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

https://doi.org/10.1007/s10884-021-09944-w

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

Riccardo Montalto¹

Received: 17 October 2020 / Revised: 2 January 2021 / Accepted: 5 January 2021 / Published online: 27 January 2021

© The Author(s) 2021, corrected publication 2021

Abstract

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on thed-dimensional torusT^d, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable inH^s(forslarge enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for t→ +∞, with an exponential rate of convergenceO(e^−αt)for any arbitraryα∈(0,1). Keywords Fluid dynamics·Navier–Stokes equation·Quasi-periodic solutions· Asymptotic and orbital stability

Mathematics Subject Classification 37K55·35Q30·76D05

Contents

1 Introduction and Main Results. . . . 1342

2 Functional Spaces . . . . 1347

2.1 Leray Projector and Some Elementary Properties of the Navier–Stokes Equation. . . . 1348

3 Construction of Quasi-Periodic Solutions . . . . 1349

3.1 Proof of Theorem 1.1 . . . . 1352

4 Orbital and Asymptotic Stability . . . . 1353

4.1 Dispersive Estimates for the Heat Propagator. . . . 1354

4.2 Proof of Proposition 4.1. . . . 1358

4.3 Proof of Theorem 1.2 . . . . 1360

A Appendix. . . . 1360

References. . . . 1361

B

1 Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy

1 Introduction and Main Results

We consider the Navier–Stokes equation for an incompressible fluid on thed-dimensional torusT^d,d≥2,T:=R/2πZ,

∂tu−u+u· ∇u+ ∇p=εf(ωt,x)

divu=0 (1.1)

whereε ∈ (0,1)is a small parameter, the frequency ω = (ω1, . . . , 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 fieldu =(u₁, . . . ,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

T^ν×T^d f(ϕ,x)dϕd x=0. (1.2) The purpose of the present paper is to show the existence and the stability of smooth quasi- periodic solutions of the Eq. (1.1). More precisely we show that if f is a sufficiently regular vector field satisfying (1.2), forεsufficiently small and forω∈R^νDiophantine¹, i.e.

|ω·| ≥ γ

||^ν, ∀∈Z^ν\{0},

for some γ ∈(0,1), (1.3)

then the Eq. (1.1) admits smooth quasi-periodic solutions (which are referred to also as invariant tori)u_ω(t,x) = U(ωt,x), p_ω(t,x) = P(ωt,x),U : T^ν ×T^d → R^d, P : T^ν×T^d →Rof sizeO(ε), oscillating with the same frequencyω∈R^νof the forcing term.

If the forcing term has zero-average inx, i.e.

T^d f(ϕ,x)d x=0, ∀ϕ∈T^ν (1.4)

then the result holds for any frequency vectorω∈R^ν, without requiring any non-resonance condition. Furthermore, we show also the orbital and the asymptotic stability of these quasi- periodic solutions in high Sobolev norms. More precisely, for any sufficiently regular initial datum which isδ-close to the invariant torus (w.r. to theH^s topology), the corresponding solution of (1.1) is global in time and it satisfies the following properties.

• Orbital stabilityFor all timest ≥ 0, the distance in H^s between the solution and the invariant torus is of orderO(δ).

• Asymptotic stabilityThe solution converges asymptotically to the invariant torus in high Sobolev norm · H_x^s ast→ +∞, with a rate of convergence which is exponential, i.e.

O(e^−αt), for any arbitraryα∈(0,1).

In order to state precisely our main results, we introduce some notations. For any vector a=(a1, . . . ,a_p)∈R^p, we denote by|a|its Euclidean norm, namely|a| :=

a₁²+ · · · +a²_p. Letd,n∈Nand a functionu∈L²(T^d,Rⁿ). Thenu(x)can be expanded in Fourier series

u(x)=

ξ∈Z^d

u(ξ)e^ix·ξ

1It is well known that a.e. frequency inR^ν(w.r. to the Lebesgue measure) is diophantine.

where its Fourier coefficientsu(ξ)are defined by u(ξ):= 1

(2π)^d

T^du(x)e^−ix·ξd x, ∀ξ∈Z^d.

