On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions

ON THE CONTROLLABILITY OF THE 2-D

INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH

THE NAVIER SLIP BOUNDARY CONDITIONS

JEAN{MICHEL CORON

Abstract. For boundary or distributed controls, we get an approxi- mate controllability result for the Navier-Stokes equations in dimension 2 in the case where the uid is incompressible and slips on the boundary in agreement with the Navier slip boundary conditions.

Keywords: Controllability, Navier-Stokes equations, Navier slip boundary conditions.

1. Introduction

Let be a bounded nonempty connected open subset of^R2of class

C

¹. Let ;^# be an open subset of ; :=

@

and let ^# be an open subset of . We assume that

;^{# #6}=

:

(1.1)

We denote by

n

the outward unit normal vector eld on ; and by

the unit tangent vector eld on ; such that (

n

) is a direct basis of^R2. The set ;^# is the part of the boundary and ^# is the part of the domain on which the controls acts. The uid that we consider is incompressible so that the velocity eld

y

satises

div

y

= 0

:

On the part of the boundary ;ⁿ;^# where there is no control the uid slips it satises

y

n

= 0 on ;ⁿ;^# (1.2)

and the Navier slip boundary condition 24]

y

+ (1^;

)

n

ⁱ

@y

ⁱ

@x

^{j +}

@y

^j

@x

ⁱ

^j = 0 on ;ⁿ;^# (1.3) where

is a constant in 0

1),

n

= (

n

¹

n

²),

= (

¹

²), and where we have used the usual summation convention. Note that the classical no-slip condition, due to Stokes,

y

= 0 (1.4)

corresponds to the case

= 1, which is not considered here. The slip boundary condition (1.3) with

= 0 corresponds to the case where there

Centre de Mathematiques et Leurs Applications, Ecole Normale Superieure de Cachan et CNRS URA 1611, 61 Av. du Pdt Wilson, 94235 Cachan Cedex, France. E-mail address:

Jean-Michel.Coron@cmla.ens-cachan.fr.

Received by the journal January 5, 1996. Accepted for publication April 15, 1996.

This work was partially supported by the GDR \Automatique" of the CNRS.

the uid slips on the wall without friction. It is the appropriate physical model for some ow problems see 16] for example. The case

² (0

1) corresponds to a case where there the uid slips on the wall with friction it is also used in models of turbulence with rough walls see, e.g., 17]. Note that in 3] F. Coron has derived rigorously the slip boundary condition (1.3) from the boundary condition at the kinetic level (Boltzmann equation) for compressible uids. Let us also recall that C. Bardos, F. Golse, and D.

Levermore have derived in 2] the incompressible Navier-Stokes equations from a Boltzmann equation.

Let us point out that, using (1.2), one sees that (1.3) is equivalent to

y

+ curl

y

= 0 on ;ⁿ;^#

with

²

C

¹(;^R) dened by

(

x

) = 2(1^;

)

(

x

)^;

1^;

⁸

x

²;

(1.5)

where

is the curvature of ; dened through the relation ^@n_@ =

:

In fact we will not use this particular character of (1.5) in our considerations Theorem 1.1 below holds for any

²

C

¹(;^R).

The problem of approximate controllability we consider is the following one: let

T >

0, let

y

⁰ and

y

¹ in

C

¹(^R2) be such that

div

y

⁰ = 0 in

(1.6)

div

y

¹ = 0 in

(1.7)

y

⁰

n

= 0 on ;ⁿ;^#

(1.8)

y

¹

n

= 0 on ;ⁿ;^#

(1.9)

y

⁰

+ curl

y

⁰ = 0 on ;ⁿ;^#

(1.10)

y

¹

+ curl

y

¹ = 0 on ;ⁿ;^#

:

(1.11) We ask whether there exist

y

²

C

¹(0

T

]^R2) and

p

²

C

¹(0

T

]^R) such that

@y @t

^;

y

+ (

y

^r)

y

+^r

p

= 0 in (^n#)0

T

]

(1.12)

div

y

= 0 in 0

T

]

(1.13)

y

n

= 0 on (;ⁿ;^#)0

T

]

(1.14)

y

+ curl

y

= 0 on (;ⁿ;^#)0

T

]

(1.15)

y

(

0) =

y

⁰ in

(1.16)

and, in an appropriate topology, we have

y

(

T

) is \close" to

y

¹

:

(1.17) That is to say, starting with the initial data

y

⁰ for the Navier-Stokes equa- tions, we ask whether there are solutions which, at a xed time

T

, approach arbitrarily closely to the given velocity eld

y

¹.

