Jean-Yves Chemin
email: chemin@ann.jussieu.fr

AND Isabelle Gallagher
email: isabelle.gallagher@math.jussieu.fr

Marius Paicu
email: marius.paicu@math.u-psud.fr

GLOBAL REGULARITY FOR SOME CLASSES OF LARGE SOLUTIONS TO THE NAVIER-STOKES EQUATIONS

Keywords:

 is that it varies slowly in one direction, though in some sense it is "well prepared" (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of [6] to an "ill prepared" situation (the norm blows up as the small parameter goes to zero). As in [4]-[6], the proof uses the special structure of the nonlinear term of the equation.

Introduction

We study in this paper the Navier-Stokes equation with initial data which are slowly varying in the vertical variable. More precisely we consider the system     

∂ t u + u • ∇u -∆u = -∇p in R + ×Ω div u = 0 u| t=0 = u 0,ε ,
where Ω = T 2 × R (the choice of this particular domain will be explained later on) and u 0,ε is a divergence free vector field, whose dependence on the vertical variable x 3 will be chosen to be "slow", meaning that it depends on εx 3 where ε is a small parameter. Our goal is to prove a global existence in time result for the solution generated by this type of initial data, with no smallness assumption on its norm.

1.1. Recollection of some known results on the Navier-Stokes equations. The mathematical study of the Navier-Stokes equations has a long history, which we shall describe briefly in this paragraph. We shall first recall results concerning the main global wellposedness results, and some blow-up criteria. Then we shall concentrate on the case when the special algebraic structure of the system is used, in order to improve those previous results.

To simplify we shall place ourselves in the whole euclidian space R d or in the torus T d (or in variants of those spaces, such as T 2 × R in three space dimensions); of course results exist in the case when the equations are posed in domains of the euclidian space, with Dirichlet boundary conditions, but we choose to simplify the presentation by not mentioning explicitly those studies (although some of the theorems recalled below also hold in the case of domains up to obvious modifications of the statements and sometimes much more difficult proofs).

1.1.1. Global wellposedness and blow-up results. The first important result on the Navier-Stokes system was obtained by J. Leray in his seminal paper [START_REF] Leray | Essai sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] in 1933. He proved that any finite energy initial data (meaning square-integrable data) generates a (possibly non unique) global in time weak solution which satisfies an energy estimate; he moreover proved in [START_REF] Leray | Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique[END_REF] the uniqueness of the solution in two space dimensions. Those results use the structure of the nonlinear terms, in order to obtain the energy inequality. He also proved the uniqueness of weak solutions in three space dimensions, under the additional condition that one of the weak solutions has more regularity properties (say belongs to L 2 (R + ; L ∞ ): this would now be qualified as a "weak-strong uniqueness result"). The question of the global wellposedness of the Navier-Stokes equations was then raised, and has been open ever since. We shall now present a few of the historical landmarks in that study.

The Fujita-Kato theorem [START_REF] Fujita | On the Navier-Stokes initial value problem I[END_REF] gives a partial answer to the construction of a global unique solution. Indeed, that theorem provides a unique, local in time solution in the homogeneous Sobolev space Ḣ d 2 -1 in d space dimensions, and that solution is proved to be global if the initial data is small in Ḣ d 2 -1 (compared to the viscosity, which is chosen equal to one here to simplify). The result was improved to the Lebesgue space L d by F. Weissler in [START_REF] Weissler | The Navier-Stokes Initial Value Problem in L p[END_REF] (see also [START_REF] Giga | Solutions in L r of the Navier-Stokes initial value problem[END_REF] and [START_REF] Kato | Strong L p -solutions of the Navier-Stokes equation in R m with applications to weak solutions[END_REF]). The method consists in applying a Banach fixed point theorem to the integral formulation of the equation, and was generalized by M. Cannone, Y. Meyer and F. Planchon in [START_REF] Cannone | Solutions autosimilaires des équations de Navier-Stokes, Séminaire[END_REF] to Besov spaces of negative index of regularity. More precisely they proved that if the initial data is small in the Besov space Ḃ-1+ d p p,∞ (for p < ∞), then there is a unique, global in time solution. Let us emphasize that this result allows to construct global solutions for strongly oscillating initial data which may have a large norm in Ḣ d 2 -1 or in L d . A typical example in three space dimensions is

u ε 0 (x) def = ε -α sin x 3 ε -∂ 2 ϕ(x), ∂ 1 ϕ(x), 0 ,
where 0 < α < 1 and ϕ ∈ S(R 3 ; R). This can be checked by using the definition of Besov norms:

∀s > 0, ∀(p, q) ∈ [1, ∞], f Ḃ-s p,q def = t s 2 e t∆ f L p L q (R + ; dt t )
.

More recently in [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF], H. Koch and D. Tataru obtained a unique global in time solution for data small enough in a more general space, consisting of vector fields whose components are derivatives of BMO functions. The norm in that space is given by

(1.1) u 0 2 BM O -1 def = sup t>0 t e t∆ u 0 2 L ∞ + sup x∈R d R>0 1 R d P (x,R) |(e t∆ u 0 )(t, y)| 2 dy, where P (x, R) stands for the parabolic set [0, R 2 ] × B(x, R) while B(x, R) is the ball centered at x, of radius R.
One should notice that spaces where global, unique solutions are constructed for small initial data, are necessarily scaling-invariant spaces: thus all those spaces are invariant under the invariant transformation for Navier-Stokes equation u λ (t, x) = λu(λ 2 t, λx). Moreover it can be proved (as observed for instance in [START_REF] Chemin | Wellposedness and stability results for the Navier-Stokes equations in R 3 to appear[END_REF]) that if B is a Banach space continuously included in the space S ′ of tempered distributions such that for any (λ,

a) ∈ R + ⋆ × R d , f (λ(• -a)) B = λ -1 f B , then • B ≤ C sup t>0 t 1 2 e t∆ u 0 L ∞ .
One recognizes on the right-hand side of the inequality the Ḃ-1 ∞,∞ norm, which is slightly smaller than the BM O -1 norm recalled above in (1.1): indeed the BM O -1 norm takes into account not only the Ḃ-1 ∞,∞ information, but also the fact that first Picard iterate of the Navier-Stokes equations should be locally square integrable in space and time. It thus seems that the Koch-Tataru theorem is optimal for the wellposedness of the Navier-Stokes equations. This observation also shows that if one wants to go beyond a smallness assumption on the initial data to prove the global existence of unique solutions, one should check that the Ḃ-1

∞,∞ norm of the initial data may be chosen large. for finite p and q) naturally provides blow-up criteria. For instance, one can prove that if the life span of the solution is finite,

then the L q ([0, T ]; Ḃ-1+ d p + 2 q p,q
) norm blows up as T approaches the blow up time. A natural question is to ask if the Ḃ-1+ d p p,q norm itself blows up. Progress has been made very recently on this question, and uses the specific structure of the equation, which was not the case for the results presented in this paragraph. We therefore postpone the exposition of those results to the next paragraph.

We shall not describe more results on the Cauchy problem for the Navier-Stokes equations, but refer the interested reader for instance to the monographs [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF] and [START_REF] Meyer | Wavelets, Paraproducts and Navier-Stokes[END_REF] for more details.

1.1.2. Results using the specific algebraic structure of the equation. If one wishes to improve the theory on the Cauchy problem for the Navier-Stokes equations, it seems crucial to use the specific structure of the nonlinear term in the equations, as well as the divergence-free assumption. Indeed it was proved by S. Montgomery-Smith in [START_REF] Montgomery-Smith | Finite-time blow up for a Navier-Stokes like equation[END_REF] (in a one-dimensional setting, which was later generalized to a 2D and 3D situation by two of the authors in [START_REF] Gallagher | Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations[END_REF]) that some models exist for which finite time blow up can be proved for some classes of large data, despite the fact that the same small-data global wellposedness results hold as for the Navier-Stokes system. Furthermore, the generalization to the 3D case in [START_REF] Gallagher | Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations[END_REF] shows that some large initial data which generate a global solution for the Navier-Stokes equations (namely the data of [START_REF] Chemin | On the global wellposedness of the 3-D Navier-Stokes equations with large initial data[END_REF] which will be presented below) actually generate a blowing up solution for the toy model.

In this paragraph, we shall present a number of wellposedness theorems (or blow up criteria) which have been obtained in the past and which specifically concern the Navier-Stokes equations. In order to make the presentation shorter, we choose not to present a number of results which have been proved by various authors under some additional geometrical assumptions on the flow, which imply the conservation of quantities beyond the scaling (namely spherical, helicoidal or axisymmetric conditions). We refer for instance to [START_REF] Ladyzhenskaya | The mathematical theory of viscous incompressible flow[END_REF], [START_REF] Leibovich | Invariant helical subspaces for the Navier-Stokes equations[END_REF], [START_REF] Ponce | Global stability of large solutions to the 3D Navier-Stokes equations[END_REF], or [START_REF] Ukhovskii | Axially symmetric flows of ideal and viscous fluids filling the whole space[END_REF] for such studies.

To start with, let us recall the question asked in the previous paragraph, concerning the blow up of the Ḃ-1+ d p p,q norm at blow-up time. A typical example of a solution with a finite Ḃ-1+ d p p,q norm at blow-up time is a self-similar solution, and the question of the existence of such solutions was actually addressed by J. Leray in [START_REF] Leray | Essai sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF]. The answer was given 60 years later by J. Neças, M. Ruziçka and V. S ¸verák in [START_REF] Neças | On Leray's self-similar solutions of the Navier-Stokes equations[END_REF]. By analyzing the profile equation, they proved that there is no self-similar solution in L 3 in three space dimensions. Later L. Escauriaza, G. Seregin and V. S ¸verák were able to prove more generally that if the solution is bounded in L 3 , then it is regular (see [START_REF] Escauriaza | On the L3,∞ solutions to the Navier-Stokes equations and backward uniqueness[END_REF]): in particular any solution blowing up in finite time must blow up in L 3 . Now let us turn to the existence of large, global unique solutions to the Navier-Stokes system in three space dimensions.

An important example where a unique global in time solution exists for large initial data is the case where the domain is thin in the vertical direction (in three space dimensions): that was proved by G. Raugel and G. Sell in [START_REF] Raugel | Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions[END_REF] (see also the paper [START_REF] Iftimie | Navier-Stokes equations in thin 3D domains with Navier boundary conditions[END_REF] by D. Iftimie, G. Raugel and G. Sell). Another example of large initial data generating a global solution was obtained by A. Mahalov and B. Nicolaenko in [START_REF] Mahalov | Global solvability of three-dimensional Navier-Stokes equations with uniformly high initial vorticity[END_REF]: in that case, the initial data is chosen so as to transform the equation into a rotating fluid equation (for which it is known that global solutions exist for a sufficiently strong rotation).

In both those examples, the global wellposedness of the two dimensional equation is an important ingredient in the proof. Two of the authors also used such a property to construct in [START_REF] Chemin | On the global wellposedness of the 3-D Navier-Stokes equations with large initial data[END_REF] an example of periodic initial data which is large in Ḃ-1 ∞,∞ but yet generates a global solution. Such an initial data is given by

u N 0 (x) def = (N u h (x h ) cos(N x 3 ), -div h u h (x h ) sin(N x 3 )),
where u h L 2 (T 2 ) ≤ C(ln N ) 1 4 , and its Ḃ-1 ∞,∞ norm is typically of the same size. This was generalized to the case of the space R 3 in [START_REF] Chemin | Wellposedness and stability results for the Navier-Stokes equations in R 3 to appear[END_REF].

Similarly in [START_REF] Chemin | Large, global solutions to the Navier-Stokes equations, slowly varying in one direction[END_REF], that fact was used to prove a global existence result for large data which are slowly varying in one direction. More precisely, if (v h 0 , 0) and w 0 are two smooth divergence free "profile" vector fields, then they proved that the initial data

(1.2) u 0,ε (x h , x 3 ) def = (v h 0 (x h , εx 3 ), 0) + (εw h 0 (x h , εx 3 ), w 3 0 (x h , εx 3 ))
generates, for ε small enough, a global smooth solution. Here, we have denoted x h = (x 1 , x 2 ). Using for instance the language of geometrical optics in the context of fast rotating incompressible fluids, and thinking of the problem in terms of convergence to the two dimensional situation, this case can be seen as a "well prepared" case. We shall be coming back to that example in the next paragraph.

