A KAM approach to the inviscid limit for the 2D Navier-Stokes equations

In this paper we investigate the inviscid limit $\nu \to 0$ for time-quasi-periodic solutions of the incompressible Navier-Stokes equations on the two-dimensional torus ${\mathbb T}^2$, with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order $O(\nu^2)$ and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems.

In this paper we consider the two dimensional Navier Stokes equations for an incompressible fluid on the two-dimensional torus T 2 , T := R/2πZ, with a small time quasi-periodic forcing term where ε ∈ (0, 1) is a small parameter, ω ∈ R d is a Diophantine d-dimensional vector, ν > 0 is the viscosity parameter, the external force f belongs to C q (T d × T 2 , R 2 ) for some integer q > 0 large enough, U = (U 1 , U 2 ) : R × T 2 → R 2 is the velocity field, and p : R × T 2 → R is the pressure.The main purpose of this paper is to investigate the inviscid limit of the Navier Stokes equation from the perspective of the KAM (Kolmogorov-Arnold-Moser) theory for PDEs, which in a broad sense is the theory of the existence and the stability of periodic, quasi-periodic and almost periodic solutions for Partial Differential Equations.More precisely, we construct quasi-periodic solutions of (1.1) converging to the quasi-periodic solutions of the Euler equation, constructed in Baldi & Montalto [4], as the viscosity parameter ν → 0 and we provide a convergence rate O(ν) which is uniform for all times.As a consequence of our result, we obtain families of initial data for which the corresponding global quasi-periodic solutions of the Navier Stokes equations converge to the ones of the Euler equation with a rate of convergence O(ν), uniformly in time.The main difficulty is that this is a singular perturbation problem, namely there is a small parameter in front of the highest order derivative.To the best of our knowledge, this is both the first result in which one exhibits solutions of the Navier Stokes equations converging globally and uniformly in time to the ones of the Euler equation in the vanishing viscosity limit ν → 0 and the first KAM result in the context of singular limit problems for PDEs.The zero-viscosity limit of the incompressible Navier-Stokes equations in bounded domains is one of the most challenging problems in Fluid Mechanics.The first results for smooth initial data (H s with s ≫ 0 large enough) have been proved by Kato [30], [31], Swann [40], Constantin [15] and Masmoudi [35] in the Euclidean domain R n or in the periodic box T n , n = 2, 3.For instance, it is proved in [35] that, if the initial velocity field u 0 ∈ H s (T n ), s > n/2 + 1, then the corresponding solutions u ν (t, x) of Navier Stokes and u(t, x) of Euler, defined on [0, T ] × T n , satisfiy and for s ′ < s u ν (t) − u(t) H s ′ (νt) It is immediate to notice that the latter estimate holds only on finite time intervals and it is not uniform in time, with the estimate of the difference u ν (t) − u(t) eventually diverging as t → +∞.For n = 2, this kind of result has been proved in low regularity by Chemin [13] and Seis [39], with rates of convergence in L 2 .We also mention same results for non-smooth vorticity.In particular, the inviscid limit of Navier Stokes equation has been addressed in the case of vortex patches in Constantin & Wu [18], [19], Abidi & Danchin [1] and Masmoudi [35], with low Besov type regularity in space.In this results one typically gets a bound only in L 2 of the form u ν (t) − u(t) L 2 (νt) α for some α > 0 .
In the case of non-smooth vorticity, the inviscid limit has been investigated by using a Lagrangian stochastic approach in Constantin, Drivas & Elgindi [16], with initial vorticity ω 0 ∈ L ∞ (T 2 ), and in Ciampa, Crippa & Spirito [14], where the initial vorticity ω 0 ∈ L p (T 2 ), p ∈ (1, +∞).When the domain has an actual boundary, the zero-viscosity limit is closely related to the validity of the Prandtl equation for the formation of boundary layers.For completeness of the exposition, we mention the work of Sammartino & Caflisch [37]- [38] and recent results by Maekawa [34], Constantin, Kukavica & Vicol [17] and Gérard-Varet, Lacave, Nguyen & Rousset [24], with references therein.The inviscid limit has been also investigated in other physical model for complex fluids, see for instance [12] for the 2D incompressible viscoelasticity system.Our approach is different and it is based on KAM (Kolmogorov-Arnold-Moser) and Normal Form methods for Partial Differential Equations.This fields started from the Ninetiees, with the pioneering papers of Bourgain [11], Craig & Wayne [20], Kuksin [32], Wayne [41].We refer to the recent review article [6] for a complete list of references on this topic.In the last years, new techniques have been developed in order to study periodic and quasi-periodic solutions for PDEs arising from fluid dynamics.For the two dimensional water waves equations, we mention Iooss, Plotnikov & Toland [27] for periodic standing waves, [10], [2] for quasi-periodic standing waves and [7], [8], [23] for quasi-periodic traveling wave solutions.We also recall that the challenging problem of constructing quasi-periodic solutions for the three dimensional water waves equations is still open.Partial results have been obtained by Iooss & Plotnikov, who proved existence of symmetric and asymmetric diamond waves (bi-periodic waves stationary in a moving frame) in [28], [29].Very recently, KAM techniques have been successfully applied also for the contour dynamics of vortex patches in active scalar equations.The existence of time quasi-periodic solutions have been proved in Berti, Hassainia & Masmoudi [9] for vortex patches of the Euler equations close to Kirchhoff ellipses, in Hmidi & Roulley [26] for the surface quasi-geostrophic (SQG) equations and in Hassainia, Hmidi & Masmoudi [25] for generalized SQG equations.All the aforementioned results concern 2D Euler equations.The quasi-periodic solutions for the 3D Euler equations with time quasi-periodic external force have been constructed in [4] and also extended in [36] for the Navier-Stokes equations in arbitrary dimension, without dealing with the zero-viscosity limit.The result of the present paper closes also the gap between these two works.
