Navier and Stokes meet Poincar\' e and Dulac

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that that the asymptotic expansion leads surprisingly to a Poincar\' e-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open issues.

the history of fluid dynamics, see [15]). However their mathematical theory of dynamical systems and physics theory of fluid mechanics have finally met more than a hundred years after their initial contributions. We will not comment on Stokes and Poincaré who are well-known scientists but make a few (may be not so well known) remarks on Dulac, and mainly on Navier.
Claude Louis Marie Henri Navier was an "X-Ponts" engineer in the jargon of Grandes Ecoles, first trained at the Ecole Polytechnique, then at the Ecole des Ponts et Chaussées, one of the Ecoles d'applications such as the Ecole des Mines (Augustin Louis Cauchy was an "X-Ponts", Henri Poincaré was an "X-Mines"). He was in the main stream of French theoretical continuum mechanics of this time.
A major figure of French Mechanics of this time, Adhémar Barré de Saint-Venant (1797-1886), also an X-Ponts, was a former student and successor of Navier. Among many other things, he derived the so-called Saint-Venant system (or shallow water system). He was the advisor and protector of Joseph Boussinesq (1842-1929) who made fundamental contributions in Fluid Mechanics, in particular on the theory of water waves.
As noticed by Olivier Darrigol in his book [15], "Navier and other Polytechnicians' efforts to reconcile theoretical and applied mechanics had no clear effect on French engineering practice. Industry prospered much faster in Britain, despite the lesser mathematical training of its engineers. Some of Navier's colleagues saw this and ridiculed the use of transcendental mathematics in concrete problems of construction. In the mid-1820s, a spectacular incident apparently justified their disdain. Navier's chef-d'oeuvre, a magnificent suspended bridge at the Invalides, had to be dismantled in the final stage of its construction".
Actually Navier was probably most famous in his time for the "disaster" of the pont des Invalides, the first suspended bridge over the Seine river. In fact Navier had mis-estimated the direction of the force exerted by the chain on the stone. This could have been corrected easily but the hostile municipal authorities decided the dismantlement of Navier bridge. Honoré de Balzac is alluding to this incident (rather ironically) in his novel Le curé de village: "La France entière a vu le désastre, au coeur de Paris, du premier pont suspendu que voulutélever un ingénieur, membre de l'Académie des Sciences, triste chute qui fut causée par des fautes que ni le constructeur du canal de Briare, sous Henri IV, ni le moine qui a bâti le Pont-Royal, n'eussent faites, et que l'Administration consola en l'appelant au Conseil Général (des Ponts et Chaussées). Les Ecoles Spéciales seraient-elles donc des fabriques d'incapacités? Ce sujet exige de longues observations".
(Translation) "All France knew of the disaster which happened in the heart of Paris to the first suspended bridge built by an engineer, a member of the Academy of Sciences, a melancholy collapse cause by blunders such as none of the ancient engineers, the man who cut the canal at Briare in Henry's IV time, or the monk who built the Pont Royal-would have made; but our administration consoled its engineer for his blunder by making him a member of the Council general. (of the Ponts et Chaussées). Are our Ecoles Spéciales producers of incapacities? This topic deserves lengthy observations".
According to Saint-Venant however, the dismantlement of the bridge was more than a local administrative deficiency: "At that time there already was a surge of the spirit of denigration, not only of the "savants" but also of science, disparaged under the name of theory opposed to practice; one henceforth exalted practice in its most material aspects, and prevented that higher mathematics could not help, as if, when it comes to results, it made sense to distinguish between the more or less elementary or transcendent procedures that led to them in an equally logical manner. Some "savants" supported or echoed these unfounded criticisms". As an aside, we note that there are some estimates of velocity fields in infinite dimensional spaces which have useful consequences, e.g. [25].
Navier was nevertheless a great scientist and, coming back to our subject, he derived what are known as the Navier-Stokes equations in a 1823 Mémoire. Further (different) derivations are due to Poisson (1831), Saint-Venant (1834) and Stokes (1843).
The last member of the quartet in title is the least famous of them. Henri Dulac (1870Dulac ( -1955, a former student of Ecole Polytechnique, was a professor at the University of Lyon and a corresponding member of the French Academy of Sciences. He was a specialist of the geometric theory of ordinary differential equations and developed in particular, after Poincaré, the theory of normal forms. 1.1. The Navier-Stokes equations for viscous, incompressible fluid flows. We now recall briefly the derivation of Navier-Stokes equations (NSE), based on conservation laws and the choice of a constitutive equation. For complete background on NSE see e.g. [43,63,64,13,24].