As a conclusion of this short (and of course incomplete) survey, let us present some results for the Navier-Stokes system with viscosity vanishing in the vertical direction. Analogous results to the classical Navier-Stokes system in the framework of small data are proved in [START_REF] Chemin | Fluids with anisotropic viscosity[END_REF], [START_REF] Iftimie | A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity[END_REF], [START_REF] Paicu | Équation anisotrope de Navier-Stokes dans des espaces critiques[END_REF] and [START_REF] Chemin | On the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations[END_REF]). To circumvent the difficulty linked with the absence of vertical viscosity, the key idea, which will be also crucial here (see for instance the proof of the second estimate of Proposition 2.1) is the following: the vertical derivative ∂ 3 appears in the nonlinear term of the equation with the prefactor u 3 , which has some additional smoothness thanks to the divergence free condition which states that

∂ 3 u 3 = -∂ 1 u 1 -∂ 2 u 2 .
1.2. Statement of the main result. In this work, we are interested in generalizing the situation (1.2) to the ill prepared case: we shall investigate the case of initial data of the form

v h 0 (x h , εx 3 ), 1 ε v 3 0 (x h , εx 3 ) ,
where x h belongs to the torus T 2 and x 3 belongs to R. The main theorem of this article is the following.

Theorem 1. Let a be a positive number. There are two positive numbers ε 0 and η such that for any divergence free vector field v 0 satisfying

e a|D 3 | v 0 H 4 ≤ η,
then, for any positive ε smaller than ε 0 , the initial data

u 0,ε (x) def = v h 0 (x h , εx 3 ), 1 ε v 3 0 (x h , εx 3 )
generates a global smooth solution of (N S) on T 2 × R.

Remarks

• Such an initial data may be arbitrarily large in Ḃ-1 ∞,∞ , more precisely of size ε -1 . Indeed it is proved in [START_REF] Chemin | Large, global solutions to the Navier-Stokes equations, slowly varying in one direction[END_REF], Proposition 1.1, that if f and g are two functions in S(T 2 ) and S(R) respectively, then

h ε (x h , x 3 ) def = f (x h )g(εx 3 ) satisfies, if ε is small enough, h ε Ḃ-1 ∞,∞ ≥ 1 4 f Ḃ-1 ∞,∞ g L ∞ .
• As in the well prepared case studied in [START_REF] Chemin | Large, global solutions to the Navier-Stokes equations, slowly varying in one direction[END_REF] and recalled in the previous paragraph, the structure of the nonlinear term will have a crucial role to play in the proof of the theorem. • The reason why the horizontal variable is restricted to a torus is to be able to deal with very low horizontal frequencies: as it will be clear in the proof of the theorem, functions with zero horizontal average are treated differently to the others, and it is important that no small horizontal frequencies appear other than zero. In that situation, we are able to solve globally in time the equation (conveniently rescaled in ε) for small analytic-type initial data. We recall that in that spirit, some local in time results for Euler and Prandtl equation with analytic initial data can be found in [START_REF] Sammartino | Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space. I. Existence for Euler and Prandtl Equations[END_REF]. In this paper we shall follow a method close to a method introduced in [2]. • We finally note that we can add to our initial data any small enough data in Ḣ 1 2 , and we still obtain the global existence of the solution. Indeed, by the results contained in [START_REF] Gallagher | Asymptotics and stability for global solutions to the Navier-Stokes equations[END_REF], if we fix an initial data which gives a global in time solution, then, all initial data in a small neighborhood, give global in time solutions.

Structure of the proof

2.1. Reduction to a rescaled problem. We look for the solution under the form

u ε (t, x) def = v h (t, x h , εx 3 ), 1 ε v 3 (t, x h , εx 3 ) .
This leads to the following rescaled Navier-Stokes system.

(RN S ε )

       ∂ t v h -∆ ε v h + v • ∇v h = -∇ h q ∂ t v 3 -∆ ε v 3 + v • ∇v 3 = -ε 2 ∂ 3 q div v = 0 v |t=0 = v 0 with ∆ ε def = ∂ 2 1 + ∂ 2 2 + ε 2 ∂ 2 3 .
As there is no boundary, the rescaled pressure q can be computed with the formula

(2.1) ∆ ε q = j,k ∂ j v k ∂ k v j = j,k ∂ j ∂ k (v j v k ).
It turns out that when ε goes to 0, ∆ -1 ε looks like ∆ -1 h . In the case of R 3 , for low horizontal frequencies, an expression of the type ∆ -1 h (ab) cannot be estimated in L 2 in general. This is the reason why we work in T 2 × R. In this domain, the problem of low horizontal frequencies reduces to the problem of the horizontal average that we denote by

(M f )(x 3 ) def = f (x 3 ) def = T 2 f (x h , x 3 )dx h .
Let us also define M ⊥ f def = (Id -M )f . Notice that, because the vector field v is divergence free, we have v 3 ≡ 0. The system (RN S ε ) can be rewritten in the following form.

(RN S ε )            ∂ t w h -∆ ε w h + M ⊥ v • ∇w h + w 3 ∂ 3 v h = -∇ h q ∂ t w 3 -∆ ε w 3 + M ⊥ (v • ∇w 3 ) = -ε 2 ∂ 3 M ⊥ q ∂ t v h -ε 2 ∂ 2 3 v h = -∂ 3 M (w 3 w h ) div(v + w) = 0 (v, w) |t=0 = (v 0 , w 0 ).
The problem to solve this sytem is that there is no obvious way to compensate the loss of one vertical derivative which appears in the equation on w h and v and also, but more hidden, in the pressure term. The method we use is inspired by the one introduced in [START_REF] Chemin | Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray[END_REF] and can be understood as a global Cauchy-Kowalewski result. This is the reason why the hypothesis of analyticity in the vertical variable is required in our theorem.

Let us denote by B the unit ball of R 3 and by C the annulus of small radius 1 and large radius 2. For non negative j, let us denote by L 2 j the space

FL 2 ((Z 2 × R) ∩ 2 j C) and by L 2 -1 the space FL 2 ((Z 2 × R) ∩ B) respectively equipped with the (semi) norms u 2 L 2 j def = (2π) -d 2 j C | u(ξ)| 2 dξ and u 2 L 2 -1 def = (2π) -d B | u(ξ)| 2 dξ.
Let us now recall the definition of inhomogeneous Besov spaces modeled on L 2 .

Definition 2.1. Let s be a nonnegative real number. The space B s is the subspace of L 2 such that

u B s def = 2 js u L 2 j j ℓ 1 < ∞.
We note that u ∈ B s is equivalent to writing u L 2 j ≤ Cc j 2 -js u B s where (c j ) is a non negative series which belongs to the sphere of ℓ 1 . Let us notice that B 3 2 is included in F(L 1 ) and thus in the space of continuous bounded functions. Moreover, if we substitute ℓ 2 to ℓ 1 in the above definition, we recover the classical Sobolev space H s . The theorem we actually prove is the following.

Theorem 2. Let a be a positive number. There are two positive numbers ε 0 and η such that for any divergence free vector field v 0 satisfying

e a|D 3 | v 0 B 7 2 ≤ η,
then, for any positive ε smaller than ε 0 , the initial data

u 0,ε (x) def = v h 0 (x h , εx 3 ), 1 ε v 3 0 (x h , εx 3 )
generates a global smooth solution of (N S) on T 2 × R.

2.2.

Definition of the functional setting.

Study of a model problem.

In order to motivate the functional setting and to give a flavour of the method used to prove the theorem, let us study for a moment the following simplified model problem for (RN S ε ), in which we shall see in a rather easy way how the same type of method as that of [START_REF] Chemin | Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray[END_REF] can be used (as a global Cauchy-Kowaleswski technique): the idea is to control a nonlinear quantity, which depends on the solution itself. So let us consider the equation ∂ t u + γu + a(D)(u 2 ) = 0, where u is a scalar, real-valued function, γ is a positive parameter, and a(D) is a Fourier multiplier of order one. We shall sketch the proof of the fact that if the initial data satisfies, for some positive δ and some small enough constant c,

u 0 X def = e δ|ξ| | u(ξ)| dξ ≤ cγ,
then one has a global smooth solution, say in the space F(L 1 ) as well as all its derivative. The idea of the proof is the following: we want to control the same kind of quantity on the solution, but one expects the radius of analyticity of the solution to decay in time. So let us introduce θ(t) the "loss of analyticity" of the solution, solving the following ODE:

θ(t) def = e (δ-λθ(t))|ξ| | u(ξ)|dξ with θ(0) = 0.
The parameter λ will be chosen large enough at the end and we will prove that δ -λθ(t) remains positive for all times. The computations that follow hold as long as that assumption is true (and a bootstrap will prove that in fact it does remain true for all times). We define the notation

u θ (t) = F -1 e (δ-λθ(t))|•| | u(t, •)| . Notice that (2.2) θ(t) = u θ (t) F (L 1 ) and θ(t) = t 0 u θ (t ′ ) F (L 1 ) dt ′ .
Taking the Fourier transform of the equation gives

| u(t, ξ)| ≤ e -γt | u 0 (ξ)| + C t 0 e -γ(t-t ′ ) |ξ| |F(u 2 )(t ′ , ξ)| dt ′ .
Using the fact that

(δ -λθ(t)) |ξ| ≤ (δ -λθ(t ′ )) |ξ -η| + (δ -λθ(t ′ )) |η| -λ|ξ| t t ′ θ(t) dt ′′ ,
we infer that

| u θ (t, ξ)| ≤ e -γt e δ|ξ| | u 0 (ξ)| + C t 0 e -γ(t-t ′ )-λ|ξ| t t ′ θ(t) dt ′′ |ξ| |F(u 2 θ )|(t ′ , ξ) dt ′ .
We note the important fact that

t 0 e -λ|ξ| t t ′ θ(t) dt ′′ |ξ| θ(t ′ ) dt ′ ≤ C λ , which is very useful in what follows. As F(ab) = (2π) -d ( a ⋆ b), we have, for any t ′ ≤ t, |F(u 2 θ )|(t ′ , ξ) ≤ sup 0≤t ′ ≤t | u θ (t ′ , •)| ⋆ | u θ (t ′ , •)|.
Recalling that F(L 1 ) is an algebra, we infer that sup

0≤t ′ ≤t | u θ (t ′ )| L 1 ≤ u 0 X + C λ sup 0≤t ′ ≤t | u θ (t ′ )| L 1 and θ(t) ≤ C γ u 0 X + sup 0≤t ′ ≤t | u θ (t ′ )| L 1 θ(t)
. This allows by bootstrap to obtain the global in time existence of the solution, as soon as the initial data is small enough; we skip the computations, as they will be presented in full detail for the case of the system (RN S ε ). 2.2.2. Functional setting. In the light of the computations of the previous section, let us introduce the functional setting we are going to work with to prove the theorem. The proof relies on exponential decay estimates for the Fourier transform of the solution. Thus, for any locally bounded function Ψ on R + × Z 2 × R and for any function f , continuous in time and compactly supported in Fourier space, we define

(f Ψ )(t) def = F -1 e Ψ(t,•) f (t, •) .
Now let us define the key quantity we wish to control in order to prove the theorem. In order to do so, let us consider the Friedrichs approximation of the original (N S) system

     ∂ t u -∆u + P n (u • ∇u + ∇p) = 0 div u = 0 u| t=0 = P n u 0,ε ,
where P n denotes the orthogonal projection of L 2 on functions the Fourier transform of which is supported in the ball B n centered at the origin and of radius n. Thanks to the L 2 energy estimate, this approximated system has a global solution the Fourier transform of which is supported in B n . Of course, this provides an approximation of the rescaled system namely

(RN S ε,n )            ∂ t w h -∆ ε w h + P n,ε M ⊥ v • ∇w h + w 3 ∂ 3 v + ∇ h q = 0 ∂ t w 3 -∆ ε w 3 + P n,ε M ⊥ v • ∇w 3 + ε 2 ∂ 3 q = 0 ∂ t v h -ε 2 ∂ 2 3 v h + P n,ε ∂ 3 M (w 3 w h ) = 0 div(v + w) = 0 (v, w) |t=0 = (v 0 , w 0 ),
where P n,ε denotes the orthogonal projection of L 2 on functions the Fourier transform of which

is supported in B n,ε def = {ξ / |ξ ε | 2 def = |ξ h | 2 + ε 2 ξ 2 3 ≤ n 2 }.
We shall prove analytic type estimates here, meaning exponential decay estimates for the the solution of the above approximated system. In order to make notation not too heavy we will drop the fact that the solutions we deal with are in fact approximate solutions and not solutions of the original system. A priori bounds on the approximate sequence will be derived, which will clearly yield the same bounds on the solution. In the spirit of [START_REF] Chemin | Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray[END_REF] (see also (2.2) in the previous section), we define the function θ (we drop also the fact that θ depends on ε in all that follows) by

(2.3) θ(t) = w 3 Φ (t) B 7 2 + ε w h Φ (t) B 7 2
and θ(0) = 0 where

(2.4) Φ(t, ξ) = t 1 2 |ξ h | + a|ξ 3 | -λθ(t)|ξ 3 |
for some λ that will be chosen later on (see Section 2.4). Since the Fourier transform of w is compactly supported, the above differential equation has a unique global solution on R + . If we prove that

(2.5) ∀t ∈ R + , θ(t) ≤ a λ ,
this will imply that the sequence of approximated solutions of the rescaled system is a bounded sequence of L 1 (R + ; Lip). So is, for a fixed ε, the family of approximation of the original Navier-Stokes equations. This is (more than) enough to imply that a global smooth solution exists.

