelle resonances[END_REF] and Faure-Sjöstrand [START_REF] Faure | Upper bound on the density of Ruelle resonances for Anosov flows[END_REF]; they were inspired by the works of Kitaev [START_REF] Kitaev | Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness[END_REF], Blank-Keller-Liverani [START_REF] Blank | Ruelle-Perron-Frobenius spectrum for Anosov maps[END_REF], Liverani [START_REF] Liverani | On contact Anosov flows[END_REF], Gouezel-Liverani [START_REF] Gouëzel | Compact locally hyperbolic sets for smooth maps: fine statistical properties[END_REF], Baladi-Tsujii [START_REF] Baladi | Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms[END_REF] that used anisotropic Sobolev/Hölder spaces adapted to the dynamics. The meromorphic extension of the zeta function for a particular case of Axiom A flow, namely the Grassmanian extension of a contact Anosov flow, was also proved by Faure-Tsujii [START_REF] Faure | The semiclassical zeta function for geodesic flows on negatively curved manifolds[END_REF].

See the introduction to [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF] for an overview of results on dynamical zeta functions and Pollicott-Ruelle resonances and the book by Baladi [Ba17] for the related case of hyperbolic maps.

The proof in this note combines [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF] with the results of Conley-Easton [START_REF] Conley | Isolated invariant sets and isolating blocks[END_REF] and Guillarmou-Mazzucchelli-Tzou [START_REF] Guillarmou | Boundary and lens rigidity for nonconvex manifolds[END_REF] which show that every locally maximal hyperbolic set has a smooth neighborhood with a strictly convex flow, as well as with Smale's spectral decomposition for Axiom A flows.

We note that just like in [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF] one can consider a more general Ruelle zeta function with a potential. In fact, the methods of [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF] apply to even more general dynamical traces and twisted zeta functions associated to the action of X on vector bundles, see [DyGu16, §5.1, Theorem 4]. Moreover, the assumption that X is C ∞ can be relaxed to C k for large k = k(C 0 ) if we only want to continue the Ruelle zeta function to a half-plane {Re λ ≥ -C 0 }. Finally, the orientability hypothesis holds in many natural cases (such as geodesic flows on orientable negatively curved manifolds) and can be removed under certain topological assumptions by using a twisted zeta function, see [GLP13, Appendix B].

Review of hyperbolic dynamics

In this section we briefly review several standard definitions and facts from the theory of hyperbolic dynamical systems. We refer the reader to [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF] for a comprehensive treatment of hyperbolic dynamics.

Assume that M is a compact C ∞ manifold without boundary and ϕ t = exp(tX) is a C ∞ flow generated by a vector field X.

Definition 2.1. Let K ⊂ M be a compact ϕ t -invariant set. We say that K is hyperbolic for the flow ϕ t , if the generator X of the flow does not vanish on K and each tangent space T x M, x ∈ K, admits a flow/stable/unstable decomposition

T x M = E 0 (x) ⊕ E s (x) ⊕ E u (x), x ∈ K such that: • E 0 (x) = RX(x); • E s (x), E u (x) depend continuously on the point x; • dϕ t (E s (x)) = E s (ϕ t (x)) and dϕ t (E u (x)) = E u (ϕ t (x)) for all x ∈ K, t ∈ R;
• for any choice of a continuous norm | • | on the fibers of T M, there exist constants C, θ > 0 such that for all x ∈ K,

|dϕ t (x)v| ≤ Ce -θ|t| |v| when t ≥ 0, v ∈ E s (x); t ≤ 0, v ∈ E u (x). (2.1)
We say ϕ t is an Anosov flow if the whole M is hyperbolic.

A fixed point x ∈ M, X(x) = 0, is called hyperbolic if the differential ∇X(x) has no eigenvalues on the imaginary axis. Hyperbolic fixed points are nondegenerate and thus isolated.

We also define the nonwandering set: Definition 2.2. [Sm67, p. 796] We call x ∈ M a nonwandering point if for every neighborhood V of x and every T > 0 there exists t ∈ R such that |t| ≥ T and ϕ t (V ) ∩ V = ∅. The set of all nonwandering points is called the nonwandering set.

The nonwandering set is closed and ϕ t -invariant, see [KaHa97, Proposition 3.3.4]. Note that each closed orbit of ϕ t lies in the nonwandering set.