1.1.Main result.We now state precisely our main result.We look for time-quasiperiodic solutions of (1.1), oscillating with time frequency ω.In particular, we look for solutions which are small perturbations of constant velocity fields ζ ∈ R 2 , namely solutions of the form where the new unknown velocity field u : Plugging this ansatz into the equation, one is led to solve with p : According to [4], we shall assume that the forcing term f is odd with respect to (ϕ, x), that is (1. 3) It is convenient to work in the well known vorticity formulation.We define the scalar vorticity v(ϕ, x) as Hence, rescaling the variable v → √ εv and the small parameter ε → ε 2 , the equation (1.2) is equivalent to and (−∆) −1 is the inverse of the Laplacian, namely the Fourier multiplier with symbol ) dx is a prime integral, we shall restrict to the space of zero average in x.Then, the pressure is recovered, once the velocity field is known, by the formula p = ∆ −1 εdivf (ωt, x) − div (u • ∇u) .For any real s ≥ 0, we consider the Sobolev spaces H s = H s (T d+2 ) of real scalar and vector-valued functions of (ϕ, x), defined in (2.1), and the Sobolev space of functions with zero space average, defined by Furthermore, we introduce the subspaces of L 2 of the even and odd functions in (ϕ, x), respectively: (1.5) We first state the result concerning the existence of quasi-periodic solutions of the Euler equation (ν = 0) for most values of the parameters [4].The statement is slightly modified for the purposes of this paper.
Theorem 1.1.(Baldi-Montalto [4]).There exists S := S(d) > 0 such that, for any S ≥ S(d), there exists q := q(S) > 0 such that, for every forcing term f ∈ C q (T d × T 2 , R 2 ) satisfying (1.3), there exists ε 0 := ε 0 (f, S, d) ∈ (0, 1) and C := C(f, S, d) > 0 such that, for every ε ∈ (0, ε 0 ), the following holds.There exists a C 1 map Moreover, there exists a constant a := a(d) ∈ (0, 1) such that sup λ∈R d+2 v(•; λ) S ≤ Cε a and, for any i = 1, . . ., d + 2, sup We now are ready to state the main result of this paper.Rougly speaking, we will prove that for any value of the viscosity parameter ν > 0 and for ε ≪ 1 small enough, independent of the viscosity parameter, the Navier-Stokes equation (1.4) admits a quasi-periodic solution v ν (ϕ, x) for most of the parameters λ = (ω, ζ) such that v ν − v e S = O(ν).This implies that v ν (ωt, x) converges strongly to v e (ωt, x), uniformly in (t, x) ∈ R × T 2 , with a rate of convergence sup (t,x)∈R×T 2 |v ν (ωt, x) − v e (ωt, x)| ν.To the best of our knowledge, this is the first case in which the inviscid limit is uniform in time.We now give the precise statement of our main theorem.
Theorem 1.2.(Singular KAM for 2D Navier-Stokes in the inviscid limit).There exist s := s(d) and µ := µ(d) > 0 such that, for any s ≥ s(d), there exists q := q(s) > 0 such that, for every forcing term satisfying the estimate As a consequence, for any value of the parameter λ ∈ O ε , the quasi-periodic solutions of the Navier Stokes equation v ν converge to the ones of the Euler equation v e in H s 0 (T d+2 ) in the limit ν → 0.
From the latter theorem we shall deduce the following corollary which provides a family of quasi-periodic solutions of Navier-Stokes equation converging to solutions of the Euler equation with rate of convergence O(ν) and uniformly for all times.The result is a direct consequence of the Sobolev embeddings.