2.3. Main steps of the proof. The proof of Inequality (2.5) will be a consequence of the following two propositions which provide estimates on v h , w h and w 3 . For technical reasons, these statements require the use of a modified version (introduced in [START_REF] Chemin | Flot de champs de vecteurs non-lipschitziens et équations de Navier-Stokes[END_REF]) of L ∞ T (B s ) spaces.

Definition 2.2. Let s be a real number. We define the space L ∞ T (B s ) as the subspace of functions f of L ∞ T (B s ) such that the following quantity is finite:

f L ∞ T (B s ) def = j 2 js f L ∞ T (L 2 j ) .
Theorem 2 will be an easy consequence of the following propositions, which will be proved in the coming sections.

The first one uses only the fact that the function Φ is subadditive.

Proposition 2.1. A constant C

(1) 0 exists such that, for any positive λ, for any initial data v 0 , and for any T satisfying θ(T ) ≤ a/λ, we have

θ(T ) ≤ ε e a|D 3 | w h 0 B 7 2 + e a|D 3 | w 3 0 B 7 2 + C (1) 0 v Φ L ∞ T (B 7 2 ) θ(T ).
Moreover, we have the following L ∞ -type estimate on the vertical component:

w 3 Φ L ∞ T (B 7 2
)

≤ e a|D 3 | w 3 0 B 7 2 + C (1) 0 v Φ 2 L ∞ T (B 7 2
)

.

The second one is more subtle to prove, and it shows that the use of the analytic-type norm actually allows to recover the missing vertical derivative on v h , in a L ∞ -type space. It should be compared to the methods described in Section 2.2.1.

Proposition 2.2. A constant C (2)
0 exists such that, for any positive λ, for any initial data v 0 , and for any T satisfying θ(T ) ≤ a/λ, we have

v h Φ L ∞ T (B 7 2
)

≤ e a|D 3 | v h 0 B 7 2 + C (2) 0 1 λ + v Φ L ∞ T (B 7 2 ) v h Φ L ∞ T (B 7 2 )
.

2.4.

Proof of the theorem assuming the two propositions. Let us assume these two propositions are true for the time being and conclude the proof of Theorem 2. It relies on a continuation argument.

For any positive λ and η, let us define

T λ def = T / max{ v Φ L ∞ T (B 7 2 )
, θ(T )} ≤ 4η ,

As the two functions involved in the definition of T λ are non decreasing, T λ is an interval. As θ is an increasing function which vanishes at 0, a positive time T 0 exists such that θ(T 0 ) ≤ 4η. Moreover, if e a|D| 3 | v 0 B 7 2 ≤ η then, since ∂ t v = P n F (v) (recall that we are considering Friedrich's approximations), a positive time T 1 (possibly depending on n)

exists such that v Φ L ∞ T 1 (B 7
2 ) ≤ 4η. Thus T λ is the form [0, T ⋆ ) for some positive T ⋆ . Our purpose is to prove that T ⋆ = ∞. As we want to apply Propositions 2.1 and 2.2, we need that λθ(T ) ≤ a. This leads to the condition

(2.6) 4λη ≤ a. From Proposition 2.1, defining C 0 def = C (1) 0 + C (2) 0 , we have, for all T ∈ T λ , v Φ L ∞ T (B 7 2
)

≤ e a|D 3 | v 0 B 7 2 + C 0 λ v Φ L ∞ T (B 7 2
)

+ C 0 v Φ 2 L ∞ T (B 7 2 )
.

Let us choose λ = 1 2C 0 • This gives v Φ L ∞ T (B 7 2
)

≤ 2 e a|D 3 | v 0 B 7 2 + 4C 0 η v Φ L ∞ T (B 7 2 )
.

Choosing η = 1 12C 0 , we infer that, for any

T ∈ T λ , (2.7) v Φ L ∞ T (B 7 2 )
≤ 3 e a|D 3 | v 0 B

The action of subadditive phases on (para)products

It will be useful to consider, for any function f , the inverse Fourier transform of | f |, defined as

f + def = F -1 | f |.
Let us notice that the map f → f + preserves the norm of all B s spaces. In all this section, Ψ will denote a locally bounded function on R + × T 2 × R which satisfies the following inequality

(3.1) Ψ(t, ξ) ≤ Ψ(t, ξ -η) + Ψ(t, η).
In all the following, we will denote by C or c universal constants, which do not depend on any of the parameters of the problem, and which may change from line to line. We will denote generically by c j any sequence in ℓ 1 (Z) of norm 1.

We shall denote by E ε f the solution of ∂ t g -∆ ε g = f with initial data equal to 0. We use also a very basic version of Bony's decomposition. Let us define (using the notation introduced in Section 2.1),

T a b def = F -1 j 2 j C∩B(ξ,2 j ) a(ξ -η) b(η)dη and R a b def = F -1 j 2 j C∩B(ξ,2 j+1 ) a(ξ -η) b(η)dη.
We obviously have ab = T a b + R b a.

The way the Fourier multiplier e Ψ acts on bilinear functionals is described by the following lemma.

Lemma 3.1. For any positive s, a constant C exists which satisfies the following properties.

For any function Ψ satisfying (3.1), for any function b in L 1 T (B s ), a positive sequence (c j ) j∈Z exists in the sphere of ℓ 1 (Z) such that, for any a in L 1

T (B 3
2 ), and any t ∈ [0, T ], we have

(T a b) Ψ (t) L 2 j + (R a b) Ψ (t) L 2 j ≤ Cc j 2 -js a(t) B 3 2 min b(t) B s , b L ∞ T (B s ) .
Proof. We prove only the lemma for R, the proof for T being strictly identical. Let us first investigate the case when the function Ψ is identically 0. We first observe that for any ξ in the annulus 2 j C, we have

F(R a b(t))(ξ) = j ′ ≥j-2 2 j ′ C∩B(ξ,2 j ′ +1 ) a(t, ξ -η) b(t, η)dη.
As B 3 2 is included in F(L 1 ), we infer that, by definition of

• L ∞ T (B s ) , R a b(t) L 2 j ≤ C a(t) B 3 2 j ′ ≥j-2 c j ′ 2 -j ′ s min b(t) B s , b L ∞ T (B s ) . Defining c j = j ′ ≥j-2 2 (j-j ′ )s c j ′ which satisfies j cj ≤ C s , we obtain (3.2) R a b(t) L 2 j ≤ Cc j 2 -js a(t) B 3 2 min b(t) B s , b L ∞ T (B s
) . The lemma is then proved in the case when the function Ψ is identically 0. In order to treat the general case, let us write that

e Ψ(t,ξ) F(R a b)(ξ) = e Ψ(t,ξ) j 2 j C∩B(ξ,2 j ) a(ξ -η) b(η)dη ≤ j 2 j C∩B(ξ,2 j ) e Ψ(t,ξ-η) a + (ξ -η)e Ψ(t,η) b + (η)dη.
This means exactly that

|F(R a b) Ψ (t, ξ)| ≤ F(R a + Ψ b + Ψ )(t, ξ).
Then, the estimate (3.2) implies the lemma.

(T a b) Ψ L ∞ T (B s ) + (R a b) Ψ L ∞ T (B s ) ≤ C a Ψ L ∞ T (B 3 2 ) b Ψ L ∞ T (B s ) and (T a b) Ψ L 1 T (B s ) + (R a b) Ψ L 1 T (B s ) ≤ C min a Ψ L 1 T (B 3 2 ) b Ψ L ∞ T (B s ) , a Ψ L ∞ T (B 3 2 ) b Ψ L 1 T (B s ) .
Proof. Taking the L ∞ norm in time on the inequality of Lemma 3.1 gives that

(T a b) Ψ L ∞ T (L 2 j ) + (R a b) Ψ L ∞ T (L 2 j ) ≤ Cc j 2 -js a L ∞ T (B 3 2 ) b L ∞ T (B s )
. which is the first inequality of the corollary. The proof of the second one is analogous.

The action of the phase Φ on the heat operator

The purpose of this section is the study of the action of the multiplier e Φ on E ε f . Let us recall that the function Φ is defined in (2.4) by Φ(t, ξ) = t

(E ε M ⊥ f ) Φ L ∞ T (B s ) ≤ C 0 g Φ L ∞ T (B s ) and (E ε M ⊥ f ) Φ L 1 T (B s ) ≤ C 0 g Φ L 1 T (B s )
where

g def = F -1 1 |ξ h | |FM ⊥ f | .
Proof. Let us write E ε in terms of the Fourier transform. We have, for any ξ ∈ (Z 2 \{0}) × R,

F (E ε f ) Φ (t, ξ) = e Φ(t,ξ) t 0 e -(t-t ′ )|ξε| 2 f (t ′ , ξ)dt ′ ,
with, as in all that follows,

|ξ ε | 2 def = |ξ h | 2 + ε 2 |ξ 3 | 2 . Thus we infer, for any ξ ∈ (Z 2 \{0}) × R, |F ((E ε f ) Φ ) (t, ξ)| ≤ t 0 e -(t-t ′ )|ξε| 2 +Φ(t,ξ)-Φ(t ′ ,ξ) F(f + Φ )(t ′ , ξ)dt ′ .
By definition of Φ, we have (see [START_REF] Chemin | Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray[END_REF], estimate ( 24))

(4.1) Φ(t, ξ) -Φ(t ′ , ξ) ≤ -λ|ξ 3 | t t ′ θ(t ′′ )dt ′′ + t -t ′ 2 |ξ h | 2 .
Thus we have, for any ξ ∈ (Z 2 \{0}) × R,

(4.2) |F ((E ε f ) Φ ) (t, ξ)| ≤ t 0 e -(t-t ′ ) 2 |ξ h | 2 -ε 2 (t-t ′ )|ξ 3 | 2 F(f + Φ )(t ′ , ξ)dt ′ . Let us define C h def = {1 ≤ |ξ h | ≤ 2} × R.
The above inequality means that we have, for any ξ in

2 j C ∩ 2 k C h , |F((E ε f ) Φ )(t, ξ)| ≤ C t 0 e -c(t-t ′ )2 2k 2 k g Φ (t ′ , ξ)dt ′ .
Taking the L 2 norm in ξ in that inequality gives

(4.3) (E ε f ) Φ (t) F L 2 (2 j C∩2 k C h ) ≤ t 0 e -c(t-t ′ )2 2k 2 k g Φ (t ′ ) L 2 j dt ′ .
By definition of the L ∞ T (B s ) norm, this gives, for any t ≤ T ,

2 js (E ε f ) Φ L ∞ T (L 2 (2 j C∩2 k C h )) ≤ Cc j g Φ L ∞ T (B s ) t 0 e -c(t-t ′ )2 2k 2 k dt ′ ≤ Cc j 2 -k g Φ L ∞ T (B s ) . Now, writing that (E ε M ⊥ f ) Φ L ∞ T (L 2 j ) ≤ ∞ k=0 E ε (f Φ ) L ∞ T (L 2 (2 j C∩2 k C h ))
gives the first inequality of the lemma.

In order to prove the second one, let us use the definition of the norm of the space B s and (4.3); this gives

j 2 js (E ε f ) Φ L 1 T (L 2 j ) ≤ j,k 2 js E ε (f Φ ) L 1 T (F L 2 (2 j C∩2 k C h )) ≤ C j,k [0,T ] 2 1 t≥t ′ e -c2 2k (t-t ′ ) 2 k c j (t ′ ) g Φ (t ′ ) B s dt ′ dt.
Integrating first in t gives

j 2 js (E ε f Φ ) L 1 T (L 2 j ) ≤ C j,k [0,T ] 2 -k c j (t ′ ) g Φ (t ′ ) B s dt.
As the index k is nonnegative, we get the second estimate of the lemma.

