beltran rojas 1995

8/10/2019 Beltran Rojas 1995

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REVISTAMATEMATICA de la

Uiversidad Complutense de Madrid

Volumen 8, nmero 1: 1995

Th e Initial Va lue Problem b r th e Equat ions

ofMagnetohydrodynamic Type in

N o n - Cyl indr ica l D om ains

M.A. RQJAS-MEDAR aid R . BELTRAN-BARRIQS

ABSTRACT. By u s i n g t h e a p e c t r a l G a l e r k i n m e t h o d , w e p r o v e t h e e x i s t-

ence of weak solutions for a system of equatics ofmagnetohydrodynamict y p e j i n o n - c y l i d r i c a l d o m a i n s .

1 . INTRODUCTION

In several situations the mation ofincompressible electrical con-du ct i n g f l n i d s ca n b e m o d e l l e d b y t h e s o called equations ofmagnetohy-drodynamics, which c o r r e s p o n d t o t h e N a v i e r - S t o k e s e q u a t i a n s c o u p l e d

withthe Maxwells equations. Iii tlie case where therei s free motion of

1991 Mathe,natics Subject Classiflcation: SSQiO, 76D99, 35005

Serviciopublicaciones Univ. Complutense. Madrid, 1995.


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2 3 0 MA. Rojas-Medar and R. Beltrn-Barrios

heavy ions, not directly dietothe electric fleid (see Schliiter [14]andPikelner [12]), these equations canbe reduced tothe following form:

9 tu+ u.Vu$It.S7It =f9V(p + ~

1% ~ pa~~+ uXlh h.Vu = gradw ,

div u = 0 ,

div h = 0 ,

together withsuitable boundary aidinitialconditions.

Here, u andIt arerespectivelytheunknown velocity and magneticfields; p is the unknown hydrostatic pressure;w is an unknown func-tion related tothe motion of heavy jons (in such way that the densityofelectric current, Jo, generated by this motion satisfles the relationrot J o = aS7w); Pm is tite density of mass of tite fluid(assmed tobe a positive constant); ~ > O is tite constant magnetic permeabilityoftite meclium; u > O is tite constant electric conductivity; i~ > 0 is tite

constant viscosity oftite fluid;fis an given externad force fleld.I n t i t i s paper we w i l l c o n s i d e r t i t e p r o b l e m o f e x i s t e n c e o f w e a k

solutions for tite problem (1.1) in a time-dependent domain of IR x(0, T),n=2,0


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The 1iti,l value Problemfor the Equatio.... 231

Next, this new initial value prablem is treatedusingGalerkinapproxi-mation. We conclude returnung to Q using the inverse of the aboyechange ofvariable. This technicaiityw as introducedby Dal Passo aidUghi [4] tostudy acertain class ofparabolicequations innon-cylindricaidomains.

Wefeel that it is apprapriate to citesomeearlier works onthe initialvalueproblem(1.1)-(1.3) aid to locate or contribution therein. Thecylindrica.l caseof(1.1)-(1.3)hasbeen studied by sorneauthors. Amongthem,letus mentionthepaper ofLassner [7], Boldrini aidRojas-Medar[2],Rojas-Medar andBoldrini [13].

Lassner [7] by making the use ofsemigroup techniques a .sthe ones inFujita and Kato[5] to showthelocal existence and uniqueness ofstrongsolution. The moreconstructivespectral Galerkinmethod was used byBoldriniandRojas-Medar [2], [13] toobtain local, global existenceanduniquenessofstrongsolutions. The aboye authors working in IR withit = 2 or 3 .

Amathematical study ofthe problem(1.1)-(1.3) inanon-cylindricaldomain was notdone, however, it hasto bepointed oit that similar time-

dependent problem bit for tIte Navier-Stokesand Boussinesqproblemshave been studied by many different authors. This is the case, for in-stance of tIte works Lions [9], [8], Fujitaand Sauer [6], deda [10], QtaniandYamada [11],C o n c a and R o j a s- M e d a r [3]. In particular, we wouldliketo emphasize that the arguments oftItementionedauthorsdemandt h a t Q(t) benondecreasingwith respectto t ( s e e L i o n s [9],p r o b l e m 1 1 . 9 ,p. 426); except tIte work ofConca aidRojas-Medar [3].

In this work, we wili adapt thetechniqe by [3]totItesystem(1.1)-(1.3).

