The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

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Introduction
We consider the Navier-Stokes equation for an incompressible fluid on the d-dimensional torus T d , d ≥ 2, T := R/2πZ, where ε ∈ (0, 1) is a small parameter, the frequency ω = (ω 1 , . . . , ω ν ) ∈ R ν is a ν-dimensional vector and f : T ν × T d → R d is a smooth quasi-periodic external force. The unknowns of the problem are the velocity field u = (u 1 , . . . , u d ) : R × T d → R d , and the pressure p : R × T d → R. For convenience, we set the viscosity parameter in front of the laplacian equal to one. We assume that f has zero space-time average, namely The purpose of the present paper is to show the existence and the stability of smooth quasiperiodic solutions of the equation (1.1). More precisely we show that if f is a sufficiently regular vector field satisfying (1.2), for ε sufficiently small and for ω ∈ R ν diophantine, i.e.
T d f (ϕ, x) dx = 0, ∀ϕ ∈ T ν (1.4) then the result holds for any frequency vector ω ∈ R ν , without requiring any non-resonance condition. Furthermore, we show also the orbital and the asymptotic stability of these quasiperiodic solutions in high Sobolev norms. More precisely, for any sufficiently regular initial datum which is δ-close to the invariant torus (w.r. to the H s topology), the corresponding solution of (1.1) is global in time and it satisfies the following properties.
• Orbital stability. For all times t ≥ 0, the distance in H s between the solution and the invariant torus is of order O(δ).
• Asymptotic stability. The solution converges asymptotically to the invariant torus in high Sobolev norm · H s x as t → +∞, with a rate of convergence which is exponential, i.e. O(e −αt ), for any arbitrary α ∈ (0, 1).
In order to state precisely our main results, we introduce some notations. For any vector a = (a 1 , . . . , a p ) ∈ R p , we denote by |a| its Euclidean norm, namely |a| := a 2 1 + . . . + a 2 p . Let d, n ∈ N and a function u ∈ L 2 (T d , R n ). Then u(x) can be expanded in Fourier series where its Fourier coefficients u(ξ) are defined by We also define the Sobolev space of functions with zero average Moreover, given a Banach space (X, · X ) and an interval I ⊆ R, we denote by C 0 b (I, X) the space of bounded, continuous functions u : I → X, equipped with the sup-norm u C 0 t X := sup t∈I u(t) X .
For any integer k ≥ 1, C k b (I, X) is the space of k-times differentiable functions u : I → X with continuous and bounded derivatives equipped with the norm In a similar way we define the spaces C 0 (T ν , X), C k (T ν , X), k ≥ 1 and the corresponding norms Notation. Throughout the whole paper, the notation A B means that there exists a constant C which can depend on the number of frequencies ν, the dimension of the torus d, the constant γ appearing in the diophantine condition (1.3) and on the C N norm of the forcing term f C N . Given n positive real numbers s 1 , . . . , s n > 0, we write A s 1 ,...,sn B if there exists a constant C = C(s 1 , . . . , s n ) > 0 (eventually depending also on d, ν, γ, f C N ) such that A ≤ CB.