The following lemma is a key one. It is here that the function θ allows the gain of the vertical derivative, in the spirit of the example presented in Section 2.2.1.

(E ε a(D)R b(D)w 3 f ) Φ L ∞ T (B 7 2 ) + (E ε a(D)T b(D)w 3 f ) Φ L ∞ T (B 7 2 ) ≤ C 1 λ + w 3 Φ L ∞ T (B 7 2
)

f Φ L ∞ T (B 7 2 )
.

Proof. We give only the proof for the first term, the second term is estimated exactly along the same lines. Let us write E ε in Fourier variables. We have

F(E ε a(D)R b(D)w 3 f ) Φ (t, ξ) = e Φ(t,ξ) t 0 e -(t-t ′ )|ξε| 2 a(ξ)F(R b(D)w 3 f )(t ′ , ξ)dt ′ .
Thus, using that |a(ξ)| ≤ C|ξ 3 |, we obtain

|F(E ε a(D)R w 3 f ) Φ (t, ξ)| ≤ C t 0 e -(t-t ′ )|ξε| 2 +Φ(t,ξ)-Φ(t ′ ,ξ) |ξ 3 | |F((R b(D)w 3 f ) Φ )(t ′ , ξ)|dt ′ .
Taking into account Inequality (4.1), we have

|F(E ε a(D)R w 3 f ) Φ (t, ξ)| ≤ C t 0 e -t-t ′ 2 |ξε| 2 -λ|ξ 3 | t t ′ θ(t ′′ )dt ′′ |ξ 3 | |F((R b(D)w 3 f ) Φ )(t ′ , ξ)|dt ′ .
Let us denote by Ψ the Fourier multiplier Ψa

def = F -1 (1 |ξ h |≤2|ξ 3 | a). If |ξ h | ≤ 2|ξ 3 |
and ξ is in 2 j C, then, we have that |ξ 3 | ∼ 2 j . Thus, we infer that, for any ξ in 2 j C,

|FΨ(E ε a(D)R b(D)w 3 f ) Φ (t, ξ)| ≤ t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ 2 j |F((R b(D)w 3 f ) Φ )(t ′ , ξ)|dt ′ .
Taking the L 2 norm gives

Ψ(E ε a(D)R b(D)w 3 f ) Φ (t, •) L 2 j ≤ t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ 2 j (R b(D)w 3 f ) Φ )(t ′ ) L 2 j dt ′ .
Using Lemma 3.1, we get

2 j 7 2 Ψ(E ε a(D)R b(D)w 3 f ) Φ (t, •) L 2 j ≤ Cc j f Φ (t) L ∞ T (B 7
2

) × t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ 2 j b(D)w 3 Φ (t ′ , •) B 3 2 dt ′ ≤ Cc j f Φ (t) L ∞ T (B 7 2
)

× t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ 2 j w 3 Φ (t ′ , •) B 7 2 dt ′ ≤ Cc j f Φ (t) L ∞ T (B 7 2
)

t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ 2 j θ(t ′ )dt ′ ≤ C λ c j f Φ (t) L ∞ T (B 7 2 )
.

By summation in j, we deduce that

(4.4) Ψ(E ε a(D)R w 3 f ) Φ L ∞ T (B 7 2
)

≤ C λ f Φ (t) L ∞ T (B 7 2 )
.

If 2|ξ 3 | ≤ |ξ h |, then, for any ξ in 2 j C, |ξ h | is equivalent to 2 j and |ξ 3 | is less than 2 j . So we infer that for any ξ in 2 j C,

|F(Id -Ψ)(E ε a(D)R b(D)w 3 f ) Φ (t, ξ)| ≤ t 0 e -c(t-t ′ )2 2j 2 j |F((R b(D)w 3 f ) Φ )(t ′ , ξ)|dt ′ . By definition of • L ∞ T (B 7 2 )
, taking the L 2 norm of the above inequality gives

2 j 7 2 (Id -Ψ)(E ε a(D)R b(D)w 3 f ) Φ (t, •) L 2 j ≤ t 0 e -c2 2j (t-t ′ ) 2 j 2 j 7 2 (R b(D)w 3 f ) Φ )(t ′ ) L 2 j dt ′ ≤ Cc j (R b(D)w 3 f ) Φ L ∞ T (B 7 2 )
.

After a summation in j, Corollary 3.1 implies that

(Id -Ψ)(E ε a(D)R b(D)w 3 f ) Φ L ∞ T (B 7 2
)

≤ C b(D)w 3 Φ L ∞ T (B 3 2
)

f Φ L ∞ T (B 7 2
)

≤ C w 3 Φ L ∞ T (B 7 2
)

f Φ L ∞ T (B 7 2 )
.

Together with (4.4), this concludes the proof of the lemma.

Lemma 4.3. A constant C 0 exists such that, for any function f with compact spectrum, we have for α in {1, 2},

E ε (ε∂ 3 ) α M ⊥ f Φ L ∞ T (B s ) ≤ C 0 f Φ L ∞ T (B s ) and E ε (ε∂ 3 ) α M ⊥ f Φ L 1 T (B s ) ≤ C 0 f Φ L 1 T (B s ) .
Proof. Let us start with the case when α = 2. Recalling (4.2), we have (for 0 < ε < 1),

ε 2 |F(E ε ∂ 2 3 f ) Φ (t, ξ)| ≤ t 0 e -ε 2 (t-t ′ ) 2 |ξ| 2 ε 2 ξ 2 3 F(f + Φ )(t ′ , ξ)dt ′ .
Writing that |ξ 3 | ≤ |ξ|, we infer that

ε 2 (E ε ∂ 2 3 f ) Φ (t) L 2 j ≤ t 0 e -cε 2 (t-t ′ )2 2j ε 2 2 2j f (t ′ ) L 2 j dt ′ .
The estimates follow directly by applying Young's inequality in t.

In the case when α = 1, we decompose f into two parts,

f = f (1) + f (2) , with f (1) = F -1 (1 ε|ξ 3 |≤|ξ h | f ).
Let us start by studying the first contribution. We simply write that

ε F E ε ∂ 3 f (1) Φ (t, ξ) ≤ t 0 e -(t-t ′ ) 2 |ξε| 2 ε|ξ 3 |F(f (1)
Φ ) + (t ′ , ξ)dt ′ ≤ t 0 e -(t-t ′ ) 2 |ξε| 2 |ξ h |F(f (1)
Φ ) + (t ′ , ξ)dt ′ which amounts exactly to the computation (4.3), with g replaced by f (1) . On the other hand, for f (2) we can write

g (2) (ξ) def = 1 |ξ h | 1 ε|ξ 3 |≥|ξ h | |FM ⊥ f (2) (ξ)| so that ε F E ε ∂ 3 M ⊥ f (2) Φ (t, ξ) ≤ t 0 e -(t-t ′ ) 2 |ξ h | 2 -ε 2 (t-t ′ )|ξ 3 | 2 ε|ξ 3 ||ξ h | g (2) (t ′ , ξ)dt ′ .
Since |ξ h | ≤ ε|ξ 3 |, we are reduced to the case when α = 2 and the conclusion comes from the fact that g

(2)

Φ B s ≤ M ⊥ f (2)
Φ B s ≤ f Φ B s . That proves the lemma.

Classical analytic-type parabolic estimates

The purpose of this section is to prove Proposition 2.1. We shall use the algebraic structure of the Navier-Stokes system and the fact that the function Φ is subadditive.

Let us first bound the horizontal component. We recall that

w h Φ (t) = e t∆ε+Φ(t,D) w h (0) -E ε M ⊥ (v • ∇w h ) Φ (t) -E ε M ⊥ (w 3 ∂ 3 v h ) Φ (t) -E ε (∇ h q) Φ (t).
We note that v • ∇w h = div h (v h ⊗ w h ) + ∂ 3 (w 3 w h ), recalling that v 3 = w 3 . On the one hand, using Lemma 4.1 and Corollary 3.1, we can write

ε E ε (div h (v h w h )) Φ L 1 T (B 7 2
)

≤ Cε (v h w h ) Φ L 1 T (B 7 2
)

≤ C v Φ L ∞ T (B 7 2
)

ε w h L 1 T (B 7 2 )
.

By definition of θ, we infer that (5.1)

ε E ε (div h (v h w h )) Φ L 1 T (B 7
2

) ≤ Cθ(T ) v h Φ L ∞ T (B 7 2 )
.

On the other hand, Lemma 4.3 and Corollary 3.1 imply that

E ε ε∂ 3 M ⊥ (w 3 w h ) Φ L 1 T (B 7 2
)

≤ C w 3 w h L 1 T (B 7 2
)

≤ Cθ(T ) v h Φ L ∞ T (B 7 2 )
. (5.2)

For the second term, we use paradifferential calculus which gives

w 3 ∂ 3 v h = T w 3 ∂ 3 v h + R ∂ 3 v h w 3 = ∂ 3 T w 3 v h -T ∂ 3 w 3 v h + R ∂ 3 v h w 3 .
Using again Lemma 4.3 and Corollary 3.1, we get

E ε ε∂ 3 M ⊥ T w 3 v h ) Φ L 1 T (B 7 2 ) ≤ C (T w 3 v h ) Φ L 1 T (B 7 2 ) ≤ C w 3 Φ L 1 T (B 7 2 ) v h L ∞ T (B 7 2 )
.

By definition of θ, we infer (5.3)

E ε ε∂ 3 M ⊥ T w 3 v h ) Φ L 1 T (B 7
2

) ≤ Cθ(T ) v h Φ L ∞ T (B 7 2 )
.

By Lemma 4.1 and Corollary 3.1, we can write that

E ε εM ⊥ T ∂ 3 w 3 v h ) Φ L 1 T (B 7 2 )
≤ Cε (T

∂ 3 w 3 v h ) Φ L 1 T (B 7
2 )

≤ Cε w 3 Φ L 1 T (B 7 2 ) v h L ∞ T (B 7
2 ) so that

(5.4)

E ε εM ⊥ T ∂ 3 w 3 v h ) Φ L 1 T (B 7
2

) ≤ Cθ(T ) v h Φ L ∞ T (B 7 2 )
, and finally along the same lines we have

(5.5) E ε εM ⊥ R ∂ 3 v h w 3 ) Φ L 1 T (B 7
2

) ≤ Cθ(T ) v h Φ L ∞ T (B 7 2
)

. Now we are left with the study of the pressure. Some of its properties are described in the following lemma.

Lemma 5.1. Let us define ∇ ε def = (∇ h , ε∂ 3 ). The following two inequalities on the rescaled pressure hold:

ε (E ε ∇ ε M ⊥ q) Φ L ∞ T (B 7 2 ) ≤ C v Φ 2 L ∞ T (B 7
2

)
and

ε (E ε ∇ ε M ⊥ q) Φ L 1 T (B 7 2
)

≤ C v Φ L ∞ T (B 7 2 )
θ(T ).

Proof. Using the formula (2.1) on the rescaled pressure and the divergence free condition on v, let us decompose it as εq = q 1,ε -q 2,ε with

q 1,ε def = 2 k=1 ∂ k ∂ ℓ ∆ -1 ε (εw k v ℓ ) + 1≤k≤2 ∂ k (ε∂ 3 )∆ -1 ε (w 3 v k ) and q 2,ε def = 2ε∂ 3 ∆ -1 ε (w 3 div h w h
). Let us start with q 1,ε . We have

∇ ε q 1,ε = 2 k=1 ∂ k 2 ℓ=1 ∇ ε ∂ ℓ ∆ -1 ε (εw k v ℓ ) + ∇ ε (ε∂ 3 )∆ -1 ε (w 3 v k ) .
As ∇ 2 ε ∆ -1 ε is a family of bounded Fourier multipliers (uniformly with respect to ε), we infer from Lemma 4.1 and Corollary 3.1 that

ε (E ε (∇ ε M ⊥ q 1,ε )) Φ L ∞ T (B 7 2 ) ≤ C v Φ 2 L ∞ T (B 7 2
) and (5.6)

ε (E ε (∇ ε M ⊥ q 1,ε )) Φ L 1 T (B 7 2 ) ≤ C v Φ L ∞ T (B 7
2 ) θ(T ). (5.7) In order to study q 2,ε , let us observe that

w 3 div h w h = R div h w h w 3 + T w 3 div h w h = R div h w h w 3 + 2 k=1 ∂ k T w 3 w k -T ∂ k w 3 w k . (5.8)
As above we get, using Lemma 4.1 and Corollary 3.1,

ε (E ε (∇ ε M ⊥ q 2,ε )) Φ L ∞ T (B 7 2 ) ≤ C v Φ 2 L ∞ T (B 7
2

)
and

ε (E ε (∇ ε M ⊥ q 2,ε )) Φ L 1 T (B 7 2
)

≤ C v Φ L ∞ T (B 7
2 ) θ(T ).