We study fluid flows in Euclidean space of dimension n = 2, 3. Let ρ denote the density of the fluid, and u its velocity.
We will consider only the case when the density ρ is constant, so that the conservation of mass reduces to div u = 0.
We refer to this as the incompressibility condition.
• Conservation of momentum (Newton's law) for a general fluid: wherep is the (scalar) pressure, T is the extra-stress tensor, andf represents body forces.
Here, we use the standard notation u · ∇ = i u i ∂ x i . When T ≡ 0, one obtains the Euler equations (1755).
• Constitutive law: For a Newtonian viscous fluid, T at the present time t is just proportional to the rate of deformation tensor D(u) = (∇u + (∇u) T )/2 at time t, that is where µ is the dynamic viscosity coefficient. (For a general non-Newtonian fluid, T can be a complicated function of the past history of the deformations).
Finally, one obtains the Navier-Stokes equations (NSE) where ν = µ/ρ is the kinematic viscosity, p =p/ρ, and f =f /ρ. For simplicity, we will just call ν viscosity, p pressure, and f body force.
The system (1.1) consists of (n + 1) equations for (n + 1) unknowns, namely, u ∈ R n and p ∈ R. It will be completed with initial and boundary conditions in our considerations.
1.2. Functional setting. We consider the following two cases of fluid flows.
The first scenario is when the fluid is confined a smooth, bounded domain Ω of R n , and the velocity satisfies the no-slip boundary condition, i.e., u = 0 on ∂Ω. We set in this case The second scenario is when (u, p) are defined in the whole space R n , but are L-periodic, for some L > 0, in all there Cartesian coordinates. Then u and p are considered as functions on the domain We usually refer to this Ω as a periodic domain, and say u and p satisfy the periodicity boundary condition on [0, L] n . By a remarkable Galilean transformation, we assume, without loss of generality, that u has zero averages over Ω, i.e., Ω u(x, t)dx = 0.
We then define the space In both cases, we will use the classical spaces: Note that notation | · | is used to denote the H-norm and the standard Euclidean norm on C n . However, its meaning will be clear in the context.
We denote the standard inner products of L 2 (Ω) k , for k ∈ N, by the same notation ·, · . The norm in the Sobolev space H m (Ω) is denoted by · m . We also denote One has the Helmholtz-Leray decomposition for the case of no-slip boundary condition, and for the case of periodicity boundary condition, We define P to be the (Leray) orthogonal projection in L 2 (Ω) n onto H. We assume at the moment that (u, p) are classical solutions of NSE. Thanks to (1.3) and (1.4), we have P (∇p) = 0. With this observation, we can reduce the unknowns of NSE from (u, p) to u only, by projecting the NSE to the space H. Having that in mind, we define the Stokes operator A by Au = −P ∆u (with the ad hoc boundary conditions), and also define the bilinear form Assume f is a potential, i.e., f = −∇ψ, then, thanks to (1.3) and (1.4) again, P f = 0. Hence, applying the Leray projection P to the NSE, and using the decomposition (1.3) or (1.4), we rewrite the NSE (1.1) in the functional form as: where u 0 is a given initial data in H. This functional form (1.5) will be the focus of our study in this paper.
1.3. Basic facts. The Stokes operator A is an unbounded, self-adjoint operator in H with domain D(A) = V ∩ H 2 (Ω) n . Its spectrum σ(A) consists of an unbounded sequence of real eigenvalues with corresponding multiplicities m 1 , m 2 , . . . , m k , . . . (See e.g. [9].) The orthogonal projection in H on the eigenspace of A corresponding to Λ j will be denoted by R j .
In the periodic case, By scaling the spatial and time variables, we can further assume, without loss of generality, It has been known since Leray's fundamental papers ( [44,45,46]) that (i) For every initial data u 0 ∈ H, problem (1.5) has a (Leray-Hopf) weak solution u (see e.g. [47,13,43,63,64,24]), that is, (1.5) in the dual space V ′ of V , and the energy inequality holds for t 0 = 0 and almost all t 0 ∈ (0, ∞), and all t ≥ t 0 .
Above, H w denotes the space H endowed with the weak topology. (vi) Any regular solution u on [0, ∞) satisfies the equation Because of property (ii) above, and that our goal is to study long-time behavior of solutions to NSE, we will, without loss of generality, mainly consider regular solutions on [0, ∞). Let R denote the set of initial data in V leading to global regular solutions. Then R is an open subset of V , and, particularly, R = V when d = 2.
Obviously, u = 0 is a trivial regular solution on [0, ∞). Hence, R contains a neighborhood of 0. However, proving or disproving that R = V when d = 3 is still an outstanding open problem.
Here afterward, we will call a regular solution u on [0, ∞), that is when u(0) ∈ R, simply a regular solution.
For a regular solution u, one has from (1.10) and the Poincaré inequality, i.e., That is |u(t)| 2 must decay exponentially as t → ∞ at the rate at least 2νΛ 1 .