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

Note that (1.12) to (1.16) have many solutions. In order to have unique- ness one needs to add extra conditions. These extra conditions are the con- trols. Various possible controls can be considered. For example, a possible choice for the controls is

y

n

on ;^#0

T

]

(1.18)

y

+ curl

y

on ;^#0

T

]

(1.19)

@y @t

^;

y

+ (

y

^r)

y

+^r

p

in ^#0

T

]

:

(1.20) More precisely, let

y

s²

C

¹(0

T

]^R2) and

p

s²

C

¹(0

T

]^R)be such that (1.12) to (1.16) hold for (

yp

) = (

y

s

p

s). Let us consider the following Cauchy problem: nd

y

²

C

¹(0

T

]^R2) and

p

²

C

¹(0

T

]^R) such that (1.12) to (1.16) hold and

y

n

=

y

s

n

on ;^# 0

T

]

y

+ curl

y

=

y

s

+ curl

y

s on ;^#0

T

]

@y @t

^;

y

+ (

y

^r)

y

+^r

p

=

@y

s

@t

^;

y

s+ (

y

s^r)

y

s+^r

p

s in ^#0

T

] has, up to an arbitrary function depending only on time added to

p

, one and only one solution which is (

yp

) = (

y

s

p

s). One can also use for the control (1.18), (1.20), and curl

y

on ;^#0

T

]

:

Another possibility for the control is (1.20) and

y

on ;^#0

T

]

:

Let

d

²

C

⁰(^R) be dened by

d

(

x

) = dist (

x

;) = Min^fj

x

^;

x

^0j

x

⁰²;^g

:

Our controllability result is

Theorem 1.1. Let

T >

0, let

y

⁰ and

y

¹ in

C

¹(

^R2) be such that (1.6), (1.7), (1.8), (1.9), (1.10), and (1.11) hold. Then, there exist a sequence (

y

^k

k

^{2 N}) of maps in

C

¹( 0

T

]^R2) and a sequence (

p

^k

k

^{2 N}) of functions in

C

¹(0

T

]^R) such that, for all

k

^{2 N}, (1.12), (1.13), (1.14), (1.15), and (1.16) hold for

y

=

y

^k and

p

=

p

^k and such that, as

k

^!+¹,

Z

d

^j

y

^k(

T

)^;

y

^1j!0

⁸

>

0

(1.21)

j

y

^k(

T

)^;

y

^1j_W^{;11( ) !}0

(1.22) and, for all compact

K

included in ;^#,

j

y

^k(

T

)^;

y

^1jL¹⁽K⁾+^jcurl

y

^k(

T

)^;curl

y

^1jL¹⁽K^{) !}0

:

(1.23) In this theorem, and throughout all this paper,

W

^;11() denotes the usual Sobolev space of rst derivatives of functions in

L

¹() and^{j j}W^{;11( )}

one of it's usual norms, for example the norm given in 1, Section 3.10].

Remark 1.2. a) The question of the approximate controllability of the Navier-Stokes equations for incompressible uids has been raised by J.-L Lions in 20] and 21] for the no-slip boundary condition

y

= 0 on (;ⁿ;^#)0

T

].

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

b) Note that (1.21), (1.22), and (1.23) are not strong enough to imply

j

y

^k(

T

)^;

y

^1jL^{2( ) !}0

:

(1.24) But, in the special case where

= 0 in ;ⁿ;^#, Theorem 1.1 still holds even if one requires also (1.24) see Remark 2.4 below.

c) Let us point out that, if ;^# = ;, there is, of course, no dierence between the slip case and the no-slip case. Theorem 1.1 is, up to our knowledge, new even if ;^# = ; in this special case the proof of Theorem 1.1 can be strongly simplied. If ;^# = ;, we get from (1.23) that

y

^k(

T

)^!

y

in the Sobolev space

H

¹() as

k

^!1, i.e. we prove approximate controllability in

H

¹(). So we give a positive answer to J.{L. Lions's conjecture 20]-21] in dimension 2 when ;^# = ;.