We now give the definition of Axiom A flows: Definition 2.3. [Sm67, §II.5, (5.1)] The flow ϕ t is Axiom A if:

(1) the nonwandering set is the disjoint union of the set F of fixed points and the closure K of the union of all closed orbits;

(2) all fixed points of ϕ t are hyperbolic;

(3) the set K is hyperbolic for the flow ϕ t .

We also define locally maximal sets and basic hyperbolic sets: Definition 2.4. We say that a compact ϕ t -invariant set K ⊂ M is locally maximal for the flow ϕ t , if there exists a neighborhood V of K such that

K = t∈R ϕ t (V ). Definition 2.5. [PaPo83, Chapter 9] A compact ϕ t -invariant set K ⊂ M is called a basic hyperbolic set † for ϕ t if
(1) K is locally maximal for ϕ t ;

(2) K is hyperbolic for ϕ t ;

(3) the flow ϕ t | K topologically transitive (i.e. contains a dense orbit); and (4) K is the closure of the union of all closed orbits of ϕ t | K .

The nonwandering set of an Axiom A flow has the following spectral decomposition: Proposition 2.6. [Sm67, §II.5, Theorem 5.2] Assume that ϕ t is an Axiom A flow and let K be given by Definition 2.3. Then we can write K as a finite disjoint union

K = K 1 • • • K N
where each K j is a basic hyperbolic set. † Strictly speaking, a single closed orbit is not considered a basic set, but we ignore this minor detail here.

See also [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF]Exercise 18.3.7] for a proof of the spectral decomposition, and [Bo75, Lemma 3.9] for the local maximality of the nonwandering set.

We finally give a few examples:

Example 2.7. Let M = R 2
x,y /2πZ 2 be the torus and ϕ t = exp(tX) where X = (sin x)∂ x + ∂ y . The nonwandering set K has two connected components, each being a single closed orbit:

K = {x = 0} {x = π}.
(2.2)

The spectral decomposition is given by (2.2). Note however that the entire M is not hyperbolic for ϕ t .

Example 2.8. Let (M, g) be a (possibly noncompact) complete Riemannian manifold of negative sectional curvature which is asymptotically hyperbolic. Put M := SM , the sphere bundle of M , and let ϕ t : M → M be the geodesic flow. Let K be the union of all geodesics which are trapped, that is their closures in SM are compact. Then K is a locally maximal hyperbolic set for ϕ t . (The noncompactness of M is not an issue since K is compact and the behavior of the flow outside of a neighborhood of K is irrelevant.) See [DyGu16, §6.3] for details and more general examples.

Proof of the theorem

We first show meromorphic continuation of the Ruelle zeta function ζ K for a locally maximal hyperbolic set: Proposition 3.1. Let K ⊂ M be a locally maximal hyperbolic set for ϕ t . Define the Ruelle zeta function ζ K by (1.1) where we only take the closed trajectories of ϕ t which lie in K. Then ζ K continues to a meromorphic function on C.

The proof of Proposition 3.1 relies on [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF]. To reduce to the case considered in [START_REF] Dyatlov | Pollicott-Ruelle resonances for open systems[END_REF] we use Lemma 3.2. Let K ⊂ M be a locally maximal hyperbolic set for ϕ t . Then there exists a neighborhood U of K in M with C ∞ boundary and a C ∞ vector field X 0 on U such that:

(1) the boundary ∂U is strictly convex with respect to X 0 in the following sense:

ρ(x) = 0, X 0 ρ(x) = 0 =⇒ X 2 0 ρ(x) < 0 (3.1)
where ρ ≥ 0 is any boundary defining function on U;

(2) X 0 = X in a neighborhood of K;

(3) K = t∈R ϕ t 0 (U) where ϕ t 0 := exp(tX 0 ) is the flow generated by X 0 .