We are now ready to state the main results of our paper. Theorem 1.1 (Existence of quasi-periodic solutions). Let s > d/2 + 1, N > 3ν 2 + s + 2, ω ∈ R ν diophantine (see (1.3)) and assume that the forcing term f is in C N (T ν × T d , R d ) and it satisfies (1.2). Then there exists ε 0 = ε 0 (f, s, d, ν) ∈ (0, 1) small enough and a constant C = C(f, s, d, ν) > 0 large enough such that for any ε ∈ (0, x)) := (U(ωt, x), P (ωt, x)) solves the Navier-Stokes equation (1.1) and If the forcing term f has zero space average, i.e. it satisfies (1.4), then the same statement holds for any frequency vector ω ∈ R ν and U(ϕ, x) satisfies Theorem 1.2 (Stability). Let α ∈ (0, 1), s > d/2 + 1, N > 3ν 2 + s + 2, u ω , p ω be given in Theorem 1.1. Then there exists δ = δ(f, s, α, d, ν) ∈ (0, 1) small enough and a constant C = C(f, s, α, d, ν) > 0 large enough such that for ε ≤ δ and for any initial datum there exists a unique global classical solution (u, p) of the Navier-Stokes equation (1.1) with initial datum u(0, x) = u 0 (x) which satisfies The investigation of the Navier-Stokes equation with time periodic external force dates back to Serrin [40], Yudovich [41], Lions [30], Prodi [36] and Prouse [37]. In these papers the authors proved the existence of weak periodic solutions on bounded domains, oscillating with the same frequency of the external force. The existence of weak quasi-periodic solutions in dimension two has been proved by Prouse [38]. More recently these results have been extended to unbounded domains by Maremonti [27], Maremonti-Padula [28], Salvi [39] and then by Galdi [19], [20], Galdi-Silvestre [21], Galdi-Kyed [22] and Kyed [32]. We point out that in some of the aforementioned results, no smallness assumptions on the forcing term are needed and therefore, the periodic solutions obtained are not small in size, see for instance [40], [41], [36], [37], [38], [28], [39]. The asymptotic stability of periodic solutions (also referred to as attainability property) has been also investigated in [27], [28], but it is only proved w.r. to the L 2 -norm and the rate of convergence provided is O(t −η ) for some constant η > 0. More recently Galdi and Hishida [23] proved the asymptotic stability for the Navier-Stokes equation with a traslation velocity term, by using the Lorentz spaces and they provided a rate of convergence which is essentially O(t − 1 2 +ε ). In the present paper we consider the Navier-Stokes equation on the d-dimensional torus with a small, quasi-periodic in time external force. We show the existence of smooth quasi-periodic solutions (which are also referred to as invariant tori) of small amplitude and we prove their orbital and asymptotic stability in H s for s large enough (at least larger than d/2 + 1). Furthermore the rate of convergence to the invariant torus, in H s , for t → +∞ is of order O(e −αt ) for any arbitrary α ∈ (0, 1). To the best of our knowledge, this is the first result of this kind. It is also worth to mention that the existence of quasi-periodic solutions, that is also referred to as KAM (Kolmogorov-Arnold-Moser) theory, for dispersive and hyperbolic-type PDEs is a more difficult matter, due to the presence of the so-called small divisors problem. The existence of time-periodic and quasi-periodic solutions of PDEs started in the late 1980s with the pioneering papers of Kuksin [33], Wayne [42] and Craig-Wayne [13], see also [34], [31] for generalizations to PDEs with unbounded nonlinearities. We refer to the recent review [7] for a complete list of references. Many PDEs arising from fluid dynamics like the water waves equations or the Euler equation are fully nonlinear or quasi-linear equations (the nonlinear part contains as many derivatives as the linear part). The breakthrough idea, based on pseudo-differential calculus and microlocal analysis, in order to deal with these kind of PDEs has been introduced by Iooss, Plotnikov and Toland [25] in the problem of finding periodic solutions for the water waves equation. The methods developed in [25], combined with a KAM-normal form procedure have been used to develop a general method for PDEs in one dimension, which allows to construct quasi-periodic solutions of quasilinear and fully nonlinear PDEs, see [2], [17], [11], [1] and references therein. The extension of KAM theory to higher space dimension d > 1 is a difficult matter due to the presence of very strong resonance-phenomena, often related to high multiplicity of eigenvalues. The fist breakthrough results in this directions (for equations with perturbations which do not contain derivatives) have been obtained by Eliasson and Kuksin [16] and by Bourgain [12] (see also , [9], Geng-Xu-You [24], Procesi-Procesi [35], Berti-Corsi-Procesi [10].) Extending KAM theory to PDEs with unbounded