LOCAL EXACT CONTROLLABILITY FOR THE … · Local exact controllability 483 space of functions in L2(Ω) whose weak derivatives of order less than or equal to m are also in L2(Ω)

Advances in Differential Equations Volume 10, Number 5, Pages 481–504

LOCAL EXACT CONTROLLABILITY FOR THEMAGNETOHYDRODYNAMIC EQUATIONS REVISITED

Viorel Barbu, Teodor Havarneanu, Catalin PopaFacultatea de Matematica, Universitatea “Al.I. Cuza”

Bdul. Carol I, Nr. 11, 700506 Iasi, Romania

S.S. SritharanDepartment of Mathematics, University of Wyoming

Laramie, WY 82071, USA

Abstract. The local exact controllability of the steady-state solutionsfor the magnetohydrodynamic equations was established by the authorsin a previous paper. However, it was subsequently found that an argu-ment given there was relying on an uncertain estimate. The purpose ofthis paper is to correct that argument.

1. Introduction

The local exact internal controllability of the steady-state solutions for themagnetohydrodynamic (MHD) equations was established by the authors in[3]. More specifically, we showed that the sufficiently smooth steady-statesolutions can be reached in a given (finite) time if we act on two controlparameters distributed on a small given subdomain and the process startsfrom initial states which are close enough to the target steady-state solution.In a few words, the method in proving that is the following: The local con-trollability of the steady-state solutions for the original MHD equations canbe reduced to the global controllability of the null solution for the linearizedMHD equations by using Kakutani’s fixed–point theorem. The null control-lability problem for the linearized system is solved by passing to the limitinto a family of appropriate optimal control problems for the same systemas a certain parameter tends to infinity. The key estimate in proving theconvergence is an observability inequality for the linearized system. Suchan inequality is obtained from so–called Carleman estimates for the Stokesand dynamo equations. To pass from the Carleman estimates to the observ-ability inequality, we have to eliminate quantities like pressure coming from

Accepted for publication: January 2005.AMS Subject Classifications: 35Q35, 76W05, 76D55, 35Q30, 35Q60, 93B05, 93C20,

93B07.

481

482 Viorel Barbu, Teodor Havarneanu, Catalin Popa, and S.S. Sritharan

both Stokes and dynamo equations. The main ingredient in doing so wasLemma 2.2.4 in [1]. Unfortunately, the proof of that lemma has been provedto be wrong.

In this paper we show how those pressure–type quantities can be removedby using (as in [4]) an argument due to O.Yu. Imanuvilov. However, toaccommodate that argument to the fixed–point technique, some more regu-larity for the target steady-state solutions is needed. In this way, the gap inthe proof of the main result in [3] is filled in. A similar approach is used in[2] for the internal controllability of the Navier–Stokes equations.

2. Main result

Let Ω be a bounded simply connected open set in R3 with boundary ∂Ωof class C2, and let T > 0 be a fixed time. We set Q = Ω × (0, T ) andconsider an open subset ω of Ω. We deal with the following controlled MHDequations:

∂y

∂t− ν∆y+(y·∇)y +∇p+∇

(12

B2)−(B·∇)B = f + χωu in Q,

∂B

∂t+ η curl(curlB)+(y·∇)B−(B·∇)y = P (χωv) in Q, (2.1)

div y = 0, div B = 0 in Q,

y = 0, B·N = 0, (curlB) × N = 0 on Σ = ∂Ω×(0, T ),

y(·, 0) = y0, B(·, 0) = B0 in Ω.

Here y = (y1, y2, y3) : Ω × [0, T ] → R3 is the velocity vector field, p :Ω×[0, T ] → R is the (scalar) pressure, and B = (B1, B2, B3) : Ω×[0, T ] → R3

is the magnetic field. The vector functions u = (u1, u2, u3) : Ω× [0, T ] → R3

and v = (v1, v2, v3) : Ω × [0, T ] → R3 are control parameters, and χω is thecharacteristic function of ω. The variables of y, p, B, u, and v are denoted byx = (x1, x2, x3) and t (belonging to Ω and [0, T ], respectively). The vectorfunction f = (f1, f2, f3) : Ω → R3 is the known density of the externalforces, and the vector fields y0 : Ω → R3 and B0 : Ω → R3 are the giveninitial velocity and magnetic fields. The operator P is the Leray projector(put there to kill the gradient part of χωv). Finally, ν and η represent thekinematic viscosity and magnetic resistivity coefficients, respectively, and Nis the unit outer normal field on ∂Ω. For the sake of simplicity and withoutloss of generality, from now on, ν and η will be supposed to be 1.

In our further statements and considerations we need several functionspaces. For each positive integer m, we denote by Hm(Ω) the Sobolev

Local exact controllability 483

space of functions in L2(Ω) whose weak derivatives of order less than orequal to m are also in L2(Ω). Similarly, H2,1(Q) is the space of functionsin L2(Q) whose first and second-order weak derivatives with respect to thespace variables x1, x2, and x3, and first-order weak derivative with respectto t belong to L2(Q), too. The space H1,1(Q) is defined similarly. Thefractional-order Sobolev space H3/2(Ω) will be required by a certain ar-gument. We denote by ((H1

0 (Ω))3)′ the dual of the space (H10 (Ω))3 (of

vector fields in (H1(Ω))3 which are zero on the boundary). Since almostall the functions involved in the equations we deal with are in fact vec-tor functions (fields), we mostly use product function spaces: (L2(Ω))3,(L2(Q))3, (H1(Ω))3, (H2(Ω))3, (H2,1(Q))3, etc., all of them endowed withthe product norms. The time-dependent function space L2(0, T ; H1(Ω))consists of all (equivalence classes of) measurable functions from (0, T ) toH1(Ω) with the square of their H1(Ω) norms integrable over (0, T ). Thespaces L2(0, T ; (L2(Ω))3), L2(0, T ; (H1(Ω))3), and L∞(0, T ; (H2(Ω))3) aredefined analogously. The first-order Sobolev spaces corresponding to thelast three time-dependent function spaces are denoted by H1(0, T ; (L2(Ω))3),H1(0, T ; (H1(Ω))3), and W 1,∞(0, T ; (H2(Ω))3), respectively. We shall alsouse the second-order versions of the first two Sobolev spaces: H2(0, T ;(L2(Ω))3) and H2(0, T ; (H1(Ω))3). The norms of all the considered spacesare denoted in the same manner: | · |(L2(Ω))3 , | · |(H1(Ω))3 , | · |(H2(Ω))3 , etc., or,simply, | · |L2 , | · |H1 , | · |H2 , etc.

It is well known that the MHD equations can be viewed as a system oftwo evolution equations in the space H of all weakly divergence–free vectorfields in (L2(Ω))3 which are tangential to the boundary in the weak sense,endowed with the L2 norm:

H = y ∈ (L2(Ω))3 : div y = 0 in Ω and y·N = 0 on ∂Ω.We also need the spaces

V1 = y ∈ (H10 (Ω))3 : div y = 0 in Ω,

V2 = B ∈ (H1(Ω))3 : div B = 0 in Ω and B·N = 0 on ∂Ω,both endowed with the H1 norm. Two linear operators are also required:The Stokes operator A1 : D(A1) → H is defined as

A1y = −P∆y for y ∈ D(A1),

where ∆ is the Laplace operator and D(A1) = (H2(Ω))3 ∩ V1 is endowedwith the H2 norm. The operator A2 : D(A2) → H is defined by

A2B = curl(curlB) for B ∈ D(A2),


where D(A2) = B ∈ (H2(Ω))3 ∩ V2 : (curlB)×N = 0 on ∂Ω is endowedwith the H2 norm.