For anys≥0, we denote byH^s(T^d,Rⁿ)the standard Sobolev space of functionsu:T^d → Rⁿequipped by the norm

uH_x^s :=

ξ∈Z^d

ξ^2s|u(ξ)|²¹

2, ξ :=max{1,|ξ|}. (1.5)

We also define the Sobolev space of functions with zero average H₀^s(T^d,Rⁿ):= u∈H^s(T^d,Rⁿ):

T^du(x)d x=0

. (1.6)

Moreover, given a Banach space(X, · X)and an intervalI⊆R, we denote byC_b⁰(I,X) the space of bounded, continuous functionsu:I→X, equipped with the sup-norm

u_Ct⁰_X:=sup

t∈Iu(t)X.

For any integerk≥1,C_b^k(I,X)is the space ofk-times differentiable functionsu:I→X with continuous and bounded derivatives equipped with the norm

u_C^k

tX :=maxn≤k∂_tⁿu_C⁰

tX.

In a similar way we define the spacesC⁰(T^ν,X),C^k(T^ν,X),k ≥1 and the corresponding norms · _Cϕ⁰X, · _Cϕ^kX(whereT^νis theν-dimensional torus). We also denote byC^N(T^ν× T^d,R^d)the space ofN-times continuously differentiable functionsT^ν×T^d →R^dequipped with the standard_C^N norm · _C^N.

Notation.Throughout the whole paper, the notationABmeans that there exists a constant Cwhich can depend on the number of frequenciesν, the dimension of the torusd, the constant γappearing in the diophantine condition (1.3) and on theC^Nnorm of the forcing termf_C^N such that A≤B. Givennpositive real numberss₁, . . . ,s_n >0, we write As1,...,sn Bif there exists a constantC =C(s₁, . . . ,s_n) >0 (eventually depending also ond, ν, γ,f_C^N) such thatA≤C B.

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 > ^3ν₂ +s+2, ω ∈R^ν diophantine (see1.3) and assume that the forcing term f is in_C^N(T^ν×T^d,R^d) and it satisfies(1.2). Then there existsε0 = ε0(f,s,d, ν) ∈ (0,1)small enough and a constant C = C(f,s,d, ν) > 0large enough such that for anyε ∈ (0, ε0) there exist U∈C¹(T^ν,H^s(T^d,R^d)), P∈C⁰(T^ν,H^s(T^d,R))satisfying

T^ν×T^dU(ϕ,x)dϕd x=0,

T^d P(ϕ,x)d x=0, ∀ϕ∈T^ν

such that(u_ω(t,x),p_ω(t,x)) := (U(ωt,x),P(ωt,x))solves the Navier–Stokes equation (1.1)and

U_Cϕ¹H_x^s,P_Cϕ⁰H_x^s ≤Cε .

If the forcing term f has zero space average, i.e. it satisfies(1.4), then the same statement holds for any frequency vectorω∈R^νand U(ϕ,x)satisfies

T^dU(ϕ,x)d x=0, ∀ϕ∈T^ν.

Theorem 1.2 (Stability)Letα∈(0,1), s >d/2+1, N > ^3ν₂ +s+2, u_ω, p_ωbe given in Theorem1.1. Then there existsδ = δ(f,s, α,d, ν)∈ (0,1)small enough and a constant C = C(f,s, α,d, ν) > 0large enough such that for ε ≤ δ and for any initial datum u₀∈H^s(T^d,R^d)satisfying

u0−u_ω(0,·)H_x^s ≤δ,

T^d

u0(x)−u_ω(0,x) d x =0

there exists a unique global solution(u,p)of the Navier–Stokes equation(1.1)with initial datum u(0,x)=u₀(x)which satisfies

u∈C_b⁰

[0,+∞),H^s(T^d,R^d)

∩C_b¹

[0,+∞),H^s−2(T^d,R^d) , p∈C⁰_b

[0,+∞),H₀^s(T^d,R) ,

T^d

u(t,x)−u_ω(t,x)

d x=0, ∀t ≥0, u(t,·)−u_ω(t,·)H_x^s,∂tu(t,·)−∂tu_ω(t,·)_Hs−2

x ,p(t,·)−p_ω(t,·)H_x^s ≤Cδe^−αt for any t≥0.