Corollary 1.3.(Uniform rate of convergence for the inviscid limit).Assume the same hypotheses of Theorem 1.2 and let s ≥ s 0 large enough, v e ∈ H s+µ 0 Let us make some remarks on the result.1) Vanishing viscosity solutions of the Cauchy problem.The time quasi-periodic solutions in Theorem 1.2 are slight perturbations of constant velocity fields ζ ∈ R 2 with frequency vector ω ∈ R d induced by the perturbative forcing term f (ωt, x).Since they exist only for most values of the parameters (ω, ζ), we obtain equivalently that the Cauchy problem associated with (1.4) (and so of (1.1)) admits a subset of small amplitude initial data of relatively large measure, with elements evolving for all time, in a eventually larger but still bounded neighbourhood in the Sobolev topology, and whose flows exhibit a uniform vanishing viscosity limit to solutions of the Cauchy problem for the Euler equations with same initial data.
2) The role of the forcing term.It is worth to note that the time quasi-periodic external forcing term F (ωt, x) in (1.1) is independent of the viscosity parameter ν > 0. Its presence ensures the existence of the time quasi-periodic Euler solution v e in Theorem 1.1, while the construction of the viscous correction v ν − v e does not rely explicitly on it: if one is able to exhibits time quasi-periodic solutions close to constant velocity fields for the free 2D Euler equation, namely (1.4) with ν = 0 and F ≡ 0, then the ones for the Navier-Stokes equation follow immediately by our scheme.To our knowledge, the only result of existence of time quasi-periodic flows for the free Euler equations on T 2 is given by Crouseilles & Faou [21], where the solutions are searched based on a prescribed stationary shear flow, locally constant around finitely many points, and propagate in time in the orthogonal direction to the shear flow.Due to the nature of their solutions, the non-resonant frequencies are prescribed as well and therefore there are no small divisors issues involved.1.2.Strategy and main ideas of the proof.In order to prove Theorem 1.2, we have to construct a solution of the Navier-Stokes equation (1.4) which is a correction of order O(ν) of the solution v e of the Euler equation (provided in Theorem 1.1 of [4]).Roughly speaking, the difficult point is the following.There are two smallness parameters which are ε, the size of the Euler solution, and ν, the size of the viscosity.If one tries to construct small solutions of the Navier-Stokes equation by using a standard fixed point argument, one immediately notes that a smallness condition of the form εν −1 ≪ 1 is needed and clearly this is not enough to pass to the inviscid limit as ν → 0. The key point is to have a smallness condition on ε which is independent of ν in such a way that one can pass to the limit as the viscosity ν → 0. We can summarize the construction into three main steps: where R : is a pseudo-differential operator of order −1.Note that the linear operator L e is obtained by linearizing the Euler equation at the solution with vorticity v e (ϕ, x).If one tries to implement a naive approach by directly using Neumann series to invert the linear operator L ν , one has to require that εν −1 ≪ 1, which is not enough to pass to the limit as ν → 0. To overcome this issue, we first implement the normal form procedure developed in [5], [4] to reduce to a diagonal, constant coefficients operator the Euler operator L e , generating an unbounded correction to the viscous term −ν∆ of size O(εν).More precisely, for most values of the parameters (ω, ζ) and for ε ≪ 1 small enough and independent of ν, we construct a bounded, invertible transformation Φ : where D ∞ and R ∞,ν have the following properties.D ∞ is a diagonal operator of the form The remainder term R ∞,ν is an unbounded operator of order two and it satisfies an estimate of the form where we denote by B(H s ), the space of bounded linear operators on H s .The estimate (1.9) is the key ingredient to invert the operator L ∞,ν in (1.7) with a smallness condition on ε uniform with respect to the viscosity parameter ν > 0. It is also crucial to exploit the reversibility structure which is a consequence of the fact that the solutions v e (ϕ, x) of the Euler equation are odd with respect to (ϕ, x).This ensures that, for any ℓ ∈ Z d , j ∈ Z 2 \ {0} , the eigenvalues µ ∞ (ℓ, j) of the diagonal operator D ∞ in (1.8) are purely imaginary (namely, the corrections r ∞ j are real).An important consequence is that the diagonal operator D ∞ − ν∆ is invertible and gains two space derivatives with an estimate for its inverse of order O(ν −1 ).Indeed, the eingenvalues of implying that D ∞ − ν∆ is invertible with inverse which gain two space derivatives, namely Thus, on one hand, (D ∞ − ν∆) −1 gains two space derivatives, compensating the loss of two space derivatives of the remainder R ∞,ν .On the other hand, the norm of (D ∞ − ν∆) −1 explodes as ν −1 as ν → 0, but this is compensated