HAL Id: hal-00962254

HAL Id: hal-00962254

https://hal.archives-ouvertes.fr/hal-00962254v1

Preprint submitted on 20 Mar 2014 (v1), last revised 15 Jan 2015 (v2)

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Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes system by a finite-dimensional

force

Vahagn Nersesyan

To cite this version:

Vahagn Nersesyan. Approximate controllability of Lagrangian trajectories of the 3D Navier-Stokes

system by a finite-dimensional force. 2014. �hal-00962254v1�

Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a

finite-dimensional force

Vahagn Nersesyan

Abstract

In the Eulerian approach, the motion of an incompressible fluid is usu- ally described by the velocity field which is given by the Navier–Stokes sys- tem. The velocity field generates a flow in the space of volume-preserving diffeomorphisms. The latter plays a central role in the Lagrangian de- scription of a fluid, since it allows to identify the trajectories of individual particles. In this paper, we show that the velocity field of the fluid and the corresponding flow of diffeomorphisms can be simultaneously approx- imately controlled using a finite-dimensional external force. The proof is based on some methods from the geometric control theory introduced by Agrachev and Sarychev.

AMS subject classifications: 35Q30, 93B05.

Keywords: Navier–Stokes system, Agrachev–Sarychev method, approx- imate controllability

Contents

0 Introduction 2

1 Preliminaries 7

1.1 Particle trajectories. . . 7 1.2 Existence of strong solutions . . . 10

2 Main results 12

2.1 Approximate controllability of the NS system . . . 12 2.2 Examples of saturating spaces. . . 15 2.2.1 Saturating spaces associated with the generators ofZ³ . . 15 2.2.2 Controls with two vanishing components. . . 21 2.2.3 6-dimensional exemple . . . 22

∗Laboratoire de Math´ematiques, UMR CNRS 8100, Universit´e de Versailles-Saint-Quentin- en-Yvelines, F-78035 Versailles, France

3 Proof Theorem 2.5 24

Bibliography 28

0 Introduction

The motion of an incompressible fluid is described by the following Navier–

Stokes (NS) system

˙

u−ν∆u+hu,∇iu+∇p=f(t, x), divu= 0, (0.1)

u(0) =u0, (0.2)

where ν > 0 is the kinematic viscosity, u = (u1(t, x), u2(t, x), u3(t, x)) is the velocity field of the fluid,p=p(t, x) is the pressure, andf is an external force.

Throughout this paper, we shall assume that the space variablex= (x1, x2, x3) belongs to the torusT³=R³/2πZ³.

The well-posedness of the 3D NS system (0.1) is a famous open problem.

Given a smooth data (u0, f), the existence and uniqueness of a smooth solution is known to hold only locally in time. One can establish global existence in the case of a small data. Global existence for a large data holds in the case of weak solution, but in that case the uniqueness is open.

The flow generated by a sufficiently smooth velocity field u gives the La- grangian trajectories of the fluid:

˙

x=u(t, x), x(0) =x0∈T³. (0.3) Since the fluid is assumed to be incompressible, for anyt≥0, the mappingφ^u_t : x07→x(t) belongs to the group SDiff(T³) of orientation and volume preserving diffeomorphisms on T³ isotopic to the identity. This group is often referred as configuration spaceof the fluid (cf. [AK98, KW09]). Thus for a sufficiently smooth data, we have a path (u(t), φ^u_t), which is defined locally in time, and its approximate controllability is the main issue addressed in this paper. We shall assume that the external force is of the following form

f(t, x) =h(t, x) +η(t, x),

wherehis the fixed part of the force (given function) and η is a control force.

To state the main result of this paper, we need to introduce some notation. Let us define the space

H :={u∈L²(T³,R³) : divu= 0, Z

T³

u(x)dx= 0}, (0.4) and denote by Π the orthogonal projection ontoH inL²(T³,R³). Consider the projection of system (0.1) ontoH:

˙

u+Lu+B(u) =h(t, x) +η(t, x), (0.5)

where L = −∆ is the Stokes operator and B(u) := Π(hu,∇iu). Let us set H_σ^k :=H^k(T³,R³)∩H, whereH^k(T³,R³) is the space of vector functionsv = (v1, v2, v3) with components in the usual Sobolev space of orderkonT³. LetE be subset of H. We shall say that system (0.5) is approximately controllable by anE-valued control, if for any ν >0, k ≥ 3, ε > 0, T > 0, u0, u1 ∈ H_σ^k, h∈L²(JT, H_σ^k−1), andψ∈SDiff(T³), there is a controlη ∈L²([0, T], E) and a solutionuof (0.5), (0.2) satisfying

ku(T)−u1kH^k(T³)+kφ^u_T −ψkC¹(T³)< ε.

The following theorem is a simplified version of our main result (see Section2.1).

Main Theorem. There is a finite-dimensional subspaceE⊂H such that(0.5) is approximately controllable by an E-valued control.

Roughly speaking, this shows that, using a finite-dimensional external force, one can drive the fluid flow (which starts at the identity) arbitrarily close to any configuration ψ ∈ SDiff(T³). Moreover, near the final position ψ(x), the particle starting fromxwill have approximately the prescribed velocityv1(x) :=

u1(ψ(x)).

