Exact boundary controllability of 3-D Euler equation

URL:http://www.emath.fr/cocv/

EXACT BOUNDARY CONTROLLABILITY OF 3-D EULER EQUATION

Olivier Glass

Abstract. We prove the exact boundary controllability of the 3-D Euler equation of incompressible inviscid fluids on a regular connected bounded open set when the control operates on an open part of the boundary that meets any of the connected components of the boundary.

R´esum´e. Nous prouvons la contrˆolabilit´e exacte fronti`ere de l’´equation d’Euler des fluides parfaits incompressibles tridimensionnels dans un domaine born´e et r´egulier, lorsque le contrˆole op`ere sur une partie ouverte du bord qui en rencontre toutes les composantes connexes.

AMS Subject Classification. 93B05, 35Q30, 76C99, 93C20.

Received December 16, 1998. Revised March 29, 1999.

1. Introduction

Let Ω be a non-empty, open, connected, bounded and regular (say C^∞-regular) subset of R³. Let Γ0 be an open and non-empty subset of its boundary ∂Ω, which meets any connected component of ∂Ω. We are interested in the exact boundary controllability of the 3-D Euler equation of inviscid incompressible fluids for (Ω,Γ0), that is, the following question: givenT >0, giveny0andy1 two solenoidal vector fields, i.e. satisfying

div y0= divy1= 0 in Ω, (1.1)

regular (in this paper,C^2,α for some H¨older coefficientα∈(0,1)) and which satisfy

y0.n=y1.n= 0 on∂Ω\Γ0, (1.2)

wherenis the outward unit normal vector field on∂Ω, does there exist a solutiony of the Euler system

∂ty+ (y.∇)y=∇pin Ω×[0, T], (1.3)

for somep∈D⁰(Ω×(0, T)) and

div y= 0 in Ω×[0, T], (1.4)

Keywords and phrases:Controllability, boundary control, Euler equation for ideal incompressible fluids.

1Universit´e Paris-Sud, Analyse num´erique et EDP, 91405 Orsay, France; e-mail:[email protected]

c EDP Sciences, SMAI 2000

with

y(x, t).n(x) = 0, ∀t∈[0, T], ∀x∈∂Ω\Γ0, (1.5) and such that

y_|t=0=y0in Ω, (1.6)

y_|t=T =y1 in Ω? (1.7)

This problem, raised by Lions in [10], was solved by Coron in [3] and [4] in the two-dimensional case. In a previous paper [6], we have sketched a proof of a solution to this problem in dimension 3 when Ω is simply connected. Here we give the details of the demonstration and prove that, as announced in [7], the result still holds when Ω is not necessarily simply connected. Actually, we prove the following result:

Theorem 1.1. Givenα∈(0,1), two functionsy0 andy1 inC^2,α(Ω;R³)satisfying (1.1) and (1.2) andT >0, then there exists a function y in the space C([0, T];C^1,α(Ω;R³))∩L^∞([0, T];C^2,α(Ω;R³)) such that (1.3) to (1.7) hold for somep∈D⁰(Ω×(0, T)).

Remark 1.2. As noticed in [4], the condition that Γ0 meets any connected component of the boundary is necessary for the exact controllability as a consequence of the Kelvin law.

Indeed, suppose that we choosey1= 0 on some connected component Γ^∗ of the boundary, which does not meet Γ0. Then the existence ofyand the Kelvin law for any loopγin this connected component of the boundary imply that

Z

γ

y0dτ = Z

˜ γ

y1dτ = 0,

where ˜γ is the loop obtained when transporting γ by the flow of y. This necessarily implies that y0|Γ^∗ is a gradient, which is not generally the case.

Now we briefly describe the method. As in [3] and [4], the steps of the proof of Theorem 1.1 are the following:

first, we prove that this question can be reduced to the problem of zero-controllability with small initial data (that isy1= 0 andky0kC^2,α(Ω;R³)< ) and small timeT.

To be more precise, we prove in section 7 that Theorem 1.1 is a consequence of the following proposition:

Proposition 1.3. There existsν >0such that ify0∈C^2,α(Ω;R³)satisfies (1.1), (1.2) andky0kC^2,α(Ω,R³)< ν, then there exists a functiony in the spaceC([0,1];C^1,α(Ω;R³))∩L^∞([0,1];C^2,α(Ω;R³))andp∈D⁰(Ω×(0,1)) satisfying (1.3) to (1.7) for y1= 0, andT = 1, and also

y= 0andp= 0, ∀t∈ 1

2,1

. (1.8)

In order to prove the last proposition, we use a method called the “return method”, used in [3] and [4] and introduced in [2] for a stabilization problem. Precisely, – since the linearized Euler equation aroundy≡0 is not controllable – we consider the linearized system around other solutions of the Euler control systemy satisfying y_|t=0=y_|t=1= 0 (a kind of “loop”). If this linearized control system is controllable, then fory0 small enough, one can hope to findyclose toyanswering to the general problem. In order to prove the existence ofy, we use a construction of solutions of the Euler system due to Bardos and Frisch (see [1]).