TItepaperi s organizedas foliows. Iii Section 2 weintroduce varjoisfunctional spaces andstatethetheorems. Section 3 aid 4 dealwiththeirproofs.

2. FUNCTION SPACES AND PRELIMINAIRES

TIte functionsin tIte paper are elther IR or IR-vauued aid wewillnotdistinguish thesetwasituations in or notations. To which case wereferwifl be clear fromthe context.


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2 3 2 MA. Rojas-Medar sud R. Beltr-B,rrios

Let =2be a baunded domain in IR (n=2) with smooth boundary0=2.We write simplyhill f o r L2(=2)-norm.T I t e i n n e r pr oduct i n L2(=2)

isdenoted by (u,y). The solenoidalfunctionspace is definedas usual

C&2=2)= {~ 6 Cr(=2) div c p =

fi(=2)= the completion ofC3~0}=2)under the L

2(=2)norm,

V3(=2)= t h e co mp l et i on of C0%(=2)under t h e fiS(=2)norm

where H(=2)denotetIte usual Sobolevspace with s 6 la.

The normand mier praductin fi(=2)andV~(=2)a r e :

(1~9) =4f19dx,u

(u,y). =~ V)H=1

fil =

I I u u ~ = (u,u)V2

w h e re ( u ~ ,v,)H is tIte standardinnerproduct offi5(=2);(V,(=2))denote6tIte toplogicaldualof V,(=2).

bu particular, wedenote

V1(=2)= V(=2) aid m i l i =Vu .

We will use other standard notations aidterrninology; for tliem,we refertoAdams[1] aid Temam [15].

Let rbe a real-valuedfuiction which is ofC-class on tIte interval[0,2] such that,

r Q o ) = m i n {r (t ) O=1=2} > O (2.1)

and

(this candition is essential).


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The Initial Value Probletnfor the Equatios... 233

TIte time-dependent space domain Q Q ) is a bounded set iii IR defined by

Q(t) = {x E IR ~ 1 < r(l), 0=1=T}where . denotes tIte usual norm u IR. Its boundary is

OQ(t) ={x E IR x I = i(t), 0=1=T}.

It is warth noting that such domains Q(t) O < t < 2, generate anon-cylindrical time-dependent domain ofIR x IR :

Q = U Q(t)x{l}.O ~ u SItj ~ h.]dxdt =Br;

Ox;af~fsodxdt,

(iii) f~[h.@ty EVhV~1+ ~ u~ ~f It1 2Iti~f u]dzdi = O

= 1 i;=1

foi ah ~ E C(Q) wilhdiv ~ = div ~ = O , tIte suifiz t denotes b I t eoperaior~ (deritativeswitItrespeet lo 1 ull sometimes also be denoledby a orsimplyby d/dt).

(iv) u ( O ) =no, It(O) =It0.mherepu a = u = ~. andy =

p pu

Remark 2.2. Iithisdefinition tIteinitialconditions (iv)Itave theusual meaning; see for exaznple,Lions [8].

TIte main result ofourarticle is


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2 3 4 MA. Rojas-Medar sud R. Beltrn-Barrios

Theorem2.3. 1ffE L2(Q), u0, I t0 E H(Q(O)), bIten lhere crisIsa u,eaksolulion ofproblera (1.1)-(1.8)forany limeinterval [0,2].

Theorem 2.4. Ifit 2 tItesolubion(u,h) obtainedun lIte TIteorera2.8isunique. Moreoteru andIt are almosteverywItereequal bofunetionseontinuousfrora[0,2] m b fi and

u(t).uo irz.H, ast*O (2.2)

It(t)*Ito infi, ast*O. (2.3)

3. PROOF OF THEOREM 2.3.

Let usintroduce thetraisformation it Q ~ U,givenby

w h e re U D x ( 0 , 2 ) ai d D= { x E I R . 1 x=1 } . S i n c e r(t) i s a C1-

f ui ct i on which v e r i f i e s ( 2 . 1 ) , w e s e e e a s i l y t h a t y is a diffeomorpItism

aid that itsinverse r1 : U Q satisfles

r(y, t)= (r(t)y, t). (3.1)

We a l s o d e f i n e

v(y,i) =

b(y,i) = h(yr(t),t),

q(y,t) =E ( y , t ) = w(yr(t),t), (3.2)J(y,t) =f(yr(t),t),

vo(y) =uo(yr(O)),

bo(y)=ho(yr(O)).