1.4.
Aim and outline of the paper. A natural question (raised by P. Lax to C. Foias) is then to ask whether or not this decay rate is optimal. In an early work, Dyer and Edmunds [17] prove that any non-trivial, regular solution u has |u(t)| 2 also bounded below by an exponential function of t. However, this answer is far from being definitive in describing the exact asymptotic behavior of a non-trivial, regular solution. In the following sections, we present the mathematical developments of the problem which lead to the asymptotic expansion and normal form theory (for NSE).
The paper is organized as follows. In section 2, the Dirichlet quotient is proved to converge, as t → ∞ to an eigenvalue of the Stokes operator. The asymptotic expansion of the regular solutions are studied. The set R is decomposed into nonlinear manifolds M k 's, which characterize the rate of the decay for the solutions. In section 3, each regular solution is proved to admit an asymptotic expansion in terms of exponential decays and polynomials in time. The application to analysis of the helicity is also presented. In section 4, we review the classical Poincaré-Dulac theory of normal forms for ODEs. In section 5, it is shown that the asymptotic expansion reduces to a normal form, which, originally, is in a Fréchet space with very weak topology. It is then studied in suitable Banach spaces. In such a weighted normed space, the normalization map is continuous and the normal form for NSE is a well-posed infinite dimensional ordinary differential equation (ODE) system. In section 6, the inverse of the normalization map is written as a formal power series in E ∞ , an appropriate topological vector subspace of C ∞ . It is then used to reduce the NSE to a Poincaré-Dulac normal form on E ∞ . In section 7, we review more related results and pose some open questions.