We give some explicit examples of finite-dimensional subspaces E which ensure the above approximate controllability property. For instance, for anyℓ∈ Z³, let {l(ℓ), l(−ℓ)}be an arbitrary orthonormal basis in {x∈R³:hx, ℓi= 0}. We show that our problem is controllable byη taking values in a space of the form

E=E(K) := span{l(±ℓ) coshℓ, xi, l(±ℓ) sinhℓ, xi:ℓ∈ K}, K ⊂Z³ (0.6) if and only ifK is a generator ofZ³(i.e., any a∈Z³ is a finite linear combina- tion of the elements ofK with integer coefficients). The simplest example of a generator ofZ³ is

K={(1,0,0),(0,1,0),(0,0,1)},

in which case dimE(K) = 12. We also establish approximate controllability of the system in question by controls having two vanishing components. More precisely, the spaceE can be chosen of the form

E= Π{(0,0,1)ζ:ζ∈ H}, (0.7) where

H:= span{sinhm, xi,coshm, xi:m∈ K}

andK:={(1,0,0),(0,1,0),(1,0,1),(0,1,1)}(i.e., dimE= 8). In (2.32) an ex- ample of a 6-dimensional subspace is given which guarantees the controllability of the 3D NS system.

The strategy of the proof of Main Theorem is based on some methods in- troduced by Agrachev and Sarychev in [AS05] and [AS06]. In that papers they prove approximate controllability for the 2D NS and Euler systems by

a finite-dimensional force. This method is then developed and generalised by several authors for various PDE’s. Rodrigues [Rod07] proves controlla- bility for the 2D NS system on a rectangle with Lions boundary conditions.

Shirikyan [Shi06, Shi07, Shi13] studies controllability for the 3D NS system on the torus and 1D viscous Burgers equation on the real line. The case of incompressible and compressible 3D Euler equations is considered by Ner- sisyan in [Ner10, Ner11], and the controllability for the 2D defocusing cubic Schr¨odinger equation is established by Sarychev in [Sar12].

All the above papers are concerned with the problem of controllability of the velocity field. The controllability of the Lagrangian trajectories of 2D and 3D Euler equations is studied by Glass and Horsin [GH10,GH12], in the case of boundary controls. For given two smooth contractible setsγ1andγ2of fluid particles which surround the same volume, they construct a control such that the corresponding flow drivesγ1arbitrarily close toγ2. In the context of our paper, a similar property can be derived from our main result. Indeed, Krygin shows in [Kry71] that there is a diffeomorphismψ∈SDiff(T³) such thatψ(γ1) =γ2. Thus we can find an E-valued control η such that φ^u_T(γ1) is arbitrarily close toγ2, and, moreover, at timeT the particles will have approximately the desired velocity.

When E is of the form (0.7), our Main Theorem is related to the recent paper [CL12] by Coron and Lissy. In that paper, the authors establish local null controllability of the velocity for the 3D NS system controlled by a distributed force having two vanishing components (i.e., the controls are valued in a space of the form (0.7), where His the space of space-time L²-functions supported in a given open subset). The reader is referred to the book [Cor07] for an introduction to the control theory of the NS system by distributed controls and for references on that topic.

Let us give a brief (and not completely accurate) description of how the Agrachev–Sarychev method is adapted to establish approximate controllability in the above-defined sense. We assume thatEis given by (0.6) for some genera- torKofZ³. Letψ∈SDiff(T³) and leth(t, x) be a smooth isotopy connecting it to the identity: h(0, x) =xand h(T, x) =ψ(x). Then ˆu(t, x) := ˙h(t, h⁻¹(t, x)) is a divergence-free vector field such that φ^u_t^ˆ(x) =h(t, x) for all t ∈[0, T]. In particular, φ^u_T^ˆ = ψ. The mapping u 7→ φ^u_T is continuous from L¹([0, T], H_σ^k) toC¹(T³), whereL¹([0, T], H_σ^k) is endowed with therelaxationnorm

|||u|||^T,k:= sup

t∈[0,T]

Z t 0

u(s)ds H^k(T³)

.

Hence we can choose a smooth vector fieldusufficiently close to ˆuwith respect to this norm, so that

u(0) =u0, u(T) =u1, kφ^u_T−ψkC¹(T³)< ε.

Then u is a solution of our system corresponding to a control η0, which can be explicitly expressed in terms of u and h from equation (0.5). In general,

this controlη0 is not E-valued, so we need to approach uappropriately with solutions corresponding toE-valued controls. To this end, we define the sets

K⁰:=K, K^j=K^j−1∪ {m±n:m, n∈ K^j−1}, j≥1.

AsKis a generator ofZ³, one easily gets that∪^j≥1K^j =Z³, hence∪^j≥1E(K^j) is dense inH_σ^k. LetPN be the orthogonal projection ontoE(K^N) inH. Then a perturbative result implies that, for a sufficiently largeN ≥1, system (0.5), (0.2) with controlPNη0 has a smooth solutionuN verifying

kuN(T)−u1kH^k(T³)+kφ^u_T^N −ψkC¹(T³)< ε.

On the other hand, if we consider the following auxiliary system

˙

u+νL(u+ζ) +B(u+ζ) =h+η (0.8) with two controlsζ andη, then the below two properties hold true

Convexification principle. For anyε >0 and any solution uj of (0.5), (0.2) with an E(K^j)-valued control η1, there are E(K^j−1)-valued controls ζ andη and a solution ˜uj−1 of (0.8), (0.2) such that

kuj(T)−u˜j−1(T)kH^k(T³)+|||uj−u˜j−1|||T,k< ε.

Extension principle. For anyε >0 and any solution ˜uj of (0.8), (0.2) with E(Kj)-valued controlsζandη, there is anE(Kj)-valued controlη2and a solutionuj of (0.5), (0.2) such that

kuj(T)−u˜j(T)kH^k(T³)+|||uj−u˜j|||T,k< ε.