Using the two operators defined before, one can write equations (2.1) (withν = η = 1) as a system of two evolution equations in H:

y′ + A1y + P (y·∇)y − P (B·∇)B = Pf + P (χωu),

B′ + A2B + (y·∇)B − (B·∇)y = P (χωv), (2.2)

y(0) = y0, B(0) = B0.

(Here and in what follows, we sometimes write y(t), B(t), u(t), and v(t)instead of y(·, t), B(·, t), u(·, t), and v(·, t), when we view them as elementsof (L2(Ω))3. The derivatives of y, B, u, and v with respect to t are denotedby y′, y′′, B′, B′′, u′, and v′, too.)

Let (ye, Be, pe) be a steady-state solution of (2.1); i.e.,

− ∆ye + (ye·∇)ye + ∇pe + ∇(12

B2e

)− (Be·∇)Be = f in Ω,

curl(curlBe) + (ye·∇)Be − (Be·∇)ye = 0 in Ω,

div ye = 0, div Be = 0 in Ω, (2.3)

ye = 0, Be·N = 0, (curlBe)×N = 0 on ∂Ω.

By a known regularity result, it is easily seen that the following linearspace is a subspace of (H4(Ω))3×(H4(Ω))3 :

X = (y0, B0) ∈ D(A1)×D(A2) : A1y0 + P (((y0 + ye)·∇)y0 + (y0·∇)ye

+ B0·(∇Be) − ((B0 + Be)·∇)B0 − (B0·∇)Be) ∈ D(A1), A2B0

+ P (((y0 + ye)·∇)B0 + (y0·∇)Be − ((B0 + Be)·∇)y0 − (B0·∇)ye)

∈ D(A2),

where B0·(∇Be) is the vector field of components B0·∂Be/∂xi, i = 1, 2, 3.

Theorem 2.1. Let Ω be a simply connected, bounded, open subset of R3,locally located on one side of its boundary ∂Ω, which is supposed to be ofclass C2, and let ω be an open subset of Ω. Let f ∈ (L2(Ω))3 and let(ye, Be, pe) ∈ (H3(Ω))3×(H3(Ω))3×H1(Ω) be a steady-state solution of (2.3).Then there is η > 0 such that for (y0, B0) ∈ (ye, Be) + X satisfying

|y0 − ye|(H4(Ω))3 + |B0 − Be|(H4(Ω))3 ≤ η


there exists (u, v, y, B, p) ∈ (H2(0, T ; (L2(Ω))3))2×(W 1,∞(0, T ; D(A1)) ∩H2(0, T ; V1))×(W 1,∞(0, T ; D(A2))∩H2(0, T ; V2))×L2(0, T ; H1(Ω)) that sat-isfies (2.1) (or (2.2)) and

y(x, T ) = ye(x), B(x, T ) = Be(x) a.e. x ∈ Ω.

We shall reduce the problem of controllability of the steady-state solution(ye, Be) for system (2.1) to that of controllability of the null solution for thesystem obtained by subtracting (2.1) and (2.3). Redenoting y − ye, B −Be,p−pe, y0−ye, and B0−Be by y, B, p, y0, and B0 respectively, the differenceof (2.1) and (2.3) looks as follows:

∂y

∂t− ∆y + (y·∇)y + (ye·∇)y + (y·∇)ye

+∇p + ∇(12

B2)

+ ∇(Be·B)−(B·∇)B − (Be·∇)B − (B·∇)Be = χωu in Q,

∂B

∂t+ curl(curlB) + (y·∇)B + (ye·∇)B + (y·∇)Be

−(B·∇)y − (Be·∇)y − (B·∇)ye=P (χωv) in Q,


y = 0, B·N = 0, (curlB)×N = 0 on Σ,

y(·, 0) = y0, B(·, 0) = B0 in Ω.

(2.4)

By Kakutani’s fixed-point theorem, the local exact controllability of the nullsolution of system (2.4) will be reduced to the global exact controllability ofthe null solution of the following linearized system:

∂y

∂t− ∆y + ((w + ye)·∇)y + (y·∇)ye + E·(∇B) + ∇(Be·B)

− ((E + Be)·∇)B − (B·∇)Be + ∇p = χωu in Q,

∂B

∂t+ curl(curlB) + P (((w + ye)·∇)B + (y·∇)Be

− ((E + Be)·∇)y − (B·∇)ye) = P (χωv) in Q,


y = 0, B·N = 0, (curlB)×N = 0 on Σ,

y(·, 0) = y0, B(·, 0) = B0 in Ω,(2.5)

where (w, E) is a fixed pair having the same regularity as (y, B) in Theorem2.1, E·(∇B) is the vector function of components E·∂B/∂xi, i = 1, 2, 3, and(y0, B0) ∈ X. On the other hand, the null controllability of system (2.5)


is equivalent (to a certain extent) to the observability of its adjoint system;that will be established in Section 4 by using Carleman–type inequalities forthe adjoint Stokes and dynamo equations.

3. Carleman inequalities for the Stokes and dynamo equations

We consider the adjoint Stokes equations with the null Dirichlet boundarycondition:

∂z

∂t+ ∆z + ∇q = g in Q,

div z = 0 in Q,

z = 0 on Σ,

(3.1)

and the adjoint of certain dynamo–type equations with the boundary con-ditions satisfied by B in (2.1):

∂C

∂t− curl(curlC) = PG in Q,

div C = 0 in Q,

C·N = 0, (curlC)×N = 0 on Σ.

(3.2)

We assume (as in the statement of Theorem 2.1) that Ω is open, bounded,and simply connected, and has the boundary ∂Ω of class C2. Let ω, ω0, andω1 be arbitrary open subsets of Ω such that ω0 ⊂⊂ ω1 ⊂⊂ ω. To expressthe Carleman inequalities, we need some auxiliary functions.

Let U be a neighborhood of ∂Ω, and set D = Ω ∩ U . We suppose that∂D = Γ0 ∪ Γ1, where Γ0 = ∂Ω and Γ1 is diffeomorphic with Γ0. One canshow that it is possible to construct (see [4]) a function ψ ∈ C2(Ω) such that

ψ > 0 in Ω, ψ = 0 on ∂Ω, |∇ψ| > 0 in Ω \ ω0, (3.3)

and having the form ψ = 1 − θ in Ω, where θ ∈ C2(D) satisfies

0 ≤ θ ≤ 1, |∇θ| > 0 in D, θ = 0 on Γ1, θ = 1 on Γ0,

and it is chosen such that, for the surface Γt = x ∈ D : θ(x) = 1 − t witht ∈ [0, T ], the following interpolation inequality holds:∫

Γt

|v|2dσ ≤ c|v|2(1−t)L2(Γ0)

|v|2tL2(Γ1)

for all v ∈ H1(D) which are harmonic in D and satisfy ∂v/∂N = 0 on Γ1,for all t ∈ [0, T ]. (The constant c > 0 is independent of t.) With the aid of


such a function ψ, we define for λ > 0 :

ϕ(x, t) =eλψ(x)

(t(T − t))2, α(x, t) =

eλψ(x) − e2λ|ψ|C(Ω)

(t(T − t))2·

On the boundary ∂Ω, ϕ and α take the values:

ϕ∗(t) =1

(t(T − t))2, α∗(t) =

1 − e2λ|ψ|C(Ω)

(t(T − t))2·

We set Qω0 = ω0×(0, T ), Qω1 = ω1×(0, T ), and Qω = ω×(0, T ).Now, we can present the two Carleman inequalities.