In the previous presentation of the problem, the control itself was not explicit. As a control, we can take for exampley.non Γ0×[0, T] and the tangent part of the vorticity where the fluid enters, that isω∧nwhere y.n <0 in Γ0(see for that [9]).

In the next section, we will present the different tools we need to introduce the particular solutiony. This function will be found in the particular potential form “∇θ”, in order for its flow to satisfy precise properties.

In the simply connected case, as in dimension 2,y has the property that any particle in Ω following the flow ofy must go out of Ω. The major difference is that in dimension 3, as in the 2-D case for the Navier-Stokes equation [5], this “∇θ” can no longer be chosen stationary. In the multi-connected case, we will have to introduce an other type of “∇θ” (which we need to append to the previous one), whose flow moves certain Jordan curves properly.

In Section 3, we define a functionF on a certain functional set, of whichy will be found as a fixed point.

Giveny neary,F associates the solution of a linear control problem relied to (1.3–1.7).

In Section 4, we prove Proposition 1.3, by showing thatF admits a fixed point which gives a solution to the non-linear problem.

Section 5 deduces Theorem 1.1 from Proposition 1.3.

Section 6 is devoted to the proof of Lemma 2.1, which corresponds to the first type of “∇θ”.

Section 7 corresponds to the second type of “∇θ” presented in Lemma 2.3.

In Sections 8 and 9, we give the details of the proofs of technical lemmas needed in Sections 5 and 6 respectively.

2. The particular solution of Euler system: y

We first set up the following lemma, which stands for any regular bounded open set ˜Ω such that ˜Ω contains Ω.

Lemma 2.1. For allain Ω, there exists θ∈C^∞( ˜Ω×[0,1];R)satisfying:

Suppθ⊂Ω˜×(0,1), (2.1)

θ= 0in Ω˜×([0,1 4]∪[3

4,1]), (2.2)

∆θ= 0in Ω×[0,1], (2.3)

∂θ

∂n = 0 on(∂Ω\Γ0)×[0,1], (2.4)

φ^∇θ(a,0,1)∈Ω˜\Ω, (2.5)

where we denote byφ^∇θ : ˜Ω×[0,1]×[0,1]−→ Ω,˜ (x, t1, t2)7→φ^∇θ(x, t1, t2) the flow of∇θ, i.e. the function which satisfies

∂φ

∂t2

=∇θ(φ, t2), (2.6)

φ(x, t1, t1) =x. (2.7)

With the help of that lemma, we will be able to single out a solution of the Euler system, which makes each part of the fluid go out Ω (far enough), and then go back the same way.

?

In the multi-connected case, we will also need another type of “∇θ”, in order to control irrotational flows which class in de Rham’s cohomology first space is not trivial. Let us describe these flows (we refer to [11], Appendix I).

We introduce, in the multi-connected case, precisely whenH1(Ω) =Z^swith s≥1,ssmooth hypersurfaces Σ1, . . .,Σs (see for example [11]) included in Ω and with boundaries in ∂Ω, the intersection being transverse,

and which neutral intersections (if needed) are also transverse.

Ω\( Ss i=1

Σi) is simply connected. (2.8)

For i ∈ {1, ..., s}, we distinguish the two sides Σi, that we denote by Σ⁺_i and Σ⁻_i . For a function f defined in Ω\Σi, which trace on Σ⁺_i may differ from the one on Σ⁻_i , one defines [f]i :=f_|Σ+

i −f_|Σ−

i considered as a function on Σi.

Then using the Lax-Milgram theorem on the functional space:

Xi:=

p∈H¹(Ω\S^s

i=1

Σi)/[p]i= constant, [pj] = 0 for j6=i

,

one easily deduces the existence of a functionq⁰_i inXi such that:

Z

Ω

∇q_i⁰.∇p= [p]i, ∀p∈Xi. This leads to the existence of a functionqi∈Xi such that:

∆qi= 0 in Ω\Σi, (2.9)

∂nqi= 0 on∂Ω, (2.10)

[qi]i = 1, (2.11)

[qi]j= 0 forj 6=i, (2.12)

[∂nqi]i= 0. (2.13)

By (2.11, 2.12) and (2.13), theQⁱ:=∇qiare inC⁰(Ω) and, in fact inC^∞(Ω) (see [11], Appendix I, Rem. 1.3.ii).

Remark 2.2. As it is known (see again [11]), any (regular) vector fieldX satisfying

curl X= 0, (2.14)

can then be written as

X =∇χ+ Xs i=1

αiQⁱ,

for someχandαi,i∈ {1, ..., s}. If we add to (2.14) the conditions:

divX = 0in Ω, X.n= 0 on∂Ω, then the previousχ is zero, and we describe only the first cohomology space.