Similarly, if, for some

"

⁰

>

0, ^f

x

²

d

(

x

)

< "

^{0g #}, we get the same result, i.e., approximate controllability in

H

¹() for the Navier boundary condition and for the no-slip boundary condition. To see it, let, for

"

^{0 2} (0

"

⁰

=

2), ⁰ = ^f

x

²

d

(

x

)

> "

^0g, let ;⁰ =

@

⁰, let

;⁰⁰

... ,;⁰_i

... , ;⁰_g be the connected components of ;, and let ;^#0 be the union of the ;⁰_i such that Min^f dist (

x

;^#)

x

²;⁰_i^g6

"

⁰. Apply Theorem 1.1 with ⁰ for , ^0\# for ^#, ;^#0 for ;^#, and

"

⁰

>

0 small enough. Then, for the (

y

^k

p

^k) given by Theorem 1.1, extend

p

^k to all of , modify in a suitable way

y

^k in ^f

x

^{2 0}

d

(

x

)

<

2

"

^0g and nally extend in a suitable way the new

y

^k to all of .

d) Of course, by density, Theorem 1.1 still holds if

y

¹ is only of class

C

¹. Moreover, as it will follow from our proof, one can also assume less regularity on

y

⁰ if one requires only that (

y

^k

p

^k) are of class

C

¹ on (0

T

]. See Remark 3.1 below for more details.

e) E. Fern!andez-Cara and J. Real in 11] and E. Fern!andez-Cara and M. Gonz!alez-Burgos in 10] have proved that, for 2-D and 3-D incom- pressible uids, the linear space spanned by the

y

(

T

) such that, for some

p

: 0

T

]^!R

one has (1.12), (1.13), (1.16), and the no-slip boundary condition, is dense, with respect to the

L

²-norm in the set of

y

¹ : ^!R2 satisfying (1.7) and

y

¹ = 0 on ;ⁿ;^#.

f) A.V. Fursikov and O.Yu Imanuvilov have proved in 13] 14] that, if

;^# = ;, then one has exact zero controllability in large time, i.e., for any

y

⁰ satisfying (1.6), there exist

T >

0,

y

and

p

satisfying (1.12) to (1.16) and

y

(

T

) = 0. In 15], they have recently obtained the same result in the more general situation where

= 0 in ;ⁿ;^#. Again if

;^# = ;, A.V. Fursikov has proved in 12] the exact zero controllability in large time in dimension 3.

g) In 9] C. Fabre has obtained, in every dimension, an approximate con- trollability of two natural \cut o" Navier-Stokes equations (with the no-slip boundary condition).

As in our proof of the controllability of the 2-D Euler equations of in- compressible perfect uids 7, 8], the strategy of the proof of Theorem 1.1 relies on a method described in 5] and 6] under the name of \the return method". This was introduced in 4] for a stabilization problem. Roughly speaking it consists in looking for ("

y p

") such that (1.12) to (1.15) hold with

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

y

= "

yp

= "

p

,

"

y

(

0) = "

y

(

T

) = 0 in

(1.25) and such that the linearized control system around ("

y p

") has a controllability in a \good" sense. With such a ("

y p

") one may hope that there exists (

yp

) {close to ("

y p

") {satisfying the required conditions, at least if

y

⁰ and

y

¹ are \small". For a suitable choice of ("

y p

"), we will see that this is indeed true, even if

y

⁰ and

y

¹are not small. Note that the linearized control system around ("

y p

") is

@z @t

^;

z

+ ("

y

^r)

z

+ (

z

^r)"

y

+^r

= 0 in (^n#)0

T

]

(1.26)

div

z

= 0 in 0

T

]

(1.27)

z

n

= 0 on (;ⁿ;^#)0

T

]

(1.28)

z

+ curl

z

= 0 on (;ⁿ;^#)0

T

]

:

(1.29) In 19] J.{L. Lions has proved that if "

y

= 0 this linear system is approxi- matively controllable in fact he has treated the no-slip case - i.e. the case where one replaces (1.28)-(1.29) by

z

= 0 on (;ⁿ;^#)0

T

] - but his proof can be easily adapted to the boundary conditions considered here. Unfortu- nately, if one takes "

y

= 0, it is not clear how to deduce from the approximate controllability of the linear system the existence of (

yp

) satisfying (1.12) to (1.17), even if

y

⁰ and

y

¹ are small, for example in a sense given by a

C

^m-norm. For this reason, we will not use ("

y p

") = (0

0), but a ("

y p

") similar to the one that we have constructed in 8] to prove the controllability of the 2-D Euler equations of incompressible perfect uids this ("

y p

") is in fact

\large" so that, in some sense, \" is small compared to \("

y

^r)+( ^r)"

y

".

Our paper is organized as follows: in Section 2 we prove Theorem 1.1 when ;^# = and then, in Section 3, deduce the general case from this particular case.