Theorem 3.1. Let Ω be a simply connected, bounded, open subset of R3,locally located on one side of its boundary ∂Ω of class C2, and let ω, ω0, andω1 be open subsets of Ω such that ω0 ⊂⊂ ω1 ⊂⊂ ω. Then there exists λ0 > 0such that for any λ > λ0 one can find s0(λ) > 0 and c(λ) > 0 such that fors > s0(λ) the following inequality holds:∫

Qe2sα

( 1sϕ

(∣∣∣∂z

∂t

∣∣∣2 +3∑

i,j=1

∣∣∣ ∂2z

∂xi∂xj

∣∣∣2) + sϕ|∇z|2 + s3ϕ3|z|2)dx dt

≤ c(λ)( ∫

Qω

e2sδα∗ |z|2dx dt +∫

Qω1

e2sδα∗q2dx dt (3.4)

+ s

∫Q

e2sα∗ϕ∗|g|2dx dt +

∫Q

e2sα|div g|2dx dt)

for all g ∈ L2(0, T ; (H1(Ω))3) and all corresponding solutions

(z, q) ∈ (H2,1(Q))3×L2(0, T ; H1(Ω))

of system (3.1), and for some δ ∈ (0, 1).Inequality (3.4) was essentially established by Imanuvilov (see Theorems

2.11 and 3.1 in [4]).Theorem 3.2. Let Ω, ω, ω0, and ω1 be open subsets of R3 as in the statementof Theorem 3.1. Then there exists λ0 > 0 such that for any λ > λ0 one canfind s0(λ) > 0 and c(λ) > 0 such that for s > s0(λ) the following inequalityholds:∫

Qe2sα

( 1sϕ

(∣∣∣∂C

∂t

∣∣∣2 +3∑

i,j=1

∣∣∣ ∂2C

∂xi∂xj

∣∣∣2) + sϕ|∇C|2 + s3ϕ3|C|2)dx dt (3.5)

≤c(λ)( ∫

Qω

e2sδα∗ |C|2dx dt +∫

Qω1

e2sδα∗r2dx dt +

∫Qe2sα(|G|2+|div G|2)dx dt

)


for all first-order linear differential operators G : (H1,1(Q))3 −→ (L2(Q))3

satisfying (for some c > 0)

|G(C)·N | ≤ c|C| on Σ for C ∈ (H2,1(Q))3, (3.6)

for all corresponding solutions C ∈ (H2,1(Q))3 of system (3.2) and r ∈L2(0, T ; H1(Ω)) which satisfy

G = PG + ∇r, (3.7)

and for some δ ∈ (0, 1).Proof. As one can see in [3] (inequality (3.58) there), one can find λ0 > 0and, for λ > λ0, s0(λ) > 0, such that the following inequality holds:∫

Qe2sα

( 1sϕ

(∣∣∣∂C

∂t

∣∣∣2 +3∑

i,j=1

∣∣∣ ∂2C

∂xi∂xj

∣∣∣2) + sλϕ|∇C|2 + s3λ3ϕ3|C|2)dx dt

≤ c(λ)( ∫

Qe2sα|PG|2dx dt +

∫Qω

e2sαs3ϕ3|C|2dx dt)

for s > s0(λ).

(3.8)By (3.7), we can replace the weighted integral of |PG|2 in (3.8) with

similar integrals of |G|2 and |∇r|2.In order to estimate the integral of |∇r|2 in a convenient way, we apply

the divergence operator to both sides of (3.7) and then multiply it by N onΣ. So, we have

∆r = div G in Q,

∂r

∂N= G·N on Σ.

(3.9)

Now we treat r in (3.9) in the same manner as C was treated in [3], taking(3.6) into account, too. We obtain the following Carleman–type estimatefor r:∫

Qe2sαsϕ|∇r|2dx dt

≤ c(λ)( ∫

Qω1

e2sαs3ϕ3r2dx dt +∫

Qe2sα|div G|2dx dt

)+

12

s3λ3

∫Q

e2sαϕ3|C|2dx dt +12

sλ

∫Q

e2sαϕ|∇C|2dx dt

(3.10)

for λ > λ0 and s > s0(λ) (where λ0 and s0(λ) are possibly greater thanthose before).


Since one can find δ ∈ (0, 1) and c(δ) > 0 such that s3ϕ3e2sα ≤ c(δ)e2sδα∗

for λ large enough, inserting (3.10) into (3.8) by means of (3.7), we obtain(3.5).

4. Proof of Theorem 2.1

First we shall study the controllability of the null solution of system (2.5).Let K be the following convex compact subset of (L2(Q))3:

K = (w, E) ∈ (W 1,∞(0, T ; D(A1)) ∩ H2(0, T ; V1))

×(W 1,∞(0, T ; D(A2)) ∩ H2(0, T ; V2)) : w(0) = y0, E(0) = B0,

µ(w, E) ≤ M,

where (y0, B0) ∈ X and

µ(w, E)= |w|W 1,∞(0,T ;D(A1))+|w|H2(0,T ;V1)+|E|W 1,∞(0,T ;D(A2))+|E|H2(0,T ;V2).

Lemma 4.1. Let Ω and ω be as in the statement of Theorem 2.1. Then thereis M > 0 such that, for any (w, E) ∈ K, there exist u, v ∈ H2(0, T ; (L2(Ω))3)and (y, B, p) ∈ (W 1,∞(0, T ; D(A1)) ∩ H2(0, T ; V1))×(W 1,∞(0, T ; D(A2)) ∩H2(0, T ; V2))×L2(0, T ; H1(Ω)) which satisfy (2.5),

y(x, T ) = 0, B(x, T ) = 0, a.e. x ∈ Ω, (4.1)

and

|u|H2(0,T ;(L2(Ω))3)+|v|H2(0,T ;(L2(Ω))3)≤ec(M2+1)(|y0|2(L2(Ω))3+|B0|2(L2(Ω))3

),

(4.2)for some positive constant c independent of (w, E) ∈ K.

Proof. Let us fix λ > λ0, where λ0 > 0 is given by Theorems 3.1 and 3.2.For ε > 0 (and s > s0(λ)) we consider the following optimal control problem:

Minimize

(Pε)12

∫Q

e−2sδα∗(|u|2 + |v|2)dx dt +

12ε

∫Ω(|y(x, T )|2 + |B(x, T )|2)dx

over all u, v ∈ (L2(Q))3, where y and B satisfy (2.5).For each ε > 0, problem (Pε) has a unique solution (uε, vε, yε, Bε, pε). We

shall show that the solution (u, v, y, B, p) of the null controllability problemfor the linear system (2.5) is the limit (in a certain norm) of the sequence(uε, vε, yε, Bε, pε) as ε → 0.