These two principles and the above-mentioned continuity property of φ^u_T with respect to the relaxation norm imply that, for any solution uj of (0.5), (0.2) with anE(K^j)-valued control η1, there is anE(K^j−1)-valued control η2

and a solutionuj−1 of (0.5), (0.2) such that

kuj(T)−u˜j−1(T)kH^k(T³)+kφ^u_T^j −φ^u_T^j−1kH^k(T³)< ε.

Combining this with the above-constructed solution uN, we get the approxi- mate controllability of (0.5) by a control valued inE(K) =E. The proofs of convexification and extension principles are strongly inspired by [Shi06].

Acknowledgments

This research was supported by the ANR grants EMAQS No ANR 2011 BS01 017 01 and STOSYMAP No ANR 2011 BS01 015 01.

Notation

We denote by T^d the standard d-dimensional torus R^d/2πZ^d. It is endowed with the metric and the measure induced by the usual Euclidean metric and the Lebesgue measure onR^d. More precisely, if Π:R^d →T^d denotes the canonical projection, we have

d(x, y) = inf{|x˜−y˜|:Πx˜=x,Πy˜=y,˜x,y˜∈R^d} for anyx, y∈T^d, d(A) = (2π)^−dd_R^d(Π⁻¹(A)∩[0,2π]^d) for any Borel subset A⊂T^d, where |x| = |x1|+· · ·+|xd|, x = (x1, . . . , xd) ∈ R^d and d_R^d is the Lebesgue measure onR^d.

L^p(T^d,R^d) andH^s(T^d,R^d) stand for spaces of vector functionsu= (u1, . . . , ud) with components in the usual Lebesgue and Sobolev spaces onT^d.

C^k,λ(T^d,R^d),k≥0,λ∈(0,1] is the space of vector functionsu= (u1, . . . , ud) with components that are continuous onT^dtogether with their derivatives up to orderk, and whose derivatives of orderkare H¨older-continuous of exponentλ, equipped with the norm

kukC^k,λ := X

|α|≤k

sup

x∈T^d|D^αu(x)|+ X

|α|=k

sup

x,y∈T^d,x6=y

|D^αu(x)−D^αu(y)| d(x, y)^λ .

H_σ^k(T^d,R^d) :=H^k(T^d,R^d)∩H andC_σ^k,λ(T^d,R^d) :=C^k,λ(T^d,R^d)∩H, whereH is given by (0.4) (withdinstead of 3). In what follows, when the space dimen- siondis 3, we shall writeL^p, H^k, . . .instead ofL^p(T³,R³), H^k(T³,R³), . . ..

C¹(T^d) is the space of continuously differentiable maps fromT^d toT^d endowed with the usual distancekψ1−ψ2kC¹(T^d), ψ1, ψ2∈C¹(T^d).

LetX be a Banach space endowed with a normk · kX andJT := [0, T]. For 1 ≤p < ∞, let L^p(JT, X) be the space of measurable functions u :JT → X such that

kukL^p(JT,X):=

Z T

0 ku(s)k^pXds p¹

<∞.

The spacesC(JT, X) andW^k,p(JT, X) are defined in a similar way. We define therelaxationnorm onL¹(JT, X) by

|||u|||T,X := sup

t∈JT

Z t 0

u(s)ds _X

.

A mapping ψ:T^d→T^d is volume-preserving if d(ψ⁻¹(A)) = d(A) for any Borel subsetA ⊂ T^d. We denote by SDiff(T^d) be the group of all diffeomor- phisms onT^dpreserving the orientation and volume and isotopic to the identity, i.e., SDiff(T^d) is the set of all functionsψ:T^d→T^d such that there is a path h∈W^1,∞(J1, C¹(T^d)) withh(0, x) =x,h(1, x) =ψ(x) for allx∈T^d, andh(t,·) is a diffeomorphism onT^d preserving the orientation and volume for allt∈J1.

1 Preliminaries

1.1 Particle trajectories

In this section, we study some existence and stability properties for the La- grangian trajectories, which are essential for the proofs of the main results.

Let us fix a time T > 0 and an integer d ≥ 1. For any vector field u ∈ L¹(JT, C¹(T^d,R^d)), we consider the following ordinary differential equation inT^d

˙

x=u(t, x). (1.1)

By standard methods, one can show that for anyy ∈T^d this equation admits a unique solution x ∈ W^1,1(JT,T^d) such that x(0) = y (e.g., see [Hal80]).

Moreover, ifφ^u_t :T^d →T^d, t∈JT is the corresponding flow sendingy to x(t), thenφ^u_t is aC¹-diffeomorphism onTⁿ and

φ^·T :L¹(JT, C¹(T^d,R^d))→C(JT, C¹(T^d)), u7→φ^u is continuous. (1.2) We shall also use the following stability property with respect to a weaker norm (cf. Chapter 4 in [Gam78]).

Lemma 1.1. For any λ∈(0,1]andR >0, there is C:=C(R, λ, T)>0 such that

kφ^u−φ^uˆkL^∞(JT,C¹(T^d)) ≤C|||u−uˆ|||^λ/2_T,C¹_(T^d_,R^d₎ (1.3) for anyu,uˆ∈L^∞(JT, C^1,λ(T^d,R^d))verifying

kukL^∞(JT,C^1,λ(T^d,R^d))+kuˆkL^∞(JT,C^1,λ(T^d,R^d))≤R.

Proof. Clearly, it suffices to prove this lemma in the case when T^d is replaced byR^d and suppu(t,·),supp ˆu(t,·)⊂Kfor some compactK⊂R^d for allt∈JT.