Limit of the Dirichlet quotients
By re-writing the energy equality (1.10) in the form it is natural to study the limit as t → ∞ of the Dirichlet quotients This is the beginning of a long process leading eventually to a normal form of NSE. One has the following results ( [26,27]). (i) lim t→∞ λ(t) = Λ(u 0 ) exists and belongs to σ(A).
(iii) There exist analytic submanifolds M k , k = 1, 2, . . . , of R having codimension m 1 + m 2 + . . . + m k such that (iv) M k is invariant by the nonlinear semi-group S(t) generated by the Navier-Stokes equation, that is We recall elementary estimates for regular solutions and large t: Let v(t) = u(t)/|u(t)|. Note that |v(t)| = 1 and v(t) 2 = λ(t). One can derive a differential equation for λ(t): It follows that dλ dt Neglecting the second term on the left-hand side, and using Gronwall lemma together with (2.1), we obtain for sufficiently large t ′ > t > 0 that Using the fact that |u(t)| and u(t) go to zero as t → ∞, and letting t ′ → ∞, then t → ∞, we obtain 0 < lim sup Thus lim t→∞ λ(t) = Λ exists and belongs to (0, ∞).
After this eigenvalue Λ is established, ones can prove in [26, Proposition 1 and Lemma 1] that It remains to deal with R Λ e νΛt u(t). We have d dt Hence, for s > t > 0: By the exponential decay (2.3) of u(t), we see that the right-hand side goes to 0 as s, t → ∞. Therefore, by Cauchy's criterion, lim t→∞ e νΛt R Λ u(t) exists and belongs to R Λ H.
The sets M k 's can be defined as level sets Then the analyticity of M k 's results from the analyticity of the mapping (t, v) → S(t)v which is due to [18].
Remark 2.1. The following remarks are in order.
(a) (L. Tartar) In the case where Ω is only bounded in one direction (so that Poincaré inequality holds), one has also that the lim t→∞ λ(t) = Λ((u 0 ) exists and belongs to the spectrum of A, which is not necessarily discrete. (b) One can prove that the rate of decay given by Λ(u 0 ) gives also the decay rate of higher Sobolev norms, and also the convergence of u(t)e tΛ(u 0 ) in all H s for any s > 0, see [29,38]. We refer to [34] for various extensions of the convergence of the Dirichlet quotients to other situations, in particular Navier-Stokes and MHD equations on compact Riemannian manifolds. (c) We recall that for any u 0 ∈ H, the NSE possesses a weak solution u which becomes regular for t sufficiently large. In this case the Dirichlet quotient λ(t) converges to an eigenvalue Λ(u(·)) of σ(A) that, by lack of uniqueness, depends a priori on the whole solution u. The invariant manifolds M k 's can also be characterized as in the next result.
Theorem 2.2 (Corollary 2 [26]). The necessary and sufficient condition for Note in (2.5) that it only requires the projection R Λ j of S(t)u 0 , not the whole S(t)u 0 .
One has further results on properties of the manifolds M k 's in the periodic case.
Theorem 2.3 (Remark 7 [26], Theorem 2 and Proposition 4 [27]). In the periodic case, each M k is a smooth analytic, truly nonlinear manifold in R, and contains a linear submanifold Proof. For the nonlinearity, we argue by contradiction. Suppose M k is linear. Then it must coincide with its tangential linear manifold at 0, which is M lin k . Together with the invariance of M k under the semigroup S(t), it follows that By construction of an explicit counter example (see [27]), this fact is shown to be not true.
The construction of the invariant linear submanifolds L k 's is based on special motions of the Navier-Stokes equations in the periodic case that we describe now.
Then any (spatial) L-periodic solution ϕ of the linear heat equation ∂ϕ ∂t − ν|k| 2 ∂ 2 ϕ ∂y 2 = 0 leads to a solution of the NSE of the form u(x, t) in (2.6) and p = constant. Clearly, such a solution satisfies Based on the above observation, we set Remark 2.2. The family of manifolds constructed in Theorem 2.3 is extended to the following more general ones, which are also used to analyze the decay of the helicity, see [19,20] and subsection 3.4 below.
Consider the periodic case in R 3 . Let a be a vector in R 3 such that its orthogonal plane has nontrivial intersection with Z 3 , this means as well as the NSE (1.5). The fact that the nonlinear term B(u(t), u(t)) vanishes in this situation can be easily verified. Therefore, M a ⊥ is an invariant linear manifold in R. Clearly, the cardinality of a ⊥ is infinite, and hence M a ⊥ is infinite-dimensional. Remark 2.4. The Dirichlet quotients (interpreted as the ratio of the enstrophy over the energy) have been used to study geophysical flows, in particular to give a precise mathematical sense (and justify) the physicists' selective decay principle: After a long time, solutions of the quasi-geostrophic equations and/or the twodimensional incompressible Navier-Stokes equations approach those states which minimize the enstrophy for a given energy. For more details we refer to [50,49,51,66,67].
Remark 2.5. In a totally different context, the Dirichlet quotients have been used in [14] to study the backward behavior of solutions to the periodic Navier-Stokes equations with (non-potential) time-independent body forces. More precisely it is proven there that the set of initial data for which the solution exists for all negative times and has exponential growth is rather rich, actually it is dense in the phase space of the NSE, answering positively a question of Bardos-Tartar [2].
Coming back to the study of NSE with potential forces, it has been proven in [28], by extending a result of Hartman ([39] chap. IX, th. 6.2) for ODE's, that the NSE have invariant manifolds with "slow" decay. More precisely, (a) While the manifolds M k 's are unique, this is not the case of the F k 's. (b) In the 2D-periodic case, one can take F 1 = R 1 H. This results from the fact that then the function t → S(t)v 2 /|S(t)v| 2 is decreasing for any nonzero v ∈ V. It is a consequence of (2.2) and the orthogonal properties Remark 2.7. We refer to [7] for construction of invariant manifolds in a rather general setting, and, in particular, to its Appendix B5 for illuminating comments on slow manifolds.
Remark 2.8. The nonlinear spectral manifolds have been used in [48] to study asymptotic stability issues for the periodic two-dimensional Navier-Stokes equations.
The results in Theorem 2.1 suggest that one can go further and look for an asymptotic expansion of the solution. This will eventually lead to the normal form.
We first introduce a technical notion on the spectrum of A. More generally, Definition 2.6. Let A be a, possibly unbounded, linear operator in a space X with spectrum σ(A).
(i) A resonance in σ(A) is a relation of the type for some Λ, Λ 1 , Λ 2 , . . . , Λ k ∈ σ(A), and some positive integers a 1 , a 2 , . . . , a k with has a resonance then we say it is resonant, otherwise nonresonant. (iv) In case A is the Stokes operator with the spectrum described in (1.6), and Λ = Λ k+1 for some k ≥ 1, then (2.7) is equivalent to We note that the periodic boundary conditions on [0, L] n always lead to resonances because Λ 2 = 2Λ 1 . On the other hand, for periodic boundary conditions on general cubes [0, It has been recently proven ( [12]) that in the case of Dirichlet boundary conditions, the spectrum of A is non-resonant generically with respect to the domain. More precisely let D 3 l be the set of bounded domains in R 3 with C l boundary equipped with a suitable topology. For any Ω ∈ D 3 l we denote by D 3 l (Ω) the Banach manifold obtained as the set of images The main result in [12] is that generically with respect to Ω ∈ D 3 5 the spectrum of A is nonresonant, in the sense that the set of domains in D 3 5 (Ω) for which the non-resonance property holds contains an intersection of open and dense subsets of D 3 5 (Ω). This result is established as a consequence of the fact that generically with respect to Ω ∈ D 3 4 , the eigenvalues of A are simple.