Let us emphasize that curlX = 0, divX = 0 in Ω andX.n= 0 on∂Ω does not imply that X = 0. Indeed, for alli ∈ {1, ..., s}, we have curlQⁱ = 0, divQⁱ = 0 in Ω and Qⁱ.n = 0 on ∂Ω, but Qⁱ 6= 0. This fact will oblige us to set up a second lemma to define our particular solutiony and to get rid of the terms “Qⁱ”. It is in particular necessary to treat the problem with (for instance) y0 = Qⁱ andy1 = 0. Roughly speaking, the following lemma solves this precise case.

Lemma 2.3. There exists ν > 0, such that for i in {1, ..., s}, there exists θⁱ ∈ C^∞( ˜Ω×[0,1];R) and ℵⁱ ∈ C^∞( ˜Ω;R³)satisfying:

Suppθⁱ⊂Ω˜×(0,1), (2.15)

θⁱ= 0 inΩ˜×([0,1 4]∪[3

4,1]), (2.16)

∆θⁱ= 0 inΩ×[0,1], (2.17)

∂θⁱ

∂n = 0on ∂Ω\Γ0×[0,1], (2.18)

Suppℵⁱ⊂Ω˜\Ω, (2.19)

and such that for anyf ∈C([0,1], C^2,α( ˜Ω;R³)) with

kf− ∇θⁱk_C([0,1]×Ω)˜ < ν, (2.20)

if we definewⁱ∈C^∞( ˜Ω×[0,1];R³) by

wⁱ(·,0) =curl(ℵⁱ) onΩ,˜ (2.21)

∂twⁱ+ (f.∇)wⁱ = (wⁱ.∇)f−wⁱdivf on Ω˜×[0,1], (2.22) and if we define the functionζⁱ in C^∞( ˜Ω×[0,1];R³)by

curlζⁱ=wⁱ inΩ×[0,1], (2.23)

div ζⁱ= 0 inΩ×[0,1], (2.24)

ζⁱ.n=∂nθⁱ on∂Ω×[0,1], (2.25)

Z

Ω

ζⁱ(0).Q^jdx= 0, (2.26)

Z

Ω

(∂tζⁱ+f∧curl(ζⁱ)).Q^jdx= 0, ∀j ∈ {1, ..., s}, (2.27) then we have

Suppwⁱ(·,1)⊂Ω˜\Ω, (2.28)

and

ζⁱ(1) =Qⁱ. (2.29)

As we will see in Section 6, y :=∇θ in this lemma will be chosen, not in terms of the flow of points, but in terms of the flow of certain Jordan curves.

We can now present what our particular solution to Euler systemy will be.

We denote byB(xi, ri) the open ball of centerxiand of radiusri, and byB(xi, ri) its closure. By Lemma 2.1, one can find by compactness of Ω a positive integerk,k points xi in Ω, k real numbersri >0 and k smooth

functions θi ∈C^∞( ˜Ω×[0,1],R), i∈ {1, ..., k}, satisfying (2.1–2.4), and an open bounded regular set Ω2 with Ω⊂Ω2 and also Ω2such that

B(xi, ri)⊂Ω,˜ (2.30)

Ω⊂^i=kS

i=1

B(xi, ri), (2.31)

φ^∇θi(B(xi, ri),0,1)∩Ω2=∅. (2.32) Let us split the time-segments [1/4,1/2] and [1/2,3/4] as follows:

ti=1 4+i 1

4k, ∀i∈ {0, ..., k}, (2.33)

t_i+¹

2 =1 4+

i+1

2 1

4k, ∀i∈ {0, ..., k−1}, (2.34)

ti=1

2+ (i−k)1

4s, ∀i∈ {k, ..., k+s}· (2.35)

We can now defineθin C^∞( ˜Ω×[0,1/2],R) by

θ(x, t) = 0, ∀(x, t)∈Ω˜×

0,1 4

, (2.36)

θ(x, t) = 8kθj(x,8k(t − tj−1)), ∀j ∈ {1, ..., k}, and∀(x, t) ∈ Ω˜ × h

tj−1, t_j−¹

2

i

, (2.37)

θ(x, t) = −8kθj(x,8k(tj − t)), ∀j ∈ {1, ..., k}, and∀(x, t) ∈ Ω˜ × h t_j−¹

2, tj

i

. (2.38) During the interval of time [¹₂,1], we defineθ by

θ(x, t) = 4sθ^j−k+1(x,4s(t−tj)), ∀j ∈ {k, ..., k+s−1}, (2.39) and∀(x, t)∈Ω˜×[tj, tj+1].

Lety:=∇θ. We remark thaty restricted to Ω×[0,1] is aC^∞ solution of (1.3–1.7) withT = 1,y0=y1= 0 and with p(x, t) =∂θ/∂t+|∇θ|²/2.