Step 1. Let us show that there is a constantC:=C(R, T)>0 such that kφ^u−φ^uˆkL^∞(JT×R^d)≤C|||u−uˆ|||^1/2_T,L^∞_(R^d₎. (1.4) Indeed, we have

kφ^u_t −φ^u_t^ˆkL^∞(R^d)=

Z t 0

(u(s, φ^u_s)−u(s, φˆ ^u_s^ˆ))ds L^∞(R^d)

≤

Z t 0

(u(s, φ^u_s)−u(s, φ^ˆu_s))ds _L∞(R^d)

+

Z t 0

(u(s, φ^u_s^ˆ)−u(s, φˆ ^u_s^ˆ))ds L^∞(R^d)

=:G1+G2. (1.5) Then

G1≤ kukL^∞(JT,C¹(R^d))

Z t

0 kφ^us−φ^us^ˆkL^∞(R^d)ds. (1.6)

To estimateG2, let us first note that for anyη >0 sup

t1,t2∈JT,|t1−t2|≤ηkφ^u_t^ˆ₁−φ^u_t^ˆ₂kL^∞(R^d)≤ηkuˆkL^∞(JT×R^d). (1.7) Taking a partitionτi=it/n, i= 0, . . . , nand using (1.7), we get

G2≤

n

X

i=1

Z τi

τi−1

(u(s, φ^u_s^ˆ)−u(s, φ^u_τ^ˆ_i−1))ds L^∞(R^d)

+

n

X

i=1

Z τi

τi−1

(ˆu(s, φ^u_s^ˆ)−u(s, φˆ ^u_τ^ˆ_i−1))ds L^∞(R^d)

+

n

X

i=1

Z τi

τi−1

(u(s, φ^u_τ^ˆ_i−1)−u(s, φˆ ^u_τ^ˆ_i−1))ds _L∞(R^d)

≤T²

n kuˆkL^∞(JT×R^d)(kukL^∞(JT,C¹(R^d))+kuˆkL^∞(JT,C¹(R^d)))

+ 2n|||u−uˆ|||T,L^∞(R^d). (1.8) If|||u−uˆ|||T,L^∞(R^d)= 0, then (1.3) holds trivially. Assume that|||u−uˆ|||T,L^∞(R^d)>

0. Choosing¹ n:= [|||u−uˆ|||^−1/2_T,L^∞_(R^d₎], we see that G2≤C|||u−uˆ|||^1/2_T,L^∞_(R^d₎.

Combining this with (1.5) and (1.6) and applying the Gronwall inequality, we obtain (1.4).

Step 2. Forj = 1, . . . , d, we have k∂jφ^u_t −∂jφ^u_t^ˆkL^∞(R^d)=

Z t 0

(h∇u(s, φ^u_s), ∂jφ^u_si − h∇u(s, φˆ ^u_s^ˆ), ∂jφ^u_s^ˆi)ds L^∞(R^d)

≤

Z t 0

(h∇u(s, φ^u_s), ∂jφ^u_s−∂jφ^u_s^ˆi)ds L^∞(R^d)

+

Z t 0

(h∇u(s, φ^u_s)− ∇u(s, φ^u_s^ˆ), ∂jφ^u_s^ˆi)ds _L∞(R^d)

+

Z t 0

(h∇u(s, φ^u_s^ˆ)− ∇u(s, φˆ ^u_s^ˆ), ∂jφ^u_s^ˆi)ds L^∞(R^d)

=:I1+I2+I3. (1.9)

It is easy to verify that there is a constantC1:=C1(R, T)>0 such that kφ^uˆkL^∞(JT,C¹(R^d))+kφ˙^uˆkL^∞(JT,C¹(R^d)) ≤C1. (1.10)

1Here [a] stands for the integer part ofa∈R.

Using this and (1.4), we get that I1≤R

Z t

0 k∂jφ^u_s−∂jφ^u_s^ˆkL^∞(R^d)ds, I2≤C1R

Z t

0 kφ^u_s−φ^u_s^ˆk^λL^∞(R^d)ds≤C2|||u−uˆ|||^λ/2T,L^∞(R^d). To estimateI3, we integrate by parts and use (1.10)

I3≤ h

Z t 0

(∇u(s, φ^u_s^ˆ)− ∇ˆu(s, φ^u_s^ˆ))ds, ∂jφ^u_t^ˆi L^∞(R^d)

+

Z t 0

h

Z s 0

(∇u(θ, φ^u_θ^ˆ)− ∇u(θ, φˆ ^u_θ^ˆ))dθ, ∂jφ˙^u_s^ˆi

ds L^∞(R^d)

≤C3 sup

s∈[0,t]

Z s 0

(∇u(θ, φ^u_θ^ˆ)− ∇u(θ, φˆ ^u_θ^ˆ))dθ L^∞(R^d)

.

Repeating the arguments used in (1.8) and using the fact that ∇uand∇ˆuare H¨older continuous with exponentλ, we obtain that

sup

s∈[0,t]

Z s 0

(∇u(θ, φ^ˆu_θ)− ∇u(θ, φˆ ^u_θ^ˆ))dθ L^∞(R^d)

≤ C4

n^λ + 2n|||u−uˆ|||T,C¹(R^d)

≤C5|||u−uˆ|||^λ/2T,C¹(R^d)

forn:= [|||u−uˆ|||^−1/2_T,C¹_(R^d₎]. Combining this with the estimates forI1,I2and (1.9), and applying the Gronwall inequality, we arrive at the required result.

By the Liouville theorem, if we assume additionally that u is divergence- free, then the flow φ^u_t preserves the orientation and the volume. Thus if u∈ L^∞(JT, C_σ¹(T^d,R^d)), thenφ^u_t ∈SDiff(T^d) for anyt∈JT. The following propo- sition shows that, using a suitable divergence-free fieldu, the flow φ^u_t can be driven approximately to any positionψ∈SDiff(T^d) at timeT.