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 [28,36–41]. The asymptotic stability of periodic solutions (also referred to asattainability property) has been also investigated in [27,28], but it is only proved with respect to theL²- norm and the rate of convergence provided isO(t^−η)for some constantη >0. More recently Galdi and Hishida [23] proved the asymptotic stability for the Navier–Stokes equation with a translation velocity term, by using the Lorentz spaces and they provided a rate of convergence which is essentiallyO(t^−12+ε). In the present paper we consider the Navier–Stokes equation on thed-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 inH^sforslarge enough (at least larger thand/2+1). Furthermore the rate of convergence to the invariant torus, in H^s, fort → +∞is of orderO(e^−αt)for any arbitraryα ∈ (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-calledsmall 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 [43] and Craig-Wayne [13], see also [31,34]

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 micro-local 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 aKAM-normal formprocedure have been used to develop a general method for PDEs in one dimension, which allows to constructquasi- periodicsolutions of quasilinear and fully nonlinear PDEs, see [1,2,11,17] and references therein. The extension of KAM theory to higher space dimensiond>1 is a difficult matter due to the presence of very strong resonance-phenomena, often related to high multiplicity of eigenvalues. The first 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 Berti-Bolle [8,9], 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 [4–6,15,18,29] 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 [14,42], in which the authors investigate quasi-periodic solutions of some PDEs with singular damping, in which the small divisors problem is avoided thanks to this damping term. 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 forlarge, but finite, timesby 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 operatorL_ω:=ω·∂_ϕ−(whereω·∂_ϕ :=_ν

i=0ωi∂_ϕi) acting on Sobolev spaces of functionsu(ϕ,x),(ϕ,x)∈T^ν×T^dwith zero average w.r. tox. Since the eigenvalues ofL_ωare iω·+ |j|²,∈Z^ν, j∈Z^d\{0}, the inverse ofL_ωgains two space derivatives, see Lemma3.2. This is suffcient to perform a fixed point argument on the map 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 Sect.3) are proved in Sect.4. More precisely we show that for any initial datumu₀which isδ-close to the quasi-periodic solutionu_ω(0,x)inH^snorm (and such that u₀−u_ω(0,·)has zero average), there exists a unique solution(u,p)such that

u(t,·)−u_ω(t,·)H_x^s =O(δe^−αt), p(t,·)−p_ω(t,·)H_x^s =O(δe^−αt) , α∈(0,1)

for anyt≥0. This is exactly the content of Theorem1.2, which easily follows from Propo- sition4.1. This Proposition is proved also by a fixed point argument on the nonlinear map defined in (4.31) in weighted Sobolev spaces_Es(see (4.14)), defined by the norm

u_Es :=sup

t≥0e^αtu(t,·)H_x^s

whereα∈(0,1)is a fixed constant. The fixed point argument relies on somedispersive-type estimatesfor the heat propagatore^t, which are proved in Sect.4.1. The key estimates are the following.

1. For anyu₀∈H^s−1(T^d,R^d)with zero average and for anyn∈N,α∈(0,1),t>0, one has

e^tu0H_x^s ≤C(n, α)t⁻ⁿ²e^−αtu0_Hs−1

x (1.7)

for some constant C(n, α) > 0 (see Lemma 4.2). This estimate states that the heat propagator gains one-space derivativeand exponential decay in time e^−αtt⁻ⁿ². Note that, without gain of derivatives onu₀, the exponential decay is stronger, namelye^−t, see Lemma4.2-(i).

2. For any f ∈Es−1

_t

0

e^(t−τ)f(τ,·)dτ

H_x^s ≤C(α)e^−αtf_Es−1 (1.8) for some constantC(α) >0 (see Proposition4.5). 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 f(t,x).