Proof. The proof follows Guillarmou-Mazzucchelli-Tzou [GMT17, Lemma 2.3]. We first use a result of Conley-Easton [CoEa71, Theorem 1.5]. Since K is locally maximal for ϕ t , there exists an open set V ⊂ M containing K such that K = t∈R ϕ t (V ). If x ∈ ∂V , then in particular x / ∈ K, thus there exists t ∈ R such that ϕ t (x) / ∈ V . Therefore V is an isolating neighborhood for ϕ t in the sense of [CoEa71, Definition 1.1] and K is an isolated invariant set in the sense of [CoEa71, Definition 1.2]. Therefore by [CoEa71, Theorem 1.5] there exists an isolating block, which is an open subset U ⊂ M with K ⊂ U with the following properties:

• U has compact closure and C ∞ boundary, that is U = {x ∈ M | ρ(x) > 0} for some function ρ ∈ C ∞ (M; R) such that dρ = 0 on ∂U = {x ∈ M | ρ(x) = 0}. • the set ∂ 0 U := {x ∈ ∂U | X 0 ρ(x) = 0} is a codimension 1 smooth submanifold of ∂U; • for each x ∈ ∂ 0 U, there exists ε > 0 such that ϕ t (x) / ∈ U for all t ∈ (-ε, ε) \ {0}. (3.2)
Now the vector field X 0 is obtained by modifying X slightly near ∂ 0 U so that we still have K = t∈R ϕ t 0 (U) and the topological (local) convexity condition (3.2) is replaced by the quadratic differential convexity condition (3.1). We refer to the proof of [GMT17, Lemma 2.3] for details. The main theorem now follows using the spectral decomposition theorem, Proposition 2.6. Indeed, if K = K 1 • • • K N is the spectral decomposition of ϕ t , then each closed trajectory of ϕ t is contained in one of the sets K j . Therefore the Ruelle zeta function (1.1) factorizes as

We now give

ζ(λ) = ζ K 1 (λ) • • • ζ K N (λ).
Since each ζ K j admits a meromorphic continuation to C by Proposition 3.1, the function ζ admits a meromorphic continuation to C as well.

  Proof of Proposition 3.1. By part (3) of Lemma 3.2, every closed trajectory of ϕ t 0 in U lies in K and thus is a closed trajectory of ϕ t . Therefore the Ruelle zeta function ζ K is equal to the Ruelle zeta function of ϕ t 0 on U. Now (U, ϕ t 0 ) is an open hyperbolic system in the sense of [DyGu16, Assumptions (A1)-(A4)]. Therefore [DyGu16, Theorem 3] applies (with potential set to 0) and gives a meromorphic continuation of ζ K to C.

Acknowledgements. This research was conducted during the period SD served as a Clay Research Fellow. CG was partially supported by the ANR project ANR-13-JS01-0006 and by the ERC consolidator grant IPFLOW.


Yann Molcard

Ourmieres

C Galusinski

C Mazoyer

S Meradji

A Molcard

Y Ourmières

INLET AND OUTLET OPEN BOUNDARY CONDITIONS FOR INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Keywords:

HAL is

Introduction

The present work deals with the choice of open boundary conditions for incompressible Navier-Stokes flows. Such a problematic is well understood for output flows even if the flow can re-enter intermittently into the domain in areas globally identified as output boundaries. The "Do nothing" boundary conditions [START_REF] Bruneau | New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result[END_REF][START_REF] Bruneau | Effective downstream boundary conditions for incompressible Navier-Stokes equations[END_REF] lead to well-posed problem and work well on numerical experiments. They are nevertheless based on a reference velocity flow and a reference pressure and such data have to be well chosen. Such a method can be extended to open boundary conditions where the flow is permanently entering but limitations can be found in the choice of the reference velocity and pressure. This point is developed in the following part, leading to the proposition of another open boundary condition avoiding as much as possible the dependence to a reference flow. Concerning the reference pressure, it appears to be a strongly constraining information as pressure can vary in a spectacular way in turbulent flows and can not be intuited.

The goal of this work is to identify pertinent informations to define the open boundary conditions both for output and input boundaries. The key point is to let a free open boundary at inlet (where the flow is durably entering) by assuming that the dynamics is driven by forces inside the studied domain. Then a criterion on energy exchange, between the studied domain and the outside will be proposed. Another scalar parameter, the flow rate, will be another pertinent information linked to the energy exchange.

Navier-Stokes equation and "do nothing" open boundary conditions.