Proposition 1.2. For any ε > 0, k > 1 +d/2, u0, u1 ∈ H_σ^k(T^d,R^d), and ψ∈SDiff(T^d), there is a vector fieldu∈C^∞(JT, Hσ^k(T^d,R^d))such that u(0) = u0, u(T) =u1, and

kφ^u_T −ψkC¹(T^d)< ε.

Proof. Step 1. We first forget about the endpoint conditionsu(0) =u0, u(T) = u1 and show that there is a divergence-free vector field ˆu ∈ C^∞(R×T^d,R^d) such that

kφ^u_T^ˆ−ψkC¹(T^d)< ε/2.

Sinceψ∈SDiff(T^d), there is a pathh∈W^1,∞(JT, C¹(T^d)) such thath(0, x) = x, h(T, x) = ψ(x) for all x ∈ T^d, and h(t,·) is a C¹-diffeomorphism on T^d preserving the orientation and the volume for all t ∈ JT. Let us define the vector field ˆu(t, x) = ˙h(t, h⁻¹(t, x)). Then we have ˆu∈L^∞(JT, C¹(T^d,R^d)) and

h(t, x) =φ^u_t^ˆ(x), t∈JT. Asφ^u_t^ˆpreserves the orientation and the volume, for any g∈C¹(T^d,R)

0 = d dt

Z

T^d

g(φ^ˆu_t(y))dy= Z

T^dh∇g(φ^u_t^ˆ(y)),φ˙^u_t^ˆ(y)idy

= Z

T^dh∇g(φ^u_t^ˆ(y)),u(t, φˆ ^ˆu_t(y))idy= Z

T^d

g(y) div ˆu(t, y)dy.

This shows that ˆu is divergence-free. Taking a sequence of mollifying kernels ρn, n ≥ 1, we consider ˆun := ρn ∗uˆ ∈ C^∞(R×T^d,R^d). Then ˆun is also divergence-free and kuˆn −ˆukL^∞(JT,C¹(T^d,R^d)) → 0 as n → ∞. By (1.2), this implies that kφ^u_T^ˆn −φ^u_T^ˆkC¹(T^d) → 0 as n → ∞. Since φ^ˆu_T = ψ, we get the required result with ˆu= ˆun for sufficiently largen≥1.

Step 2. By the Sobolev embedding, H^k ⊂ C¹(T^d), k > 1 +d/2 (e.g., see [Ada75]). For any δ > 0, we take an arbitrary u ∈ C^∞(JT, H_σ^k(T^d,R^d)) satisfying

u(0) =u0, u(T) =u1, ku−ˆukL¹(JT,C¹(T^d,R^d))< δ.

Then by Step 1 and (1.2), we have

kφ^u_T −ψkC¹(T^d)≤ kφ^u_T−φ^u_T^ˆkC¹(T^d)+kφ^u_T^ˆ −ψkC¹(T^d)

< ε/2 +ε/2 =ε for sufficiently smallδ >0.

1.2 Existence of strong solutions

In what follows, we shall assume thatd= 3, k≥3, andν = 1. In this section, we prove a perturbative result on existence of strong solutions for the evolution equation

˙

u+Lu+B(u) =g, (1.11)

whereB(a, b) = Π{ha,∇ib}andB(a) =B(a, a). Along with (1.11), we consider the following more general equation

˙

u+L(u+ζ) +B(u+ζ) =g. (1.12) Let us fix anyT >0 and introduce the spaceXT,k:=C(JT, H_σ^k)∩L²(JT, H_σ^k+1) endowed with the norm

kukXT ,k:=kukL^∞(JT,H^k)+kukL²(JT,H^k+1).

The following result is a version of Theorem 1.8 and Remark 1.9 in [Shi06] and Theorem 2.1 in [Ner10] in the case of the 3D NS system in the spacesH^k, k≥3.

For the sake of completeness, we give all the details of the proof, even though it is very close to the proofs of the previous results.

Theorem 1.3. Suppose that for some functions uˆ0 ∈ H_σ^k,ζˆ∈ L⁴(JT, H_σ^k+1), and ˆg ∈ L²(JT, H_σ^k−1) problem (1.12), (0.2) with u0 = ˆu0, ζ = ˆζ, and g = ˆg has a solutionuˆ ∈ XT,k. Then there are positive constants δ and C depending only on

kζˆkL⁴(JT,H^k+1)+kˆgkL²(JT,H^k−1)+kˆukXT ,k

such that the following statements hold.

(i) Ifu0∈H_σ^k, ζ∈L⁴(JT, H_σ^k+1), andg∈L²(JT, H_σ^k−1)satisfy the inequality ku0−uˆ0k^k+kζ−ζˆkL⁴(JT,H^k+1)+kg−ˆgkL²(JT,H^k−1)< δ, (1.13) then problem (1.12), (0.2) has a unique solution u∈ XT,k.

(ii) Let

R:H_σ^k×L⁴(JT, H_σ^k+1)×L²(JT, H_σ^k−1)→ XT,k

be the operator that takes each triple (u0, ζ, g) satisfying (1.13) to the solutionu of (1.12), (0.2). Then

kR(u0, ζ, g)− R(ˆu0,ζ,ˆ g)ˆk^XT ,k≤C ku0−uˆ0k^k +kζ−ζˆkL⁴(JT,H^k+1)+kg−ˆgkL²(JT,H^k−1)

.