We also remark that the constantsC(n, α),C(α)appearing in the estimates (1.7), (1.8) tend to∞whenα →1. This is the reason why it is not possible to get a decayO(e^−t)in the asymptotic stability estimate provided in Theorem1.2.

The latter two estimates allow to show in Proposition4.9that the mapdefined in (4.31) is a contraction. The proof of Theorem1.2is then easily concluded in Sect.4.3.

It is also worth to mention that our methods does not cover the zero viscosity limitμ→0, whereμis 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 whenμ→0. Actually, it would be very interesting to study thesingular perturbation problemforμ→0 and to see if one is able to recover the quasi-periodic solutions of the Euler equation constructed in [3].

As a concluding remark, we mention that the methods used in this paper also apply to other parabolic-type equations with some technical modifications. For instance, one could prove the existence of quasi-periodic solutions (Theorem1.1) for a general fully nonlinear parabolic type equation of the form

∂tu−u+mu+N(x,u,∇u,∇²u)=εf(ωt,x), m>0

whereN is a smooth nonlinearity depending also on the second derivatives ofuand which is at least quadratic w.r. to(u,∇u,∇²u). Indeed, as we explained above, this can be done by a fixed point argument, by inverting the operatorL_ω:=ω·∂ϕ−+m. The inverse of this operator gains two space derivatives and hence it compensates the fact that the nonlinearity has a loss of two space-derivatives.

We prefer in this paper to focus on the Navier–Stokes equation for clarity of exposition and since it is a very important physical model.

2 Functional Spaces

In this section we collect some standard technical tools which will be used in the proof of our results. Foru=(u₁, . . . ,u_n)∈H^s(T^d,Rⁿ), one has

uH_x^s maxi=1,...,nuiH_x^s. (2.1) The following standard algebra lemma holds.

Lemma 2.1 Let s>d/2and u, v∈H^s(T^d,Rⁿ). Then u·v∈H^s(T^d,R)(where·denotes the standard scalar product onRⁿ) andu·vH_x^s suH_x^svH_x^s.

We also consider functions

T^ν→L²(T^d,Rⁿ), ϕ→u(ϕ,·) which are in L²

T^ν,L²(T^d,Rⁿ)

. We can write the Fourier series of a function u ∈ L²

T^ν,L²(T^d,Rⁿ) as

u(ϕ,·)=

∈Z^ν

u(,·)e^i·ϕ (2.2)

where

u(,·):= 1 (2π)^ν

T^νu(ϕ,·)e^−i·ϕdϕ∈L²(T^d,Rⁿ), ∈Z^ν. (2.3) By expanding also the functionu(,·)in Fourier series, we get

u(,x)=

j∈Z^d

u(,j)e^ij·x,

u(,j):= 1 (2π)^ν+d

T^ν+du(ϕ,x)e^−i·ϕe^−ij·xdϕd x, (,j)∈Z^ν×Z^d

(2.4)

and hence we can write

u(ϕ,x)=

∈Z^ν

j∈Z^d

u(,j)e^i·ϕe^ij·x. (2.5)

For anyσ,s≥0, we define the Sobolev spaceH^σ

T^ν,H^s(T^d,Rⁿ)

as the space of functions u∈L²

T^ν,L²(T^d,Rⁿ)

equipped by the norm u_σ,s ≡ uH_ϕ^σH_x^s :=

∈Z^ν

2σu()H_x^s

¹₂

=

∈Z^ν

j∈Z^d

2σ j^2s|u(,j)|²¹₂ . (2.6) Similarly to (2.1), one has that foru=(u1, . . . ,u_n)∈H^σ

T^ν,H^s(T^d,Rⁿ)

uσ,s max_i=1,...,nu_iσ,s. (2.7)

Ifσ > ν/2, then H^σ

T^ν,H^s(T^d,Rⁿ)

is compactly embedded in C⁰

T^ν,H^s(T^d,Rⁿ) ,

and u_Cϕ⁰_Hx^s _σ uH_ϕ^σH_x^s. (2.8)

Moreover, the following standard algebra property holds.