by the fact that R ∞,ν is of order O(εν).Therefore, recalling (1.9), one gets a bound Hence, by Neumann series, for ε ≪ 1 small enough and independent of ν, the operator L ∞,ν is invertible and gains two space derivatives, with estimate L −1 ∞,ν (−∆) B(H s ) s ν −1 .By (1.7), we deduce that L ν is invertible as well and satisfies, for ε ≪ 1 and for any ν > 0, First order approximation for the viscosity quasi-periodic solution and fixed point argument.Once we have a good knowledge for properly inverting the operators L ν and L ε , we are ready to construct quasi-periodic solutions of the Navier Stokes equation converging to the Euler solution u e as ν → 0. First, we define an approximate solution v app = v e + νv 1 which solves the equation (1.4) up to order O(ν 2 ).By making a formal expansion with respect to the viscosity parameter ν, we ask v e to solve the equation at the zeroth order O(ν 0 ), namely the Euler equation, whose existence is provided by Theorem 1.1, and v 1 to solve the linear equation at the first order O(ν), that is L e v 1 = ∆v e .This procedure leads to a loss of regularity due to the presence of small divisors, appearing in the inversion of the linearized Euler operator L e in (1.6), which satisfies an estimate of the form L −1 e h s s h s+τ for some τ ≫ 0 large enough.On the other hand, this is not a problem in our scheme since it appears only twice: first, in the construction of the quasi-periodic solution v e , but it has already been dealt in Theorem 1.1; second, in the definition indeed of v 1 .We overcome this issue by requiring v e to be sufficiently regular.The final step to prove Theorem 1.2 is to implement a fixed point argument for constructing solutions of the form v = v e + νv 1 + ψ, where the quasi-periodic correction ψ lies in the ball ψ s ≤ ν.It is crucial here that v e + νv 1 is an approximate solution up to order O(ν 2 ): indeed, the fixed point iteration asks to invert linearized operator at the Euler solution L ν , which has a bound of order O(ν −1 ) (recall (1.11)), and in this way the new term ends up to be of order O(ν) as desired.The good news here is that, at this stage, no small divisors are involved and, consequently, no losses of derivatives, which would have made the fixed point argument not applicable otherwise.2D vs. 3D.It is worth to conclude this introduction by making some comments on the 3D case, that it is not covered by the method developed in this paper.In the present paper, we construct global in time quasi-periodic solutions for the two dimensional Navier-Stokes equations converging uniformly in time to global quasi-periodic solutions of the two-dimensional forced Euler equation.The three dimensional is much harder.The biggest obstacle is that the reversible structure is not enough to deduce that the spectrum of the linearized Euler operator after the KAM reducibility scheme is purely imaginary.Indeed, as in [4], the reduced Euler operator This block matrix could have eigenvalues µ 1 (j), µ 2 (j), µ 3 (j) of the form µ i (j) = iζ • j + εr i (j), i = 1, 2, 3, with real part different from zero, in particular with Re(r i (j)) = 0 for some i = 1, 2, 3.This seems to be an obstruction to get a lower bound like (1.10) with a gain of two space derivatives, which holds uniformly in ε and for any value of the viscosity parameter.More precisely, one gets a lower bound on the eigenvalues of the form It is therefore not clear how to bound the latter quantity by Cν|j| 2 without linking ε and ν which prevent to pass to the limit as ν → 0 (independently of ε).
Outline of the paper.The rest of this paper is organized as follows.In Section 2 we introduce the functional setting and some general lemmata that we will employ in the other sections.In Section 3 we formulate the nonlinear functional F ν in (3.1), whose zeroes correspond to quasi-periodic solutions of the equation (1.4), together with the linearized operators that we have to study.In Section 4 we implement the normal form method on of the linearized Euler and Navier Stokes operators operator L ε , L ν in (1.6): first, we regularize to constant coefficients the highest and lower orders, in Sections 4.1 and 4.2 up to sufficiently smoothing orders; then, in Sections 4.3-4.4we prove the full KAM reducibility scheme.We shall prove that the normal form transformations conjugate the linearized Navier Stokes operator to a diagonal one plus a remainder which is unbounded of order two and has size O(εν).This normal form procedure is uniform w.r. to the viscosity parameter since it requires a smallness condition on ε which is independent of the viscosity ν > 0.Then, in Section 5 we show the invertibility of the operator L ν (and also L e ) that will be used in Section 6 for the construction of the first order approximate solution and in Section 7 for the fixed point argument.Finally, the proof of Theorem 1.2 is provided in Section 8, together with the measure estimates proved in Section 8.1.