2. Proof of Theorem 1.1 when ^{;# =}

In this section, we assume that ;^# = hence, by (1.1), ^{# 6}=.

Let us prove a slightly stronger result than Theorem 1.1 this will be useful when we will study the case ;^{# 6}=. Let

²

C

¹(;^R) be such that

Z

;

= 0

:

(2.1)

We are going to prove that Theorem 1.1 still holds if, in the statement of this theorem, one replaces (1.8), (1.9) and (1.14) respectively by

y

⁰

n

=

on ;ⁿ;^#

(2.2)

y

¹

n

=

on ;ⁿ;^#

(2.3)

y

(

xt

)

n

(

x

) =

(

x

)

⁸(

xt

)²(;ⁿ;^#)0

T

]

:

(2.4) Note that Theorem 1.1 corresponds to the case

= 0 and that

is given (it is not a control).

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

Of course, reducing if necessary , we may assume without loss of gen- erality that

^#

(2.5)

^# = ^#

:

(2.6)

Let us assume the following proposition, whose proof is given in Appendix A,Proposition 2.1. There exists a constant

C

and there exists a function

"

C

: (0

+¹) ^! (0

+¹)

such that, for all

" >

0

there exist

^{" 2}

C

¹( 0

T

]^R) and "

y

^"2

C

¹(0

T

]^R2) satisfying

^" = 0 in (^n#)0

T

]

(2.7)

@

^"

@n

^{= 0 on ;0}

T

]

(2.8)

Support

^" is included in

T=

4

T

)

(2.9) Support

y

"^" is included in

T=

4

T

)

(2.10)

@

^"

@

+

@

²

^"

@

²

6

"

on ;0

T

]

(2.11)

"

y

^"=^r

^" in (^n#)0

T

]

(2.12)

div

y

"^"= 0 in 0

T

]

(2.13) and such that the following property holds: for all (

z

⁰

z

¹) in

C

¹(^R2)² such that

div

z

⁰ = 0 in

(2.14)

div

z

¹ = 0 in

(2.15)

z

⁰

n

=

on;

(2.16)

z

¹

n

=

on;

(2.17)

there exist

z

^" =

Z

^"(

z

⁰

z

¹) in

C

¹( 0

T

]^R2) and

^" = #^"(

z

⁰

z

¹) in

C

¹(0

T

]^R) satisfying

@z

^"

@t

^{+ ("}

y

^"r)

z

^"+ (

z

^"r)"

y

^"+^r

^"= 0 in (^n#)0

T

]

(2.18)

^"= 0 in 0

T=

4]

(2.19)

z

^"=

z

⁰ in 0

T=

4]

(2.20)

z

^"(

xT

) =

z

¹(

x

)

⁸

x

² such that

d

(

x

)^>

"

(2.21)

div

z

^" = 0 in 0

T

]

(2.22)

z

^"(

xt

)

n

(

x

) =

(

x

)

⁸(

xt

)²;0

T

]

(2.23)

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

j

z

^"(

T

)^j_L^{1( )6}

C

^j

z

^0j_L^{2( )}+^j

z

^1j_L^{2( )}

+^jcurl

z

^0j_L^{1( )}+^jcurl

z

^1j_L^{1( )}

(2.24)

j

z

^"j_C^{3( 0}_T^{]R2) 6}

C

^"(

"

)^j

z

^0j_C^{4( R2)}+^j

z

^1j_C^{4( R2)}

:

(2.25) Before going into the details of the proof, let us rst briey explain how we use Proposition 2.1 to construct solutions (

yp

) to our controllability problem. We choose

>

0 \very small". During the interval of time 0

(1^;

)

T

] we use no control: on this interval of time (

yp

) is a solution of (1.12), (1.13), (1.15) (1.16), and (2.4) with ^# = and ;^# = . During the interval of time (1^;

)

TT

] we decompose (

yp

) in the following way

y

= "

y

+

z

+

R p

= "

p

+

+

q

where

the map "

y

is obtained by the following scaling of "

y

^"

"

y

(

xt

) = 1

y

^""

x t

^;(1;

)

T

and the function "

p

is dened by

"

p

(

xt

) =^;1

²

@

^"

@t

+ 12^jr

^"j2

x t

^;(1;

)

T

where "

y

^" and

^" are dened in Proposition 2.1. Let us emphasize that ("

y p

") satises (1.12), (1.13), and (1.14) moreover it satises \almost"

(1.15), at least if

"

is small enough (see in particular (2.11)).

the functions

z

and

are obtained by scaling

z

^" and

^"in the following way

z

(

xt

) =

z

^"

x t

^;(1;

)

T

(

xt

) = 1

^"

x t

^;(1;

)

T

where

z

^" and

^" are dened in Proposition 2.1 by taking

z

⁰=

y

(

(1^;

)

T

)

z

¹ =

y

¹

:

(

Rq

) is a correction term dened so that (

yp

) is a solution of (1.12), (1.13), (1.15), and (2.4), with (1^;

)

TT

] instead of 0

T

].