By the Pontryagin maximum principle, the solution (uε, vε, yε, Bε, pε) ofproblem (Pε) satisfies the following conditions:

uε = χωe2sδα∗zε, vε = χωe2sδα∗

Cε a.e. in Q, (4.3)

where (zε, Cε, qε) is the solution of the adjoint of system (2.5):

∂zε

∂t+ ∆zε + ((w + ye)·∇)zε − zε·(∇ye)

− ((E + Be)·∇)Cε − Cε·(∇Be) + ∇qε = 0 in Q,

∂Cε

∂t− curl(curlCε)+P (((w + ye)·∇)Cε+Cε·(∇ye)

−((E+Be)·∇)zε+zε·(∇Be)+(zε·∇)E)=0 in Q,

div zε = 0, div Cε = 0 in Q,

zε = 0, Cε·N = 0, (curlCε)×N = 0 on Σ,

zε(·, T ) = −1ε

yε(·, T ), Cε(·, T ) = −1ε

Bε(·, T ) in Ω,

(4.4)

where zε·(∇ye), Cε·(∇Be), Cε·(∇ye), and zε·(∇Be) are defined as E·(∇B)in (2.5).

Using (2.5), (4.3), and (4.4), we have∫Qω

e2sδα∗(|zε|2 + |Cε|2)dx dt +

1ε

(|yε(T )|2L2 + |Bε(T )|2L2)

= −∫

Ωzε(x, 0)·y0(x)dx −

∫Ω

Cε(x, 0)·B0(x)dx

≤ |zε(0)|L2 |y0|L2 + |Cε(0)|L2 |B0|L2 .

(4.5)

In what follows, for simplicity, we shall omit the ε subscript after z, C,and q. We can view equations (4.4) as (3.1), (3.2) with

g = −((w + ye)·∇)z + z·(∇ye) + ((E + Be)·∇)C + C·(∇Be),G = −((w + ye)·∇)C − C·(∇ye) + ((E + Be)·∇)z − z·(∇Be) − (z·∇)E.

Now we should couple inequality (4.5) with Carleman inequalities (3.4) and(3.5). But, in doing so, it would be very difficult to remove q and r there. Apossible way to surpass this difficulty is indicated by Imanuvilov in [4]: Weneed to pass from z, C, q, r, g, and G to the new functions z, C, q, r, g, andG, defined as

z(t) =∫ t

T2

z(τ)dτ, q(t) =∫ t

T2

q(τ)dτ, g(t) =∫ t

T2

g(τ)dτ,


C(t) =∫ t

T2

C(τ)dτ, r(t) =∫ t

T2

r(τ)dτ, G(t) =∫ t

T2

G(τ)dτ,

where r satisfies (3.7) with G given before.Integrating equations (4.4) from T/2 to t and taking (3.7) into account,

we obtain

∆z + ∇q = g − z + z(·, T

2)

in Q,

∆C + ∇r = G − C + C(·, T

2)

in Q,

div z = 0, div C = 0 in Q,

(4.6)

where g and G are given by

g(t) = −(ye·∇)z + z·(∇ye) + (Be·∇)C + C·(∇Be) − (w·∇)z + (E·∇)C

+∫ t

T2

(∂w

∂τ·∇

)z dτ −

∫ t

T2

(∂E

∂τ·∇

)C dτ,

G(t) = −(ye·∇)C − C·(∇ye) + (Be·∇)z − z·(∇Be) − (w·∇)C + (E·∇)z

− (z·∇)E +∫ t

T2

(∂w

∂τ·∇

)C dτ −

∫ t

T2

(∂E

∂τ·∇

)z dτ +

∫ t

T2

(z·∇)∂E

∂τdτ.

Since z = ∂z/∂t and C = ∂C/∂t, we also have

∂z

∂t+ ∆z + ∇q = g + z

(·, T

2)

in Q,

∂C

∂t− curl(curlC) + ∇r = G + C

(·, T

2)

in Q,

div z = 0, div C = 0 in Q,

z = 0, C·N = 0, (curlC)×N = 0 on Σ.

(4.7)

Equations (4.7) are just equations (3.1) and (3.2) with their right–hand sidesmodified in an obvious way, taken together. Besides, inequality (3.6) withG replaced by G + C(·, T/2) is satisfied (see the comments in [3]). So, theCarleman inequalities (3.4) and (3.5) give∫

Qe2sα

( 1sϕ

(∣∣∣∂z

∂t

∣∣∣2 +∣∣∣∂C

∂t

∣∣∣2 +3∑

i,j=1

(∣∣∣ ∂2z

∂xi∂xj

∣∣∣2 +∣∣∣ ∂2C

∂xi∂xj

∣∣∣2))+ sϕ(|∇z|2 + |∇C|2) + s3ϕ3(|z|2 + |C|2)

)dx dt (4.8)


≤ c(λ)( ∫

Qω

e2sδα∗(|z|2 + |C|2)dx dt +

∫Qω1

e2sδα∗(q2 + r2)dx dt

+ s

∫Q

e2sα∗ϕ∗

(|g|2 +

∣∣∣z(·, T

2

)∣∣∣2)dx dt

+∫

Qe2sα

(|G|2 + |div g|2 + |div G|2 +

∣∣∣C(·, T

2

)∣∣∣2)dx dt

for s > s0(λ). Now, without loss of generality, we may suppose that∫ω1

q(x, t)dx = 0 and∫

ω1

r(x, t)dx = 0 for t ∈ [0, T ].

Otherwise, we replace q and r by

q − (meas ω1)−1

∫ω1

q dx and r − (meas ω1)−1

∫ω1

r dx.

Then, q and r satisfy the same conditions:∫ω1

q(x, t)dx = 0 and∫

ω1

r(x, t)dx = 0 for t ∈ [0, T ],

so, by Proposition 1.2 in [6], we have∫Qω1

e2sδα∗(q2 + r2)dx dt

≤ c

∫ T

0e2sδα∗

(|∇q|2((H1

0 (ω1))3)′ + |∇r|2((H10 (ω1))3)′

)dt.

(4.9)

To be able to estimate the right–hand side of (4.9), we split equations (4.6)into two parts:

∆z1 + ∇q1 = g − z + z(·, T

2)

in ω,

∆C1 + ∇r1 = G − C + C(·, T

2)

in ω,

div z1 = 0, div C1 = 0 in ω,

z1 = 0, C1 = 0 on ∂ω,

(4.10)

and∆z2 + ∇q2 = 0 in ω,

∆C2 + ∇r2 = 0 in ω,

div z2 = 0, div C2 = 0 in ω,

(4.11)

where z2 = z − z1, C2 = C − C1, q2 = q − q1, and r2 = r − r1.


By equations (4.10) we have

|∇q1|((H10 (ω1))3)′

≤ |z1|(H1(ω1))3 + |g|((H10 (ω1))3)′ + |z|((H1

0 (ω1))3)′ +∣∣z(

·, T

2)∣∣

((H10 (ω1))3)′ ,

|∇r1|((H10 (ω1))3)′

≤ |C1|(H1(ω1))3 + |G|((H10 (ω1))3)′ + |C|((H1

0 (ω1))3)′ +∣∣C(

·, T

2)∣∣

((H10 (ω1))3)′ .

Using the well-known estimate for the weak solution of the Stokes equationswith null boundary condition, we obtain

|∇q1|((H10 (ω1))3)′≤c

(|g|((H1

0 (ω1))3)′ + |z|(L2(ω))3+∣∣z(

·, T

2)∣∣

(L2(ω))3

), (4.12)

|∇r1|((H10 (ω1))3)′≤c

(|G|((H1

0 (ω1))3)′ + |C|(L2(ω))3+∣∣C(

·, T

2)∣∣

(L2(ω))3

).