Lemma 2.2 Let σ > ^ν₂, s > ^d₂, u, v ∈ H^σ

T^ν,H^s(T^d,Rⁿ)

. Then u · v ∈ H^σ

T^ν,H^s(T^d,R)

andu·vσ,sσ,suσ,svσ,s.

For anyu∈L²(T^d,Rⁿ)we define the orthogonal projectionsπ0andπ₀^⊥as π0u:= 1

(2π)^d

T^du(x)d x=u(0) and π₀^⊥u:=u−π0u=

ξ∈Z^d\{0}

u(ξ)e^ix·ξ. (2.9)

According to (2.9), (2.5), every functionu∈L²

T^ν,L²(T^d,Rⁿ)

can be decomposed as u(ϕ,x)=u₀(ϕ)+u_⊥(ϕ,x) ,

u₀(ϕ):=π0u(ϕ)=

∈Z^ν

u(,0)e^i·ϕ, u_⊥(ϕ,x):=π₀^⊥u(ϕ,x)=

∈Z^ν

j∈Z^d\{0}

u(,j)e^i·ϕe^ij·x.

(2.10)

Clearly ifu∈H^σ

T^ν,H^s(T^d,Rⁿ)

,σ,s≥0, then

u₀∈H^σ(T^ν,R^d) and u0_σ≤ u_σ,0≤ u_σ,s, u_⊥∈H^σ

T^ν,H₀^s(T^d,Rⁿ)

and u⊥σ,s ≤ uσ,s, u_σ,s= u0_σ+ u_⊥σ,s.

(2.11)

We also prove the following lemma that we shall apply in Sect.4.

Lemma 2.3 Letσ > ν/2, U ∈ H^σ

T^ν,H^s(T^d,Rⁿ)

andω ∈R^ν. Defining u_ω(t,x) :=

U(ωt,x),(t,x) ∈ R×T^d, one has that u_ω ∈ C_b⁰

R,H^s(T^d,Rⁿ)

andu_ωC⁰_tHs

x σ

Uσ,s.

Proof By the Sobolev embedding (2.8), and using that the mapR→T^d,t→ωtis contin- uous, one has thatu_ω∈C_b⁰

R,H^s(T^d,Rⁿ) and u_ωC⁰

tH_x^s ≤ U_Cϕ⁰H_x^s σ UH_ϕ^σH_x^s.

2.1 Leray Projector and Some Elementary Properties of the Navier–Stokes Equation We introduce the space of zero-divergence vector fields

D0(T^d):= u∈L²(T^d,R^d):div(u)=0

(2.12) where clearly the divergence has to be interpreted in a distributional sense. TheL²-orthogonal projector on this subspace ofL²(T^d,R^d)is called theLerayprojector and its explicit formula is given by

L:L²(T^d,R^d)→D0(T^d) ,

L(u):=u+ ∇(−)⁻¹div(u) (2.13)

where the inverse of the laplacian (on the space of zero average functions)(−)⁻¹is defined by

(−)⁻¹u(x):=

ξ∈Z^d\{0}

1

|ξ|²u(ξ)e^ix·ξ. (2.14) By expanding in Fourier series, the Leray projectorLcan be written as

L(u)(x)=u(x)+

ξ∈Z^d\{0}

ξ

|ξ|²ξ·u(ξ)e^ix·ξ. (2.15) By the latter formula, one immediately deduces some elementary properties of the Leray projectorL. One has

T^dL(u)(x)d x=

T^du(x)d x, ∀u∈L²(T^d,R^d) (2.16) and for any Fourier multiplier,u(x)=

ξ∈Z^d(ξ)u(ξ)e^ix·ξ, the commutator [L, ] =L−L=0. (2.17) Moreover

L(u)H_x^s uH_x^s, ∀u∈H^s(T^d,R^d) , L(u)_σ,s u_σ,s, ∀u∈H^σ

T^ν,H^s(T^d,R^d)

. (2.18)

For later purposes, we now prove the following Lemma.