ACKNOWLEDGEMENTS.The authors warmly thank Gennaro Ciampa, Nader Masmoudi and Eugene Wayne for many stimulating discussions on the topic and on related results.The work of the author R. M. is supported by the ERC STARTING GRANT "Hamiltonian Dynamics, Normal Forms and Water Waves" (HamDyWWa), project number: 101039762.R. M. is also supported by INDAM-GNFM.The work of the author L.F. is supported by Tamkeen under the NYU Abu Dhabi Research Institute grant CG002.

NORMS AND LINEAR OPERATORS
In this section we collect some general definitions and properties concerning norms and matrix representation of operators which are used in the whole paper.
Notations.In the whole paper, the notation A s,m B means that A ≤ C(s, m)B for some constant C(s, m) > 0 depending on the Sobolev index s and a generic constant m.We always omit to write the dependence on d, which is the number of frequencies, and τ , which is the constant appearing in the non-resonance conditions (see for instance (4.32)).We often write u = even(ϕ, x) if u ∈ X and u = odd(ϕ, x) if u ∈ Y (recall (1.5)).For a given Banach space Z, we recall that B(Z) denotes the space of bounded operators from Z into itself.
2.1.Function spaces.Let a : T d × T 2 → C, a = a(ϕ, x), be a function.then, for s ∈ R, its Sobolev norm a s is defined as where a(ℓ, j) (which are scalars, or vectors, or matrices) are the Fourier coefficients of a(ϕ, x), namely We denote, for E = C n or R n , In the paper we use Sobolev norms for (real or complex, scalar-or vector-or matrixvalued) functions u(ϕ, x; ω, ζ), (ϕ, x) ∈ T d × T 2 , being Lipschitz continuous with respect to the parameters λ := (ω, ζ) ∈ R d+2 .We fix once and for all, and define the weighted Sobolev norms in the following way.
For any N > 0, we define the smoothing operators (Fourier truncation)

(Smoothing).
The smoothing operators Π N , Π ⊥ N satisfy the smoothing estimates ) be a linear operator.Such an operator can be represented as where, for j, j ′ ∈ Z 2 , the matrix element R j ′ j is defined by We also consider smooth ϕ-dependent families of linear operators T d → B(L 2 (T 2 )), ϕ → R(ϕ), which we write in Fourier series with respect to ϕ as According to (2.8), for any ℓ ∈ Z d , the linear operator If the operator R is invariant on the space of functions with zero average in x, we identify R with the matrix Definition 2.4.(Diagonal operators).Let R be a linear operator as in (2.7)-(2.10).
We define D R as the operator defined by For the purpose of the Normal form method for the linearized operator in Section 4, it is convenient to introduce the following norms that take into account the order and the off-diagonal decay of the matrix elements representing any linear operator on L 2 (T d+2 ).
Directly from the latter definition, it follows that m,s ′ .We now state some standard properties of the decay norms that are needed for the reducibility scheme of Section 4.3.If a ∈ H s , s ≥ s 0 , then the multiplication operator M a : u → au satisfies (2.13) m ′ ,s+|m| .(iii) Let s ≥ s 0 and R ∈ OPM 0 s .Then, for any integer n ≥ 1, R n ∈ OPM 0 s and there exist constants C(s 0 ), C(s) > 0, independent of n, such that Then there exists δ(s) ∈ (0, 1) small enough such that, if |R| Lip(γ) −m,s 0 ≤ δ(s), then the map Φ = Id + R is invertible and the inverse satisfies the estimate The proofs of the first two items use similar arguments.We only prove item (ii).We start by assuming that both R and Q do not depend on the parameter λ.
The matrix elements for the composition operator RQ follow the rule Using that ℓ, j − j ′ s s ℓ − k, j − i s + k, i − j ′ s , one gets where We start with estimating (A).By the elementary inequality i m j ′ −m m j ′ − i |m| , the Cauchy-Schwartz inequality and having the series By similar arguments, one gets (B) m |Q| 2 s+|m|,m ′ |R| 2 s 0 ,m and hence the claimed estimate follows by taking the supremum over j ′ ∈ Z 2 in (2.14).If we reintroduce the dependence on the parameter λ, the estimate for the Lipschitz seminorm follows as usual by taking two parameters λ 1 , λ 2 and writing R(λ iii) The claim follows by an induction argument and item (ii).
iv) The claim follows by a Neumann series argument, together with item (iii).
v) The claims are a direct consequence of the definition of the matrix decay norm in Definition 2.5.
We recall the definition of the set of the Diophantine vectors in a bounded, measurable set Λ ⊂ R d+2 .Given γ, τ > 0, we define ) m,s+2τ +1 .Moreover, if R is invariant on the space of zero average functions, also Ψ is invariant on the space of zero average functions.