Let us estimate the ((H10 (ω1))3)′ norms of g and G in (4.12). Using suitable

estimates for the trilinear form

b(u, v, w) =∫

Ω((u·∇)v)·w dx

(see [5]) and the fact that ue, Be ∈ (H3(Ω))3, we have∣∣∣ ∫Ω

g·χ dx∣∣∣ =

∣∣∣ − b(ye, z, χ) + b(χ, ye, z) + b(Be, C, χ) + b(χ, Be, C)

− b(w, z, χ) + b(χ, C, E) +∫ t

T2

b(∂w

∂τ, z, χ

)dτ −

∫ t

T2

b(∂E

∂τ, C, χ

)dτ

∣∣∣≤ c

(|z|(L2(ω1))3 + |C|(L2(ω1))3 +

∣∣∣ ∫ t

T2

(|z|(L2(ω1))3 + |C|(L2(ω1))3

)dτ

∣∣∣ )×|χ|(H1

0 (ω1))3 a.e. on (0, T ), for all χ ∈ (H10 (ω1))3. (4.13)

Analogously,∣∣∣ ∫Ω

G·χ dx∣∣∣

≤ c(|z|(L2(ω1))3 + |C|(L2(ω1))3 +

∣∣∣ ∫ t

T2

(|z|(L2(ω1))3 + |C|(L2(ω1))3

)dτ

∣∣∣ )×|χ|(H1

0 (ω1))3 a.e. on (0, T ), for all χ ∈ (H10 (ω1))3. (4.14)


Inserting the estimates of the ((H10 (ω1))3)′ norms of g and G coming from

(4.13) and (4.14) into inequalities (4.12), one obtains

|∇q1|((H10 (ω1))3)′ + |∇r1|((H1

0 (ω1))3)′

≤ c(|z|(L2(ω))3 + |C|(L2(ω))3 +

∣∣∣ ∫ t

T2

(|z|(L2(ω))3 + |C|(L2(ω))3

)dτ

∣∣∣+|z|(L2(ω))3 + |C|(L2(ω))3 +

∣∣∣z(·,T

2

)∣∣∣(L2(ω))3

+∣∣∣C(

·,T2

)∣∣∣(L2(ω))3

)a.e. on (0, T ).

(4.15)

Applying the divergence operator to (4.11), we have

∆q2 = 0 in ω,

∆r2 = 0 in ω,

and so, (4.11) can become

∆2z2 = 0 in ω,

∆2C2 = 0 in ω.

A well–known interior estimate for biharmonic functions yields

|z2|(H2(ω1))3 + |C2|(H2(ω1))3 ≤ c(|z2|(L2(ω))3 + |C2|(L2(ω))3

)≤ c

(|z|(L2(ω))3 + |C|(L2(ω))3 + |z1|(L2(ω))3 + |C1|(L2(ω))3

).

(4.16)

From (4.10), using (4.15), we derive

|z1|(H1(ω))3 + |C1|(H1(ω))3

≤ c(|z|(L2(ω))3 + |C|(L2(ω))3 +

∣∣∣ ∫ t

T2

(|z|(L2(ω))3+|C|(L2(ω))3

)dτ

∣∣∣+|z|(L2(ω))3 + |C|(L2(ω))3 +

∣∣∣z(·,T

2

)∣∣∣(L2(ω))3

+∣∣∣C(

·,T2

)∣∣∣(L2(ω))3

)a.e. on (0, T ).

(4.17)

By (4.11), we have

|∇q2|((H10 (ω1))3)′ + |∇r2|((H1

0 (ω1))3)′ ≤(|z2|(H1(ω1))3 + |C2|(H1(ω1))3

),

which together with (4.16) and (4.17) gives

|∇q2|((H10 (ω1))3)′ + |∇r2|((H1

0 (ω1))3)′ (4.18)


≤ c(|z|(L2(ω))3 + |C|(L2(ω))3 +

∣∣∣ ∫ t

T2

(|z|(L2(ω))3 + |C|(L2(ω))3

)dτ

∣∣∣+ |z|(L2(ω))3 + |C|(L2(ω))3 +

∣∣∣z(·,T

2

)∣∣∣(L2(ω))3

+∣∣∣C(

·,T2

)∣∣∣(L2(ω))3

)almost everywhere on (0, T ). Thus, using (4.9), (4.15), and (4.18), since α∗

is increasing on [0, T/2] and decreasing on [T/2, T ], we have∫Qω1

e2sδα∗(q2 + r2)dx dt (4.19)

≤ c( ∫

Qω

e2sδα∗(|z|2+|C|2)dx dt +

∣∣∣z(·,T

2

)∣∣∣2(L2(ω))3

+∣∣∣C(

·,T2

)∣∣∣2(L2(ω))3

).

Now let us estimate∫Q e2sα(|div g|2 + |div G|2)dx dt in (4.8). We have

div g = −3∑

i,j=1

∂wi

∂xj

∂zj

∂xi+

3∑i,j=1

∂Ei

∂xj

∂Cj

∂xi−

3∑i,j=1

∂(ye)i

∂xj

∂zj

∂xi+

3∑i,j=1

∂(Be)i

∂xj

∂Cj

∂xi

+ ∇z·∇ye + z·∆ye + ∇C·∇Be + C·∆Be

+3∑

i,j=1

∫ t

T2

∂

∂xj

(∂wi

∂τ

)∂zj

∂xidτ −

3∑i,j=1

∫ t

T2

∂

∂xj

(∂Ei

∂τ

)∂Cj

∂xidτ,

div G = −3∑

i,j=1

∂wi

∂xj

∂Cj

∂xi−

3∑i,j=1

∂(ye)i

∂xj

∂Cj

∂xi+

3∑i,j=1

∂(Be)i

∂xj

∂zj

∂xi

−∇C·∇ye − C·∆ye −∇z·∇Be − z·∆Be

+3∑

i,j=1

∫ t

T2

∂

∂xj

(∂wi

∂τ

)∂Cj

∂xidτ.

It follows that∫Q

e2sα(|div g|2 + |div G|2

)dx dt ≤ c

∫Q

e2sα((

1 + |∇w|2 +∣∣∣∇∂w

∂t

∣∣∣2)|∇z|2

+(1 + |∇w|2 + |∇E|2 +

∣∣∣∇∂w

∂t

∣∣∣2 +∣∣∣∇∂E

∂t

∣∣∣2)|∇C|2 (4.20)

+ (|∆ye|2 + |∆Be|) (|z|2 + |C|2))dx dt.


By the Sobolev imbedding theorem, we have

|w|2L6 + |∇w|2L6 +∣∣∣∂w

∂t

∣∣∣2L6

+∣∣∣∇∂w

∂t

∣∣∣2L6

+|E|2L6 + |∇E|2L6 +∣∣∣∂E

∂t

∣∣∣2L6

+∣∣∣∇∂E

∂t

∣∣∣2L6

≤ cµ2(w, E),

which enables us to estimate the right–hand side of (4.20) as similar expres-sions in [3] (see inequalities (4.16) and (4.19) there) and obtain∫

Qe2sα

(|div g|2 + |div G|2

)dx dt (4.21)

≤ c(λ)( ∫

Qe2sα

(s2ϕ2(|z|2 + |C|2) + |∇z|2 + |∇C|2

)dx dt

+ µ2(w, E)∫

Qe2sα

(sϕ(|∇z|2 + |∇C|2)

+1sϕ

3∑i,j=1

(∣∣∣ ∂2z

∂xi∂xj

∣∣∣2 +∣∣∣ ∂2C

∂xi∂xj

∣∣∣2))dx dt

).