For the present study, the Navier-Stokes equations solving an incompressible viscous flow is used in an adimensional formulation,

Re ∂u ∂t + u.∇u -∆u + ∇p = F ∀(t, x) ∈ R + × Ω, (1)
∇.u = 0 ∀(t, x) ∈ R + × Ω, (2)
where u denotes the velocity, p the pressure, F is a source term and Re is the Reynolds number. in the present work. The ways to close the system with an OBC leading to a well-posed problem are not so numerous. A natural approach is to impose a normal constraint to the boundaries but the inertia term (a nonlinear transport term) plays an important role when the flow enters into the domain and has then to be taken into account. Therefore, in [START_REF] Bruneau | New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result[END_REF][START_REF] Bruneau | Effective downstream boundary conditions for incompressible Navier-Stokes equations[END_REF], the authors proposed an inhomogeneous Dirichlet boundary condition at the inlet and a nonlinear Robin boundary condition at the outlet. The boundary Γ o is then splitted in two parts,

Γ o = Γ in ∪ Γ out .
A reference flow and a reference pressure are then chosen, (u r , p r ) and the boundary condition writes

u = u r on Γ in (3)
∂ n v -qn + Re 2 (u • n) -v = 0, on Γ out , (4)
with u = u r + v, p = p r + q, where x -= max(0, -x). The problem (1),( 2), ( 3) and (4) studied in [START_REF] Bruneau | New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result[END_REF] is then well-posed. The choice of (u r , p r ) on Γ out is important for the simulations and works well when chosen from the solution at the previous time step.

In order to let free the flow at the inlet, the first idea is to use the OBC on Γ out for the boundary Γ in . On this boundary Γ in , where the flow has to be lastingly entering, the nonlinear Robin condition is durably applied to the reference flow up to a perturbation due to the reference pressure. For high Reynolds numbers, this boundary condition degenerates to an unknown Dirichlet boundary condition on Γ in . This inhomogeneous Dirichlet boundary condition is unknown since the pressure can be far from the reference pressure and the velocity far from the reference velocity u r concomitantly.

We then focus on OBCs that avoid the choice of a reference flow, and most importantly, a reference pressure. This is possible thanks to a splitting technique widely used to approach incompressible Navier-Stokes equations [START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF]. This is done for downstream OBCs, for example, in [START_REF] Dong | A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows[END_REF]. At one time step, a velocity prediction is computed thanks to the momentum equation, for a given pressure. Then, the predicted velocity is not divergence free and a projection onto the free divergence field allows the construction of a pressure for the next time step. For the prediction step, a free boundary condition is necessary only for the velocity. For the projection step defining the pressure, no reference pressure is necessary. In the absence of OBC, this projection is unique and orthogonal, but with an OBC, it is not the case and a wide choice of projection onto the free divergence field is possible. Our choice will be detailed in section 2. We first exhibit pertinent informations that will be the starting point to construct the correct projection.

Criteria to control the flow

As discussed in the previous section, when the OBC can be splitted between a region of durably ingoing flow Γ in and a region of globally outgoing flow Γ out , the first goal is to avoid a Dirichlet boundary condition on Γ in . A flow rate can then be imposed instead. This information can not close the system, as it is simply a scalar information which can be pertinent and imposed as a property of the projected velocity on the free divergence field. This lets a lot of latitude to the projection. Before detailing the projection, let us introduce another pertinent scalar parameter, the energy exchange between the studied domain and the outside. To define such an energy, let us remark that on a bounded domain Ω with ∂Ω = Γ w , the solution to [START_REF] Bruneau | New efficient boundary conditions for incompressible Navier-Stokes equations: a well-posedness result[END_REF] verifies

Ω ∇p • u dx = - Ω p ∇ • u dx + Γw p u • n dγ = 0.
This shows that ∇p can be seen as a force that does not work globally on such a domain or an infinite domain. This means that, for a subdomain Θ ⊂ Ω, the quantity

Θ ∇p • u dx = - Ω\Θ ∇p • u dx
is exactly the energy exchange, due to the pressure, between the domain Θ and Ω\Θ.

Returning to a domain Ω with walls and OBCs, the term

e = Ω ∇p • u dx = Γo p u • n dγ (5)
defines the energy exchange between the domain Ω and the outer domain. A pertinent scalar criterion can be to prohibit the energy exchange so that the flow dynamics is driven by the force inside Ω without additional energy due to the outside dynamics.

Because this condition seems difficult to impose, we choose to work with a criterion on the flow rate and to adapt this flow rate in order to modulate the energy e.