The asymptotic expansion
In this section we obtain asymptotic expansions, as time tends to infinity, for regular solutions of NSE. These expansions are of the following type.
Definition 3.1. Let X be a real vector space.
(a) An X-valued polynomial is a function t ∈ R → d n=1 a n t n , for some d ≥ 0, and a n 's belonging to X.
(b) When (X, · ) is a normed space, a function g(t) from (0, ∞) to X is said to have the asymptotic expansion n=1 is a strictly increasing sequence of positive numbers, g n (t)'s are X-valued polynomials, if for all N ≥ 1, there exists ε N > 0 such that Throughout, A is the Stokes operator.

The non-resonant case. Assume σ(A) is non-resonant.
Theorem 3.2 (Theorem 2 [29]). Let u be a regular solution. For each N ∈ N, one has the expansion in H: More precisely, one has in this case The rather technical proof is by induction on N, see details in [29].

The resonant case. Assume σ(A) is resonant.
Theorem 3.3 (Theorem 4 [29]). Let u be a regular solution with initial data u 0 ∈ R. For any N ∈ N, one has the asymptotic expansion in H: Moreover, R k W Λ j (t), for k = j and the coefficients of order Proof. The proof is also technical and by induction on N. We merely sketch the main steps.
• First step. We recall that u(t) = O(e −νΛ 1 t ). Ones can prove the limit in V : and then establish for some δ > 0, ∀m ≥ 0. We apply the projector R λ k and obtain the equation for w N,k = R Λ k w N : where p N,k is a polynomial in t. This equation is an ODE of the type: where p(t) is a polynomial. When either α ≥ 0, or α < 0 with lim t→∞ (e αt w(t)) = 0, there exists a polynomial solution q(t) of , ∀m ≥ 1, which proves the induction step.
• Note that, in dealing with the higher norms · m , the proof in [29] estimates d (j) u/dt j m for all j ≥ 1.
Remark 3.2. The first coefficient W µ j (t) which is not identically zero in the expansion corresponds to µ j = Λ(u 0 ). In this case it is constant in t and belongs to R Λ(u 0 ) H.
Notation. Based on Theorems 3.2 and 3.3, and according to Definition 3.1, we have which we will simply write 3.3. The asymptotic expansion in Gevrey spaces. Theorem 3.3 has been recently improved in [40] where it is proved in particular that the asymptotic expansion in the 3Dperiodic case actually holds in all Gevrey classes. Consider the periodic case (1.2) with n = 3 and assume (1.8). Recall that one has the properties (1.9). We first describe the relevant Gevrey classes.
For α, σ ∈ R, and u = k∈Z 3 \{0}û (k)e ik·x , we define and the Gevrey class The next theorem improves Theorem 3.3 for the periodic case for any weak solution. By working with the Gevrey norms, the proof in [40] can avoid the estimates of d (j) u/dt j m for all j ≥ 0 and m ≥ 0.
Before moving to the normal form theory for the NSE, we review other applications of the Dirichlet quotients techniques and asymptotic expansions.

3.4.
Application: asymptotic behavior of the helicity. It turns out that the techniques developed to study the Dirichlet quotients can be used to obtain information on the asymptotic behavior of the helicity for Navier-Stokes equations with potential forces. This is the object of the papers [19,20]. We will focus on the results of [19] where the (3D, periodic) deterministic case is considered. (Interested readers can read [20] which deals with the statistical case).
For a regular solution u of the NSE, the helicity is a scalar quantity defined by In the inviscid case (ν = 0) the invariance of the helicity was noticed by Moreau [56]. The first thorough study of the helicity and of its density for inviscid incompressible flows was carried out by Moffatt [54], who gave in particular a connection of the helicity to the topological invariants and dynamics of the vortex tubes as well as the first examples of physically relevant fluid flows with non-zero helicity. There is a general agreement that helicity plays an important role in magneto-hydro dynamics, but not in the dynamics of neutral flows (that is, solutions of the Euler or the Navier-Stokes equations). However, theoretical, empirical and numerical evidence indicate that the helicity can provide insights into the nature of the fluid flows, at least in the case when the viscosity is small and this motivates the present study.
In the periodic case with our choice of Ω in (1.2) (n = 3), we recall from (1.7) that the first eigenvalue of the Stokes operator A is Λ 1 = 4π 2 /L 2 , and the other ones are among nΛ 1 , with n ∈ N. The previous results on the limit of the Dirichlet quotients together with Cauchy-Schwarz inequality imply that the helicity of a regular solution tends to zero as t → ∞, at least with a rate 2νΛ 1 n 0 , where n 0 depends on the initial data. However, due to possible changes of sign and cancellations in u · ω, this does not imply that it has the same decay rate 2νΛ 1 n 0 , as that of the energy. In particular, it was not known whether the helicity could change sign or vanish infinitely times as t → ∞. An answer to those questions is found in [19].
We further define related quantities |ω(x, t)| 2 dx (rate of energy dissipation/viscosity) These entities satisfy the following (balance) equations: We now assume (1.8), and recall that (1.9) holds true. Proof. Using asymptotic expansion of u, we derive for the helicity that where the φ j 's are polynomials in t ∈ R. If one of φ j 's is not a zero polynomial, then we obtain case (i). Otherwise, H(t) decays to zero, as t → ∞, faster than any exponential functions. This fact itself cannot yield conclusion for case (ii). More properties of the solution u and helicity H are needed. For those, we complexify the NSE in time and denote the resulting solution and helicity by u(ζ) and H(ζ), for the complex time ζ ∈ C. These functions are proved to be analytic and bounded in a domain E which, see [19,Propositions 8  The next theorem shows that the case where the helicity is non-zero is generic. Theorem 3.6 (Theorem 3.2 [19]). Let R 1 and R 0 be the sets of initial data u 0 ∈ R corresponding to cases (i) and (ii) in Theorem 3.5, respectively. Then R 1 is open and dense in R while R 0 is closed and contains an infinite union of linear, closed infinite dimensional manifolds.
The asymptotic decay of helicity is precisely described in the next result. Note that one has the norm relation u(t) = |ω(t)|, hence, If α 0 = 0 in the previous theorem, one is in case (i) of Theorem 3.5 (helicity decays) with d = 0 and h 0 = n 0 ∈ σ(A). The situation where α 0 = 0 is considered in the next theorem. Theorem 3.8 (Theorem 3.5 [19]). For any n ∈ σ(A) and M > 0, there exists an initial data u 0 ∈ R such that one is in case (i) of Theorem 3.5 with n 0 = n and h 0 ≥ n 0 + M, and such that H(t) |u(t)| 2 = O(e −2M t ) when t → ∞. Moreover, there exist solutions whose helicity satisfies the condition lim t→∞ H(t)t −d e 2h 0 t exists and is not zero, where d > 0 or h 0 is not an eigenvalue of A.
Comments on the proofs of Theorems 3.6-3.8.
• The properties of the set R 0 of initial data leading to an identically zero helicity result from a study of the spectrum of the curl operator and of a global stability result of NSE in 3D ( [60]). • Examples of linear submanifolds of R 0 are, among others, the family M a ⊥ presented in Remark 2.2.