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TheInitial Value Problem< oc the Equations... 235

u s i n g ( 3 . 2 ) , t I t e s y s t e m o n Q (1.1)-(1.3) i s transformed l i t o t I t e

tv+Zr(t) .~

/1

0~ + r(t)

O br(t)

9v 1

r(t) ~ 8v

4t)Z

1

v ( 0 , y ) = t,o(y)

b(O,y) =b o ( y )

on tIte cylindrical domain U =D > < (0,2).

Qn tIte otherhand, let us set

D v1

c(v,w)= E~j=i

a(v~w)=J 9vj

B(u,v,w) =

Otii j

oyi

0v1

w1dy

for vector-valueddetined.

functions u, y aid wfor whicIt tIte integrals arewell-

Bysystem:

O b _

o y e(3.3)

0v

(3.4)

(3.5)

(3.6)


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236 MA. Rojea-Medar ad R.Beltr-Barrios

The n ot a ti on of w e a l c s o l u t i on f o r ( 3 . 3 ) - ( 3 . 6 ) i s c o m p l e t e l y s i m i l a r

t o t I t e on es f o r ( 1 . 1 ) - ( 1 . 3 ) .

To p r ov e t I t e e x i s t e n c e o f s o l u ti o n s o f t I t e transformed s y s t e m ( 3 . 3 ) -

( 3 . 6 ) we w i l l u s e t I t e s p e c t r a l G a l e r k i n m e t h o d . T h a t i s , we lix . sand we consider the Hilbert special basis {w(xfl42

1 ofT~8(D), whoseelements we w i l I c h o o s e a s t h e s o l u t i o n o f t I t e f o l l o w i n g apectralproblem

(w~,v)~ =A (u9,v) Vv 6 1/8(D).

L et Vk be the subspace of 18(D) spanned by {w,. .. ,wk}. F ore v e r y k=Y , we d e f i n e appr oxi mati ons ~ b oftiandb , respectively, bym e a i s o f t h e f o l l o w i n g hite expansions:

kyk => j 3 C.k(t)W(X) 6 Vk t E (0,2)

i=1

aid

k

= Zdk(t)u,t(x) 6Vk te (0,2)i=1

where the coefficients (elk) aid (d~k) will be calculatedin such a waythat y andb solve tIte following approximationsof system(3.3)-(3.6):

1/a(vk ~pjt r(t)

r(t)r(t)

(3.7)

1 1(b4)+ a(b,iP)+ B(v,b,t,b) B(bk,vk,t)[r(t)]

2 r(t) rQ)

(3.8)

r(t) ~

r(t) c(b


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The Iitial V,lue Problem for the Equstio.... 237

for ad J ~,4, 6V,y(O ) = t4; b(O) = (3.9)

where vt . y0 aid b ~ . b 0 ji H(D) as k-+00.

Equatians ( 3 . 7 ) , ( 3 . 8 ) a i d ( 3 . 9 ) i s a s y s tem o f o r di n a r y d i f f e r e t i a l

eq ua ti on s f o r t I t e c o e f f i c i e n t functios ei~(t) aid d1~(l), whjch d e f i n e s0< aid b k in a interval [O,1,4. We w i l l s h o w s o m e a p r i o r i es ti ma te si n d e pe n d e n t o f kaid 1 , u orderto take 4 = 2 . A l s o , we w i l l p r ov e t h a t(yk,bk) c o n v e r g e s u a p p r o p r i a t e s e i s e t o a s o h u t i o n (v,b) of ( 3 . 3 ) - ( 3 . 6 ) .We p r a v e t I t e f o l l o w i g l e m m a .

Lemnia 3 . 1 . TIte transformed system (8.8,)-


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238 M.A. Rojas-Medarsud R. Beltrn-Barrios

Now, w e u s e Hlder aid Young inequalities to estimate tIte right-Itaid s i d e o f t I t e aboy e inequality, weobtain

lFkl =~1 h 1 12 + ~ I h v l I ~ +C~r(t)h2a2Iy1~ 0 < 1 1 2

+C r ( t ) 2 y I j .o b 2 + [(t)]2 Vv 2

6

+ [(t)]2

wItece, taking = /2 aid 6 = -y/2, we arriveat tIteinequality

d 1(uVv2+ yVb2)~ (a ~ v ~ + j b 2 ) +

=]~i2 + (c r ; (0 2 a 2 1 1 Y io o + 1 1 0 < 1 1 2 + cr(t)fly~.~hlbIr


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The Initial Value Problemfor the Equ,tios... 2 3 9

Duetothe choice of4 and b~ , thereexists Cindependent ofksuch

that 1 4 1 1=

Cvo, [ b ~ =

Cjbo, moreover fII J (s )1 1

2 d

8 islimite forO < t< 2, we conclude, by usig Growallsinequality, that

+ b(t)2 +f ~ (vVv(s)2+ y I hV b ( s )11 2 )ds=e

1 .