Proof. We use the following standard estimates for the bilinear formB

kB(a, b)k^k≤Ckak^kkbk^k+1 fork≥2, (1.14)

|(B(a, b), L^kb)| ≤Ckak^kkbk²k fork≥3 (1.15) for anya∈H_σ^k and b∈H_σ^k+1 (see Chapter 6 in [CF88]). We are looking for a solution of (1.12), (0.2) of the formu= ˆu+w. We have the following equation forw:

˙

w+L(w+η) +B(w+η,uˆ+ ˆζ) +B(ˆu+ ˆζ, w+η) +B(w+η) =q,

w(0, x) =w0(x), (1.16)

wherew0=u0−uˆ0,η=ζ−ζ, andˆ q=g−g. Setting ˜ˆ B(u, v) =B(u, v)+B(v, u), we get that

˙

w+Lw+B(w)+ ˜B(w, η)+ ˜B(w,u)+ ˜ˆ B(w,ζ) =ˆ q−(Lη+B(η)+ ˜B(ˆu, η)+ ˜B(ˆζ, η)), (1.17) Using (1.14), we see that for anyε >0, we can chooseδ >0 in (1.13) such that

kw0k^k+kq−(Lη+B(η) + ˜B(ˆu, η) + ˜B(ˆζ, η))kL²(JT,H^k−1)< ε.

Then a standard existence result implies that system (1.17), (1.16) has a solu- tionw∈ X^T,k (see [Tay97]).

To prove (ii), we multiply (1.17) byL^kwand use estimates (1.14) and (1.15) 1

2 d

dtkwk²k+kwk²k+1≤C

kwk³k+kwkk+1kwkk kηkk+kuˆkk+kζˆkk +kwkk+1 kqkk−1+kηkk+1+kηkk(kηkk−1+kuˆkk+kζˆkk)

.

This implies that 1

2 d

dtkwk²k+1

2kwk²k+1≤C1

kwk³k+kwk²k kηk²k+kuˆk²k+kζˆk²k

+h

kqk²k−1+kηk⁴k+1+kηk²k(kuˆk²k+kζˆk²k)i .

Integrating this inequality and setting A:=kw0k²k+

Z T 0

hkqk²k−1+kηk⁴k+1+kηk²k(kuˆk²k+kζˆk²k)i dt, we obtain

kwk²k+ Z t

0 kwk²k+1≤2A+ 2C1

Z t 0

kwk³k+kwk²k kηk²k+kuˆk²k+kζˆk²k

dt.

(1.18) The Gronwall inequality gives that

kwk²k≤C2A+C2

Z t

0 kw(s,·)k³kds,

whereC2 depends only onkˆukL²(JT,H^k)+kζˆkL²(JT,H^k). Another application of the Gronwall inequality implies that

kw(t)k^k≤ C2A

1−C₂²At≤2C2A for any t≤ 1

2C₂²A. (1.19) Let us choose δ > 0 so small that _2C¹2

2A ≥ T. Using (1.19) and (1.18), and choosingδ >0 sufficiently small, we get for anyt∈JT

kwk²k+ Z t

0 kwk²k+1≤C3A≤C4 kw0k²k+kηk²L⁴(JT,H^k+1)+kqk²L²(JT,H^k−1)

.

This completes the proof of the theorem.

2 Main results

2.1 Approximate controllability of the NS system

In this section, we state the main results of this paper. Let us fix anyT > 0 andk≥3, and consider the NS system

˙

u+Lu+B(u) =h(t) +η(t), (2.1)

u(0, x) =u0(x), (2.2)

where h ∈ L²(JT, H_σ^k−1) and u0 ∈ H_σ^k are given functions and η is a control taking values in a finite-dimensional space E⊂H_σ^k+1. We denote by Θ(h, u0) the set of functions η ∈ L²(JT, H_σ^k−1) for which (2.1), (2.2) has a solution u in X^T,k. By Theorem1.3, Θ(h, u0) is an open subset ofL²(JT, H_σ^k−1). Recall thatR(·,·,·) is the operator defined in Theorem1.3. To simplify notation, we writeR(·,·) instead ofR(·,0,·). The embedding H³⊂C^1,1/2 implies that the flowφ^R(u_t ^0,h+η)is well defined for anyη∈Θ(h, u0) andt∈JT. We set

YT,k:=XT,k∩W^1,2(JT, H_σ^k−1).

We shall use the following notion of controllability.

Definition 2.1. Equation (2.1) is said to be controllable at time T by an E- valued control if for anyε >0 and anyϕ∈YT,kthere is a controlη∈Θ(h, u0)∩ L²(JT, E) such that

kR^T(u0, h+η)−ϕ(T)k^k+|||R(u0, h+η)−ϕ|||^T,k+kφ^R(u0,h+η)−φ^ϕkL^∞(JT,C¹)< ε, whereu0=ϕ(0) and|||·|||^T,k:=|||·|||T,H^k.

Let us recall some notation introduced in [AS05,AS06], and [Shi06]. For any finite-dimensional subspaceE ⊂ H_σ^k+1, we denote byF(E) the largest vector spaceF ⊂H_σ^k+1 such that for anyη1∈F there are vectors² η, ζ¹, . . . , ζⁿ ∈E satisfying the relation

η1=η−

n

X

i=1

B(ζⁱ). (2.3)

AsEis a finite-dimensional subspace andB is a bilinear operator, the set of all vectorsη1 ∈H_σ^k+1 of the form (2.3) is contained in a finite-dimensional space.

It is easy to see that if subspacesG1, G2 ⊂H_σ^k+1 are composed of elementsη1

of the form (2.3), then so doesG1+G2. Thus the space F(E) is well defined.

We defineEj by the rule

E0=E, Ej =F(Ej−1) for j≥1, E∞=

∞

[

j=1

Ej. (2.4) Clearly,Ejis a non-decreasing sequence of subspaces. We say thatEissaturat- inginH_σ^k−1 ifE∞ is dense in H_σ^k−1. The following theorem is the main result of this paper.