Together with estimates (5.6) and (5.7), this concludes the proof of the lemma.

The above Lemma 5.1, together with estimates (5.1) to (5.4), implies that (5.9)

ε w h L 1 T (B 7 2 )
≤ ε e

a|D 3 | w h 0 B 7 2 + C (1) 0 v Φ L ∞ T (B 7
2 ) θ(T ).

Let us prove the estimates on the vertical component. It turns out that it is better behaved because of the special structure of the system. Indeed, thanks to the divergence free condition, almost no vertical derivatives appear in the equation of w 3 : we have (since

w 3 = v 3 ) (5.10) ∂ t w 3 -∆ ε w 3 = -v h • ∇ h w 3 + w 3 div h w h -ε 2 ∂ 3 q.
The Duhamel formula reads

w 3 (t) = e t∆ε w 3 (0) + E ε M ⊥ (w 3 div h w h -v h • ∇ h w 3 )(t) -E ε M ⊥ (ε 2 ∂ 3 q)(t).
Applying the Fourier multiplier e Φ(t,D) to the above relation gives

(5.11) w 3 Φ (t) = e t∆ε+Φ(t,D) w 3 (0)+ E ε M ⊥ (w 3 div h w h -v h •∇ h w 3 ) Φ (t)-E ε M ⊥ ε 2 ∂ 3 q Φ (t)
. Using (5.8) and then Lemma 4.1 and Corollary 3.1, we get

E ε M ⊥ (w 3 div h w h ) Φ L ∞ T (B 7 2 ) ≤ C w 3 Φ L ∞ T (B 7 2
)

w h Φ L ∞ T (B 7 2
) and (5.12)

E ε M ⊥ (w 3 div h w h ) Φ L 1 T (B 7 2 ) ≤ C w 3 Φ L 1 T (B 7 2
)

w h Φ L ∞ T (B 7 2 )
. (5.13) Writing that

v h • ∇ h a = 2 k=1 T v k ∂ k a + R ∂ k a v k = -T div h w h a + 2 k=1 ∂ k T v k a + R ∂ k a v k
and using Lemma 4.1 and Corollary 3.1, we get

E ε M ⊥ (v h • ∇ h w 3 ) Φ L ∞ T (B 7 2 ) ≤ C w 3 Φ L ∞ T (B 7 2
)

w h Φ L ∞ T (B 7 2
)

and

E ε M ⊥ (v h • ∇ h w 3 ) Φ L 1 T (B 7
2 )

≤ C w 3 Φ L 1 T (B 7 2
)

w h Φ L ∞ T (B 7 2
)

.

Together with estimates (5.12) and (5.13), and Lemma 5.1, this gives

w 3 L 1 T (B 7 2
)

≤ e a|D 3 | w 3 0 B 7 2 + C (1) 0 v Φ L ∞ T (B 7
2 ) θ(T ) and

w 3 Φ L ∞ T (B 7 2
)

≤ e a|D 3 | w 3 0 B 7 2 + C (1) 0 v Φ 2 L ∞ T (B 7 2
)

.

Together with (5.9), this concludes the proof of Proposition 2.1.

The gain of one vertical derivative on the horizontal part

In this section we shall prove Proposition 2.2. The proof will be separated into two parts: first we shall consider the case of the horizontal average v h Φ , and then the remainder w h Φ .

6.1. The gain of one vertical derivative on the horizontal average. We shall study in this section the equation on the horizontal average of the solution. We emphasize that in the equation on v we cannot recover the vertical derivative appearing in the force term by the regularizing effect. The fundamental idea to gain a vertical derivative is to use the analyticity of the solution and therefore to estimate v Φ . The lemma is the following.

Lemma 6.1. A constant C 0 exists such that, for any positive λ, for any initial data v 0 , and for any T satisfying θ(T ) ≤ a/λ, we have

v h Φ L ∞ T (B 7 2
)

≤ e a|D 3 | v h 0 B 7 2 + C 0 1 λ + v Φ L ∞ T (B 7
2 ) v h Φ L ∞ T (B 7 2 )
.

Proof. The horizontal average v satisfies (6.1)

∂ t v -ε 2 ∂ 2 3 v = -∂ 3 M (w 3 w h ) and v |t=0 = v 0 . Let us define G def = -∂ 3 M (w 3 w k ).
Writing the solution of (6.1) in terms of the Fourier transform, we get, using (4.1) with ξ h = 0,

|F(v Φ )(t, ξ)| ≤ |Fv 0 (ξ)|e a|ξ 3 | + t 0 e -λ|ξ 3 | t t ′ θ(t ′′ )dt ′′ |F(G Φ )(t ′ , ξ)|dt ′ .
Then, taking the L 2 j norm, we infer that

(6.2) v Φ (t) L 2 j ≤ e a|D 3 | v 0 L 2 j + t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ G Φ (t ′ ) L 2 j dt ′ . Now, let us estimate G Φ (t ′ ) L 2 j
. For any function a, using the fact that the vector field w is divergence free, let us write that

∂ 3 (w 3 a) = ∂ 3 T w 3 a + R a w 3 = ∂ 3 T w 3 a + R ∂ 3 a w 3 -R a div h w h = ∂ 3 T w 3 a + R ∂ 3 a w 3 - 2 ℓ=1 ∂ ℓ R a w ℓ + 2 ℓ=1 R ∂ ℓ a w ℓ . (6.3) Thus, we infer that G = -∂ 3 M T w 3 w k -M R ∂ 3 w k w 3 + 2 ℓ=1 R ∂ ℓ w k w ℓ - 2 ℓ=1 ∂ ℓ R w 3 w ℓ = -∂ 3 M T w 3 w k -M R ∂ 3 w k w 3 + 2 ℓ=1 R ∂ ℓ w k w ℓ . (6.4)
Now, let us study FM (T a b) Φ and FM (R a b) Φ for two functions a and b which have 0 horizontal average. As the two terms are identical, let us study the first one. By definition, we have

F T a b)(t, (0, ξ 3 )) = j 2 j C∩B((0,ξ 3 ),2 j ) a((0, ξ 3 ) -η) b(η)dη.
As θ(T ) ≤ λ -1 a, by definition of Φ we have, for any η ∈ (Z 2 \{0}) × R, Φ(t, (0, ξ 3 )) ≤ Φ(t, (0, ξ 3 -η 3 )) + Φ(t, (0, η 3 ))

≤ -2t Inequality (3.1) then implies that, for any t ∈ [0, T ], (6.5)

2 j 7 2 G Φ (t) L 2 j ≤ Cc j v h Φ L ∞ T (B 7
2 )

2 j w 3 Φ (t)

B 7 2 + e -2t 1 2 v Φ L ∞ T (B 7 2 )
.

Then, by definition of θ, Inequalities (6.2) and (6.5) imply that

2 j 7 2 (v Φ )(t) L 2 j ≤ 2 j 7 2 e a|D 3 | v 0 L 2 j + Cc j v Φ L ∞ T (B 7
2

)
t 0 e -cλ2 j t t ′ θ(t ′′ )dt ′′ 2 j θ(t

′ )dt ′ + v Φ L ∞ T (B 7
2

)
t 0 e -2t ′ 1 2 dt ′ .

This gives

2 j 7 2 v Φ L ∞ T (L 2 j ) ≤ 2 j 7 2 e a|D 3 | v 0 L 2 j + Cc j v h Φ L ∞ T (B 7
2 )

1 λ + v Φ L ∞ T (B 7 2 )
.

Taking the sum over j concludes the proof of the lemma.

6.2. The gain of the vertical derivative on the whole horizontal term. Now let us estimate the rest of the horizontal term, that is w h

Φ L ∞ T (B 7 2 )
. As in Section 6.1, the function θ will play a crucial role. Lemma 6.2. A constant C 0 exists such that, for any λ, for any initial data v 0 , and for any T satisfying θ(T ) ≤ a/λ, we have

w h Φ L ∞ T (B 7 2
)

≤ e a|D 3 | w h 0 B 7 2 + C 0 1 λ + v Φ L ∞ T (B 7 2 ) v h Φ L ∞ T (B 7 2 )
.

Proof. The Duhamel formula writes

w h (t) = e t∆ε w h (0) -E ε div h (v h ⊗ v h )(t) -E ε M ⊥ ∂ 3 (w 3 v h )(t) -E ε (∇ h q)(t).
(E ε div h (v h ⊗ v h )) Φ L ∞ T (B 7
2

) ≤ C (v h ⊗ v h ) Φ L ∞ T (B 7 2
)

≤ C v h Φ 2 L ∞ T (B 7 2 )
. (6.6) Then using (6.3) we get, thanks to Leibnitz formula,

M ⊥ ∂ 3 (w 3 v k ) = M ⊥ ∂ 3 T w 3 v k + F k with F k def = M ⊥ R ∂ 3 v k w 3 - 2 ℓ=1 ∂ ℓ R v k w ℓ -R ∂ ℓ v k w ℓ .
Thanks to Lemma 4.1 and Corollary 3.1, we get

(E ε M ⊥ F k ) Φ L ∞ T (B 7 2
)

≤ (F k ) Φ L ∞ T (B 7 2
)

≤ C v Φ L ∞ T (B 7
2

) v h Φ L ∞ T (B 7 2
)

.

Together with Lemma 4.2, this gives (6.7)

E ε M ⊥ ∂ 3 (w 3 v h ) Φ L ∞ T (B 7 2
)

≤ C 0 1 λ + v Φ L ∞ T (B 7 2
)

v h Φ L ∞ T (B 7 2 )
. Now let us study the pressure term. Formula (2.1) together with the divergence free condition leads to the decomposition q = q h (w h ) + q 3 (v) with

q h (w h ) def = ∆ -1 ε (div h w h ) 2 + 1≤k,ℓ≤2
∂ k w ℓ ∂ ℓ w k and (6.8)

q 3 (v) def = ∆ -1 ε 1≤ℓ≤2
∂ 3 v ℓ ∂ ℓ w 3 . (6.9)

For the first term we use Bony's decomposition in order to obtain

∂ k w ℓ ∂ ℓ w k = T ∂ k w ℓ ∂ ℓ w k + R ∂ ℓ w k ∂ k w ℓ .
Then the Leibnitz formula implies that (6.10)

∂ k w ℓ ∂ ℓ w k = ∂ ℓ T ∂ k w ℓ w k + ∂ k R ∂ ℓ w k w ℓ -T ∂ ℓ ∂ k w ℓ w k -R ∂ k ∂ ℓ w k w ℓ .
On the other hand, again by paradifferential calculus, we can write that

(div h w h ) 2 = T div h w h div h w h + R div h w h div h w h = div h T div h w h w h + R div h w h w h - 2 k=1 T ∂ k div h w h w k + R ∂ k div h w h w k . (6.11) Then Lemma 4.1 implies that (E ε ∇ h q h (w h )) Φ L ∞ T (B 7 2 ) ≤ C 0 (M ⊥ q h (w h )) Φ L ∞ (B 7 2 )
. Using Corollary 3.1 and the fact that the operators ∇ h ∆ -1 ε M ⊥ and ∆ -1 ε M ⊥ are bounded (uniformly in ε) Fourier multipliers, we obtain (6.12)

E ε (∇ h q h (w h )) Φ L ∞ T (B 7 2
)

≤ C 0 v h Φ 2 L ∞ T (B 7 2 )
.

For the second term, let us decompose q 3 (v) in the following way:

∂ 3 v ℓ ∂ ℓ w 3 = T ∂ 3 v ℓ ∂ ℓ w 3 + R ∂ ℓ w 3 ∂ 3 v ℓ = ∂ ℓ T ∂ 3 v ℓ w 3 + ∂ 3 R ∂ ℓ w 3 v ℓ -T ∂ 3 ∂ ℓ v ℓ w 3 -R ∂ 3 ∂ ℓ w 3 v ℓ .
Using now Lemma 4.1 together with Corollary 3.1 and Lemma 4.2, we obtain (6.13)

E ε (∇ h q 3 (v)) Φ L ∞ T (B 7 2
)

≤ C 0 1 λ + v Φ L ∞ T (B 7
2 )

v h L ∞ T (B 7 2 )
.

The expected result is obtained putting together estimates (6.12) and (6.13) on the pressure with estimates (6.6) and (6.7) on the nonlinear terms.