TItus, for al k we Itave that 0 and b exists globally in 1 aid

aid (b) remains bouded in L00(O,2;fi(D)) r l L

2(O,2;V(D)) as

00. TItenext s t ep o f t I t e p r o o f c o n s i s t s o f p r o v i n g t h a t (v~) aid (b ~ )bouded in L2(O,2;(1

8(D))). lo tItis end,let us flx some otations

(vk)

are

=

=

=

We wili prove that (Q)v),bounded in L

2(0,2;(V5(D))).

[rQ)]2a(v,u)

rQ) e(v,u)

B()

(C(tft#9, (E(t)vk) and (E(flb) are

Indeed,forah u 6 14, wehave

1IAI = [r(t)]2

e [r(t)]2

whence, we have

I I A ( t ) v I I ( v ( D ) ) v = sup Il e


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2 40 M.A.Roj,s-Medar ad R. Beltr-Barrios

cansequently

Since 6 C ([O ,2 ]) and (y) is bounded in L2(O,T;V(D)), wededucethat

[r(t)]2 Slv(t)j2dt< L

and (A(t)v) i s t h e r e f o r e bouded j n L2(O,2;(V.(D))). Aadogously,we shaw that (C(t)v) js bounded in L2(O,T;V

8(D))). To prove theb o un d e d n e ss of (E(t)v) jn t I t e space L2(O,2;(V~(D))) we will usetItefollowing interpolation result w h o s e p r oo f can b e faunA l x i L i a n a [8]:

Lemma 3.2. If(y) is boundedin

L2(O,2;V(D)) flLw(O,2;fi(D)),

tIten (y) is also boundedu n L4 (O,2;LP(D)), u,Itere 1 = 1 _ 1~> 2 2n

Us i ng t h j s Lemma, we conclude t h a t

r(t)h114IILPIIViIILPIIyII

e y

_ r(t)

since 1 + 1 + = 1 and tIte Soholev embedding fi91(fl) ~ L(D).1 p 1

TItis imply

J I E(t)v~,(D> ),dt


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The Initial Value Problemter the Equatio.... 2 41

Sjnce E C ( [O ,2 ] ) , we ca co n cl ud e t I t a t (E(t)v) is boindedSimilary, we prove tItat (E(t)b) is bounded uji L2(0,2;(14(D))).

Now, we consider P,: fi * ydefinedby

l e

P~u =Z(u,wt)wt=1

14(D)14(D). Itis

c fi aid y C 14(D), we can consider F ,, : 14(D) *easyto see tItat P i, E L(14(D),V

8(D)), (L(X,Y) denotet h e s p a c e of t I t e b o u n d e d op e r a t or o f X i n t o Y) hence

*

defined by =

l P k ~ I I=FkI=1.lies in L((V8(D)),(14(D))) and

We also observe tItat tIte autofunctionsw a r e i n v a r j a n t s by Pi,, i.e.,

Pi,(wt) = w1.

From j t an d ( 3 . 7 ) w e co n cl ud e t h a t

a(vP,w) =


14/23


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The Initial value Prblem for the Equatione... 2 43

tItis imply

T

c(fT[J)]2ve~y)I/2(fT [(i)]2 b llt~) . < e

Similarly, we prove thatfi(b,v) isbounded u L2(O,T;(V,(D))).

T h e r e f o r e , a r g u i g a s ji tIte book ofLions [8, p. 76] andmaking u s e o f the Aubin-Lios Lemma withB

0 = V(fl), P o = 2 , -B ~ =(14(D)), Pi =2 aid B =fi(D) (see Theorem 1 . 5 . 1 aid Lemma 1 . 5 . 2of tIte aboye baak, p. 58), we can conclude that tItere exist y, b 6L

2(O,2;V(D)) such tItat, up to a subsequence wItich we sha]I denoteagain by the suffixl e, tItere hold

l eU *271

b ...4bj in L2 (O ,2 ; V(D)) aid L00(O,2;H(D))wealdy aid

: } L2(O,T;fi(D)) strongly, as kle 1

~ *v~

-+ b t in L2(O,T;(V8(D)) wealdy, as le 400.