perturbations in higher space dimension is one of the main open problems in the field. Up to now, this has been achieved only in few examples, see [29], [15], [18], [6], [4], [5] and recently on the 3D Euler equation [3] which is the most meaningful physical example. For the Navier-Stokes equation, unlike in the aforementioned papers on KAM for PDEs, the existence of quasi-periodic solutions is not a small divisors problem and it can be done by using a classical fixed point argument. This is due to the fact that the Navier-Stokes equation is a parabolic PDE and the presence of dissipation avoids the small divisors. In the same spirit, it is also worth to mention [14], in which the authors investigate quasiperiodic solutions of nonlinear wave equations with dissipation. We also point out that the present paper is the first example in which the stability of invariant tori, in high Sobolev norms, is proved for all times (and it is even an asymptotic stability). This is possible since the presence of the dissipation allows to prove strong time-decay estimates from which one deduces orbital and asymptotic stability. In the framework of dispersive and hyperbolic PDEs, the orbital stability of invariant tori is usually proved only for large times by using normal form techniques. The first result in this direction has been proved in [26]. In the remaining part of the introduction, we sketch the main points of our proof. As we already explained above, the absence of small divisors is due to the fact that the Navier-Stokes equation is a parabolic PDE. More precisely, this fact is related to invertibility properties of the linear operator , the inverse of L ω gains two space derivatives, see Lemma 3.2. This is suffcient to perform a fixed point argument on the map Φ defined in (3.13) from which one deduces the existence of smooth quasi-periodic solutions of small amplitude. The asymptotic and orbital stability of quasi-periodic solutions (which are constructed in Section 3) are proved in Section 4. More precisely we show that for any initial datum u 0 which is δ-close to the quasi-periodic solution u ω (0, x) in H s norm (and such that u 0 − u ω (0, ·) has zero average), there exists a unique classical solution (u, p) such that , α ∈ (0, 1) for any t ≥ 0. This is exactly the content of Theorem 1.2, which easily follows from Proposition 4.1. This Proposition is proved also by a fixed point argument on the nonlinear map Φ defined in (4.31) in weighted Sobolev spaces E s (see (4.14)), defined by the norm u Es := sup 1. For any u 0 ∈ H s−1 (T d , R d ) with zero average and for any n ∈ N, α ∈ (0, 1), t > 0, one has for some constant C(n, α) > 0 (see Lemma 4.2). This estimate states that the heat propagator gains one-space derivative and exponential decay in time e −αt t − n 2 . Note that, without gain of derivatives on u 0 , the exponential decay is stronger, namely e −t , see Lemma 4.2-(i).

For any
for some constant C(α) > 0 (see Proposition 4.5). This estimate states that the integral term which usually appears in the Duhamel formula (see (4.31)) gains one space derivative w.r. to f (t, x) and keeps the same exponential decay in time as We also remark that the constants C(n, α), C(α) appearing in the estimates (1.7), (1.8) tend to ∞ when α → 1. This is the reason way it is not possible to get a decay O(e −t ) in the asymptotic stability estimate provided in Theorem 1.2. The latter two estimates allow to show in Proposition 4.9 that the map Φ defined in (4.31) is a contraction. The proof of Theorem 1.2 is then easily concluded in Section 4.3.
As a concluding remark, it is also worth to mention that our methods does not cover the zero viscosity limit µ → 0, where µ is the usual viscosity parameter in front of the laplacian (that we set for convenience equal to one). Indeed some constants in our estimates become infinity when µ → 0. Actually, it would be very interesting to study the singular perturbation problem for µ → 0 and to see if one is able to recover the quasi-periodic solutions of the Euler equation constructed in [3].
Acknowledgements. The author is supported by INDAM-GNFM. The author warmly thanks Dario Bambusi, Luca Franzoi, Thomas Kappeler and Alberto Maspero for their feedbacks.

Functional spaces
In this section we collect some standard technical tools which will be used in the proof of our results.
The following standard algebra lemma holds.
We also consider functions which are in L 2 T ν , L 2 (T d , R n ) . We can write the Fourier series of a function By expanding also the function u(ℓ, ·) in Fourier series, we get and hence we can write (2.5) For any σ, s ≥ 0, we define the Sobolev space Similarly to (2.1), one has that for u = (u 1 , . . . , (2.8) Moreover, the following standard algebra property holds.