3. The application F

In this section,we use this particular solution to single out the applicationF, the fixed point of which gives a solution for Proposition 1.3. For that purpose we first introduce a certain functional setXν.

Letµ: [0,1]→[0,1]C^∞-regular, such that





0≤µ≤1 in [0,1], µ= 1 in [0,1/8], µ= 0 in [1/4,1].

(3.1)

Then the setXν forν >0 small enough, is defined as:

Xν =n

u∈C⁰([0,1], C^2,α(Ω;R³))/ divu= 0, in Ω ku−ykC⁰(Ω×[0,1])< ν,

u(x, t).n(x) =µ(t)y0(x).n(x) +y.non∂Ω×[0,1]

o· (3.2)

The value ofF will be a solution to a certain linear controllability problem inC([0,1], C^2,α(Ω)).

We introduce a linear operatorπ which extends functions defined on Ω to functions defined on ˜Ω, and with support in Ω2. We will require also for it to send continuously C^{[λ],λ−[λ]}(Ω;R³) into C^{[λ],λ−[λ]}( ˜Ω;R³), for all λ∈[0,3)\N.

Now we define the applicationF. Foru∈Xν, we set

˜

u=y+π(u−y). (3.3)

ThenF(u) will be a solution of the following problem:

F(u)(·,0) =y0in Ω, (3.4)

F(u)(·,·).n= 0 on [0,1]×(∂Ω\Γ0), (3.5)

divF(u) = 0 in [0,1]×Ω, (3.6)

and if we setω:= curl(F(u)), then it should satisfy Z

Ω

(∂tF(u) + ˜u∧ω).Qⁱ= 0 in [0,1], ∀i∈ {1, . . . , s}, (3.7)

∂tω+ (˜u.∇)ω= (ω.∇)˜uin [0,1]×Ω. (3.8) The controllability problem is to find aF(u) such that

F(u)(1,·) = 0 in Ω. (3.9)

Of course, this linear problem becomes “close” to the Euler problem asω approaches curlu.

3.2. Preliminaries

Before makingF explicit, we introduce some notations.

For a regular open bounded subset E ofR³, we denote byk · k^i,α,E fori∈Nandα∈(0,1), the usual norm forC^i,α(E) and byk · k^i,Ethe usual norm forCⁱ(E).

We introduce a partition of unity adapted to the open covering of Ω by the open setsB(xi, ri) (described in (2.30–2.32)), that is some functionsκi∈C₀^∞( ˜Ω; [0,1]) such that

Suppκi⊂B(xi, ri), (3.10)

i=kX

i=1

κi≡1 in Ω. (3.11)

In this section, we will frequently use the following lemma, of which we postpone the demonstration to Section 3.4.

Lemma 3.1. LetU be a function inC⁰([0, T], C^2,α( ˜Ω,R³)), andW0 be a function inC^1,α( ˜Ω,R³). LetW be a function inC⁰([0, T], C^1,α( ˜Ω,R³))defined by the following system

(

W(·,0) =W0 inΩ,˜

∂tW+ (U.∇)W = (W.∇)U −(divU)W in Ω˜×[0, T]. (3.12) Then for all t∈[0, T], one has

divW(·, t) = 0. (3.13)

Moreover, if divU = 0inΩandW0=curlV0 in Ω, then there exists V ∈C⁰([0, T], C^2,α( ˜Ω,R³))such that for all t∈[0, T]

W(t) =curlV(t)inΩ. (3.14)

3.3. Construction of F

We now give an explicit formulation ofF(u). Letu∈Xν forν small enough (sayν < ν0 with ν0< ν). We associate ˜udefined by (3.3).

We defineF(u) by its curl ω in Ω and by “coordinate”λi with respect to the functionsQⁱ. We define the functions ω andλi in a first step, during the times [0,1/2], and then we define them in the interval [1/2,1].

Along the construction ofF, we will allow ourselves to reduceν0 in order to makeF correctly defined.

We introduce a first functionω^∗ in C⁰([0,1], C^1,α( ˜Ω;R³)). We defineω^∗ by the relations (

ω^∗(·,0) = curlPk

i=1(κiπ(y0))

in ˜Ω,

∂tω^∗+ (˜u.∇)ω^∗= (ω^∗.∇)˜u−(div ˜u)ω^∗ in ˜Ω×(0,1). (3.15) By Lemma 3.1, ω^∗(·,1/4) is a curl in Ω: let us say

ω^∗(·,1/4) = curlW in Ω, (3.16)

withW ∈C^2,α( ˜Ω).