7 2 .B 7 2 + e a|D 3 | w 3 0 B 7 2+B 7 2 + 2 e a|D 3 | w 3 0 B 7 2 . 7 2 + 2 e a|D 3 | w 3 0 B 7 2 7 2

 727377327377 Propositions 2.1 and 2.2 imply that, for all T ∈ T λ , θ(T ) ≤ ε e a|D 3 | w h 0 C 0 ηθ(T ). This implies that θ(T ) ≤ 2ε e a|D 3 | w h 0 If 2ε e a|D 3 | w h 0 B ≤ η and e a|D 3 | v h 0 B ≤ η, then the above estimate and Inequality (2.7) ensure (2.5). This concludes the proof of Theorem 2.

Corollary 3 . 1 .

 31 If s is positive, we have, for any function Ψ satisfying (3.1),

1 2

 1 |ξ h | + a|ξ 3 | -λθ(t)|ξ 3 |.This action is described by the following lemma.

Lemma 4 . 1 .

 41 A constant C 0 exists such that, for any function f with compact spectrum, we have, for any s,

Lemma 4 . 2 .

 42 Let a(D) and b(D) be two Fourier multipliers such that |a(ξ)| ≤ C|ξ 3 | and |b(ξ)| ≤ C|ξ| 2 . We have

Lemma 4 . 1 and

 41 Corollary 3.1 imply that

  To conclude this paragraph, let us remark that the fixed-point methods used to prove local in time wellposedness for arbitrarily large data (such results are available in Banach spaces

	in which the Schwartz class is dense, typically	Ḃ-1+ d p p,q

  Applied to(6.4), this implies thatFG Φ (t, ξ) ≤ |ξ 3 |F T (w 3 Φ ) + (w k Φ ) + (t, (0, ξ 3 ))

			1 2 + Φ(t, ((0, ξ 3 ) -η) + Φ(t, -η).
	Thus we have	|(FM (T a b) Φ )(t, ξ)| ≤ e -2t	1 2 (FM T a + Φ	b + Φ )(t, ξ).
		+ e -2t	1 2 F R (∂ 3 w k Φ ) + (w 3 Φ ) + +	2	R (∂ ℓ w k Φ )
					ℓ=1

+ (w ℓ Φ ) + (t, (0, ξ 3 )).

Acknowledgments

The authors wish to thank Vladimir S ¸verák for pointing out the interest of this problem to them. They also thank Franck Sueur for suggesting the analogy with Prandlt's problem.


Giuseppe Cucinotta

Lorenzo Poggini

Niccolò Giaconi

Alberto Cini

Mathieu Gonidec

Matteo Atzori

Enrico Berretti

Alessandro Lavacchi

Maria Fittipaldi

Aleksandr I Chumakov

Rudolf Rüffer

Patrick Rosa

Matteo Mannini

Space charge-limited current

Keywords: Spin crossover, transport measurements, hybrid device, FIB-STEM, synchrotron Mössbauer spectroscopy, molecular magnetism, molecular spintronics

INTRODUCTION

Nowadays molecule-based materials represent a good alternative to conventional inorganic semiconductor materials for the development of innovative devices as a consequence of their rich tunability of the molecular properties. [1][START_REF] Xin | Concepts in the Design and Engineering of Single-Molecule Electronic Devices[END_REF][START_REF] Vilan | Large-Area, Ensemble Molecular Electronics: Motivation and Challenges[END_REF] Moreover, the processability and the light weight of organic materials permit the production of flexible devices, [START_REF] Kim | Reversible Switching Phenomenon in Diarylethene Molecular Devices with Reduced Graphene Oxide Electrodes on Flexible Substrates[END_REF][START_REF] Bhimanapati | Recent Advances in Two-Dimensional Materials beyond Graphene[END_REF] a key aspect for next generation devices. Spin crossover (SCO) complexes, are molecular compounds able to reversibly switch their physical properties upon application of external stimuli (temperature, light-irradiation, applied magnetic and electric fields, pressure, etc.). This is associated to a switch of the spin state between two magnetic states (low-spin, LS, and high-spin, HS) of a coordinated metal ion. These inorganic complexes have been proposed as active materials in functional devices with reversible magnetic and electrical response. [START_REF] Molnár | Spin Crossover Nanomaterials: From Fundamental Concepts to Devices[END_REF][START_REF] Bartual-Murgui | Spin-Crossover Metal-Organic Frameworks: Promising Materials for Designing Gas Sensors[END_REF][START_REF] Linares | Pressure and Temperature Spin Crossover Sensors with Optical Detection[END_REF][START_REF] Ruiz | Charge Transport Properties of Spin Crossover Systems[END_REF][START_REF] Lefter | Current Switching Coupled to Molecular Spin-States in Large-Area Junctions[END_REF][START_REF] Kumar | Emerging Trends in Spin Crossover (SCO) Based Functional Materials and Devices[END_REF][START_REF] Poggini | Vertical Tunnel Junction Embedding a Spin Crossover Molecular Film[END_REF][START_REF] Poggini | Temperature-Induced Transport Changes in Molecular Junctions Based a Spin Crossover Complex[END_REF][START_REF] Ruben | Sublimable Spin Crossover Complexes: From Spin-State Switching to Molecular Devices[END_REF] To the best of our knowledge, only a few reported studies deal with the integration of SCO materials in electrical and electromechanical devices, [START_REF] Molnár | Spin Crossover Nanomaterials: From Fundamental Concepts to Devices[END_REF][START_REF] Lefter | Charge Transport and Electrical Properties of Spin Crossover Materials: Towards Nanoelectronic and Spintronic Devices[END_REF] since the preparation of high-quality thin films and electronically optimized hybrid architectures is critical. Most studies have been focused on rather thick films of highly insulating compounds that may operate in the hopping regime in order to limit problems associated to the device preparation, such us percolation and short-circuits issues. [START_REF] Lefter | Current Switching Coupled to Molecular Spin-States in Large-Area Junctions[END_REF] However, entering in the domain of spintronics, interfacial phenomena play a crucial role. In this context, it is important to increase the general knowledge on devices based on thinner molecular layers. [START_REF] Cinchetti | Activating the Molecular Spinterface[END_REF] Some of us have previously studied [START_REF] Poggini | Vertical Tunnel Junction Embedding a Spin Crossover Molecular Film[END_REF][START_REF] Poggini | Temperature-Induced Transport Changes in Molecular Junctions Based a Spin Crossover Complex[END_REF] the transport properties of vertical junctions incorporating very thin films of SCO complexes formulated as [Fe(HB(trz)3)2] (HB(trz)3 = tris(1H-1,2,4-triazol-1-yl)borohydride) and [Fe(H2B(pz)2)2(phen)] (H2B(pz)2 = bis(pyrazol-1-yl)borohydride, phen = 1,10-phenanthroline), demonstrating that different transport mechanisms can occur. This has been done by the use of a vertical junction architecture, [START_REF] Poggini | Vertical Tunnel Junction Embedding a Spin Crossover Molecular Film[END_REF][START_REF] Poggini | Temperature-Induced Transport Changes in Molecular Junctions Based a Spin Crossover Complex[END_REF] based on a soft eutectic GaIn liquid top electrode (EGaIn) [START_REF] Chiechi | Eutectic Gallium-Indium (EGaIn): A Moldable Liquid Metal for Electrical Characterization of Self-Assembled Monolayers[END_REF] allowing a gentle contact with ultra-thin films. In this context, we have also recently shown that the ferrous complex [Fe(qnal)2] (Hqnal = quinoline-naphtaldehyde) is an ideal target toward the integration of molecular spin crossover in vertical devices, since it can be deposited as very smooth high-quality ultra-thin films by thermal sublimation while maintaining its bulk properties with a slight shift in the spin crossover temperature T1/2 (ca. 10 K) observed at ca. 220 K. [START_REF] Atzori | Thermal and Light-Induced Spin Transition in a Nanometric Film of a New High-Vacuum Processable Spin Crossover Complex[END_REF] In this paper, we report the development of an Ag//[Fe(qnal)2]//LiF//Au multilayer device to study the mechanism which determine the current flowing across these junctions as a function of the temperature, with the support of gas-phase DFT calculations, electrochemistry, magnetization measurements, synchrotron Mössbauer source (SMS) spectroscopy and focused ion beam scanning transmission electron microscopy (FIB-STEM). LiF was used as an intercalation layer between the molecular film and the top electrode to prevent unwanted short-circuits across the molecular layer and to modulate the work function of the metallic electrodes. [START_REF] Prada | Work Function Changes Induced by Deposition of Ultrathin Dielectric Films on Metals: A Theoretical Analysis[END_REF][START_REF] Braun | Energy-Level Alignment at Organic/Metal and Organic/Organic Interfaces[END_REF] We show that in this device, there is a change in the temperature dependence of current density at the temperature at which the switching between HS and LS states occurs. More precisely, we show that the system is in a low conductance state, with no evident temperature dependence when the compound is in the LS state. When it switches to the HS state, the device then switches to a high conductance state characterized by the presence of thermal activation mechanisms at higher temperature. In addition, the analysis of the current density dependence on applied voltage reveals that the electric transport is described by a transition from space charge limited current (SCLC) with shallow traps [START_REF] Lampert | Volume-Controlled Current Injection in Insulators[END_REF] at low temperature to SCLC with an exponential trap distribution 21 at high temperature.

RESULTS AND DISCUSSION

[Fe(qnal)2] were synthesized following the literature procedures [START_REF] Tisato | Synthesis and Characterization of Technetium(V) and Rhenium(V) Oxo-Complexes with Schiff-Base Ligands Containing the ONN Donor-Atom Set. Molecular Structure of Trans-Dichloro-Oxo[1-(8′-Quinolyliminomethyl)-2-Naphtholato-NN′O] Technetium(V)[END_REF][START_REF] Kuroda-Sowa | Abrupt Spin Transitions and LIESST Effects Observed in FeII Spin-Crossover Complexes with Extended π-Conjugated Schiff-Base Ligands Having N4O2 Donor Sets[END_REF] and the details are reported in Supporting Information and in Methods part (see Synthesis Section, Figure S1 andS2). [Fe(qnal)2] shows a molecular structure in which a hexacoordinated Fe(II) ion features a slightly distorted octahedral coordination geometry (insert in Figure 1). Thanks to four Nitrogen atoms and two Oxygen donor atoms, the qnal ligand field provides an appropriate strength to stabilize the diamagnetic LS state at low temperature (below 150 K), and the paramagnetic HS state at high temperature (above 250 K). [START_REF] Atzori | Thermal and Light-Induced Spin Transition in a Nanometric Film of a New High-Vacuum Processable Spin Crossover Complex[END_REF][START_REF] Phan | A Simple Approach for Predicting the Spin State of Homoleptic Fe(II) Tris-Diimine Complexes[END_REF] The low temperature phase is characterized by an Fe II ion featuring an octahedral coordination geometry which is only slightly distorted by the coordination with the ligand, with average MN(O) distances of ca. 1.94 Å, while in the high temperature phase, the coordination octahedron is highly distorted and associated to longer MN(O) distances, on average ca. 2.11 Å. This is associated to the partial promotion of d-electrons located in bonding/nonbonding orbitals (t2g) to those having an antibonding character (eg) when passing from the low to the high temperature phase. [START_REF] Atzori | Functional Molecular Materials An Introductory Textbook[END_REF] Consequently, the total volume of the unit cell reversibly changes from 1568 Å 3 (low temperature) to 1619 Å 3 (high temperature) without loss of crystallinity.Bulk magnetic measurements of [Fe(qnal)2] (see Figure 1) show an abrupt SCO behavior with a T1/2 (225 + 1 K) [START_REF] Atzori | Thermal and Light-Induced Spin Transition in a Nanometric Film of a New High-Vacuum Processable Spin Crossover Complex[END_REF] close to that reported in the seminal work of Kuroda-Sowa and coworkers (218 K). [START_REF] Kuroda-Sowa | Abrupt Spin Transitions and LIESST Effects Observed in FeII Spin-Crossover Complexes with Extended π-Conjugated Schiff-Base Ligands Having N4O2 Donor Sets[END_REF] Magnetic measurements on a microcrystalline powder of the same complex isotopically enriched in [START_REF] Rassolov | 6-31G * Basis Set for Atoms K through Zn[END_REF] Fe, [ 57 Fe(qnal)2] relies as well in an abrupt transition at 225 + 1 K (see Figure S2). This thermal behavior has been confirmed by the analysis of standard transmission Mössbauer spectra of bulk [ 57 Fe(qnal)2], as reported in Figure 1 (green dots), pointing out a T1/2 = 225 + 4 K. The complete series of the Mössbauer spectra as a function of temperature is reported in Figure S3.