Proof. By the Sobolev embedding (2.8), and using that the map R → T d , t → ωt is continuous, one has that u ω ∈ C 0 b R, H s (T d , R n ) and

Leray projector and some elementary properties of the Navier-Stokes equation
We introduce the space of zero-divergence vector fields where clearly the divergence has to be interpreted in a distributional sense. The L 2orthogonal projector on this subspace of L 2 (T d , R d ) is called the Leray projector and its explicit formula is given by where the inverse of the laplacian (on the space of zero average functions) (−∆) −1 is defined by (2.14) By expanding in Fourier series, the Leray projector L can be written as (2.15) By the latter formula, one immediately deduces some elementary properties of the Leray projector L. One has and for any Fourier multipier Λ, Λu(x) = ξ∈Z d Λ(ξ) u(ξ)e ix·ξ , the commutator For later purposes, we now prove the following Lemma.
Proof. Proof of (i). By integrating by parts, Proof of (ii). For u = (u 1 , . . . , u d ), v = (v 1 , . . . , v d ), the vector field u · ∇v is given by Then the claimed statement follows by (2.7) and the algebra Lemma 2.2.

Construction of quasi-periodic solutions
We look for quasi periodic solutions u ω (t, x), p ω (t, x) of the equation (1.1), oscillating with frequency ω = (ω 1 , . . . , ω ν ) ∈ R ν , namely we look for u ω (t, and by projecting on the space of zero divergence vector fields, one gets a closed equation for U of the form where we recall the definitions (2.12), (2.13). According to the splitting (2.10) and by applying the projectors π 0 , π ⊥ 0 to the equation (3.3) one gets the decoupled equations and Then, since ω is diophantine (see (1.3)) and using that (3.6) By (2.11) and using (1.3), one gets the estimate We now solve the equation (3.5) by means of a fixed point argument. To this aim, we need to analyze some invertibility properties of the linear operator and assume that g has zero divergence. Then there exists a unique u : Proof. By (2.5), we can write iω · ℓ + |j| 2 u(ℓ, j)e iℓ·ϕ e ij·x .
We now implement the fixed point argument for the equation (3.5) (to simplify notations we write U instead of U ⊥ ). For any σ, s, R ≥ 0, we define the ball and we define the nonlinear operator The following Proposition holds.
is a classical solution of (3.3) and it satisfies T ν ×T d U(ϕ, x) dϕ dx = 0. The unique solution with zero average in x of the equation (3.2) is given by Hence, P ∈ C 0 b T ν , H s 0 (T d , R d ) and The claimed estimate on P then follows since f σ,s−1 Note that if f has zero average in x, one has that The equation (3.4) reduces to ω · ∂ ϕ U 0 = 0. Hence the only solution U = U 0 + U ⊥ of (3.3) with zero average in x is the one where we choose U 0 = 0 and hence U = U ⊥ . The claimed statement has then been proved.

Orbital and asymptotic stability
We now want to study the Cauchy problem for the equation (1.1) for initial data which are close to the quasi-periodic solution (u ω , p ω ), where u ω (t, x) := U(ωt, x), p ω (t, x) := P (ωt, x) (4.1) and the periodic functions U ∈ C 1 T ν , H s (T d , R d ) , P ∈ C 0 T ν , H s 0 (T d , R d ) are given by Theorem 1.1. We then look for solutions which are perturbations of the quasi-periodic ones (u ω , p ω ), namely we look for solutions of the form Plugging the latter ansatz into the equation (1.1), one obtains an equation for v(t, x), q(t, x) of the form If we take the divergence in the latter equation we get the equation for the pressure q(t, x) By using the Leray projector defined in (2.13), we then get a closed equation for v of the form We prove the following Proposition 4.1. Let s > d/2 + 1, α ∈ (0, 1). Then there exists δ = δ(s, α, d, ν) ∈ (0, 1) small enough and C = C(s, α, d, ν) > 0 large enough, such that for any ε ∈ (0, δ) and for any initial The Proposition above will be proved by a fixed point argument in some weighted Sobolev spaces which take care of the decay in time of the solutions we are looking for. In the next section we shall exploit some decay estimates of the linear heat propagator which will be used in the proof of our result.