Next it is easy to see that∫Q

e2sα|G|2dx dt ≤∫

Qe2sα

((1 + |∇E|2 +

∣∣∣∇ ∂E

∂t

∣∣∣2)|z|2+

(1 + |E|2 +

∣∣∣∂E

∂t

∣∣∣2)|∇z|2 +(1 + |w|2 +

∣∣∣∂w

∂t

∣∣∣2)|∇C|2)dx dt.

So, in the same way as before (see inequalities (4.17) and (4.18) in [3], too),one obtains∫

Qe2sα|G|2dx dt ≤

∫Q

e2sα(|z|2 + |∇z|2 + |∇C|2

)dx dt (4.22)

+ c(λ)µ2(w, E)∫

Qe2sα

(sϕ(|∇z|2 + |∇C|2)

+1sϕ

3∑i,j=1

(∣∣∣ ∂2z

∂xi∂xj

∣∣∣2 +∣∣∣ ∂2C

∂xi∂xj

∣∣∣2))dx dt.

Finally, let us see the term s∫Q e2sα∗

ϕ∗|g|2dx dt. As in [3] (see inequality(4.21) there) we have

|g|L2 ≤ c(µ(w, E) + 1

)(|z|H1 + |C|H1 +

∣∣∣ ∫ t

T2

|z|H1dτ∣∣∣ +

∣∣∣ ∫ t

T2

|C|H1dτ∣∣∣)


almost everywhere on (0, T ), whence, using the monotonicity of e2sα∗ϕ∗, we

obtain

s

∫ T

0e2sα∗

ϕ∗|g|2L2dt ≤ c(µ(w, E)+1)2s∫ T

0e2sα∗

ϕ∗(|z|2H1 +|C|2H1)dt. (4.23)

Then, passing from z and C to esα∗(ϕ∗)1/2z and esα∗

(ϕ∗)1/2C, and arguingas in [3], one obtains∫

Qe2sα∗

ϕ∗(|∇z|2 + |∇C|2)dx dt

≤ c(λ)(µ2(w, E) + 1)s∫

Qe2sα∗

(ϕ∗)52 (|z|2 + |C|2)dx dt

+c

s

(∣∣∣z(·, T

2

)∣∣∣2L2

+∣∣∣C(

·, T

2

)∣∣∣2L2

).

(4.24)

Combining (4.23) and (4.24), we have

s

∫Q

e2sα∗ϕ∗|g|2dx dt ≤ c(λ)(µ2(w, E) + 1)2s2

∫Q

e2sαϕ3(|z|2 + |C|2)dx dt

+ c(∣∣∣z(

·,T2

)∣∣∣2L2

+∣∣∣C(

·, T

2

)∣∣∣2L2

). (4.25)

Now we put (4.19), (4.21), (4.22), and (4.25) in the appropriate places in in-equality (4.8). Next we first enlarge s and then diminish M (in the definitionof K) in order that the terms in the left–hand side which contain the firstand second-order derivatives of z and C should absorb the correspondingterms in the right–hand side. In this way, we obtain∫

Qe2sα

( 1sϕ

(∣∣∣∂z

∂t

∣∣∣2 +∣∣∣∂C

∂t

∣∣∣2 +3∑

i,j=1

(∣∣∣ ∂2z

∂xi∂xj

∣∣∣2 +∣∣∣ ∂2C

∂xi∂xj

∣∣∣2))+ sϕ

(|∇z|2 + |∇C|2

)+ s3ϕ3

(|z|2 + |C|2

))dx dt

≤ c(λ)( ∫

Qω

e2sδα∗(|z|2 + |C|2)dx dt +

∣∣z(·, T

2)∣∣2

(L2(Ω))3+

∣∣C(·, T

2)∣∣2

(L2(Ω))3

),

which produces the following inequality:∫Q

e2sα 1sϕ

(|z|2 + |C|2)dx dt ≤ c(λ)( ∫

Qω

e2sδα∗(|z|2 + |C|2)dx dt

+∣∣∣z(

·, T

2

)∣∣∣2(L2(Ω))3

+∣∣∣C(

·, T

2

)∣∣∣2(L2(Ω))3

). (4.26)


In the same manner as in [4], we can eliminate

|z(·, T/2)|2(L2(Ω))3 + |C(·, T/2)|2(L2(Ω))3

in (4.26) to obtain an observability inequality for system (4.4):∫Q

e2sα 1sϕ

(|z|2 + |C|2)dx dt ≤ c(λ)∫

Qω

e2sδα∗(|z|2 + |C|2)dx dt. (4.27)

From (4.27), as in [3], one can obtain

|z(0)|2L2 + |C(0)|2L2 ≤ c(s)ec(M2+1)

∫Qω

e2sδα∗(|z|2 + |C|2)dx dt. (4.28)

Taking (4.5), (4.3), and (4.28) together, we have∫Q(|uε|2 + |vε|2)dx dt +

1ε(|yε(T )|2L2 + |Bε(T )|2L2)

≤ c(s)ec(M2+1)(|y0|2L2 + |B0|2L2).(4.29)

So, uε and vε are bounded in (L2(Q))3.The next step is to prove that uε and vε are bounded in H1(0, T ; (L2(Ω))3).

To this purpose, we pass from zε and Cε to zε = e2sδα∗zε and Cε = e2sδα∗

Cε.We have:

∂zε

∂t+ ∆zε + ((w + ye)·∇)zε − zε·(∇ye)

−((E+Be)·∇)Cε−Cε·(∇Be) = −e2sδα∗∇qε+2sδdα∗

dtzε in Q,

∂Cε

∂t− curl(curl Cε) + P (((w + ye)·∇)Cε + Cε·(∇ye)

−((E + Be)·∇)zε + zε·(∇Be) + (zε·∇)E) = 2sδdα∗

dtCε in Q,

div zε = 0, div Cε = 0 in Q,

zε = 0, Cε·N = 0, (curl Cε)×N = 0 on Σ,

zε(·, T ) = zε(·, 0) = 0, Cε(·, T ) = Cε(·, 0) = 0 in Ω.(4.30)

(The final conditions in (4.30) follow from the fact that (yε, Bε) ∈ C([0, T ];(L2(Ω))6) via the final conditions in (4.4), which ensures that (zε, Cε) ∈C([0, T ]; (L2(Ω))6), too.)


We multiply the two equations in (4.30) by zε and Cε, respectively, andthen integrate over Q. Adding the obtained equalities, we have∫ T

0|∇zε|2L2dt +

∫ T

0|curl Cε|dt

= −∫ T

0b(zε, ye, zε)dt −

∫ T

0b(zε, Be, Cε)dt − 2sδ

∫ T

0

dα∗

dt|zε|2L2dt

+∫ T

0b(Cε, ye, Cε)dt +

∫ T

0b(Cε, Be, zε)dt

+∫ T

0b(zε, E, Cε)dt − 2sδ

∫ T

0

dα∗

dt|Cε|2L2dt.

(4.31)

Now using known estimates for the trilinear form b together with Young’sinequality, we can convert equality (4.31) into the inequality∫ T

0(|∇zε|2L2 + |curl Cε|2L2)dt

≤ c

∫ T

0(|zε|2L2 + |Cε|2L2)dt + csδ

∫ T

0|α∗|(ϕ∗)

12 (|zε|2L2 + |Cε|2L2)dt.