The Poincaré-Dulac theory of normal forms
This section briefly reviews the Poincaré-Dulac theory of normal forms. This is, of course, a classical topic in dynamical systems, initiated by Poincaré and, later, Dulac (see [16,59]) to analyze the dynamics of a nonlinear system of ODEs in the neighborhood of a singular point. We refer to Arnold's book [1] for a modern treatment.
The next theorem is extracted from Poincaré's thesis (1879).
Theorem 4.1 (Poincaré's thesis 1879 [58]). If the eigenvalues of the matrix A are nonresonant, the equation is a homogeneous polynomial of degree d in R n , reduces to the linear equation by a formal change of variable is a homogeneous polynomial of degree d.
(Translation) ... the remainder of the thesis is a little confused and shows that the author was still unable to express his ideas in a clear and simple manner. Nevertheless, considering the great difficulty of the subject and the talent demonstrated, the faculty recommends that M. Poincaré be granted the degree of Doctor with all privileges.
The resonant case was treated in Dulac's thesis (1912).

A normalization map and a normal form for NSE
One can use the generating part of the asymptotic expansion to construct a normalization of the NSE.
We first define the Fréchet space Theorem 5.1 (Theorem 3, Corollaries 1 and 2 [29]). Define the mapping W : R → S A by Then: (i) W is analytic and one-to-one.
(iii) u 0 ∈ M k if and only if the first k components of W (u 0 ) vanish.
In this non-resonant case, v(t) = W (u(t)) is, thanks to (ii), a solution of the linear Navier-Stokes equations in the large space S A , i.e., Thus, the mapping W transforms the (nonlinear) NSE (1.5) to the linear one (5.1), which is the case of Theorem 4.1. Therefore, we call W a normalization map, even though it is not a formal series.

5.2.
The resonant case. Assume σ(A) is resonant. In view of the structure of the asymptotic expansion in this case, considering the following mapping W is natural.
Theorem 5.2 (Theorem 5 [29]). The mapping W : R → S A given by is analytic and one-to-one.
We will see that this mapping W also plays the role of a normalization map. First, we find important polynomials that are essential in transforming the NSE into a normal form.
With these polynomials P j 's, we are ready to rewrite the NSE under the transformation W (u(t)). Theorem 5.3 (Theorem 6 [29]). Let u 0 ∈ R and u(t) = S(t)u 0 be the corresponding global solution of the NSE. The (S A -valued) function v(t) = W (u(t)) satisfies the equation where the polynomials P l and P j are defined as in Lemma 5.1.
From the relation (5.2), one can prove also that each monomial in B(v) is resonant, i.e., if M(v 1 , v 2 , .., v k j ) is a (nonzero) monomial in B of degree m 1 , . . . , m k j in v 1 , . . . , v k j respectively, and M ∈ R j H, then Thus, W transforms NSE (1.5) to (5.3), which satisfies the resonance condition as in Theorem 4.3. Therefore, we, again, call W a normalization map, and equation (5.3) a normal form of NSE.
Although the normal form (5.3) is nonlinear in S A , it can be solved by successive integration of an infinite set of non-homogeneous linear differential equations in R k H, k = 1, 2, . . . , each one having an already known non-homogeneous part.
Remark 5.1. Minea ([53]) shows that this type of normalization, when applied to ODEs, is a normalization in the sense of Bruno ([6]).