Lemma 2.4 (i) Let u, v∈H¹(T^d,R^d)and assume thatdiv(u)=0, then u· ∇v,L(u· ∇v) have zero average.

(ii) Letσ > ν/2, s > d/2, u ∈ H^σ

T^ν,H^s(T^d,R^d)

,v ∈ H^σ

T^ν,H^s+1(T^d,R^d) . Then u· ∇v∈H^σ

T^ν,H^s(T^d,R^d)

andu· ∇v_σ,sσ,su_σ,sv_σ,s+1. Proof of(i) By integrating by parts,

T^dL(u· ∇v)d x^(2.16)=

T^du· ∇vd x= −

T^ddiv(u)vd x=0.

Proof of(ii) Foru=(u1, . . . ,ud),v=(v1, . . . , vd), the vector fieldu· ∇vis given by u· ∇v=

u· ∇v1,u· ∇v2, . . . ,u· ∇vd

.

Then the claimed statement follows by (2.7) and the algebra Lemma2.2.

3 Construction of Quasi-Periodic Solutions

We look for quasi periodic solutionsu_ω(t,x), p_ω(t,x)of the Eq. (1.1), oscillating with frequencyω=(ω1, . . . , ω_ν)∈R^ν, namely we look foru_ω(t,x):=U(ωt,x),p_ω(t,x):=

P(ωt,x)whereU:T^ν×T^d →R^dandP:T^ν×T^d →Rare smooth functions. This leads to solve a functional equation forU(ϕ,x),P(ϕ,x)of the form

ω·∂_ϕU−U+U· ∇U+ ∇P=εf(ϕ,x)

divU =0. (3.1)

If we take the divergence of the first equation in (3.1), one gets P=div

εf −U · ∇U

(3.2) and by projecting on the space of zero divergence vector fields, one gets a closed equation forUof the form

ω·∂ϕU−U+L(U· ∇U)=εL(f), U(ϕ,·)∈D0(T^d) (3.3) where we recall the definitions (2.12), (2.13). According to the splitting (2.10) and by applying the projectorsπ0, π₀^⊥to the Eq. (3.3) one gets the decoupled equations

ω·∂ϕU₀(ϕ)=εf₀(ϕ) (3.4)

and

ω·∂ϕU_⊥−U_⊥+L(U_⊥· ∇U_⊥)=εL(f_⊥) . (3.5) Then, sinceωis diophantine (see1.3) and using that

T^ν f0(ϕ)dϕ=

T^ν×T^d f(ϕ,x)dϕd x^(1.2)= 0,

(f(0,0)=0) the averaged equation (3.4) can be solved explicitely by setting U0(ϕ):=(ω·∂ϕ)⁻¹f0(ϕ)=

∈Z^ν\{0}

f(,0)

iω· e^i·ϕ. (3.6)

By (2.11) and using (1.3), one gets the estimate

U0σ ≤εγ⁻¹f0σ+ν ≤εγ⁻¹fσ+ν,0. (3.7) Remark 3.1 (Non resonance conditions)The diophantine condition (1.3) on the frequency vectorωis used only to solve the averaged equation (3.4). In order to solve the Eq. (3.5) on the space of zero average functions (with respect tox) no resonance conditions are required.

We now solve the Eq. (3.5) by means of a fixed point argument. To this end, we need to analyze some invertibility properties of the linear operator

L_ω:=ω·∂ϕ−. (3.8)

Lemma 3.2 (Invertibility ofL_ω)Letσ,s≥0, g∈H^σ

T^ν,H₀^s(T^d,R^d)

and assume that g has zero divergence. Then there exists a unique u:=L⁻¹_ω g∈H^σ

T^ν,H₀^s+2(T^d,R^d) with zero divergence which solves the equation L_ωu=g. Moreover

uσ,s+2 ≤ gσ,s. (3.9)

Proof By (2.5), we can write L_ωu(ϕ,x)=

∈Z^ν

j∈Z³\{0}

iω·+ |j|²

u(,j)e^i·ϕe^ij·x.