(4.32)

Multiplying equations (4.30) by ∂zε/∂t and ∂Cε/∂t, respectively, integratingover Q and again using suitable estimates for b and Young’s inequality, weobtain ∫ T

0

(∣∣∣∂zε

∂t

∣∣∣2L2

+∣∣∣∂Cε

∂t

∣∣∣2L2

)dt

≤ c( ∫ T

0(|zε|2L2 + |Cε|2L2)dt +

∫ T

0(|zε|2H1 + |Cε|2H1)dt

+sδ

∫ T

0|α∗|ϕ∗(|zε|2L2 + |Cε|2L2)dt

).

(4.33)

Coupling (4.32) and (4.33), and using the fact that y → (∫Ω |∇y|2dx)1/2 and

B → (∫Ω |curlB|2dx)1/2 are norms on V1 and V2, respectively, which are

equivalent to the usual norms of V1 and V2 (see [5]), we have∫ T

0

(∣∣∣∂zε

∂t

∣∣∣2L2

+∣∣∣∂Cε

∂t

∣∣∣2L2

)dt ≤ csδ

∫ T

0|α∗|ϕ∗(|zε|2L2 + |Cε|2L2)dt,

whence∫Q

(∣∣∣∂uε

∂t

∣∣∣2 +∣∣∣∂vε

∂t

∣∣∣2)dx dt ≤ csδ

∫ T

0|α∗|ϕ∗e4sδα∗

(|zε|2L2 + |Cε|2L2)dt


≤ c

∫Q

e4sδ′α∗(|zε|2 + |Cε|2)dx dt (4.34)

for 0 < δ′ < δ (and possibly larger λ > 0 and s > 0). Now we take (4.34)together with inequality (4.29) considered for δ′ instead of δ. In this way,we have obtained∣∣∣∂uε

∂t

∣∣∣2(L2(Q))3

+∣∣∣∂vε

∂t

∣∣∣2(L2(Q))3

≤ c(s)ec(M2+1)(|y0|2L2 + |B0|2L2

). (4.35)

Let us now prove the boundedness of ∂2uε/∂t2 and ∂2vε/∂t2 in (L2(Q))3.We set ζε = ∂ζε/∂t and Γε = ∂Cε/∂t, and differentiate the two equations of(4.30) with respect to t. So, we have

∂ζε

∂t+ ∆ζε +

(∂w

∂t·∇

)zε + ((w + ye)·∇)ζε − ζε·(∇ye)

−(∂E

∂t·∇

)Cε − ((E + Be)·∇)Γε − Γε·(∇Be)

= −2sδd2α∗

dt2e2sδα∗∇qε − e2sδα∗∇∂qε

∂t

+2sδd2α∗

dt2zε + 2sδ

dα∗

dtζε in Q,

∂Γε

∂t− curl(curl Γε)

+P((∂w

∂t·∇

)Cε + ((w + ye)·∇)Γε + Γε·(∇ye)

−(∂E

∂t·∇

)zε − ((E + Be)·∇)ζε + ζε·(∇Be)

+(ζε·∇)E + (zε·∇)∂E

∂t

)= 2sδ

d2α∗

dt2Cε + 2sδ

dα∗

dtΓε in Q.

(4.36)We multiply the two equations (4.36) by ∂ζε/∂t and ∂Γε/∂t, respectively,then integrate over Q, and add the results. Next, we use appropriate esti-mates for b, Young’s inequality and the fact that |dα∗/dt|2 ≤ c(λ)(ϕ∗)3 and|d2α∗/dt2|2 ≤ c(λ)(ϕ∗)4. Thus, it follows after some calculation that∫

Q

(∣∣∣∂ζε

∂t

∣∣∣2 +∣∣∣∂Γε

∂t

∣∣∣2)dx dt (4.37)

≤ c(s)( ∫ T

0(|ζε|2H1 + |Γε|2H1)dt +

∫ T

0(|zε|2H1 + |Cε|2H1)dt


+∫ T

0(ϕ∗)3(|ζε|2L2 + |Γε|2L2)dt +

∫ T

0(ϕ∗)4(|zε|2L2 + |Cε|2L2)dt

).

Let us estimate the terms in the right–hand side of (4.37). First we multi-ply equations (4.36) by ζε and Γε, respectively, and integrate over Q. So,after some calculation involving suitable estimates for b and the estimatesof dα∗/dt and d2α∗/dt2, one obtains∫ T

0(|∇ζε|2L2 + |curl Γε|2L2)dt

≤ c

∫ T

0(|zε|2L2 + |Cε|2L2)dt + c(λ)s

∫Q(ϕ∗)4(|zε|2 + |Cε|2)dx dt

+c(λ)s∫

Q(ϕ∗)3(|ζε|2 + |Γε|2)dx dt.

(4.38)

Multiplying equations (4.30) by (ϕ∗)3∂zε/∂t = (ϕ∗)3ζε and (ϕ∗)3∂Cε/∂t =(ϕ∗)3Γε, integrating over Q, and performing the same calculation as before,we have∫

Q(ϕ∗)3

(∣∣∣∂zε

∂t

∣∣∣2 +∣∣∣∂Cε

∂t

∣∣∣2)dx dt =∫

Q(ϕ∗)3(|ζε|2 + |Γε|2)dx dt (4.39)

≤ c(λ)s( ∫ T

0(ϕ∗)

72 (|zε|2H1 + |Cε|2H1)dt +

∫Q(ϕ∗)5(|zε|2 + |Cε|2)dx dt

).

Next, multiplying (4.30) by (ϕ∗)7/2zε and (ϕ∗)7/2Cε, respectively, we simi-larly have∫ T

0(ϕ∗)

72 (|zε|2H1 + |Cε|2H1)dt ≤ c(λ)s

∫Q(ϕ∗)5(|zε|2 + |Cε|2)dx dt. (4.40)

Setting (4.37) through (4.40) together, we finally obtain∫Q

(∣∣∣∂2uε

∂t2

∣∣∣2 +∣∣∣∂2vε

∂t2

∣∣∣2)dx dt ≤ c(s)∫

Q(ϕ∗)5e4sδα∗

(|zε|2 + |Cε|2)dx dt

≤ c(s)∫

Qe4sδ′α∗

(|zε|2 + |Cε|2)dx dt (4.41)

for 0 < δ′ < δ (and possibly larger λ > 0 and s > 0). So, taking (4.41)together with inequality (4.29) in which δ has been replaced by δ′, we have∣∣∣∂2uε

∂t2

∣∣∣2(L2(Q))3

+∣∣∣∂2vε

∂t2

∣∣∣2(L2(Q))3

≤ c(s)ec(M2+1)(|y0|2L2 + |B0|2L2). (4.42)


Inequalities (4.29), (4.35), and (4.42) taken together give

|uε|2H2(0,T ;(L2(Ω))3) + |vε|2H2(0,T ;(L2(Ω))3) +1ε

(|yε(·, T )|2L2 + |Bε(·, T )|2L2)

≤ c(s)ec(M2+1)(|y0|2L2 + |B0|2L2) (4.43)

for sufficiently small positive M , where c(s), c, and M are independent of ε.In the same way as in [3], from (4.43) one obtains the boundedness of

(yε, Bε) in (L∞(0, T ; D(A1))∩H1(0, T ; V1))×(L∞(0, T ; D(A2)) ∩ H1(0, T ;V2)) :

|yε|2L∞(0,T ;D(A1)) + |yε|2H1(0,T ;V1) + |Bε|2L∞(0,T ;D(A2)) + |Bε|2H1(0,T ;V2)