We define then the functionsw^lin C⁰([1/4,1/2], C^1,α( ˜Ω;R³)) by the equations:

(

w^l(·,1/4) = curl(κlπ(W)) in ˜Ω,

∂tw^l+ (˜u.∇)w^l= (w^l.∇)˜u−(div ˜u)w^l in ˜Ω×[1/4,1/2]. (3.17)

Of course, we have the relation fort∈[1/4,1/2]

ω^∗= Xl=k

l=1

w^l. (3.18)

Let us now build ω : ˜Ω×[0,1] −→ R³, continuous in the variable t from [0,1]\{t_i−¹

2, i ∈ {1, . . . , k}} into C^1,α( ˜Ω;R³), continuous at the right of eacht_i−¹

2 and with a limit in C^1,α( ˜Ω;R³) at the left of each t_i−¹

2 (for i∈ {1, ..., k}). We will extractF(u) from thisω.

Let us defineω this way :

ω(x, t) =ω^∗(x, t) in ˜Ω×

0,1 4

, (3.19)

then for t∈[1/4,1/2]:

∂tω + (˜u.∇)ω = (ω.∇)˜u − (div ˜u)ω, in 1

4, t¹

2

_i=kS₋₁

i=1

t_i−¹

2, t_i+¹

2

∪

t_k−¹

2,1 2

× Ω.˜ (3.20) Thus to define ω properly, we have yet to define it at times t_i−¹

2. We do it in order that at time ti only Pk

l=i+1w^l(x, ti) stays on Ω, instead of ω^∗(x, ti). For that, we simply have to considerω at timet⁻_i−1 2

. Let us suppose by induction that, one has

ω(·, t⁻_i−1 2

) = Xk

l=i

w^l(·, t_i−¹

2). (3.21)

Relations (2.32) and (3.3) imply that forku−ykC⁰(Ω×[0,1]) small enough (that is, for a suitable choice ofν0), one has

φ^˜u

B(xi, ri),0, t_i−¹

2

∩Ω =∅.

But by (3.10) and (3.17), at time 0, the support ofwⁱ is included inB(xi, ri). It follows from the form (3.17) that the support ofwⁱ follows the flow of ˜u. We deduce that

Suppwⁱ

t_i−¹

2,·

∩Ω =∅. Then, we just have to define

ω x, t⁺_i−1

2

= Xk l=i+1

w^l

·, t_i−¹

2

, (3.22)

with the convention

ω(x, t⁺_k−1 2

) = 0. (3.23)

So (3.20) and (3.22) do completely defineω fortin [1/4,1/2]. Note that by (3.17, 3.20) and (3.22), one gets ω(·, t) =

Xk l=i+1

w^l(·, t), ∀t∈h

ti−¹2, ti+¹₂

i

. (3.24)

This way, we get that the restriction ofωto Ω×[0,¹₂] isC([0,¹₂], C^1,α(Ω))-regular and that we have in Ω×[0,¹₂] the relation

∂tω+ (˜u.∇)ω= (ω.∇)˜u. (3.25)

Furthermore, by Lemma 3.1, ω stays divergence-free in ˜Ω×[0,¹₂].

We want to definev inC([0,¹₂], C^1,α(Ω;R³)) by

curlv=ω in Ω×

0,1 2

, (3.26)

divv= 0 in Ω×

0,1 2

, (3.27)

v.n=µ(t)y0.n+y.nin ∂Ω×

0,1 2

, (3.28)

Z

Ω

v.Qⁱdx= 0, ∀t∈

0,1 2

· (3.29)

But to prove that it is possible, let us point out that, for the existence of such a v, we need, in addition to divω= 0, the fact thatω is a curl in Ω. This is proved also by Lemma 3.1.

By the way, we remark that the relation (3.29) is necessary to obtain the unicity ofv.

Now, we can see that any

v⁰ :=v+ Xs j=1

λj(t)Q^j(x), (3.30)

still satisfies (3.26, 3.27) and (3.28), for any choice ofλi. We chooseλi, and hencev⁰ such that Z

Ω

v⁰(0).Qⁱdx= Z

Ω

y0.Qⁱdx, (3.31)

Z

Ω

(∂tv⁰+ ˜u∧ω).Qⁱdx= 0, ∀i∈ {1, ..., s},∀t∈

0,1 2

. (3.32)

Note that this is made possible because the matrix Z

Ω

Qⁱ.Q^jdx

1≤i≤s,1≤j≤s

,

is invertible, as the (Qi) is a free family.