We also define the projection Π 0 on the space of zero average functions as In particular, for any m, s ≥ 0, We finally mention the elementary properties of the Laplacian operator −∆ and its inverse (−∆) −1 acting on functions with zero average in x: By Definition 2.5, one easily verifies, for any s ≥ 0, (2.21) 2.3.Real and reversible operators.We recall the notation introduced in (1.5), that is, for any function u(ϕ, x), we write u ∈ X when u = even(ϕ, x) and u ∈ Y when u = odd(ϕ, x).
Definition 2.9.(i) We say that a linear operator Φ is reversible if Φ : X → Y and Φ : Y → X.We say that Φ is reversibility preserving if Φ : X → X and Φ : Y → Y .
(ii) We say that an operator Φ : It is convenient to reformulate real and reversibility properties of linear operators in terms of their matrix representations.
Lemma 2.10.A linear operator R is :

THE NONLINEAR FUNCTIONAL AND THE LINEARIZED NAVIER STOKES OPERATOR AT THE EULER SOLUTION
We shall show the existence of solutions of (1.4) by finding zeroes of the nonlinear operator ) with ∇ ⊥ as in (1.4), and, without loss of generality, F = ∇ × f has zero average in space, namely We consider parameters (ω, ζ) in a bounded open set Ω ⊂ R d × R 2 ; we will use such parameters along the proof in order to impose appropriate non resonance conditions.
In this section and Section 4 we assume the following ansatz, which is implied by Theorem 1.1: there exists S ≫ 0 large enough such that v e (•; where a(ϕ, x) is the function defined by with ∇ ⊥ as in (1.4) and R(ϕ) is a pseudo-differential operator of order −1, given by Using that div(∇ ⊥ h) = 0 for any h, the operators a • ∇, R L ν and L e leave invariant the subspace of zero average function, with implying that We always work on the space of zero average functions and we shall preserve this invariance along the whole paper.
The goal of next two sections is to invert the whole linearized Navier Stokes operator L ν obtained by linearizing the nonlinear functional F ν (v) in (3.1) at any quasi-periodic solution v e (ϕ, x)| ϕ=ωt provided by Theorem 1.1 by requiring a smallness condition on ε which is independent of the viscosity parameter ν > 0. This is achieved in two steps.First, we fully reduce to a constant coefficient, diagonal operator the linearized Euler operator L e in (3.3).This is done in Section 4, in the spirit of [4], [5], by combining a reduction to constant coefficients up to an arbitrarily regularizing remainder with a KAM reducibility scheme.We check step by step that this normal form procedure, when applied to the full operator L ν in (3.3), just perturbs the unbounded viscous term −ν∆ by an unbounded pseudo differential operator of order two that "gain smallness", namely it is of size O(νε), see (5.1)-(5.2).In Section 5, we use this normal form procedure in order to infer the invertibility of the operator L ν uniformly with respect to the viscosity parameter, namely by imposing a smallness condition on ε that is independent of ν.The inverse of the Navier Stokes operator is bounded from H s 0 , gains two space derivatives and it has size O(ν −1 ) (see Proposition 5.4), whereas the inverse of the linearized Euler operator L e loses τ derivatives, due to the small divisors (see Proposition 5.5).The invertibility of the linearized Euler operator L e is used to construct the approximate solution in Section 6 and the invertibility of the linearized Navier Stokes operator L ν is used to implement the fixed point argument of Section 7.