Note that since j=0, one has that

|iω·+ |j|²| =

|ω·|²+ |j|⁴≥ |j|². (3.10)

Hence, the equationL_ωu=gadmits the unique solution with zero space average given by u(ϕ,x):=L⁻_ω¹g(ϕ,x)=

∈Z^ν

j∈Z^d\{0}

g(,j)

iω·+ |j|²e^i·ϕe^ij·x (3.11) Clearly if div(g)= 0 and then also div(u)= 0. We now estimateu_σ,s+2. According to (2.6), (3.11), one has

u²_σ,s+2=

∈Z^ν

j∈Z^d\{0}

2σ j^2(s+2) |g(,j)|²

|iω·+ |j|²|²

(3.10)

≤

∈Z^ν

j∈Z^d\{0}

2σ|j|^2(s+2)|j|⁻⁴|g(,j)|²= g²_σ,s

which proves the claimed statement.

We now implement the fixed point argument for the Eq. (3.5) (to simplify notations we write Uinstead ofU_⊥). For anyσ,s,R≥0, we define the ball

B_σ,s(R):= U∈H^σ

T^ν,H₀^s(T^d,R^d)

:div(U)=0, U_σ,s ≤R

. (3.12)

and we define the nonlinear operator (U):=L⁻¹_ω L

εf −U· ∇U

, U ∈B_σ,s(R) . (3.13) The following Proposition holds.

Proposition 3.3 (Contraction for)Letσ > ν/2, s >d/2+1, f ∈ C^N(T^ν×T^d,R^d), N > σ+s−2. Then there exists a constant C_∗=C_∗(f, σ,s) >0large enough andε0= ε0(f, σ,s)∈(0,1)small enough, such that for anyε ∈(0, ε0), the map:Bσ,s(C_∗ε)→ Bσ,s(C∗ε)is a contraction.

Proof LetU ∈ Bσ,s(C∗ε). We apply Lemmata2.4-(i),3.2from which one immediately

deduces that

T³(U)d x=0, div

(U)

=0. (3.14)

Moreover

(U)_σ,s=L⁻_ω¹L

εf −U· ∇U

σ,s

(2.18),(3.9)

εf −U · ∇U

σ,s−2

εfσ,s−2+ U· ∇Uσ,s−1.

Note that since f ∈C^N with N > σ +s−2, one has thatf_σ,s−2 f_C^N. In view of Lemma2.4-(ii), using thatσ > ν/2,s−1>d/2, one gets that

(U)_σ,s≤C(f,s, σ )

ε+ U_σ,s−1U_σ,s

≤C(f,s, σ )

ε+ U²_σ,s for some constantC(f,s, σ ) >0. Using thatUσ,s ≤C_∗ε, one gets that

(U)σ,s ≤C(f,s, σ )ε+C(f,s, σ )C_∗²ε²≤C_∗ε provided

C_∗≥2C(f,s, σ ) and ε≤ 1 2C(f,s, σ )C∗.

Hence:B_σ,s(C_∗ε)→B_σ,s(C_∗ε). Now letU₁,U₂∈B_σ,s(C_∗ε)and we estimate (U1)−(U2)=L⁻¹_ω L

U₁· ∇U1−U₂· ∇U2

.

One has that

(U1)−(U2)_σ,s ≤L⁻_ω¹L

(U1−U2)· ∇U1

σ,s+L⁻_ω¹L

U2· ∇(U1−U2)

σ,s (2.18),(3.9) ,Lemma2.4

s,σ U1−U2_σ,sU1_σ,s+ U1−U2_σ,sU2_σ,s

≤C(s, σ )

U₁σ,s+ U₂σ,s

U₁−U₂σ,s

for some constantC(s, σ ) >0. SinceU1,U2∈B_σ,s(C_∗ε), one then has that (U1)−(U2)_σ,s≤2C(s, σ )C_∗εU1−U2_σ,s≤ 1

2U1−U2_σ,s

providedε≤ _{4C(s,σ )C}¹ _∗ .Henceis a contraction.