We are now able to define ω in the time-interval (¹₂,³₄). We consider i ∈ {k, . . . , k+s−1}. Let ˜wⁱ ∈ C([ti, ti+1], C^1,α(Ω;R³)) be defined by

˜

wⁱ(ti) = curlℵⁱ, (3.33)

∂tw˜ⁱ+ (˜u.∇) ˜wⁱ= ( ˜wⁱ.∇)∇u˜−u(div ˜˜ wⁱ) in ˜Ω×[ti, ti+1], (3.34)

where ℵⁱ is defined in Lemma 2.3 (see (2.21)). Here precisely, will be needed the fact that ν0 < ν, in such a way that

Supp ˜wⁱ(·, ti+1)⊂Ω˜\Ω. (3.35)

Then, we define the applications ζⁱ (which differ from those in Lemma 2.3 by the time intervals only), for i= 1, . . . , s, respectively on the intervals [ti+k−1, ti+k] by the relations:

curlζⁱ= ˜wⁱ in Ω×[ti+k−1, ti+k], (3.36) divζⁱ= 0 in Ω×[ti+k−1, ti+k], (3.37) ζⁱ.n=∂nθⁱ on∂Ω×[ti+k−1, ti+k], (3.38)

Z

Ω

ζⁱ(tk+i−1).Q^jdx= 0, (3.39)

Z

Ω

(∂tζⁱ+u∧curl(ζⁱ)).Q^jdx= 0, ∀j∈ {1, ..., s}, ∀t∈[ti+k−1, ti+k]. (3.40) We can then define:

ω(x, t) =−λi

1 2

˜

wⁱ(x, t) in [ti, ti+1)×Ω,˜ ∀i∈ {1, ..., s}· (3.41) As in [¹₄,¹₂],ωis continuous in variabletfromSk+s−1

i=k (ti, ti+1) toC^1,α( ˜Ω);R³, and with a limit in (C^1,α( ˜Ω);R³) at the left and at the right of eachti (fori∈ {k+ 1, ..., s+k−1}). Moreover, also as in [¹₄,¹₂], it is continuous from [1/2,3/4] intoC^1,α(Ω);R³.

Now we extend formula (3.26–3.29) to the whole interval [0,1]:









curlv=ω in Ω×[0,1], divv= 0 in Ω×[0,1],

v.n=µ(t)y0.n+y.nin ∂Ω×[0,1], R

Ωv.Qⁱdx= 0, ∀t∈[0,1],

(3.42)

and also extend formula (3.32) (and hence extend the functionsλi), in addition to (3.31):

Z

Ω

(∂tv+ Xj=s j=1

λ⁰j(t)Q^j+ ˜u∧ω).Qⁱdx= 0, ∀i∈ {1, ..., s}, ∀t∈[0,1]. (3.43)

We can now defineF(u) from Ω×[0,1] intoR³:

F(u) :=









v+Ps

i=1λi(t)Qⁱ(x) in Ω×

0,3 4

, 0 in Ω×

3 4,1

.

(3.44)

By this way, the functionF is correctly defined.

By (3.24) and (3.35) we get that

curlF(u)∈C⁰([0,1];C^1,α(Ω);R³). (3.45)

By (3.36–3.40, 3.41) and (3.43), we get that λi

3 4

= 0.

Together with (3.45), this proves that

F(u)∈C([0,1], C^2,α(Ω);R³) (3.46)

and that the relation (3.25) holds in Ω×[0,1]. Moreover, F(u) is obviously a solution to the controllability problem (3.9).

3.4. Proof of Lemma 3.1

As it can be seen from (3.12), divW satisfies the equation

∂t(divW) + (U.∇)(divW) =−(divU)(divW). (3.47) The point (3.13) is hence clear.

To get the second point (3.14), we need more that (3.13). Let us indeed introduce the following family of special functions of Ω. Let us consider, when ∂Ω has many connected components (that is in the case where H2(Ω)6= 0), the set of functionsP^j constructed as follows. We note the connected components of∂Ω: γ0,. . ., γs, and we defineP^j:=∇p^j for anyj ∈ {1, .., s}wherep^j is defined by the relations





∆p^j = 0 in Ω, p_|∂Ω= 0 on (∂Ω\γj), p_|∂Ω= 1 onγj.

(3.48) It is well known that a solenoidal vector field on Ω can be written as the sum of the curl of a vector field and of a linear combination of theP^j.

Consequently, if Ω is an open set such thatH2(Ω) 6= 0, in order that (3.14) occurs, we need, besides the divergence free condition, the following relations to hold:

Z

Ω

W(·, t).P^jdx= 0, ∀t∈[0, T], ∀j∈ {1, .., s}, (3.49) where we definedsandP^j in (3.48). These relations (3.49) are true fort= 0, sinceW(·,0) = curlV0. We want to show this property stays true after t= 0. To prove it, we compute (indicesj forP^j are dropped),

d dt

Z

Ω

W.∇pdx =R

Ω{(U.∇)W}.∇pdx−R

Ω{(W.∇)U}.∇pdx=R

ΩUⁱW_i^jpjdx−R

ΩWⁱ.U_i^jpjdx,

where we denote derivations by lower indices and vector coordinates by upper exponents. Then, integrating by parts, we obtain, since divU = divW = 0,

d dt

R

ΩW.∇pdx = Z

∂Ω

(WⁱU^jpjnⁱ−W^jpjUⁱnⁱ)dσ. (3.50) Aspis constant on each connected component of∂Ω,∇pis normal to the boundary everywhere on the boundary.