NORMAL FORM REDUCTION OF THE OPERATOR L ν
In this section we reduce to a constant coefficients, diagonal operator the operator L ν in (3.3) up to an unbounded remainder of order two which is of size O(εν).First, we deal with the conjugation of the transport operator in Section 4.1, which is the highest order term in the operator L e .In Section 4.2, the lower order terms after the previous conjugation are regularized to constant coefficients up to a remainder of arbitrary smoothing matrix decay and up to an unbounded remainder of order two and size O(εν).Then, in Sections 4.3-4.4we perform the full KAM reducibility for the regularized version of the operator L e .In particular, in Section 4.3 the n-th iterative step of the reduction is performed and in Section 4.4 the convergence of the scheme is proved via Nash-Moser estimates to overcome the loss of derivatives coming from the small divisors.The linearized Navier Stokes operator is then reduced to a diagonal operator plus an unbounded operator of order two and size O(εν) in (5.1)-(5.2).This is the starting point for its inversion in Section 5 From now on, the parameters γ ∈ (0, 1) and τ > 0, characterizing the set Λ(γ, τ ) in (2.15) of the Diophantine frequencies in a given measurable set Λ, are considered as fixed and τ is chosen in (8.2) and γ at the end of Section 8, see (8.6).Therefore we omit to recall them each time.Moreover, from now on, we denote by DC(γ, τ ), the set of Diophantine frequencies in Ω ε , where the set Ω ε is provided in Theorem 1.1, namely Λ(γ, τ ) with Λ = Ω ε .We repeat the definition for clarity of the reader: 4.1.Reduction of the highest order term.First, we state the proposition that allows to reduce to constant coefficients the highest order operator where we recall, by (3.4), that Π 0 a = 0 and div(a) = 0.The result has been proved in Proposition 4.1 in [4] (see also [22] for a more general result of this kind).We restate it with clear adaptation to our case and we refer to the former for the proof.one gets the conjugation Furthermore, α, α are odd(ϕ, x) and the maps A, A −1 are reversibility preserving, satisfying the estimates We remark that the assumptions of Proposition 4.1 are satisfied by the ansatz (3.2) and by the choice of the parameter γ > 0 at the end of Section 8 in (8.6).In particular, the smallness condition (4.2) becomes εγ −1 = ε 1− a 2 ≪ 1, which is clearly satisfied since a ∈ (0, 1) and for ε sufficiently small.
In order to study the conjugation of the operator L ν : 3) under the transformation A, we need the following auxiliary Lemma.Lemma 4.2.Let S > s 0 + σ + 2 (where σ is the constant appearing in Proposition 4.1).Then there exists δ := δ(S, τ, d) ∈ (0, 1) small enough such that if (3.2), (4.2) are fulfilled, the following hold: Then, for any s 0 ≤ s ≤ S − σ, the operator A ⊥ : H s 0 → H s 0 is invertible with bounded inverse given by ) and let R a be the linear operator defined by R a : h(ϕ, x) → ∇a(ϕ, x) • ∇ ⊥ h(ϕ, x).
(iv) For any s 0 ≤ s ≤ S − σ − 2, the operator P ∆ is invertible, with inverse of the form

Proof. PROOF OF (i).
For any u ∈ H s (T d+2 ), we split u = Π ⊥ 0 u + Π 0 u, where Π ⊥ 0 u ∈ H s 0 = H s 0 (T d+2 ) and Π 0 u ∈ H s ϕ := H s (T d ).Since A is an operator of the form (4.4), one has that Ah = h if h(ϕ) does not depend on x.This implies that AΠ 0 = Π 0 .Similarly, one can show that A −1 Π 0 = Π 0 and therefore We now show that the operator The claimed statement then follows.PROOF OF (ii).First, we note that, given a function h(ϕ, x) and integrating by parts since div(∇ ⊥ h) = 0 for any function h.Moreover it is easy to see that R a Π 0 = 0.This implies that the linear operator R a is invariant on the space of zero average functions.Then, using also item (i), one has that where M ∇a denotes the multiplication operator by ∇a.A direct calculation shows that the operator A −1 M ∇a A = M g , where the function g(ϕ, y) := ∇a(ϕ, y + α(ϕ, y)) = {A −1 ∇a}(ϕ, y).The estimates (2.13), (4.5) imply that Moreover, one computes Using the trivial fact that |Π ⊥ 0 | 0,s ≤ 1, the formula (4.8), the estimates (4.9), (4.10), together with the composition Lemma 2.6-(ii), imply the claimed bound.PROOF OF (iii).Since Π 0 ∆ = ∆Π 0 = 0, one computes

and hence, a direct calculation shows that
where By Lemma 2.3 and the estimate (4.3), we have, for any s 0 ≤ s ≤ S − σ − 2 PROOF OF (iv).We write where Id 0 is the identity on the space of the L 2 zero average functions.By Lemma 2.6-(ii), estimates (2.21), (4.6), one obtains that |(−∆) −1 R ∆ | Lip(γ) 0,s s εγ −1 .Hence, by the smallness condition in (4.2) and by Lemma 2.6-(iv), one gets that The claimed statement then follows since we have using again Lemma 2.6-(ii).
We conclude this section by conjugating the whole operator L defined in (3.3) by means of the map A constructed in Proposition 4.1.

4.2.
Reduction to constant coefficients of the lower order terms.In this section we diagonalize the operator L where Q ∈ OPM −1 s is a diagonal operator, R (2) belongs to OPM −M s and R (2) The operators Q, R (2) are real, reversible and Q, R (2) , R ν leave invariant the space of functions with zero average in x.
and, for any n = 1, . . ., M − 1 and for any (ω, ζ) ∈ DC(γ, τ ), where L n has the form where n ∈ OPM −(n+1) s and they satisfy The operators L n , Z n and R n are real, reversible and leave invariant the space of zero average functions in x.