Now, tIte ext step is to take thelimit. But, once tIte aboye convergence results Itave beenestablisIted, tItis is standard pr ocedur e ai dit follows the sanie patter as u Lions [8, p. 76-77]. Consequetly, we


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244 MA. Rojas-Medar ami R. Beltrn-Barrios

s h a i l o m i t j t and we w i l i d i r e c t l y deduce t h a t

Ta(v,ek) +

1. B(b b , ~) =fT a(J, ~) +r(t)

a(v,4,) + - yi [r(t)]25 1

a(b,4,) +

l o r ( t )

a

r(t)

J I c(b,~)o r(t)for ah 4> ,4 , 6 < 2 1 (U ) such tItat div ~ =div 4 , =O . So, the Lemmaisproved.

To conclude t h e p r oo f ofTIteorem, let usconsider a test functions o E C1(U) suchtItat div s ~ =O ,anddefine

y,t) = [r(l)]~so(yr(t),t).

Iiitegratingby parts,

f 2 a E jj ]e v ~ o = f 2

~[r(j)]fl(y, s o )

2

-f [r(i)]2 a(v, #) +fJ I

T

o

aid

(3.10)~T r(t)

a ~ c(v,t)

fi( 3 . 1 1 )

-2

a(v,qS)


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2 46 MA. Rojas-Medar ad R . Beltr-Bnrios

Let us nowconsider tIte traisformation i-1 : U * Q which jsd e f i n e d b y ( 3 . 1 ) . Weobserve that its Jacobianis[r(t)]TI. Consequently,by change ofvariables ji t I t e i n t e g r a l s , ( 3 . 1 3 ) an d ( 3 . 1 4 ) become

1 a~son

+ ~ J I vVuJLp~ n

> 3 .11,1=1 2

+ tJ I

u

5 u1

O x 5

It5 =J I

and

J I It . ~a > 3+fj~ yVIt~S7~1 1,5=1

J I ni q Ox5TI

h.+>3J I It ~ 0@ iO x 5 ~1 = O

wItich proves that (u,It) is a weak solutionmappigs

L2(0,2 ; V(D)) *

-.4 u(x,t) = ___

b(y,t) > h(x,i) =

of the problem; since tIte

aid

L2(0,2;H(D)) 4

L2(0,2 ;H(Q(t))

v(y,t) 4 n(x,t) =

b(y,t) > It(x,t) =

L2(O, T;V(Q(tfl)

r(t)1)

b(),t)


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Tbe lniti,I Value Problemfor the Equationa... 2 47

are smooth bijectios of class C1, itfollows that

u,It E L2(O,2;V(Q(t))) nL00(O,2;H(Q(t))).

Finally, a standard arguments sItows that u(O) = u0 aid It(O) =I t0

(see remark2.2). TItisfinished tIte proofoftIte Theorem.

4. PROOF OF THEOREM 2.4.

We first prove tIteregiilarity result. We observe that tIte proofof

tIteaboyetheorem sItown that u E L

2(0, 2 ;y);consequently, applying

Lemma 1.2jnTeman [15], p. 260,we obtain thatu is almosteverywhereequal to a fnction continuousfrom[0,2]lito fi.

Thus,u E C ( [ O , 2 ] ; H )

and(2.2) follows easjly. Analogously it is proved tIte continuityof It aid(2.3).

We a l s o r e c a i l that Lemma 1.2 u Temam [15], p. 260-261, asserts

thattIte equatios below holds:

d~ uQ)2=2,

d

TItese resultswill be used ji tItefollowigproofofuniqueness wItich we

will start ow.Consider t h a t ( u i , I t i ) and(u

2, I t 2 ) a r e t w o solutjonsof tIteproblem( 1 . 1 ) - ( 1 . 3 ) w i th t I t e s a m e f aid uo,Ito aid define tIte differences w =


20/23


21/23

me liti,I Value Problen, for Ihe Equatio.... 2 4 9

I4ow, we observe that

aB(w,u1,w)=aCwjj5~huui

< a~Cjwj Ih w lli I l u iIii

2

~ IIw Ih +C , .4 a )a I Iw I I2 I Iu iI I ~

where we used the Lemma 3.3 u Temam [15], p. 291, together with

Hlder aidYoung inequahities.Aalogously, we can prove

2B(v,ui,v)=~ t 4 ~ +C . , Iy l2 I lu ih ~ ,B(v,Iti,w)=~ lwl~ +

6

B(w,h2,v)=~ w ~ +

I y [~ +C,c,(a)(I lv12+aIlw I2)ItItill~

~h h v ll 2 +C ~ ,~ (a ) ( l v l I2 + awj2)h2~.