We can deduce from this fact, that (W.n)(U.∇p) = (W.∇p)(U.n) on the boundary. The term on the right hand side of (3.50) is thus 0, so (3.49) stays true for all times. From that, we deduce thatW(·, t) is a curl in Ω.

4. Proof of Proposition 1.3

The goal of this section is to prove that F admits a fixed point, and then to prove that it gives a proper solution to Proposition 1.3.

The first part of this proof is thus to find a set invariant byF.

We denote byB(B) the ball inC⁰([0,1], C^2,α(Ω;R³)) with radius B and center 0. Then this invariant set will be found as a certainXν∩ B(B), for properB andν.

In a first step, we prove the following proposition:

Proposition 4.1. For any B > 0, there exists ν0 and ν1, such that if one has ky0k^1,α,Ω < ν1, then for all ν < ν0, for allu∈Xν∩ B(B), one has F(u)∈Xν.

In a second step we prove this proposition:

Proposition 4.2. There exists ν2 > 0, such that if ky0kC^2,α(Ω;R³) ≤ ν2, and if we define the sequences of functions(y^m)m≥0∈(C⁰([0,1], C^2,α(Ω;R³)))^Nand (ω^m)m≥0∈(C⁰([0,1], C^1,α(Ω;R³)))^Nas follows:





y⁰(x, t) =µ(t)y0(x) +y(x, t), y^m+1=F(y^m),

ω^m+1 defined as previously onΩ˜×[0,1],

(4.1)

then the sequence(y^m)m≥0 is bounded inC⁰([0,1], C^2,α(Ω;R³)), the bound depending only on Ωandν2. A fortiori, we will be able to findB and ν such that

F(Xν∩ B(B))⊂Xν∩ B(B).

The last step of the proof of Proposition 1.3 is then to establish that F has a fixed point solution to the non-linear controllability problem.

The proofs of these propositions will require a technique introduced by Bardos and Frisch in [1]. Particularly, we will use the following lemma:

Lemma 4.3. ([1], Lem. 1) Letu,v andgbe three functions of regularityC⁰([0, T], C^1,α( ˜Ω,R³)), satisfying the relations

∂u

∂t + (v.∇)u=g,v.n_|∂Ω˜×[0,T] = 0. (4.2) Then we have on[0, T]

d

dt⁺kuk0,α,Ω˜ ≤ k∂tuk0,α,Ω˜ ≤ kgk0,α,Ω˜+αk∇vk0,α,Ω˜.kuk0,α,Ω˜. (4.3)

4.2. Proof of Proposition 4.1

In the sequel, we will denote byC, C⁰,C1 andC2different positive constants depending only on Ω.

In this section, we will mark each object introduced in the previous section and corresponding to the m-th iteration of the operatorF in the construction of the sequence (y^m)m≥0 by a lower indexm.

As previously, we will first considert in the set [0,1/2], and thent in the interval [1/2,1].

When considering thew^mintroduced in Section 3, we will no longer use the upper indexl (corresponding to thel-th ballB(xl, rl)) in order not to confuse with the index m corresponding to thism-th iteration. All the assertions aboutwwill be valid for any upper index. We will do the same with the index “i” inλi. Also, when considering a time-dependent function f :=f(x, t), we will make no difference betweenkf(·, t)kand kfk(t), whatever spatial norm we use.

We first get an estimate onω^∗_m+1. By (3.15) and Lemma 4.3, one easily gets for t∈[0,1]

d

dt⁺kω_m+1^∗ k^0,α,Ω(t)≤(2 +α)ky˜^mk1,α,Ω˜(t)kω_m+1^∗ k^0,α,Ω(t). (4.4) With Gronwall’s lemma, we deduce from (4.4) that fort∈[0,1]

kω_m+1^∗ k^0,α,Ω(t)≤ kω^∗_m+1k^0,α,Ω(0)e^{(2+α)tky˜mkC0([0,1],C1,α(Ω);R3)}. (4.5) We do the same with the equation (3.17), and get by Lemma 4.3 the estimate fort∈[1/4,1/2]

d

dt⁺kwm+1k0,α,Ω(t)≤(2 +α)ky˜^mk1,α,Ω˜kwm+1k0,α,Ω(t). (4.6) We easily deduce from that and from Gronwall’s lemma that fort∈[1/4,1/2]

kwm+1k^0,α,Ω(t)≤ kwm+1k^0,α,Ω 1

4

e^{(2+α)(t−14)ky˜mk}C0 ([0,1],C1,α(Ω);R3 ), (4.7) from which we get, with (3.24), that fort∈[1/4,1/2]

kω^m+1k^0,α,Ω(t)≤kkω^mk^0,α,Ω 1

4

e^{3(t−14)k˜ymk}C0 ([0,1],C1,α(Ω);R3 ), (4.8) from what we deduce with (3.15) and (4.5) that fort∈[0,1/2]

kω^m+1k^0,α,Ω(t)≤kky0k^1,α,Ωe^{3tky˜mkC0([0,1],C1,α(Ω);R3)}. (4.9)