We assume that the claimed statement holds for some n ∈ {0, . . ., M − 2} and we prove it at the step n + 1.Let us consider a transformation T n+1 = Id + K n+1 where By applying again Lemma 2.6-(ii) and the estimates (4.14), (4.15) (using also that |∆| 2,s ≤ 1 for any s), one gets that R (2) ν satisfies (4.17) for any s 0 ≤ s ≤ S − σ M −1 .The proof of the claimed statement is then concluded.4.3.KAM reducibility.In this section we perform the KAM reducibility scheme for the operator obtained by neglecting the small viscosity term −ν∆ + R e .More precisely we consider the operator where the diagonal operator Q and the smoothing operator R (2) are as in Proposition 4.4.Given τ, N 0 > 0, we fix the constants where [4τ ] is the integer part of 4τ and M, σ M −1 are introduced in Proposition 4.4.By Proposition 4.4, replacing s by s + β in (4.17) and having Q = diag j∈Z 2 \{0} q 0 (j) diagonal, one gets the initialization conditions for the KAM reducibility, for any Proposition 4.6 (Reducibility).Let S > s 0 +Σ(β).There exist N 0 := N 0 (S, τ, d) > 0 large enough and δ := δ(S, τ, d) ∈ (0, 1) small enough such that, if (3.2) holds and then the following statements hold for any integer n ≥ 0. (S1) n There exists a real and reversible operator defined for any λ ∈ Λ γ n , where we define Λ γ 0 := DC(γ, τ ) for n = 0 and, for n ≥ 1, For any j ∈ Z 2 \ {0}, the eigenvalues µ n (j) = µ n (j; λ) are purely imaginary and satisfy the conditions and the estimates The operator R n is real and reversible, satisfying, for any s 0 ≤ s ≤ S − Σ(β), for some constant C * (s) = C * (s, τ ) > 0.
When n ≥ 1, there exists an invertible, real and reversibility preserving map Moreover, for any for some constant C(s, β) > 0.
(S2) n For all j ∈ Z 2 \ {0}, there exist a Lipschitz extension of the eigenvalues µ n (j; Proof.PROOF OF (S1) 0 , (S2) 0 .The claimed properties follow directly from Proposition 4.4, recalling (4.27), (4.29) and the definition of Λ γ 0 := DC(γ, τ ).PROOF OF (S1) n+1 .By induction, we assume the the claimed properties (S1) n , (S2) n hold for some n ≥ 0 and we prove them at the step n + 1.Let Φ n = Id + Ψ n where Ψ n is an operator to be determined.We compute where ).Our purpose is to find a map Ψ n solving the homological equation where D Rn is the diagonal operator as per Definition 2.4.By (2.9) and (4.31), the homological equation (4.41) is equivalent to . Therefore, we define the linear operator Ψ n by which is the solution of (4.42).
Proof.To simplify notations, in this proof we drop the index n.Since λ = (ω, ζ) ∈ Λ γ n+1 (see (4.32)), one immediately gets the estimate For any . By (4.34), (4.32) one has The latter estimate (recall (4.43)) implies also that Since M > 4τ by (4.28), recalling Definition 2.5, the latter estimates imply that and similarly, using also that ℓ, j Hence, we conclude the claimed bounds in (4.44).Finally, since R is real and reversible, by Lemma 2.10 and the properties (4.33) for µ(j), we deduce that Ψ is real and reversibility preserving.

( 1 )
Analysis of the linearized Navier-Stokes equation at the Euler solution and estimates for the inverse operators; (2) Construction of the first order approximation for the viscous solution up to errors of order O(ν 2 ); (3) A fixed point argument around the approximated viscous solution leading to the desired full solution of the Navier-Stokes equation.Inversion of the linearized operator at the Euler solution.The essential ingredient is to analyse the linearized Navier-Stokes operator at the Euler solution u e , namely one has to linearize (1.4) at the Euler solution u e (ϕ, x).This leads to study a linear operator of the form L ν := L e − ν∆, ,

Definition 2 .
5. (Matrix decay norm and the class OPM m s ).Let m ∈ R, s ≥ s 0 and R be an operator represented by the matrix in (2.10).We say that R belongs to the class OPM m s if is a solution of the Euler equation satisfying v e Lip(γ) S S ε a ≪ 1 , a ∈ (0, 1) , S > S (3.2) where S := S(d) is the minimal regularity threshold for the existence of quasi-periodic solutions of the Euler equation provided by Theorem 1.1.We want to study the linearized operator L ν := dF ν (v e ) at the solution of the Euler equation v e .where F ν (v) is defined in (3.1).The linearized operator has the form L ν = L e − ν∆ ,