Theorem 2.2. Assume that E is a finite-dimensional subspace of H_σ^k+1 and h∈L²(JT, H_σ^k−1). If E is saturating in H_σ^k−1, then (2.1) withη ∈C^∞(JT, E) is controllable at timeT.

We have the following two corollaries of this result.

2 The integernmay depend onη¹.

Corollary 2.3.Under the conditions of Theorem2.2, ifEis saturating inH_σ^k−1, then for any ε > 0, u0, u1 ∈ H_σ^k, and ψ ∈ SDiff(T³) there is a control η ∈ Θ(h, u0)∩C^∞(JT, E)such that

kR^T(u0, h+η)−u1k^k+kφ^R(u_T ^0,h+η)−ψkC¹< ε.

Let us denote by VPM(T³) the set of all volume-preserving mappings fromT³ to T³. According to Corollary 1.5 in [BG03] and Theorem 2.1 in [Shn85], we have that VPM(T³) is the closure of SDiff(T³) inL^p(T³) for anyp∈[1,+∞).

Thus we get the following result.

Corollary 2.4.Under the conditions of Theorem2.2, ifEis saturating inH_σ^k−1, then for any ε > 0, p∈ [1,+∞), u0, u1 ∈ H_σ^k, and ψ ∈ VPM(T³) there is a controlη∈Θ(h, u0)∩C^∞(JT, E)such that

kR^T(u0, h+η)−u1k^k+kφ^R(u_T ^0,h+η)−ψk^Lp< ε.

The rest of this subsection is devoted to the proofs of Theorem2.2and Corol- lary2.3. They are based on the following result which is proved in Section3.

Theorem 2.5. Under the conditions of Theorem 2.2, for anyε >0,u0∈H_σ^k, andη1∈Θ(h, u0)∩L²(JT, E1)there is η∈Θ(h, u0)∩C^∞(JT, E)such that

kR^T(u0, h+η1)−R^T(u0, h+η)k^k+|||R(u0, h+η1)− R(u0, h+η)|||^T,k +kφ^R(u0,h+η1)−φ^R(u0,h+η)kL^∞(JT,C¹)< ε.

Proof of Theorem2.2. Let us take anyε >0, δ >0, andϕ∈YT,k. Then η0:= ˙ϕ+Lϕ+B(ϕ)−h

belongs to Θ(u0, h) andϕ(t) =R^t(u0, h+η0) for anyt∈JT, whereu0=ϕ(0).

SinceE∞is dense in H_σ^k−1, we have that

kPENη0−η0kL²(JT,H^k−1)→0 asN → ∞.

By Theorem1.3, for sufficiently largeN, we havePENη0∈Θ(h, u0) and kR(u0, h+PENη0)−ϕk^XT ,k < δ.

By (1.2), we can chooseδ >0 so small that

kφ^{R(u0,h+PENη0)}−φ^ϕkL^∞(JT,C¹)< ε.

ApplyingN times Theorem2.5, we complete the proof of Theorem2.2.

Proof of Corollary 2.3. Let us take anyε >0,ψ∈SDiff(T³), andu0, u1∈H_σ^k. By Proposition 1.2, there is a vector field u∈ C^∞(JT, H_σ^k) such that u(0) = u0, u(T) =u1, and

kφ^u_T −ψkC¹ < ε. (2.5)

Applying Theorem2.2withϕ=u, we find a controlη∈Θ(h, u0)∩C^∞(JT, E) such that

kRT(u0, h+η)−u(T)kk+kφ^R(u_T ^0,h+η)−φ^u_TkC¹< ε.

Combining this with (2.5), we get the required result.

2.2 Examples of saturating spaces

In this section, we provide three types of examples of saturating spaces which ensure the controllability of the 3D NS system in the sense of Definition2.1.

2.2.1 Saturating spaces associated with the generators of Z³

Let us first introduce some notation. For anyℓ∈Z³_∗, let us define the functions cℓ(x) =l(ℓ) coshℓ, xi, sℓ(x) =l(ℓ) sinhℓ, xi, (2.6) where{l(ℓ), l(−ℓ)}is an arbitrary orthonormal basis in

ℓ^⊥ :={x∈R³:hx, ℓi= 0}.

Then cℓ and sℓ are eigenfunctions of L and the family {cℓ, sℓ}ℓ∈Z³∗ is an or- thonormal basis inH. Letc0=s0= 0. For any subsetK ⊂Z³, we denote

E(K) := span{cℓ, c−ℓ, sℓ, s−ℓ:ℓ∈ K}. (2.7) WhenK is finite, the spacesEj(K) and E∞(K) are defined by (2.4) withE = E(K). We denote byZ³_K the set of all vectorsa∈Z³which can be represented as finite linear combination of elements ofK with integer coefficients. We shall say thatK ⊂Z³ is agenerator ifZ³_K =Z³. The following theorem provides a characterisation of saturating spaces of the form (2.7).

Theorem 2.6. For any finite setK ⊂Z³, we have the equality

E(Z³_K) =E∞(K). (2.8)

Moreover,E(K)is saturating inHif and only ifKis a generator ofZ³. IfE(K) is saturating inH, then it is saturating inH_σ^k for anyk≥0.

In [Rom04] a similar result is conjectured in the case of finite-dimensional approximations of the 3D NS system and a proof is given for the saturating property ofE(K) whenK={(1,0,0),(0,1,0),(0,0,1)}³. A 2D version of The- orem2.6 is established in [HM06]. In that case, the setK is a generator ofZ² containing at least two vectors with different Euclidian norms (the reader is referred to the original paper for the exact statement). The proof in the 3D case, as well as the statement of the result, differ essentially from the 2D case.