Splitting method and OBC

Let us recall, for simplicity, the projection method of order one, based on an order one time scheme with a time step δt. The upperscript n refers to the n th time iteration. The prediction step consists in finding the predicted velocity u solving

Re u -u n δt + u n .∇u n -∆ u + ∇p n = F ∀x ∈ Ω, (6)
with additional boundary conditions on the velocity variable u, detailed herafter. The correction step consists in finding the pressure p n+1 and then the velocity u n+1 solution to

Re u n+1 -u δt -+∇(p n+1 -p n ) = 0, ∀x ∈ Ω, (7)
∇.u n+1 = 0 ∀x ∈ Ω, (8)
with additional boundary conditions on q = p n+1 -p n . It is worth noting that the u n+1 solution to (7) is a projection of the predicted velocity u onto the free divergence vector field, fixed by the choice of the boundary condition on q. By applying the operator ∇• to (7), the pressure difference q, solves the Poisson problem Re δt ∇ • u = ∆q, with the inhomogenous Neuman boundary condition we have chosen,

∂ n q = 0, on Γ w , ∂ n q = α u • n + β| u • n| on Γ o , (9)
where the constants α and β are fixed with two scalar informations, the desired flow rate d of u n+1 and the compatibility condition of the Poisson problem. This leads to

α = 1 + d δ out + δ in d out δ in -d in δ out , β = d d out + d in d out δ in -d in δ out , (10)
where

d in = Γin u • ndγ, d out = Γout u • ndγ, δ in = Γin | u • n|dγ, δ out = Γout | u • n|dγ.
The choice (9) is motivated by modulations of u • n on boundaries to construct u n+1 with the appropriate flow rate. The boundary conditions (9) (10) then closes the problem (7) with without introducing a reference pressure. A variant of the system (7) (9), although less natural, is to gradually modulate the flow rate even if introducing a possible compressibility,

Re δt ∇ • u -(1 -θ) Ω ∇ • u dx = ∆q, ∂ n q = 0, on Γ w , ∂ n q = θ (α u • n + β| u • n|) on Γ o , (11)
with 0 ≤ θ ≤ 1.

The next remaining task is then to close the boundary condition of ( 6). We choose a Robin like condition on open boundaries,

u = 0 on Γ w (12) ∂ n ( u -u r ) + k( u • n) -( u -u r ) = 0, on Γ o , (13)
where the positive constant k can be chosen to zero and a zero reference flow u r can be used. When k is a large value, it enforces, where the flow enters the domain, the velocity to the reference flow u r which flow rate should be d for compatibility. The next section devoted to numerical experiments, allows discussions on the influence of θ. In what follows, the parameter k will be kept to zero with a trivial reference flow u r = 0. It is important to note that no reference pressure arises and only a reference flow rate drives the flow.

Experiments

In the following, the computations are done with the computational code introduced in [START_REF] Galusinski | Problematic of Incompressible Bifluid Navier-Stokes Equations Approached on Non Body Fitted Meshes[END_REF], in a classical geometry for test case: a 2D channel with a circular obstacle or a 3D channel with a ball as obstacle. The obstacle diameter value is equal to half of the channel section length value. Namely, the obstacle is taken into account, on a staggered grid [START_REF] Francis | Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface[END_REF], with an order two approximation, even if the grid is non body fitted [START_REF] Galusinski | Problematic of Incompressible Bifluid Navier-Stokes Equations Approached on Non Body Fitted Meshes[END_REF]. The time discrete model (6) ( 7) is closed with the OBCs (11) (13) parametrized with θ and k. The Reynolds number is chosen high (∼ 10 5 ) for 2D simulations to increase physical instabilities in order to bring out the robustness of the OBCs, even if the grid size is not fine enough to produce relevant flows. The Reynolds number is chosen of order 10 3 for 3D simulations.

The force F is chosen to a fixed value along the direction of the channel. At the inlet before the obstacle, the flow is expected laminar driven by such forces and we are concerned with flow rate which are compatible with such forces. We then choose the flow rate so that the energy exchange ( 5) is sufficiently small. We also will investigate some experiments with larger flow rates in order to evaluate the augmented energy exchange. The figure 1 is an example of an expected f low driven by the external force F with low energy exchanges. The reference flow rate for low energy exchange is 0.0275 for our geometry and force F. We first investigate low value of θ, namely θ = 1 400 , with an imposed flow rate equal to 0.0275. Even if θ is low, it is sufficient to lead the flow close to the desired flow rate and to respect the free divergence condition. The energy exchange oscillates around zero, leading to a low energy exchange, revealing a flow rate compatible with forces F. The scale of the energy is proportional to the Reynolds number. It is also worth noting that the transitional regime occurs until time around 0.8. The influence of the expected flow rate value is now studied. Two simulations are analyzed: the first one with a reference flow rate of low energy exchange, and the second one with this flow rate multiplied by two. In the first time (before 0.5), flow rates and energy go through a transitional regime (figure 3): during this regime, the flow rate of low energy exchange has less important energy instabilities, that quickly stabilize. Then, after this transitional regime, both flow rates remain close to the expected flow rates (figure4). Regarding the energy, the figure 4 highlights the fact that a larger flow rate induces a more energetic flow during the whole simulation. In the previous simulations, no reference flows are used and k = 0 in (13). When taking k large and a reference flow at the inlet, similar flows to those associated to the Figures 1 and2 are obtained and values of θ are not sensitive.