5.3.
Further results in the 3D periodic case. The papers [21,22] aim to answer the following questions.
• When does the asymptotic expansion actually converge?
• In what natural normed spaces is the normal form a well-behaved infinite-dimensional system of ODEs? • What is the range of the normalization map? Partial answers to those questions are established in the 3D periodic case, namely: • In paper ( [21]): Construction of a suitable Banach space S ⋆ A ⊂ S A on which the normal form is a well-posed system near the origin. The norm ū ⋆ ofū = (u n ) ∞ n=1 ∈ S ⋆ A is where (ρ n ) ∞ n=1 is a sequence of positive weights. • In paper ( [22]): choice of a suitable set of weights ρ n such that the normalization map W : R → S ⋆ A is continuous and such that the normal form of the NSE is well-posed in the entire space S ⋆ A .
For N ∈ N, we denote by R N the projection from H onto the eigenspace of A corresponding to N in case N ∈ σ(A), and set R N = 0 in case N ∈ σ(A).
We note that the definition of R N is only different from that in subsection 1.3 by the change of the index. This aims to unify calculations and make them more efficient in lengthy proofs.
The definitions of polynomials P j 's and the normal form (5.3) can be expressed more explicitly as follows. We recall that the asymptotic expansion (3.3) for a regular solution u of the NSE with initial data u 0 ∈ R is where q j (t)'s are polynomials in t with values in V, and are unique polynomial solutions of the following ODEs , q l (t)), for j > 1.
Ones can verify that each monomial in B satisfies the resonance condition, see (5.4). Therefore, (5.7) is a normal form of NSE in S A under the transformation The solution of the normal form (5.7) with initial data ξ 0 = (ξ 0 n ) ∞ n=1 ∈ S A is precisely (R n q n (t, ξ 0 )e −nt ) ∞ n=1 . We denote this solution by S normal (t)ξ 0 .
Next, we investigate the convergence of the asymptotic expansion. We introduce and make use of the following construction of regular solutions. It is motivated by the asymptotic expansion itself.
We decompose the initial data u 0 in V as We find the solution u(t) of the form where for each n, (5.10) du n (t) dt + Au n (t) + B n (t) = 0, t > 0, with initial condition System (5.10) will be called the extended NSE. We denote by S ext (t) the semigroup generated by solutions of this system. It turns out that such a construction (5.8)-(5.11), indeed, produces regular solutions of the form (5.9) for NSE with the initial condition (5.8). The following existence theorem is a special case of Corollary 3.5 [21] with specific parameter ρ 0 > 1 = ρ.
Let (u n (t)) ∞ n=1 be the solutions to (5.10) and (5.11), then u(t) = ∞ n=1 u n (t) is the regular solution to the NSE with initial data u 0 = ∞ n=1 u 0 n , for t ∈ [0, T ), for some T > 0. We now make the connections between the solutions of the extended NSE with the asymptotic expansions of solutions of NSE.
If a regular solution u(t) has the expansion ∞ n=1 W n (t, u 0 )e −nt , then formally we wish for Therefore, we set u 0 n = W n (0, u 0 ) in the extended NSE. Then solutions u n (t) of the extended NSE are exactly W n (t, u 0 )e −nt . Hence, conclusion in Theorem 5.4 on u n (t), helps us make conclusion on ∞ n=1 W n (t, u 0 )e −nt . First, we have a small initial data result.
Note in Theorem 5.6 that it is not known whether the sum ∞ n=1 W n (t, u 0 )e −nt converges to a solution in short time.
Although the conclusions in Theorems 5.5 and 5.6 are satisfactory, it is not known whether the condition (5.12) or (5.13) holds true for a non-zero u 0 . At the moment, we do not know whether ∞ n=1 W n (0, u 0 ) and ∞ n=1 W n (t, u 0 ) converge in V , i.e., with respect to the norm · . However, we hope to obtain some convergence in weaker norms. Therefore, we study, in the following, the asymptotic expansions with a different approach, which uses suitable weighted normed spaces.
Theorem 5.11 (Theorem 4.2 [22] and Theorem 7.4 [21]). For each t ≥ 0, the map ξ ∈ S ⋆ A → S normal (t)ξ ∈ S ⋆ A is continuous. In particular, there exists ε 0 > 0 such that ifξ,χ ∈ S ⋆ A and ξ ⋆ , χ ⋆ < ε 0 , then According to Theorem 5.11, the normal form (5.7) is a well-posed system of ODEs in the infinite dimensional Banach space S ⋆ A . In other words, the semigroup S normal (t), t > 0, generated by the solutions of the normal form (5.7) leaves invariant the whole space S ⋆ A . Furthermore, we establish the continuity (but not necessarily Lipschitz continuity on the entire S ⋆ A ) of each S normal (t) as a map form S ⋆ A to S ⋆ A , which means that the normal form is a well-posed system. Theorem 5.12 (Theorems 5.9 and 5.21 [22]). The normalization map W is a continuous function from R to S ⋆ A . We summarize our results stated above in the commutative diagram ( Figure 1) in which all mappings are continuous. The complete proofs of Theorems 5.10, 5.11 and 5.12 are lengthy and technical and giving them exceeds the scope of this survey. We refer the reader to the papers [21,22] for details. They involve the complexification of NSE and extended NSE, for which the solutions are analytic in the complex time in a large domain of the form (3.4) and (3.5). With appropriate transformation to transfer them to the half plane, and utilizing some Phragmen-Linderlöff estimates, we can obtain recursive estimates for each step W n (u 0 ), q n (0, W (u 0 )), q n (ζ, W (u 0 )), and u(ζ) − n j=1 q j (ζ, W (u 0 ))e −jζ for complex time ζ. They, of course, depend on the weights ρ n 's. Then the sum, say, ∞ n=1 ρ n q n (0, W (u 0 )) is convergent when ρ n 's are chosen specifically and decay to zero extremely fast.