Note that, by construction,

y

(

T

) =

z

^"(

xT

) +

R

and

z

^"(

T

) is \close" to

y

¹ if

"

is \small". So it su$ces to check that

R

is

\small". We will prove that this is indeed the case if \

"

is small and

is very small". Rouhgly speaking the reasons are the following. For

\very small" the rst leading term of

@

("

y

+

z

)

@t

^;("

y

+

z

) + (("

y

+

z

)^r)("

y

+

z

) +^r("

p

+

) is the term of order 1

=

², which is

@ y

"

@t

^{+ ("}

y

^r)"

y

+^r

p

"

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

and the second leading term is the term of order 1

=

, which is

@z @t

^;"

y

+ ("

y

^r)

z

+ (

z

^r)"

y

+^r

:

By construction these two terms vanish. Moreover "

y

+

z

satises (2.4) and, for

\small", the leading term of "

y

+

z

, which is "

y

, saties \almost" (1.15) if

"

is \small". Note that, in the case where

= 0 on ;, "

y

saties (1.15) exactly for all

"

. This is why, as mentioned above in b) of Remark 1.2, we get a better convergence in this case see also a) of Remark 2.4 below.

Let us now give the details of the proof. Let

y

in

C

¹(0

T

]^R2) and

p

in

C

¹(0

T

]^R) be such that

@y

@t

^;

y

+ (

y

^r)

y

+^r

p

= 0 in 0

T

]

(2.26)

div

y

= 0 in 0

T

]

(2.27)

y

n

=

on ;0

T

]

(2.28)

y

+ curl

y

= 0 on ;0

T

]

(2.29)

y

(

0) =

y

⁰ in

:

(2.30)

The existence (and uniqueness up to an arbitrary function depending only on time added to

p

) can be proved by standard techniques see 18, Chapitre 1, Th!eor%eme 6.10] for the case

= 0,

= 0, and simply connected see also 23] for 3-D Navier-Stokes equations.

Let

² (0

1

=

2] and let

Q

= (1^;

)

TT

]

:

For

" >

0

let "

y

^{" 2}

C

¹(

Q

^R2)

z

^{" 2}

C

¹(

Q

^R2)

p

"^{" 2}

C

¹(

Q

^R)

^{" 2}

C

¹(

Q

^R) be de- ned by

"

y

^"(

xt

) = 1

y

^""

x t

^;(1;

)

T

(2.31)

z

^"(

xt

) =

Z

^"(

y

(

(1^;

)

T

)

y

¹)

x t

^;(1;

)

T

(2.32)

"

p

^"(

xt

) =^;1

²

@

^"

@t

+ 12^jr

^"j2

x t

^;(1;

)

T

(2.33)

^"(

xt

) = 1

^#"(

y

(

(1^;

)

T

)

y

¹)

x t

^;(1;

)

T

(2.34)

for all (

xt

)²

Q

. Note that, by (2.10) and (2.31),

"

y

^"= 0 in (1^;

)

TT

^;(3

=

4)

T

]

(2.35)

and, by (2.20) and (2.32),

z

^"(

xt

) =

y

(

x

(1^;

)

T

)

⁸(

xt

)²(1^;

)

TT

^;(3

=

4)

T

]

:

(2.36) Similarly, by (2.9) and (2.33),

"

p

^"= 0 in (1^;

)

TT

^;(3

=

4)

T

]

(2.37)

and, by (2.19) and (2.34),

^"= 0 in (1^;

)

TT

^;(3

=

4)

T

]

:

(2.38)

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

Let

F

^" : 0

T

]^!R be dened by

F

^"= 0 in 0

(1^;

)

T

)

(2.39)

F

^" =

@

@t

^("

y

^"+

z

^")^;"

y

^"+ ("

y

^"r)

z

^"+ (

z

^"r)"

y

^"

+ ("

y

^"r)"

y

^"+^r

p

"^"+^r

^" in

Q

:

(2.40) Then {see in particular (2.35) to (2.40) {

F

^{" 2}

C

¹(0

T

]^R2)

:

For

a

= (

a

¹

a

²) ^{2 R2}

let

a

^? = (^;

a

²

a

¹). By standard techniques {see also (2.35) {, one shows the existence of

y

^" in

C

¹(0

T

]^R2) and

p

^" in

C

¹(0

T

]^R) such that

@y

^"

@t

^;

y

^"+ (

y

^"r)

y

^"+^r

p

^"= 0 in 0

(1^;

)

T

]

(2.41)

@y

^"

@t

^;

y

^"+ (

y

^"r)

y

^"+^r

p

^"=

F

^"

+ (curl "

y

^")(

y

^";

y

"^";

z

^")^? in (1^;

)

TT

]

(2.42)

div

y

^"= 0 in 0

T

]

(2.43)

y

^"

n

=

on ;0

T

]

(2.44)

y

^"

+ curl

y

^" = 0 on ;0

T

]

(2.45)

y

^"(

0) =

y

⁰ in

:

(2.46)

For simplicity, let us write

y p z y

"

p F Q

" instead of

y

^"

p

^"

z

^{" "}

"

y

^"

p

"^"

F

^"

Q

:

From (2.7), (2.12), (2.6), (2.31), and (2.33), we have

@ y

"

@t

^;"

y

+ "

y

^r

y

"+^r

p

"= 0 in (^n#)(1^;

)

TT

]

(2.47) curl "

y

= 0 in (^n#)(1^;

)

TT

]

:

(2.48) From (2.18), (2.31), (2.32), and (2.34), we get

@z @t

^{+ ("}

y

^r)

z

+ (

z

^r)"

y

+^r

= 0 in (^n#)(1^;

)

TT

]

which, with (2.40) and (2.47), implies that

F

= 0 in (^n#)(1^;

)

TT

]

:

(2.49) From (2.41), (2.42), (2.48), and (2.49), we obtain (1.12). From (2.43) to (2.46), we obtain (1.13), (1.15), (1.16), and (2.4). So, in order to nish the proof, it remains only to check that, given a compact

K

, given

>

0 and

>

0, we have, for a suitable choice of

"

and

,

Z

d

^j

y

(

T

)^;

y

^1j

<

(2.50)

j

y

(

T

)^;

y

^1jW^{;11( )}

<

(2.51)

jcurl

y

(

T

)^;curl

y

^1jL¹⁽K⁾

< :

(2.52)

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

Let us rst point out that, by (2.26) to (2.30), (2.41), and (2.43) to (2.46), we have

y

=

y

on 0

(1^;

)

T

]

:

(2.53) Let

R

²

C

¹(

Q

^R2) and

q

²

C

¹(

Q

^R) be dened by

R

=

y

^;

y

"^;

z

(2.54)

q

=

p

^;

p

"^;

:

(2.55)

By (2.40), (2.42), (2.54), and (2.55), we have

@R @t

^;

R

+ ((

R

+ "

y

+

z

)^r)

R

+ (

R

^r)("

y

+

z

)

;

z

+ (

z

^r)

z

^;(curl "

y

)

R

^?+^r

q

= 0 in

Q:

(2.56) From (2.13), (2.22), (2.31), (2.32), (2.43), and (2.54), we get

div

R

= 0 in

Q:

(2.57)

From (2.8), (2.12), (2.23), (2.31), (2.32), (2.44), and (2.54), we obtain, with

I

= (1^;

)

TT

],

R

n

= 0 on ;

I:

(2.58)

From (2.7), (2.12), (2.5), (2.31), (2.32), (2.45), and (2.54), we get, with

!

= curl

R

R

+

!

=^;

y

"

^;

z

^;curl

z

on ;

I:

(2.59) From (2.10), (2.31), and (2.54), we get

R

=

y

^;

z

on ^f

T

^g

:

(2.60)

We x a compact

K

and two real numbers

>

0 and

>

0. By (2.21), (2.24), and (2.32), there exists

"

⁰

>

0 such that, for any

"

²(0

"

⁰] and for any

²(0

1

=

2],

Z

d

^j

z

(

T

)^;

y

^1j

< =

2

j

z

(

T

)^;

y

^1j_W^{;11( )}

< =

2

curl

z

(

T

) = curl

y

¹ on

K:

Hence, by (2.60), in order to get (2.50), (2.51), and (2.52), it su$ces to check that, for suitable choices of

"

²(0

"

⁰] and of

²(0

1

=

2],

Z

d

^j

R

(

T

)^j

< =

2

(2.61)

j

R

(

T

)^j_W^{;11( )}

< =

2

(2.62)

j

!