Earlier reports evidenced that [Fe(qnal)2] exhibits the possibility to induce the SCO through lightirradiation (Light-Induced Excited Spin State Trapping, LIESST) at cryogenic temperature ( = 531 nm) in bulk [START_REF] Kuroda-Sowa | Abrupt Spin Transitions and LIESST Effects Observed in FeII Spin-Crossover Complexes with Extended π-Conjugated Schiff-Base Ligands Having N4O2 Donor Sets[END_REF] as well as once it is deposited as a thin film. [START_REF] Atzori | Thermal and Light-Induced Spin Transition in a Nanometric Film of a New High-Vacuum Processable Spin Crossover Complex[END_REF] It was also shown that after a thermal treatment at ca. 460 K at ambient pressure and under high vacuum condition, a release of the crystallization CH2Cl2 molecule leads to the formation of the unsolvated form [Fe(qnal)2], which is accompanied with a modification of the T1/2. [START_REF] Atzori | Thermal and Light-Induced Spin Transition in a Nanometric Film of a New High-Vacuum Processable Spin Crossover Complex[END_REF][START_REF] Kuroda-Sowa | Abrupt Spin Transitions and LIESST Effects Observed in FeII Spin-Crossover Complexes with Extended π-Conjugated Schiff-Base Ligands Having N4O2 Donor Sets[END_REF] SMS spectroscopy study was performed initially on a thick film obtained by dropcasting a concentrated solution of [ 57 Fe(qnal)2] in CH2Cl2 and subsequently on a sublimated film. SMS spectra of the [ 57 Fe(qnal)2] dropcast sample, whose average thickness, evaluated with the SMS fitting, was 360 ± 20 nm, are reported in Figure 2a, evidencing that a complete spin conversion is encountered, as in the bulk (Figure S3). At 290 K, a HS state characterized by an isomer shift 𝜹 = 0.890 + 0.003 mm/s and a quadrupole splitting ∆𝑬 𝑸 = 2.496 + 0.006 mm/s is present, while at 3.0 K a LS state with 𝜹 = 0.381 + 0.001 mm/s and ∆𝑬 𝑸 = 1.198 + 0.001 mm/s is found. The hyperfine values of the HS and LS states are comparable with those found using standard Mössbauer Spectroscopy for [ 57 Fe(qnal)2] in the bulk phase (see Table S1).

Moving to the sublimated 100 nm thick film (Figure 2b), it can be noticed that at 3.0 K there is still a high contribution of HS, exemplified by the persistence in the spectrum of a third line above 2 mm/s. Similar hyperfine values are found for the HS and LS states, although characterized by different Gaussian distributions (see Table S1). Moreover, as one can notice from the fitting of the temperature trend (see Figure S4) the SCO efficiency is of the order of 20 %. A reduction of the SCO efficiency when molecules are deposited on a substrate is not unexpected and can be associated to the interaction with the substrate as well as to a different organization of the sublimated molecules with respect to the bulk phase. [START_REF] Poggini | Surface Effects on a Photochromic Spin-Crossover Iron( Ii ) Molecular Switch Adsorbed on Highly Oriented Pyrolytic Graphite[END_REF][START_REF] Gopakumar | Spin-Crossover Complex on Au(111): Structural and Electronic Differences between Mono-and Multilayers[END_REF][START_REF] Ossinger | Vacuum-Evaporable Spin-Crossover Complexes in Direct Contact with a Solid Surface: Bismuth versus Gold[END_REF][START_REF] Atzori | Thermal and Light-Induced Spin Transition in a Nanometric Film of a New High-Vacuum Processable Spin Crossover Complex[END_REF][START_REF] Ossinger | Spin Transition of an Iron(II) Organoborate Complex in Different Polymorphs and in Vacuum-Deposited Thin Films: Influence of Cooperativity[END_REF][START_REF] Poggini | Room Temperature Control of Spin States in a Thin Film of a Photochromic Iron(Ii) Complex[END_REF] A similar efficiency is encountered after LIESST with  = 531 nm (for 10 minutes with a total power 4 mW, Figures 2c-d) where after the light irradiation at 6.0 K the system reaches a HS fraction of 90 + 6 % (starting from an HS fraction of 70 + 4 % before irradiation) suggesting that the same fraction of molecules that can be converted with temperature can be also switched optically at low temperature. We have previously observed a similar behavior on another SCO compound on the same substrate. [START_REF] Cini | Synchrotron-Based Mössbauer Spectroscopy Characterization of Sublimated Spin Crossover Molecules[END_REF] In order to have a complete picture of the expected functionality of [Fe(qnal)2] in a device, and before assembling a vertical architecture embedding it, we performed gas-phase DFT calculations (see Methods for details) for [Fe(qnal)2] in both spin states and estimated the HOMO and LUMO orbital energies using Koopman´s theorem. [START_REF] Koopmans | Über Die Zuordnung von Wellenfunktionen Und Eigenwerten Zu Den Einzelnen Elektronen Eines Atoms[END_REF] As mentioned earlier, charge transport through thin dielectric films, in general, can occur via a variety of mechanisms. It follows, as we have shown recently, that varying the nature of the SCO compounds in vertical devices can lead to significant changes in the temperature dependent characteristics, as some compounds present clear tunnellike behaviors while others exhibit thermally activated transport. In order to properly describe the occurring processes, it is important to get reasonable estimates of the energy of the molecular orbitals involved in charge transport (in both HS and LS states). Our DFT analysis as expected for Fe(II) complexes, [START_REF] Nijhuis | Mechanism of Rectification in Tunneling Junctions Based on Molecules with Asymmetric Potential Drops[END_REF] evidences that the HOMO orbitals of both states (-4.66 and -4.49 eV, for the LS and the HS states, respectively) are much closer in energy to the tabulated work functions of the Ag (-4.26 eV) and LiFAu (-4.37 eV) electrodes [START_REF] Prada | Work Function Changes Induced by Deposition of Ultrathin Dielectric Films on Metals: A Theoretical Analysis[END_REF][START_REF] He | Affecting an Ultra-High Work Function of Silver[END_REF] (see Figure 3a) than the respective LUMO orbitals (-3.24 and -3.19 eV, respectively). This points out that charge transport in [Fe(qnal)2] is likely to be dominated by hole transport. It is noteworthy that the estimated HOMO energies for [Fe(qnal)2]LS and [Fe(qnal)2]HS, are quite high values when compared with other Fe(II) compounds, [START_REF] Poggini | Vertical Tunnel Junction Embedding a Spin Crossover Molecular Film[END_REF][START_REF] Poggini | Temperature-Induced Transport Changes in Molecular Junctions Based a Spin Crossover Complex[END_REF] and suggest that [Fe(qnal)2] could better act as a conducting layer than other systems previously used to assemble hybrid devices. [START_REF] Poggini | Vertical Tunnel Junction Embedding a Spin Crossover Molecular Film[END_REF][START_REF] Poggini | Temperature-Induced Transport Changes in Molecular Junctions Based a Spin Crossover Complex[END_REF] To ascertain the accuracy of energy estimations, we performed in parallel room-temperature solution electrochemistry experiments (see Figure S5) to get an independent estimation of the HOMO orbital energy of the HS state (that should be populated almost quantitatively at room temperature). The Fe(II)/Fe(III) redox couple was therefore measured to be E1/2 = -0.62 vs. Fc/Fc + in dichloromethane containing 0.1M nBu4PF6 as supporting electrolyte. Assuming that EHOMO ≈ ENHE + E1/2vs.NHE, we extracted a value of -4.52 eV for the HOMO of [Fe(qnal)2]HS which is in excellent agreement with the values calculated by DFT.

As bottom electrodes for the vertical device, we used Ag template-stripped patterned layers which provide high-quality metal-films with a very low roughness. [START_REF] Weiss | Si/SiO2-Templated Formation of Ultraflat Metal Surfaces on Glass, Polymer, and Solder Supports: Their Use as Substrates for Self-Assembled Monolayers[END_REF] This was crucial to limit short-circuits problems that are also often associated to the roughness of the metal films and the inhomogeneities in the molecular film. However, in this case, the latter has been excluded because of the high-quality ultra-thin films of [Fe(qnal)2], as estimated by the AFM analysis (Figure S6) of the deposited material. This analysis evidenced a nice defect-free SCO deposit excluding the presence of pin-holes on the surface and Vollmer-Weber growth (figure S6c) and an RMS roughness of about. 1.1 + 0.1 nm over an area of 23 m 2 . The thickness of the "active" layer of [Fe(qnal)2] deposited on top of the Ag electrodes was about 50 nm (effective thickness equal to 47 nm estimated by an AFM test, see Methods). The stacked devices were completed, as described in the Methods section and reported in Figure S7, with the deposition of a ca. 35 nm thick Au crossbar. To avoid gold percolation through the molecular film that could lead to short-circuits in the final device, a 10nm LiF intercalation layer was evaporated before gold deposition.

One of the devices used for the transport measurements (see below) was a posteriori characterized morphologically by STEM imaging on a lamella, extracted from a device surface using Focused Ion Beam methods. The nature of this procedure enabled the stratigraphic study of this ultra-thin stacks of heterogeneous materials, allowing the precise thickness determination of the single layers and excluding significative interdiffusion among them. In Figure 3b a STEM Dark Field image of the device section is reported. Layer thicknesses have been determined as described in Methods. Electric transport properties of the hybrid devices have been studied connecting the Ag and Au stripes and measuring the current flowing as a function of applied voltages by varying the temperature between 135 and 310 K. Considering the rather good energy alignment between reference values for the work functions of Ag and LiF/Au, [START_REF] Prada | Work Function Changes Induced by Deposition of Ultrathin Dielectric Films on Metals: A Theoretical Analysis[END_REF][START_REF] He | Affecting an Ultra-High Work Function of Silver[END_REF] and the estimated HOMO orbital energies for both [Fe(qnal)2]HS and [Fe(qnal)2]LS (see Figure 3a), the collected data have been analyzed according to SCLC model. [START_REF] Rose | Space-Charge-Limited Currents in Solids[END_REF] SCLC usually occurs when charges are injected from a metallic electrode to a dielectric such as an insulator or a wide-gap semiconductor: at the interface a space charge region arises which controls the current through the medium. For low voltages and low charge injection, electric current is mainly due to free charges present in the dielectric and follows an ohmic behaviour [START_REF] Lampert | Volume-Controlled Current Injection in Insulators[END_REF] :

𝐽 = 𝜎 Ω 𝑉 𝑑 = 𝑒𝑛 0 𝜇 𝑉 𝑑 (1)
being e the electric charge, n0 the free electron density, μ the electron mobility and d the thickness of the dielectric. When the density of injected charges starts becoming comparable to n0, SCLC injection takes place and, for ideal dielectrics, the current density is described [START_REF] Mott | Electronic Processes in Ionic Crystals[END_REF] by 𝐽 ∝ 𝑉 2 𝑑 3 ⁄ . Actually, real dielectrics are characterized by the presence of lattice defects, impurities, etc. that can give rise to the presence of electron traps, i.e. states inside the energy gap that can capture electrons by removing them from the conduction band and so reducing the current flow.

Taking into account also the presence of electron traps, the relation linking J on voltage depends on the energy distribution of traps. [START_REF] Lampert | Volume-Controlled Current Injection in Insulators[END_REF] In particular we consider the cases of shallow traps and of exponentially distributed traps. In the former, the energy of the traps is close to the edge of the conduction band (Ec -Et << kBT) and J is given by:

𝐽 = 𝜎 𝑆𝑇 𝑉 2 𝑑 3 = 9 8 𝜇𝜀𝜗 𝑉 2 𝑑 3 , 𝜗 = 𝑁 𝑐 𝑁 𝑡 𝑒𝑥𝑝 (- 𝐸 𝑐 -𝐸 𝑡 𝑘 𝐵 𝑇 ) (2)
with  representing the fraction of free electrons obtained as the ratio between the number of states at the bottom of conduction band, Nc, and the number of trap states, Nt.

