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Texte intégral

(1)

by a Klein-Gordon equation in spae dimension 2

byMathieuColin

UniversiteParis-Sud,UMRdeMathematiques

Bat425,91405OrsayCedex,FRANCE

Mathieu.Colinmath.u-psud.fr

tel. 01.69.15.57.85

Abstrat.Inthisartile,westudythenonlinearplasmawaveequation

"

2

2

u

"

t 2

+2i u

"

t +u

"

=

1

p

1+ju

"

j 2

1

!

u

"

+ (

p

1+ju

"

j 2

)

p

1+ju

"

j 2

u

"

withinitial datau

"

(;0)=u

"

0 ()2H

8

(R 2

),

t u

"

(;0)=u

"

1 ()2H

7

(R 2

). WeshowthattheCauhyproblem

isloally well-posedonaninterval[0;T℄where thetimeT isindependentof"ifu

"

1

issmallenough. Then,

wedemonstratethestrongonvergeneofu

"

towardsthesolutionuofanonlinearrelativistiShrodinger

equationas"goesto0.

1 Introdution

ThenonlinearShrodinger equationstatedin R 3

2i u

t +

? u

u

p

1+juj 2

? (

p

1+juj 2

)+(1

1

p

1+juj 2

)u=0: (1)

is known to desribe properly the self-hanneling of a high power ultra-short laser pulse in matter (see

for example [3℄, [4℄, [5℄, [6℄, [13℄ and [7℄ for more referenes). Here, we denote

?

=

2

x 2

+

2

y 2

and

u=u(x;y;z;t). Theaimofthispaperisto explainoneoftheapproximationswhihleadtothisequation.

FirstfromMaxwellsystemdesribingthepropagationofhighpowerultra-shortlaseringaseousmediumwe

anderivethefollowingwaveequation(see[7℄)

A=k 2

p

1+k 2

p

!

A; (2)

where A isthe eletromagnetivetoreld, = p

1+jAj 2

isthe Lorentz fatorand =

2

t 2

. Then,

aordingto physial observations (see [4℄),equation (2) anbe simplied ifweuse theapproximationin

whihtheomplexamplitudeAisslowlyvaryingoverdistanesontheorderofawavelengthinthediretion

ofpropagationzandovertimesontheorderoftheperiodofthehigh-frequenyeldosillations. Itisthen

naturaltodene

A=A

"

(x;y;z;t)=u

"

(x;y;"z;"t)e i(k z !t)

where

"=

!

p;0

!

;

dependsontheparametersoftheplasma(!

p;0

denotestheunperturbedplasmafrequeny).

(2)

? u

"

+"

2

2

u

"

z 2

"

2

2

u

"

t 2

+2i p

1 "

2 u

"

z +2i

u

"

t

=

1

"

1

u

"

+ (

? +"

2

2

z )

"

"

u

"

(3)

where

"

= p

1+ju

"

j 2

. Intheapproximationofunderdenseplasmas,oneanassumethat "1.

Then,takingformally "=0in (3)and performing ahangeofvariables givethenonlinearShrodinger

equation(1). It isthennaturaltothink thatthesolutionsu

"

of (3)ouldonvergetowardsthesolutionsu

ofequation(1).

This kind of questions onstitute an ative domain of researh in Mathematis. For example, in [2℄,

T. Colinand L.Bergeinvestigated theso-alled Langmuir waveenvelopeapproximationwhih onsists in

takingthelimit" !0inthenonlinearplasmawaveequation

"

2

2

E

"

t 2

2i E

"

t E

"

=f(jE

"

j 2

)E

"

:

Under spei assumptions on f, the authors show the onvergene of E

"

towards the solution E of the

orrespondingShrodingerequation. In[1℄,theauthorsprovethesamekindofresultsforasetofequations

desribingtheinertialregimeofthestrongLangmuirturbulene,namely

"

2

2

E

"

t 2

2i E

"

t E

"

= n

"

E

"

1

2

2

n

"

t 2

n

"

=jE

"

j 2

:

In[14℄, theauthorsshowthatthesolutionsofthenonlinearKlein-Gordonequationanbedesribedby

usingasystemoftwoouplednonlinearShrodingerequationsas tendstoinnity.

In[9℄,T.GallayandG.ShneiderobtaintheKPequation

2

2

A

xt +

2

A 2

x 2

+

4

A

x 4

+

2

A

y 2

=0

asanapproximationfortheBoussinesqequation

2

t 2

=+(

2

)+

2

t 2

:

Inordertojustifyrigorouslythederivationof(1)from(3),wehavetondasequeneu

"

solutionof(3)

whihsatisesthefollowing:

u

"

existsonaninterval[0;T℄whereT isindependentof"

u

"

isbounded inH s

(R 3

)withabound independentof"

theinitial valuesdonottendto zerowith".