(

T

)^j_L¹⁽_K⁾

< :

(2.63) From (2.10), (2.20), (2.31), (2.32), (2.53), and (2.54), we have

R

= 0 on ^f(1^;

)

T

^g

:

(2.64)

Let us denote by

C

j(

"

)

j

^>1

various positive constants which may depend on

Ty

⁰

y

¹

:::

^#

and

"

, but are independent of

in (0

1

=

2] and of

s

in

I

. Furthermore let us denote by

C

j

j

^>1

various positive constants which

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

may depend on

Ty

⁰

y

¹

:::

, but are independent of

in (0

1

=

2]

of

s

in

I

, and of

"

in (0

1]

:

For example, from (2.11), (2.12), (2.25), (2.31), (2.32), and (2.59), we get the existence of

C

¹

>

0 and

C

¹(

"

)

>

0 such that, for all

"

²(0

1] and for all

²(0

C

¹(

"

)^;1],

j

!

^jL^1(;(1;)Ts^]6

C

¹

"

^+j

R

^j_L^{1( (1;)}_Ts^]

⁸

s

²

I:

(2.65) Taking the curl of (2.56), we get, using (2.13), (2.22), (2.31), (2.32), and (2.57),

@! @t

^;

!

+ ((

R

+ "

y

+

z

)^r)

!

+ (

R

^r)curl

z

;(curl

z

) + (

z

^r)(curl

z

) = 0 in

Q:

(2.66) For

s

²

I

let

I

⁰= (1^;

)

Ts

]

Q

⁰=

I

⁰

:

In order to obtain a pointwise estimate on

!

one uses the following lemma, whose proof is given in Appendix B,

Lemma 2.2. Let

f

²

C

¹(0

+¹)0

+¹)) be such that

f

(

s

) =

s

⁸

s

²0

1]

(2.67)

0⁶

f

⁰⁶1 in 0

+¹)

(2.68)

f

_{= 32 in 2}

+¹)

:

(2.69)

Let

²(0

1) and

²(0

1) be such that

+

= 1

:

(2.70)

Then there exists a positive real number

C

such that, for any

t

²(0

C

^;1]

for any

²

C

¹(0

t

]^R)

for any

X

²

C

¹(0

t

]^R2)

and for any

Y

²

C

¹(0

t

]^R2) such that

@ @t

^;

+ ((

X

+

Y

)^r)

⁶0 in 0

t

]

(2.71)

(

0)⁶0 in

(2.72)

⁶1 on ;0

t

]

(2.73)

@Y @n

6

C

^;1

t

^;1 on;0

t

]

(2.74)

Y

n

= 0 on ;0

t

]

(2.75)

we have, for all

x

in and all

t

in (0

t

],

(

xt

)⁶(exp

Ct

^j

X

^j2_L^{1( 0}_t^]))(exp

Ct

^32jr2

Y

^jL^{1( 0}t^]))

exp^; 1

(

Ct

)

f d

₍

Ct

²)

:

(2.76)

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

In (2.76) as well as in the remaining part of this paper

jr

2

Y

^j_L^{1( 0}_t^])=

@

²

Y

@x

²¹

L^{1( 0}t^])

+

@

²

Y

@x

¹

@x

²

L^{1( 0}t^])+

@

²

Y

@x

²²

L^{1( 0}t^])

:

We take

²(0

C

¹(

"

)^;1] and we apply this lemma with

X

(

xt

) = (

R

+

z

)(

xt

+ (1^;

)

T

)

(2.77)

Y

(

xt

) = "

y

(

xt

+ (1^;

)

T

)

(2.78)

(

xt

) = 1

C

^1;_" +^j

R

^jL¹⁽Q⁰⁾

!