By using tite aboyeinequalities u (4.1), weget

d

w ( a 1 w 1 1 2 +I I v I 2 ) +v u I w Ii~ + rU v I I ~< C ~ (a w 2 + II v h l2 )( h h u i l~ + h 1 j ~ + h2j~),

whereC is a positive costant that oly depend on u, -y , a.

By integratung u t i m e , t h e u s e ofGronwahls iequality, we obtadn

aw(fl2+ h I v c lt ) l 1 2=(aw(0)l2 + l Iv(O ) l l2)e ~(t>


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250 MA. Rojas-Medar and R. Beltrn-Barrios

whereso(t) =(f~(ui~+It~+It2flds < +oo,foreveryt 6 [0,2].Thislastiequality,impliesw(t) =v(t) = O . Hence u1 = u2 and I t 1 =I t2aidtheuniqueness is proved. Thiscompletes theproofoftheTheorem.

References

[1]Adams, R.A.: S o b o l e v Spaces, Academic Press, New York, 1975.

[2]Boldrinj, J.L. and Rojas-Medar, M.A.: Qn a system ofevolubionequations ofmagnebohydrodynamic bype. To appear ji Matemtica Con-temporanea.

[3]Conca, C. and Rojas-Medar, M.A.: TIte inilial value probhem forlIte Boussinesq equabions in a lime-dependen domain, InformeInterno,M.A.-993-B-402, Universidad de Chile, Chile, 1993, submitted.

[4]Val Passo, R. et Ughl, M.: Problme de Diriehel pourune classedquations parabohiquesnonhinairesdansdesouverts non cyhindriques,

C.R. Acad. Sc. Paris Srie 1 , 308 (1989), 555-558.

[5] Fujita, H.aid Kato, T.: Qn tIte Navier-Stokes muid valueproblem,1, Arch Rational Mech. Anal., 16 (1964), 269-315.[6] Fujita, II. and Sauer, N.: (onstruelion ofweak solutions of lIteN a v i e r - S b o k e s e q u a b i o n s i n a non-ey l i n d r i c a l domain, hill. Amer. Math.Soe., 75 (1969), 465-468.

[7] Lasser, O.: U ber einrand-anfangswerl-probhem der magnebohydro-dinamik, Arch. Rationad Mech. Anal., 25 (1967), 388-405.

[8]Lions, 3.L.: Uneremarque surles problmes devohubionnonhinairesdais des domainsn o n eglindriques, R e y . RoimaineMath. Pires Appl.,9 (1964), 11-18.

[9] Lios, J.L.: Quelques mihodes de rsohution desproblmesauz hm-iles non-hineaires, finad Gaithier-Villars, Paris, 1969.

[10]eda, K.: Qn bIte inibial problem br lIte heal conveebion equationofBoussinesq approximabion in a bime-dependent domain, Proc. JapanAcad. Ser. A. Math Sci., 25 (1988), 143-146.

[11] tanj, M. aidYamada, Y.: Qn lIte Navier-Sbokes equations u non-cyhindrical domains: Arz approach by tIte subdifferenlial operabor b Iteory,

F a c . S c i . Unjv. Tokyo Sect lA, 25 (1978), 185-204.


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The Initial Value Problern for th e Eqnations... 2 51

[12] Pikelner, S.B.: Gnundiander denKosmisc/zen dllcbrodynamic, Mos-cou, 1966.

[13] Rojas-Medar, M.A. ai d Boldrini, J.L.: Global stmng sohutions ofequations ofmagneboItydnodynamic lype,Relatrio de Pesquisa, R.P. 53 ,IMECC-UNICAMP,Brazil, 1993, submitted.

[14]Schliiter, A.: Ognamie des plasmas, 1 aid II, Z. Naturforsch. Sa,(1950), 72-78; 6a., (1951), 73-79.

[15]Temam, 1 1 . . : Navier-Sbokes equations, North-Holland,Amsterdam,Rey. Edit., 1979.

UNICAMP rMECC, CPSO6S Departamentode Matemtica,

13081-970, Campinas, SP Universidad de Tarapach

Brazil Casilla 7D,Anca

Chile

Recibido: 2 7 d c Octubre de 1993

beltran rojas 1995

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