In the following 3D simulations, larger values of θ are used, leading to very low variations on the flow rate, as shown in Figure 5. The value θ = 1 should lead to an exact flow rate but, as the divergence of the velocities is not exactly zero in simulations, very low variations on the flow rate occur. In this case, the energy exchange is negative and small compared to the Reynolds number value. This means that the reference flow rate is slightly greater than a zero exchange energy flow rate. As depicted in Figure 5, we can observe that the energy exchange is very weakly influenced by θ for 1 10 ≤ θ ≤ 1, suggesting that flow rate variations are negligible. For θ = 1 10 or θ = 1, time t = 1, the velocity solution of the 3D channel is introduced in Figures 7 and8, for the same flow rate. Figure 7 shows the velocity solution for a long channel and Figure 8 shows it for a shorter channel . Few differences can be observed on velocities because of the large time value chosen and the sensitivity of the flow.

Conclusions

The new boundary condition introduced here allows to construct a flow driven by forces inside the geometry and by a given flow rate taking into account interactions with the outer domain. A notion of energy exchange has been introduced. The flow rate can be tuned to control the energy exchange between the domain and the outer domain. It has been evidenced that the exchange energy is strongly linked to the flow rate. If the model allows to impose a flow of reference at inlet, we prefer to eliminate the influence of a flow of reference by choosing k = 0 in the boundary condition (13). The flow is then only driven by a scalar, the flow rate of reference. Furthermore, the model depends on an artificial parameter θ in (11) which allows to obtain precisely the desired flow rate for θ close to 1 or to oscillate around the desired flow rate for θ close to 0.

In the continuity of this work, in a future development, another OBC could be proposed in order to ensure a free (or given) energy exchange in a direct way (without given flow rates) and should be compared to this one. Finally, our aim is to implement such OBCs in the context of ocean modelling for the simulation of reduced sea areas when the flow dynamics is potentially driven by local forces such as the wind. In such a case, the region considered as Γ in and Γ out are not as simple as the ones in a channel, but Γ in can be identified as the upstream frontier and Γ out as the downstream frontier, possibly changing or swapping with time.

Figure 1 :

 1 Figure 1: Example of expected flows.

Figure 2 :

 2 Figure 2: Flow rate relative error (left plot) and energy (right plot) with θ = 1 400

Figure 3 :

 3 Figure 3: Reference flow rate influence (0.0275 (red color) and 0.055 (green color))on the observed flow rate (left plot) and the energy exchange (right plot).

Figure 4 :

 4 Figure 4: Reference flow rate influence (0.0275 (red color) and 0.055 (green color))on the observed flow rate and the energy exchange, skipping the transitional regime.

Figure 5 :

 5 Figure 5: Influence of θ, θ = 1 (green plots) and θ = 1 10 (red plots).

Figure 6 :

 6 Figure 6: Flow rate relative error multiplied by θ on the left figure, energy exchange on the right figure. Plots related to a truncated channel are in green. Plots related to a standard channel are in red. Dashed lines represent θ = 1, and full lines represent θ = 1 10 .

Figure 7 :

 7 Figure 7: Streamlines and two isovalues of a spanwise component of the velocity: θ = 1 10 (top), θ = 1 (bottom).

Figure 8 :

 8 Figure 8: Streamlines and two isovalues of a spanwise component of the velocity: θ = 1 10 (top and middle), θ = 1 (bottom). Grid size is finer on the middle figure (90 nodes in the cross section of the channel instead of 60).

Acknowledgments: This work was granted access to the HPC resources of Aix-Marseille Universit financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program Investissements d'Avenir supervised by the Agence Nationale de la Recherche.