≤ c(|y0|H2 + |B0|H2). (4.44)

Let us now prove that (yε, Bε) is bounded in

(W 1,∞(0, T ; D(A1)) ∩ H2(0, T ; V1))×(W 1,∞(0, T ; D(A2)) ∩ H2(0, T ; V2)),

too. Differentiating (2.5) with respect to t, we have

dy′εdt

+ A1y′ε + P ((w′·∇)yε + ((w + ye)·∇)y′ε + (y′ε·∇)ye

− (E′·∇)Bε − ((E + Be)·∇)B′ε − (B′

ε·∇)Be

+ E′·(∇Bε) + E·(∇B′ε) = P (χωu′

ε) in (0, T ),

dB′ε

dt+ A2B

′ε + P ((w′·∇)Bε + ((w + ye)·∇)B′

ε + (y′ε·∇)Be

− (E′·∇)yε − ((E + Be)·∇)y′ε − (B′ε·∇)ye) (4.45)

= P (χωv′ε) in (0, T ),

y′ε(0) = −A1y0 − P (((y0 + ye)·∇)y0 + (y0·∇)ye + B0·(∇Be)

− ((B0 + Be)·∇)B0 − (B0·∇)Be)

B′ε(0) = −A2B0 − P (((y0 + ye)·∇)y0 + (y0·∇)Be

− ((B0 + Be)·∇)y0 − (B0·∇)ye).

We remark that

y′ε(0) ∈ D(A1) and B′ε(0) ∈ D(A2), (4.46)

because (w, E) ∈ K. By Theorem III.3.1 in [7], from (4.46) we have

A121

dy′εdt

∈ C((0, T ]; (L2(Ω))3), A122

dB′ε

dt∈ C((0, T ]; (L2(Ω))3). (4.47)


Now we successively multiply the two equations in (4.45) scalarly in H by y′εand B′

ε, dy′ε/dt and dB′ε/dt, A1y

′ε and A2B

′ε, and A1dy′ε/dt and A2dB′

ε/dt,respectively. Using (4.47), (4.43), and (4.44), the fact that (w, E) ∈ K and(ye, Be) ∈ (H3(Ω))6, the interpolation inequality

|y|H

32≤ c|y|

12

H1 |y|12

H2 for y ∈ (H2(Ω))3,

and appropriate estimates for the trilinear form b (which can be taken from[5]), in the same way as in [3] (see inequalities (4.61) and (4.62) there), weobtain

|y′ε(t)|2H2 + |B′ε(t)|2H2 +

∫ t

0(|y′′ε |2H1 + |B′′

ε |2H1)dτ

≤ c(|y0|H2 + |B0|H2 + |y′ε(0)|H2 + |B′ε(0)|H2) for t ∈ [0, T ].

(4.48)

Taking (4.48) together with the initial conditions in (4.43), for (y0, B0) ∈ X,we have

|y′ε|2L∞(0,T ;D(A1)) + |B′ε|2L∞(0,T ;D(A2)) + |y′ε|2H1(0,T ;V1) + |B′

ε|2H1(0,T ;V2)

≤ c(|y0|H4 + |B0|H4). (4.49)

From (4.44) and (4.49), we have (on a subsequence of ε)(yε, Bε)−→(y, B) weak star in W 1,∞(0, T ; (H2(Ω))6),

strongly in H1(0, T ; V1×V2) ∩ C1([0, T ], H2),(∂yε

∂t, ∂Bε

∂t

)−→

(∂y

∂t, ∂B

∂t

)weakly in H1(0, T ; V1×V2),

curlBε−→curlB weakly in L2(Σ),

for some (y, B) ∈ (W 1,∞(0, T ; D(A1))∩H2(0, T ; V1))×(W 1,∞(0, T ; D(A2))∩H2(0, T ; V2)).

Arguing as in [3], we obtain that

|pε|2L2(0,T ;H1(Ω)) ≤ c(|y0|2H1 + |B0|2H1

)if

∫Ω

pεdx = 0.

Thus, for some p ∈ L2(0, T ; H1(Ω)), we have (on a subsequence of ε)∇pε −→ ∇p weakly in L2(Q).

Finally, from (4.43), we have (on a subsequence of ε)(uε, vε) −→ (u, v) weakly in H2(0, T ; (L2(Ω))6)

for some (u, v) ∈ H2(0, T ; (L2(Ω))6).


So, we can pass to limits in (2.5) (with y = yε, B = Bε, p = pε, u = uε,and v = vε) as ε −→ 0 to obtain that (y, B, p, u, v) also satisfies (2.5).Letting ε tend to 0 in (4.43), too, we derive that y(·, T ) = B(·, T ) = 0almost everywhere in Ω, and u and v satisfy (4.2). This finishes the proof ofLemma 4.1.

Let us define the map S : K −→ (L2(Q))3×(L2(Q))3 by

S(w, E) =

(y, B) ∈ (W 1,∞(0, T ; D(A1)) ∩ H2(0, T ; V1))

×(W 1,∞(0, T ; D(A2)) ∩ H2(0, T ; V2)) :(y, B) is the solution of (2.5) corresponding to (w, E) ∈ Kand to some (u, v) ∈ H2 (0, T ; (L2(Ω))6) satisfying (4.2)in Lemma 4.1 such that y(T ) = 0, B(T ) = 0 a.e. in Ω

.

We remark that S(K) ⊂ K if (y0, B0) ∈ X and |y0|H4 + |B0|H4 is sufficientlysmall because, by estimates like (4.44) and (4.49), there exists η > 0 suchthat for (y0, B0) ∈ X with |y0|H4 + |B0|H4 < η we have µ(y, B) ≤ M.Arguing as in [3] by using Kakutani’s fixed-point theorem, we conclude thatthere exists (y∗, B∗) ∈ K such that (y∗, B∗) ∈ S(y∗, B∗). Therefore, y∗ andB∗ together with their u∗ and v∗ solve the null controllability problem forsystem (2.4). The proof of Theorem 2.1 is finished.

References

[1] V. Barbu, Controllability of parabolic and Navier–Stokes equations, Sci. Math. Jpn.,56 (2002), 143–211.

[2] V. Barbu, On local controllability of the Navier–Stokes equations, Adv. DifferentialEquations, 8 (2003), 1481–1498.

[3] V. Barbu, T. Havarneanu, C. Popa, and S.S. Sritharan, Exact controllability for themagnetohydrodynamic equations, Comm. Pure Appl. Math., 56 (2003), 732–783.

[4] O.Yu. Imanuvilov, On exact controllability for the Navier-Stokes equations, ESAIMControl Optim. Calc. Var., 3 (1998), 97–131.

[5] M. Sermange and R. Temam, Some mathematical questions related to the MHD equa-tions, Comm. Pure Appl. Math., 36 (1983), 635–664.

[6] R. Temam, “Navier-Stokes Equations, Theory and Numerical Analysis,” Third edi-tion, Studies in Mathematics and Its Applications, 2, North–Holland, Amsterdam,1984.

[7] W. von Wahl, “The Equations of Navier–Stokes and Abstract Parabolic Equations,”Vieweg & Sohn, Braunschweig, 1985.

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LOCAL EXACT CONTROLLABILITY FOR THE … · Local exact controllability 483 space of functions in L2(Ω) whose weak derivatives of order less than or equal to m are also in L2(Ω)