If the traps present an exponential energy distribution N t (E)=𝑁 𝑡0 exp (-

𝐸 𝑐 -E k B T t
) the current density is described by the equation: 37

𝐽 = 𝜎 𝐸 𝑉 𝑙+1 𝑑 2𝑙+1 = 𝑒𝜇𝑁 𝑐 ( 𝜀 𝑒𝑁 𝑡0 𝑇 𝑡 ) 𝑙 𝑉 𝑙+1 𝑑 2𝑙+1 , 𝑙 = 𝑇 𝑡 𝑇 > 1 ( 3
)
where the trap distribution temperature Tt is a parameter related to the energy width of the exponential trap distribution kBTt. b )

The J-V curves collected at different temperatures in the range 135 -310 K for one of the devices realized are reported in Figure 4a. All the J-V characteristics, as expected in presence of SCLC, show an ohmic behavior J ~ V at low voltages and a super-linear dependence J ~ V α (α ≥ 2) at high voltages. Linear fits of ln(J)-ln(V) curves in the ohmic region, constraining slope to 1 (see Figure S8), allowed to obtain from the intercept the values of  that, once reported as a function of 1/T (see Figure 4b), revealed the presence of two distinct regimes characterized by a different dependence of  on temperature (labelled I and III in Figure 4b) separated by a transition region (II). In the first range (I, T ≤ 175 K) no temperature dependence is present, indicating that the measured current is due to free-charges present in the material with no contribution of thermally generated free charges. On the contrary, an Arrhenius type activation mechanism is observed in range III (T  230 K) the fit of which provides an activation barrier of  = (304  3) meV. Referring to Equation (1), the temperature trend of  was ascribed to n0, 𝜎(𝑇) ∝ 𝑛 0 (𝑇) = 𝑛 ̅ 0 𝑒𝑥𝑝 (-

𝜙 𝑘 𝐵 𝑇 ),
and  was consequently considered as the activation barrier for thermally generated free-charges.

Furthermore, given the good agreement between the temperature ranges I, II, III characterizing the (T) behavior with the LS-HS temperature dependence of [Fe(qnal)2] (see Figure 1), we link the transport properties observed in interval I and III to the LS and HS states of [Fe(qnal)2]

respectively. The results coming from the analysis of the currents measured at higher voltages revealed also different trends occurring in the temperature intervals I, II and III. For temperatures lower than 175 K the current showed a quadratic dependence on voltage (see Figure S9) in accordance with a presence of shallow traps (Equation 2) while in the temperature ranges II and III a more than quadratic dependence is observed, which is consistent with the presence of an exponential trap distribution. In particular, from Equation 3 it follows that ln(J)=ln ( σ E d 2l+1 ) +(l+1) ln(V) so the slope of a linear fit in a log-log plot (see Figure S10) allowed the determination of the width of the exponential distribution kBTt. The presence of two different energy distributions of traps (shallow and exponential) can be explained as due to the different packing of the LS and HS states for [Fe(qnal)2], and the ensuing different overlap between the aromatic moieties of the ligand in the molecules which are probably crucially associated to the transport mechanism. [START_REF] Kuroda-Sowa | Abrupt Spin Transitions and LIESST Effects Observed in FeII Spin-Crossover Complexes with Extended π-Conjugated Schiff-Base Ligands Having N4O2 Donor Sets[END_REF][START_REF] Ossinger | Spin Transition of an Iron(II) Organoborate Complex in Different Polymorphs and in Vacuum-Deposited Thin Films: Influence of Cooperativity[END_REF] In Figure 5 are reported the values of the trap distribution temperature Tt obtained from the analysis of J-V characteristics in II and III as a function of temperature. For temperature range I, since no trap distribution is involved in transport mechanism, no temperature trap distribution is defined (see Equation 2) so a fictious values Tt = T was considered imposing to Equation 3 a quadratic voltage dependence to resemble that of Equation 2 Magnetization data were corrected for the Teflon TM sample holder and for the diamagnetic contribution as deduced by using Pascal's constants table. [START_REF] Bain | Diamagnetic Corrections and Pascal's Constants[END_REF] Standard transmission Mössbauer characterization. Preliminary Mössbauer measurements on a powder sample of [ 57 Fe(qnal)2] were performed by means of a standard Mössbauer setup in transmission geometry. Mössbauer spectra were collected by means of a Kr-CO2 proportional counter, Fast TM electronics for gamma ray spectroscopy and a Wissel TM spectrometer, which was run in sinusoidal acceleration mode (vmax = 2.7 mm/s or 4.0 mm/s) and calibrated by using a standard metal iron foil. Measurements were carried out between 80 K and 290 K by using a He based Oxford TM flux cryogenic system. The γ-ray source was a 25-mCi [START_REF] Rassolov | 6-31G * Basis Set for Atoms K through Zn[END_REF] Co in rhodium matrix with Lamb-Mössbauer factor f = 0.66, as measured by applying the method described previously. [START_REF] Spina | A Straightforward Experimental Method to Evaluate the Lamb-Mössbauer Factor of a 57Co/Rh Source[END_REF] The intensity of the radiation on the sample, having a focal spot ~ 1 cm 2 , is of the order of 10 4 photons per second. The sensitivity of the setup is limited to bulk samples. [START_REF] Gütlich | Mossbauer Spectroscopy and Transition Metal Chemistry : Fundamentals and Application[END_REF] and monochromatization is achieved via a multi-step optics including the nuclear resonant monochromator obtained with a 57 FeBO3 single crystal. 43,[START_REF] Potapkin | The 57 Fe Synchrotron Mössbauer Source at the ESRF[END_REF] Full width at half maximum (FWHM) of the 57 Fe-resonant line was set at a value approximately three times larger than for a radioactive source (FWHM mean value ~ 0.34 mm/s), obtaining an intensity of about 1.5×10 4 photons per second. The spot size was ca. 18 μm in both dimensions. The radiation is fully recoilless (it does not contain any background), implying f ~ 1. The Mössbauer spectra were recorded by collecting the radiation reflected by the surface in a grazing incidence geometry (about 0.1 degree). This feature, together with the extreme focusing of the beam, provides a thousand-fold amplification factor in the effective thickness (a dimensionless parameter that takes into account the number of Mössbauer active nuclei encountered by the radiation along its path in the sample) with respect to a standard experiment performed at normal or close to normal incidence. Further details can be found in two recent papers in which some of us exploited SMS for the characterization of molecular ultra-thin layer deposits. [START_REF] Cini | Synchrotron-Based Mössbauer Spectroscopy Characterization of Sublimated Spin Crossover Molecules[END_REF][START_REF] Cini | Mössbauer Spectroscopy of a Monolayer of Single Molecule Magnets[END_REF] Samples for this characterization were a 100 nm film of [ 57 Fe(qnal)2] grown on a thick polycrystalline gold film deposited on silicon with a 5 nm Ti adhesion layer (SSENS, Enschede, NL), and as a check against the bulk conventional Mössbauer spectra, a dropcast film of the compound on the same substrate obtained from a dichloromethane solution. The thickness of this sample was evaluated from the effective thickness obtained from the fit of the SMS spectra and comparing it with the one obtained for the 100 nm film (the thickness of the film was determined by AFM). In particular, a ratio of effective thicknesses equal to 3.6 was found, therefore pointing to an average thickness of 320 ± 20 nm for the dropcast. Spectra were measured at different temperatures in the range 3.0 K -290 K using the superconducting Heexchange gas cryo-magnetic system. For the Light-Induced Excited Spin State Trapping (LIESST) measurements, the sample holder was modified in order to accommodate a couple of InGaAsP LEDs (Roithner LaserTechnik GmbH, nominal optical power = 2 mW with 545 nm nominal wavelength, decreasing to 531 nm at 10 K).

Mössbauer data analysis. The SMS spectra were interpreted by means of a fitting procedure based on the evaluation of the transmission integral function. [START_REF] Cini | Synchrotron-Based Mössbauer Spectroscopy Characterization of Sublimated Spin Crossover Molecules[END_REF][START_REF] Cini | Mössbauer Spectroscopy of a Monolayer of Single Molecule Magnets[END_REF] At each temperature, the absorption cross-section of the sample was considered as the superposition of two contributions associated with an HS and a LS state, characterized by different values of the hyperfine parameters (isomer shift 𝜹 and quadrupole splitting ∆𝑬 𝑸 ). Moreover, Gaussian distributions of the LS and HS quadrupole splitting were supposed. A proper fitting of the dropcast and sublimated sample required to include impurity species. Their contribution to the spectra is of the order of 10 % and 20 % for the dropcast and 100 nm thick sample, respectively. The same fitting procedure was used to interpret the spectra of the bulk sample measured with standard Mössbauer spectroscopy, except for replacing in the transmission integral function the line shape of the source and the sample effective thickness with those of the standard Mössbauer setup. [START_REF] Cini | Synchrotron-Based Mössbauer Spectroscopy Characterization of Sublimated Spin Crossover Molecules[END_REF] The core of the fitting procedure is represented by the LMDIF routine of the MinPack library (www.netlib.org). Moreover, the standard deviations on the fit parameters are evaluated from the diagonal terms of the variancecovariance matrix, which is a fit procedure output.

Electrochemical analysis. The electrochemical experiments were carried out using an Autolab PGSTAT101 potentiostat. The experiments were performed under an inert atmosphere of argon using a three-electrode configuration in a single compartment cell. The electrodes consisted of a platinum disc as working electrode, a silver wire coated with silver chloride as reference electrode, and a platinum wire as counter electrode. A 0.1 M solution of nBu4PF6 (Fluka, electrochemical grade) in dichloromethane (Honeywell Riedel-de Haën, purissimum) was used as supporting electrolyte. Standard cyclic voltammetry experiments (Figure S5) and square wave voltammetry experiments (Figure S5) were performed at 25, 50, 100, and 200 mV s -1 , and were done both with and without ferrocene as an internal reference. The potentials are quoted versus the Fc/Fc + redox couple. For the estimation of the ionization potentials, we used the literature value E1/2(Fc/Fc + ) = 0.640 V vs. NHE.

Theoretical modelling. The orbital energies were estimated from density functional theory calculations with the Gaussian09 46 program using the restricted and unrestricted B3LYP functionals [START_REF] Becke | Density-Functional Exchange-Energy Approximation with Correct Asymptotic Behavior[END_REF][START_REF] Lee | Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density[END_REF][START_REF] Vosko | Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis[END_REF][START_REF] Stephens | Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields[END_REF] for the low spin and high spin complexes, respectively. The geometry of the complexes were first pre-optimized with the LanL2DZ basis set, [START_REF] Dunning | Gaussian Basis Sets for Molecular Calculations[END_REF][START_REF] Hay | Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for the Transition Metal Atoms Sc to Hg[END_REF][START_REF] Wadt | Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for Main Group Elements Na to Bi[END_REF][START_REF] Hay | Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for K to Au Including the Outermost Core Orbitals[END_REF] followed by a geometry optimization with the 6-31G(d,p) basis set, [START_REF] Binning | Compact Contracted Basis Sets for Third-Row Atoms: Ga-Kr[END_REF][START_REF] Blaudeau | Extension of Gaussian-2 (G2) Theory to Molecules Containing Third-Row Atoms K and Ca[END_REF][START_REF] Rassolov | 6-31G * Basis Set for Atoms K through Zn[END_REF][START_REF] Rassolov | 6-31G* Basis Set for Third-Row Atoms[END_REF] and finally single point energy calculations were performed with the 6-31++G(d,p) basis set. [START_REF] Frisch | Self-consistent Molecular Orbital Methods 25. Supplementary Functions for Gaussian Basis Sets[END_REF] Device preparation. Ultraflat Ag metal stripes were prepared using a combination of a standard lift-off and template stripping processes. A 300 nm thick layer of LOR-3B lift-off resin

TABLE OF CONTENTS

Figure 1 .

 1 Figure 1. Comparison between bulk SQUID magnetic characterization (red dots), Mössbauer

Figure 2 .

 2 Figure 2. a) Synchrotron Mössbauer source spectra of the dropcast sample of [ 57 Fe(qnal)2] at 3.0

Figure 3 .

 3 Figure 3. a) Electronic configuration of the junctions in open circuit for the Ag and LiF/Au

Figure 4 a

 4 Figure 4 a) log-log plots of J-V characteristics at different temperatures (temperature colour-

𝑇Figure 5 5 K

 55 Figure 5 Trend of the Tt parameter obtained from J-V analysis (black dots) as function of

  photons per second. The sensitivity of the setup is limited to bulk samples.[START_REF] Gütlich | Mossbauer Spectroscopy and Transition Metal Chemistry : Fundamentals and Application[END_REF] Approximately 20.3 mg/cm 2 of [ 57 Fe(qnal)2]•were used for the measurements.

(

  photoresist (S1818, Shipley) was then spin-coated on top of the LOR3B at 4000 rpm for 30 s and

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally. NG and MA synthetized the molecular system and their [START_REF] Rassolov | 6-31G * Basis Set for Atoms K through Zn[END_REF]