In view of Theorem2.6, the following simple criterion is useful for construct- ing saturating spaces (see Section 3.7 in [Jac85]).

3In that case one has dimE(K) = 12. In Proposition 2.14, we give an example of a 6-dimensional saturating space.

Theorem 2.7. A set K ⊂ Z³ is a generator if and only if the greatest com- mon divisor of the set {det(a, b, c) : a, b, c ∈ K} is 1, where det(a, b, c) is the determinant of the matrix with rowsa, b andc.

The proof of Theorem2.6is deduced from the following auxiliary result.

Proposition 2.8. Assume that W ⊂ Z³ is a finite set containing a linearly independent family{p, q, r} ⊂Z³. Then for any non-parallel vectorsm, n∈ W we haveAm±n,Bm±n⊂E5(W), where

Aℓ:= span{cℓ, c−ℓ}, Bℓ:= span{sℓ, s−ℓ}, ℓ∈Z³_∗.

Proof of Proposition 2.8. We shall confine ourselves to the proof of the inclu- sionA^m+n ⊂E5(W). The other conclusions of the proposition are checked in the same way.

Step 1. We shall write m∦nwhen the vectorsm, n∈R³ are non-parallel.

For anym, n ∈ W such that m∦ n, let us denote by d:=d(m, n) one of the two unit vectors belonging tom^⊥∩n^⊥. Let us show that

dcoshm±n, xi, dsinhm±n, xi ∈E1(W). (2.9) Indeed, for any a ∈ R³_∗, let us denote by Pa the orthogonal projection in R³ ontoa^⊥. Then we have

Π(acoshl, xi) = (Pla) coshl, xi, Π(asinhl, xi) = (Pla) sinhl, xi

for any l ∈ Z³_∗. Combining this with some trigonometric identities and the definition ofB, one gets that

2B(acoshm, xi+bsinhn, xi) = coshm−n, xiPm−n(ha, nib− hb, mia) + coshm+n, xiPm+n(ha, nib+hb, mia),

(2.10) for anya∈m^⊥andb∈n^⊥(see Step 1 of the proof of Proposition 2.8 in [Shi06]).

This implies that

2B(bcoshn, xi+asinhm, xi) =−coshm−n, xiPm−n(ha, nib− hb, mia) + coshm+n, xiPm+n(ha, nib+hb, mia).

(2.11) Taking the sum of (2.10) and (2.11), we obtain that

coshm+n, xiPm+n(ha, nib+hb, mia) =B(acoshm, xi+bsinhn, xi)

+B(bcoshn, xi+asinhm, xi). (2.12) Let us fix any λ∈R and choose in this equalitya =dand hb, mi =λ. This choice is possible sincem∦n. Then we have

λdcoshm+n, xi=B(dcoshm, xi+bsinhn, xi) +B(bcoshm, xi+dsinhn, xi).

Sinceλ ∈R is arbitrary, from the definition ofE1(W) we get thatdcoshm+ n, xi ∈E1(W). To prove thatdcoshm−n, xi ∈E1(W), it suffices to replaceb by−bin (2.11), take the sum of the resulting equality with (2.10):

coshm−n, xiPm−n(ha, nib− hb, mia) =B(acoshm, xi+bsinhn, xi) +B(−bcoshn, xi+asinhm, xi),

(2.13) and choosea=dandhb, mi=−λ

λdcoshm−n, xi=B(dcoshm, xi+bsinhn, xi) +B(−bcoshn, xi+dsinhn, xi).

The fact that dsinhm±n, xi ∈ E1(W) is proved in a similar way using the following identities

2B(acoshm, xi+bcoshn, xi) = sinhm−n, xiPm−n(ha, nib− hb, mia)

−sinhm+n, xiPm+n(ha, nib+hb, mia), 2B(asinhm, xi+bsinhn, xi) = sinhm−n, xiPm−n(ha, nib− hb, mia)

+ sinhm+n, xiPm+n(ha, nib+hb, mia). Step 2. Let us take any vectorr ∈ W such that E :={m, n, r} is a basis in R³ (E is not necessarily a generator ofZ³). This choice is possible, by the conditions of the proposition. For any α, β, γ ∈ R, we shall write (α, j, k)E

instead ofαm+βn+γr. Since

(1,1,0)E = (1,0,0)E+ (0,1,0)E =m+n, we get from Step 1 that

d(m, n) cosh(1,1,0)E, xi ∈E1(W).

Applying (2.12), we obtain for anyb∈(0,0,1)^⊥_E

cosh(1,1,1)E, xiP(1,1,1)E(hd(m, n),(0,0,1)Eib+hb,(1,1,0)Eid(m, n))

=B(d(m, n) cosh(1,1,0)E, xi+bsinh(0,0,1)E, xi)

+B(bcosh(0,0,1)E, xi+d(m, n) sinh(1,1,0)E, xi)∈E2(W). (2.14) Let us define the set

G:={hd(m, n),(0,0,1)Eib+hb,(1,1,0)Eid(m, n) :b∈(0,0,1)^⊥_E}. Sincehd(m, n),(0,0,1)Ei 6= 0,G is a two-dimensional subspace ofR³ contained in (1,1,−1)^⊥_E. This shows thatG= (1,1,−1)^⊥_E. Let us assume that

(1,1,1)E ∈/ (1,1,−1)^⊥_E. (2.15) This assumption implies that the orthogonal projectionP(1,1,1)EGcoincides with (1,1,1)^⊥_E and proves that A(1,1,1)E ⊂ E2(W). Similarly, one can show that B(1,1,1)E ⊂E2(W). Finally, writing

(1,1,0)E = (1,1,1)E−(0,0,1)E