We now want to get (4.9) for the rest of the time [1/2,1], and by the way obtain an estimate on theλ^m_i . By (3.42) and (3.43) we have on [0,1]

d

dt⁺|λ^m_i | ≤Ckω^m+1k^0,Ωky^mk^0,Ω. (4.10) We deduce that fort∈[0,1/2]

|λ^m+1_i (t)| ≤C1ky0k^1,α,Ωky^mkC⁰([0,1],C¹(Ω);R³)e^3ky˜mkC0([0,1],C1 (Ω);R3). (4.11) As for (4.5), one can deduce that for alli∈ {k, . . . , k+s−1}and for allt∈[ti, ti+1), one has

kω^m+1k^0,α,Ω(t)≤ kω^m+1k^0,α,Ω(ti)e^{(2+α)(t−ti)ky˜mk}C0 ([0,1],C1,α(Ω);R3 ). With (3.41) and (4.11), we get that for t∈[1/2,3/4]

kω^m+1k^0,α(t)≤C2ky0k^1,α,Ωky^mkC⁰([0,1],C¹(Ω))e^3ky˜mkC0([0,1],C1 (Ω);R3). (4.12)

We can deduce from it, with (3.42–3.44, 4.9, 4.11) and that fort∈[0,1]

kF(u)−yk^0,Ω(t)≤C(ky^mk^1,α,Ω)ky0k^1,α,Ω, (4.13) where C(·) is an increasing, positive real-valued, numerical function.

So we have proved that F is well defined on Xν and that for any B >0, there exists ν1 =ν1(B)>0 such that for anyy0 satisfyingky0k^2,α,Ω< ν < ν1, one has

F(Xν∩ B^{C([0,1],C1,α(Ω;R3))}(B))⊂Xν,

where we have denoted byB^{C([0,1],C1,α(Ω;R3))}(B) the 0-centered open ball inC([0,1], C^1,α(Ω;R³)) with radius B, at least ifν0 andν1 (depending onB) are small small enough.

4.3. Proof of Proposition 4.2

Let us consider the sequence (y^m)m≥0∈(C⁰([0,1], C^2,α(Ω;R³)))^Nby (4.1).

In a first step, we just deal with the boundedness of the sequence (y^m) in the spaceC⁰([0,1], C^1,α(Ω;R³)).

We will come back to the boundedness inC⁰([0,1], C^2,α(Ω;R³)) at the end of this section.

Let us denote byCi,i≥1, various constants which do not depend onm. Combining (4.9) and (4.12), one can get for anyt∈[0,1] that

kω^m+1k^0,α,Ω≤C3ky0k^1,α,Ωe^3kymkC0 ([0,1],C1,α(Ω);R3 )

1 +ky^mkC⁰([0,1],C^1,α(Ω);R³)

. (4.14)

By (4.10) and (4.12), one as also in [0,1]

|λ^m+1_i | ≤C4ky0k^1,α,Ωe^{3kymkC0([0,1],C1,α(Ω);R3)}

1 +ky^mk²_C0([0,1],C^1,α(Ω);R³)

. (4.15)

On another side, by (3.42) and (3.44), one can find some constants such that, for anyt∈[0,1], ky^m+1k^1,α,Ω(t)≤C7ky0k^1,α,Ω+C8kω^m+1k^0,α(t) +C9

X

i

kλikC⁰([0,1]). (4.16)

We deduce from (4.14, 4.15) and (4.16) that for everyt∈[0,1]

ky^m+1k^1,α,Ω(t)≤C10ky0k^1,α,Ω

1 +e^{3kymkC0([0,1],C1,α(Ω);R3)} 1 +ky^mk²_C0([0,1],C^1,α(Ω);R³)

. (4.17)

Note that this is made valid in [3/4,1] because of the trivial form ofy^m in this time segment.

We want to deduce from (4.17) that, reducingν0 if necessary, one can get

ky^mkC⁰([0,1],C^1,α(Ω);R³)≤2kykC⁰([0,1],C^1,α(Ω);R³), ∀m∈N. (4.18) The proof of (4.18) is done by induction.

We check (4.18) form= 0. Asy⁰=µ(t)y0+y, (4.18) is satisfied ifν0<kyk^1,α.

We now suppose that (4.18) is satisfied for a fixedm, and show it is still valid at rankm+ 1.

We impose ν0 in order that C10ν0

1 +e^6kykC0 ([0,1],C1,α(Ω);R3) 1 + 4kyk²_C0([0,1],C^1,α(Ω);R³)

<2kykC⁰([0,1],C^1,α(Ω);R³). (4.19)