We fix α ∈ (0, 1) and for any s ≥ 0, we define the space (4.14) Clearly The following elementary lemma holds: Then the product u · ∇v ∈ E s and u · ∇v Es s u Es v C 0 , div(u) = 0 and v ∈ E s+1 . Then the product u · ∇v ∈ E s and u · ∇v Es s u C 0 Then u · ∇v ∈ E s and u · ∇v Es s u Es v E s+1 . Proof. The item (i) is very elementary and it follows in a straightforward way by the definition (4.14) and by recalling the estimate (2.18) on the Leray projector L. We prove the item (ii). By Lemma 2.4-(i), one has that u(t) · ∇v(t) has zero average in x. Moreover u · ∇v = u · ∇v 1 , . . . , u · ∇v d therefore, since s > d/2, by Lemma 2.1 and using (2.1) one has that for any t ∈ [0, +∞) (4.20) Passing to the supremum over t ≥ 0 in the left hand side of (4.20), we get the claimed statement. The item (iii) follows by similar arguments and the item (iv) follows by applying items (i) and (ii).
We now prove some estimates for the heat propagator e t∆ in the space E s . For t > 1, by applying Lemma 4.2-(ii) (for n = 1), one gets that Hence the claimed statement follows by (4.21), (4.22) passing to the supremum over t ≥ 0.
The main result of this section is the following Proposition Then u ∈ E s and u Es α f E s−1 . (4.24) The proof is split in several steps. The first step is to estimate the integral in (4.23) for any t ∈ [0, 1]. Lemma 4.6. Let t ∈ [0, 1], f ∈ E s−1 and u defined by (4.23). Then Proof. Let t ∈ [0, 1]. Then (4.25) By making the change of variables z = t − τ , one gets that and hence in view of (4.25), one gets for any t ∈ [0, 1], which is the claimed stetement.
For t > 1, we split the integral term in (4.23) as and we estimate separately the two terms in the latter formula. More precisely the first term is estimated in Lemma 4.7 and the second one in Lemma 4.8.
Proof. Let t > 1. One then has (4.26) By choosing n = 4 and by making the change of variables z = t − τ , on gets that The latter estimate, together with the estimate (4.26) imply the claimed statement. Proof. One has By making the change of variables z = t − τ , one gets The latter estimate, together with (4.27) imply the claimed statement.
Proof of Proposition 4.5.
Proof of Proposition 4.1 concluded. By Proposition 4.9, using the contraction mapping theorem there exists a unique v ∈ B s (δ) which is a fixed point of the map Φ in (4.31). By the functional equation v = Φ(v), one deduces in a standard way that and hence v is a classical solution of the equation (4.5). By (4.16) and using the trivial fact that ∆v E s−2 ≤ v Es  Therefore using that v ∈ B s (δ) ( v Es ≤ δ) and by (4.33), u ω C 0 t H s x s δ, one gets, for δ small enough, the estimate ∂ t v E s−2 s δ and the claimed statement follows by recalling (4.16).

Proof of Theorem 1.2
In view of Proposition 4.1, it remains only to solve the equation (4.4) for the pressure q(t, x). The only solution with zero average of this latter equation is given by q := (−∆) −1 div u ω · ∇v + v · ∇u ω + v · ∇v . (4.40) Using that (−∆) −1 diva Es a E s−1 for any a ∈ E s , one gets the inequality q Es u ω · ∇v E s−1 + v · ∇u ω E s−1 + v · ∇v E s−1 . Hence arguing as in (4.39), one deduces the estimate q Es s δ. The claimed estimate on q then follows by recalling (4.16) and the proof is concluded (recall that by (4.2), v = u − u ω , q = p − p ω ). Moreover f ′ (y) = y n−1 e −y n − ζy therefore f admits a global maximum at y = n/ζ. This implies that max y≥0 f (y) = f (n/ζ) = (n/ζ) n e −n and the lemma follows.