(

xt

+ (1^;

)

T

)

;

t

^j

R

^jL¹⁽Q^0)jrcurl

z

^jL¹⁽Q^0);

t

^j;curl

z

+ (

z

^r)curl

z

^jL¹⁽Q⁰⁾

(2.79)

_{= 14}

(2.80)

_{= 34}

(2.81)

t

=

s

^;(1^;

)

T

(⁶

T

)

:

(2.82) Using (2.66), and (2.77) to (2.79), we get (2.71). From (2.64) and (2.79), we get (2.72). From (2.65) and (2.79), we get (2.73). Note that, by (2.7), (2.11), (2.12), (2.31), and (2.78), (2.74) holds if, for some

C

²

>

0,

" < C

^2;1 and for some

C

²(

"

)

>

0,

² (0

C

²(

"

)^;1]. From (2.8), (2.12), (2.31), and (2.78), we get (2.75). Moreover, by (2.31), (2.78), (2.81), and (2.82), we have, for some

C

³(

"

)

>

0,

t

^{3 =2jr2}

Y

^jL^{1( 0}t^])61

⁸

t

²0

t

]

⁸

²(0

C

³(

"

)^;1]

:

(2.83) We choose a function

f

²

C

¹(0

+¹)0

+¹)) satisfying (2.67) to (2.69) and apply Lemma 2.2 together with (2.25), (2.32), (2.77), and (2.83) we get the existence of

C

⁴

>

0 and

C

⁴(

"

)

>

0 such that, if

"

² (0

C

^4;1], if

²(0

C

⁴(

"

)^;1], and if

^j

R

^j2_L¹⁽_Q⁰⁾⁶1

(2.84) then, for all (

xt

) in (1^;

)

Ts

]

j

!

(

xt

)^j6

C

⁴(

"

)(

^j

R

^jL¹⁽Q⁰⁾+

) +

C

⁴

"

^+j

R

^j_L¹⁽_Q⁰⁾exp ^; 1

(

C

⁴

)

f d

²⁽

x

) (

C

⁴

)

:

(2.85) In order to deduce estimates on

R

from (2.85), we use the following lemma, whose proof is given in Appendix C,

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.

Lemma 2.3. Let

f

²

C

(0

+¹)0

+¹)) be such that

f

(

s

) =

s

⁸

s

²0

1]

(2.86)

1⁶

f

(

s

)⁶

s

⁸

s

²1

+¹)

:

(2.87) Let

² (0

1] and let

² 0

1) be such that (2.70) holds. Then there exists a positive real number

C

such that, for any

A

in 1

+¹), for any

B

in 0

+¹), for any

B

in 0

+¹), and for any

in

C

¹(^R) such that

²

L

¹() and satisfying

j

^j6

B

+

B

exp(^;

A

f

(

A d

²)) in

(2.88)

= 0 on ;

(2.89)

one has

jr

^j6

C B

+

B

p

A

in

(2.90)

j

d

^r

^jL^{1( )6}

C B

+

B

A

^{8 2}(0

1]

(2.91)

j

^j6

C B

+

B

A

in

:

(2.92)

Let us rst nish the proof of Theorem 1.1 when is simply connected (with ;^# = , and with (2.2), (2.3), and (2.4) instead of (1.8), (1.9), and (1.14) ). Then one can write

R

=^r?

'

:= (^r

'

)^?

(2.93)

where

'

²

C

¹(^R) satises

'

=

!

in

(2.94)

'

= 0 on ;

:

(2.95)

Then, it follows from (2.85) and Lemma 2.3 {see (2.90) {that there exist

C

⁵

>

0 and

C

⁵(

"

)

> C

⁴(

"

) such that, if

"

²(0

C

^5;1), if

²(0

C

⁵(

"

)^;1], and if (2.84) holds,

j

R

^jL¹⁽Q⁰⁾⁶

C

⁵

"

p

:

Hence, if

"

²(0

C

^5;1

=

2] and if

²(0

C

⁵(

"

)^;1], then

(2.84)^{) p}

^j

R

^jL¹⁽Q⁰⁾⁶

C

⁵

"

^{6 1}₂

:

(2.96)

Note that, by (2.64), (2.84) holds if

s

is close enough to (1^;

)

T

. Hence, by (2.96), (2.84) holds for all

s

²

I

if

"

² (0

C

^5;1

=

2] and

² (0

C

⁵(

"

)^;1] therefore, by (2.96),

j

R

^j_L¹⁽_Q⁾⁶

C

⁵

"

p

⁸

"

²(0

C

^5;1

=

2]

⁸

²(0

C

⁵(

"

)^;1]

:

(2.97) Using (2.85), (2.97), and Lemma 2.3 {more precisely (2.91) and (2.92) {, we have shown the existence of

C

⁶

>

0 and

C

⁶(

"

)

> C

⁵(

"

) such that, for any

"

in (0

C

^6;1], any

in (0

C

⁶(

"

)^;1]

and any in (0

1],

j

!

^j6

C

⁶

"

in

K

(1^;

)

TT

]

Esaim: Cocv, May 1996, Vol. 1, pp. 35-75.