3.1 Proof of Theorem1.1

Proposition 3.3 implies that for σ > ν/2, s > ^d₂ +1, there exists a unique U_⊥ ∈ H^σ(T^ν,H₀^s(T^d,R^d)),U_⊥σ,s σ,s ε which is a fixed point of the map defined in (3.13). We fixσ:=ν/2+2 andN > ^3ν₂ +s+2. By the Sobolev embedding property (2.8), sinceσ−1> ν/2, one gets that

U_⊥∈C_b¹

T^ν,H₀^s(T^d,R^d)

, U_⊥Cϕ¹_Hx^s s ε (3.15) andU_⊥is a solution of the Eq. (3.5). Similarly, by recalling (3.4), (3.6), (3.7), one gets that

U₀∈C¹(T^ν,R^d), U0_C¹_ϕ ≤εγ⁻¹f3ν

2+2,0≤εγ⁻¹f_C^N,

T^νU₀(ϕ)dϕ=0 (3.16) andU₀ is a solution of the Eq. (3.4). HenceU = U₀+U_⊥ ∈ C¹

T^ν,H^s(T^d,R^d) is a solution of (3.3) and it satisfies

T^ν×T^dU(ϕ,x)dϕd x =0. The unique solution with zero average inxof the Eq. (3.2) is given by

P:=(−)⁻¹div

U· ∇U−εf .

Hence,P∈C_b⁰

T^ν,H₀^s(T^d,R^d) and P_Cϕ⁰H_x^s

(2.8),σ=ν/2+2

Pσ,sσ,sεfσ,s−1+ U· ∇Uσ,s−1 Lemma2.4

σ,s εfσ,s−1+ U²_σ,s.

The claimed estimate onPthen follows sincefσ,s−1f_C^N,Uσ,s≤C_∗ε. Note that if f has zero average inx, one has that

f0(ϕ)=π0f(ϕ)= 1 (2π)^d

T^d f(ϕ,x)d x=0, ∀ϕ∈T^ν.

The Eq. (3.4) reduces toω·∂ϕU₀=0. Hence the only solutionU =U₀+U_⊥of (3.3) with zero average inx is the one where we chooseU₀ = 0 and henceU =U_⊥. The claimed statement has then been proved.

4 Orbital and Asymptotic Stability

We now want to study the Cauchy problem for the Eq. (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 functionsU ∈C¹

T^ν,H^s(T^d,R^d)

,P∈C⁰

T^ν,H₀^s(T^d,R^d)

are given by Theorem1.1. 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

u(t,x)=u_ω(t,x)+v(t,x), p(t,x)=p_ω(t,x)+q(t,x) . (4.2) Plugging the latter ansatz into the Eq. (1.1), one obtains an equation forv(t,x),q(t,x)of

the form

∂tv−v+u_ω· ∇v+v· ∇u_ω+v· ∇v+ ∇q=0

div(v)=0. (4.3)

If we take the divergence in the latter equation we get the equation for the pressureq(t,x)

−q=div

u_ω· ∇v+v· ∇u_ω+v· ∇v

. (4.4)

By using the Leray projector defined in (2.13), we then get a closed equation forvof the

form

∂tv−v+L

u_ω· ∇v+v· ∇u_ω+v· ∇v

=0

div(v)=0. (4.5)

We prove the following

Proposition 4.1 Let s >d/2+1,α ∈(0,1). Then there existsδ =δ(s, α,d, ν)∈ (0,1) small enough and C=C(s, α,d, ν) >0large enough, such that for anyε∈(0, δ)and for any initial datumv0∈H₀^s(T^d,R^d)withv0H_x^s ≤δ, there exists a unique global solution

v∈C_b⁰

[0,+∞),H₀^s(T^d,R^d)

∩C¹_b

[0,+∞),H₀^s−2(T^d,R^d)

(4.6) of the Eq.(4.5)which satisfies

v(t,·)H_x^s,∂tv(t,·)_Hs−2

x ≤Cδe^−αt, ∀t ≥0. (4.7)

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.