Navier and Stokes meet Poincaré and Dulac
It was not totally clear that the normal form theory for the NSE derived in section 5 could be related to the Poincaré-Dulac theory presented in section 4. It turns out to be the case, at least in the periodic case.
We consider thus the periodic case and use the same notation as in subsection 5.3. In that subsection, (5.7) is a normal form of NSE in a suitable Banach space S ⋆ A . This space, however, is too big to make a link with the concrete (formal series) approach of the Poincaré-Dulac theory. Such link was finally established in [23]. In short, • The system (5.7), indeed, provides a Poincaré-Dulac normal form of the NSE, and is obtained by a (formal) explicit change of variables. The change of variables is a formal series expansion of the inverse of the normalization map W . • Each homogeneous term in the formal series is well-defined in suitable Sobolev spaces.
We present below the precise results and provide main ideas and techniques in their proofs. The following topological vector space will be essential in our study It is endowed with the topology generated by the family of norms |A α · | for all α ≥ 0.
First, we give an explicit definition of a normal form for the NSE, which is an analogue to classical ones by Poincaré and Dulac reviewed in section 4. We start with homogeneous polynomials and resonant monomials in infinite-dimensional spaces.
, is a monomial of degree α k i > 0 in ξ k i where i = 1, 2, . . . , m, α k 1 + . . . + α km = d and k 1 < k 2 < . . . < k m , if it can be represented as Although the definition of resonant monomials in Definition 6.1 is more abstract than that in Definition 4.2, they are essentially the same, see details in [ To establish a normal form theory for the NSE, according to Definition 6.2, we need to identify the framework E, the normal form (6.3), and the formal change of variable (6.4).
The framework. We will use the space E ∞ defined by (6.1).
The normal form. The natural candidate for the normal form is (5.7). However, we must rewrite it in the power series form.
Let P [d] j (ξ) and B [d] j (ξ) denote the sum of all homogeneous monomials of degree d of P j (ξ) and B j (ξ), respectively. Then the series j P j (ξ) converge in E ∞ to continuous polynomials P [d] (ξ) and B [d] (ξ), respectively, see Theorem 6.5 below.
The system (5.7) is rewritten in the formal power series as Inheriting the spectral property (5.4), each polynomial B [d] (ξ) in (6.5) can be verified to be resonant. Hence, system (6.5) is a potential Poincaré-Dulac normal form, except that it is missing a power series change of variable.
The formal change of variable. We already know that NSE reduces to (6.5) by the transformation ξ = W (u). Therefore, u should be W −1 (ξ). Of course, W −1 is not rigorously defined on E ∞ and, additionally, not expressed in the power series form. To resolve these, we heuristically argue that Note that P [1] (ξ) = ξ. Thus, the formal change of variable would be The change of variable (6.6) is considered as the formal inverse of the normalization map W .
It turns out that, these arguments can be made rigorous and we obtain the following result. Theorem 6.3 (Theorem 4.9 [23]). The system (6.5) is a Poincaré-Dulac normal form in E ∞ for the NSE (1.5), and is obtained by the formal change of variable (6.6).
The proof of Theorem 6.3 relies on recursive formulas giving the homogeneous terms of the normal form. The main tool to estimate their Sobolev norms is the following family of homogeneous gauges [[ξ]] d,n .

Final comments
7.1. Other related results.
(1) The recent paper [41] obtains the asymptotic expansion of the same type for weak solutions of NSE in periodic domains with exponentially decaying (non potential) outer body forces satisfying an asymptotic expansion of the type f n (t)e −nt in appropriate (Gevrey type) functional spaces.
(2) When Ω = R n , because of lack of the Poincaré inequality, the situation is drastically different and the decay rate is only algebraic. The techniques and their proofs are quite different than those used for the bounded domains. We refer to [61,52,33,4,5,65] and the references therein. In particular, Kukavica and Reis ( [42]) obtain a precise space-time asymptotic of smooth solutions in a weighted space. (3) Section 7 of [29] focuses on the viscous Burgers equation and the Minea system. In the case of the viscous Burgers equation, the normalizing mapping W can be explicitly computed in terms of the Cole-Hopf transform (4) The asymptotic expansion is also established for dissipative wave equations by Shi in [62].

7.2.
Some open issues. We indicate below a few open questions related to the topics considered in the present paper.
(1) The classical Poincaré-Dulac theory for ODEs has a second part concerning the convergence of the formal series which give the change of variables, and the convergence of the series in the normal form (see [1]). The extension of these convergence results to the NSE seems to be an open problem. (2) Complete our knowledge of the normalization map and of the normal form.
(3) What happens in 3D when the initial data u 0 ∈ R is near the boundary ∂R in case R = V ? (4) It is very likely that the normal form theory studied here extends to the NSE posed on a compact Riemann manifold (e.g. the Euclidean sphere in R 3 ) (see [34] for results on the asymptotic decay). In particular, all specific results obtained in the periodic case should have a counterpart in this framework.