Unfortunately,thisprogramistooambitiousforthemoment. That iswhywewillnotstudyhereequation

(3). Indeed,thepreseneof"

2

infrontof 2

z

givesrisetomanyproblems. Forexample,inordertoderivean

energyestimateonequation(3),wehavetoonsiderthenormjj(1

"

) s

2

jj

L 2

(R 3

)

where

"

= 2

x +

2

y +"

2

2

z ,

whihpreventsfrominjetingthefuntionsu

"

inL 1

(R 3

)withaonstantindependentof".

(3)

where allthetermsdependingonz havebeendropped

"

2

2

u

"

t 2

+2i u

"

t +u

"

=

1

"

1

u

"

+

"

"

u

"

: (4)

Here

"

= p

1+ju

"

j 2

. Formally,taking"=0in (4)givestherelativistiShrodingerequation(1) .

WerstsolvetheCauhyproblem(4)withinitial ondition

u

"

(x;y;0)=u

"

0

(x;y);

t u

"

(x;y;0)=u

"

1

(x;y); (5)

keepingin viewthethree assertionsof ourprogramstatedabove. Arst approahouldbeto treat(4)as

anonlinearKlein-Gordonequation. Indeed,ifwedenote

v

"

(x;y;t)=u

"

(x;y;t)e i

1

"

2 t

aneasyalulationshowsthat v

"

isasolutionof

"

2

2

v

"

t 2

+v

"

1

"

2 v

"

=

1

"

1

v

"

+

"

"

v

"

(6)

withinitial ondition

v

"

(x;y;0)=u

"

0 (x;y)

t v

"

(x;y;0)=u

"

1

(x;y;z)+i 1

"

2 u

"

0 (x;y):

Equation (6) satises the hypothesis of Klainerman and Pone [12℄. We an dedue that if the initial

data are small enough in an adequate spae, the Cauhy problem (6) admits a unique global solution

v

"

2C([0;+1[;H s

(R 2

))\C 1

([0;+1[;H s 2

(R 2

))where s is an integersuÆiently large. Bak to u

"

, we

anshowthat (4)admitsauniquesolutionu

"

2C([0;+1[;H s

(R 2

))\C 1

([0;+1[;H s 2

(R 2

))iftheinitial

data issmallenough. Theproblemisthat,in doingso,it iseasytosee thatthepreseneof theterm 1

"

2 v

"

imposestov

"

andsotou

"

totendto0inthespaeL 1

(0;T;H s

(R 2

))as"goesto0whihisinontradition

withourprogram. Thus,wehavetosolvetheCauhyproblem(4)withinitialondition(5)inanotherway.

Aseondapproahouldbeto applytheusual energyestimates(see [12℄)toequation(4). Indeed,itis

possibleto provethetwofollowingassertions

The Cauhy problem (4) is loally well posed in L 1

(0;T

"

;H s

(R 3

)) for any initial data u

"

0 , u

"

1 with

T

"

=

"

2

(jju

"

0 jj

H s

(R 3

) +jju

"

1 jj

H s 1

(R 3

) )

.

The Cauhy problem (4) is loally well posed in L 1

(0;T

"

;H s

(R 3

)) with T = 1 if jju

"

0 jj

H s

(R 3

) +

jju

"

1 jj

H s 1

(R 3

)

h(") whereh(")tendstozerowith".

Thesetwoassertionsarenotveryusefultoourproblem. Indeed,itislearthatintherstone,thetimeT

"

tendsto0with"whereasin theseondone,theinitialdatau

"

0

tendsto zerowith".

Then,werstprovethefollowingtheorem

Theorem1.1 Assume that u

"

0 2 H

8

(R 2

) and u

"

1 2 H

7

(R 2

). Then there exists Æ

"

> 0 (Æ

"

2 1

"

2

) and

M>0independent of "anddepending onlyonthe initial datas suhthatif jju

"

1 jj

H 7

(R 2

) Æ

"

,there existsa

timeT independentof "suhthat (4)admits auniquesolutionu

"

satisfying

u

"

2L 1

(0;T;H 8

(R 2

))\C([0;T℄;L 2

(R 2

)):

Furthermore, wehave

sup

t2[0;T℄

jju

"

jj 2

H 8

(R 2

) +"

2

jj

t u

"

jj 2

H 6

(R 2

) +"

2

jj 2

t u

"

jj 2

H 6

(R 2

)

M:

(4)

We have to notie that u

1

tends to 0 very rapidly with ". In order to get estimates on u

"

whih

are independent of ", we will onsider (4) not asa wave equation but as a perturbation of a quasilinear

Shrodinger equation. Thismeans that in equation (1), theterm"

2

2

t u

"

will betreatedasaperturbative

term. Forthatpurpose,wewillusethemethodintroduedin[8℄,namelywerewriteequation(4)asasystem

in (u

"

;u

"

) t

. Then,wedierentiatethe equationwithrespetto spaeand timeto obtainanewsystemin

u

0 ,..,u

5

(wedropforonvenienetheindex ")where

u

0

=u

"

; u

j

=

j u

"

81j 2; u

3

=e q(ju

0 j

2

)

u

"

; u

4

=

t u

"

; u

5

= 2

t u

"

:

Thefuntionqplaystheroleofagaugetransformwhihallowstotreattheequationssatisedbyu

0 ,u

1 ,u

2 ,

andu

3

exatlyaswehavetreatedequations2:9, 2:10and 2:11of[8℄. Conerningu

4 andu

5

wedonotuse

gaugetransformsbeausetheirrespetiveequationswillbetreatedinadierentway,whihdoesnotsuppose

to integratebyparts thetermsinludingtherstderivativesofu

4 andu

5

. Indeed,to estimateu

4 andu

5 ,

weusethelassialenergyestimatesonwaveequations, namelywemultiplytheequations respetivelyby

(u

4 )

t and(u

5 )

t

andweintegratetherealpartoverR 2

. Finally,theproofofTheorem1:1willfollowfrom a

lassialxed-pointTheorem.

EquippedwithTheorem(1.1),weanstudytheonvergeneofu

"

toasolutionuof(1)withinitialdata

u(x;y;0)=u

0

(x;y)under theonditionthat

u

"

0

!

"!0 u

0

in H 6

(R 2

) and u

"

1

!

"!0

0 in H 7

(R 2

)

Morepreisely,wehavethefollowing

Theorem1.2 Let(u

"

0

)and(u

"

1

)betwoboundedsequenesrespetively inH 8

(R 2

)andH 7

(R 2

)satisfying

u

"

0

!

"!0 u

0

in H 6

(R 2

) and u

"

1

!

"!0

0 in H 7

(R 2

):

Wesupposefurthermorethatjju

"

1 jj

H 7

(R 2

) 2

1

"

2

. Letu

"

bethesolutionof(4)withinitialonditionu

"

(;0)=

u

"

0 (),

t u

"

(;0)=u

"

1

()given by Theorem 1:1. Letuthe solution of(1) withinitial ondition u(;0)=u

0 ()

given by Theorem 1:1 of [8 ℄Then, thereexists atime T >0independent of " suh that u

"

andu existon

[0;T℄for any" and

u

"

u !

"!0

0 in L 1

(0;T;H 6

(R 2

)):

Theproofusesanenergyestimatewhihreadsexatlyastheoneof[8℄.

Notationand funtion spaes:Asusual,wedenotebyL p

(R 2

)theLebesguespae

L p

(R 2

)= n

u2S 0

(R 2

) = jjujj

p

<+1 o

where

jjujj

p

= Z

R N

ju(x)j p

dx

1

p

if1p<+1

and

jjujj

1

=ess.sup

ju(x)j;x2R N

:

WedenetheSobolevspaeH s

(R 2

)asfollows

H s

(R 2

)=

u2S 0

(R 2

) = jjujj 2

H s

(R 2

)

= Z

R 2

(1+jj 2

) s

jbu()j 2

d <+1

where bu()istheFouriertransformofu. LetC(I;E)bethespaeofontinuousfuntionsfromaninterval

I of R to aBanah spae E. For 1 j 2, we set

j

=

x

j

and by onvention

0

u = u. Fork 2 N 2

,

(5)

k =(k

1

;k

2

),wedenote k

u= 1

1

2

2

uandjkj=k

1 +k

2

. As usual, the oeÆientsofamatrix Dwill be

denoted by D ij

. The notation D(D r

V) means that the matrix D depends on k

V for jkj r. Dierent

positiveonstantsmightbedenotedbythesameletterC. WealsodenotebyRe(u)andIm(u)therealpart

andtheimaginarypartofuand[a℄isused todenotetheintegerpartofa.

Insetion2,wewilltransformequation(4)intoasystemtowhihweanapplytheappropriateenergy

method. Then,in setion3,weproveTheorem 1:1onerningtheCauhyproblem (4). Finally, insetion

4,weproveTheorem1:2.

2 Transformation of equation (4)

Inthissetion, wetransformequation(4)into asystemto whih weanapply ourenergymethod. Wex

a

"

0 2H

8

(R 2

)anda

"

1 2H

7

(R 2

). Following[8℄, equation(4)anberewrittenasasystemin(u

"

;u

"

) t

"

2

(u

"

)

tt

(u

"

)

tt

+2i

(u

"

)

t

(u

"

)

t

+A(u

"

)

u

"

u

"

u"

2

"

jru

"

j 2 u"

4 4

"

jrju

"

j 2

j 2

+u

"

g(ju

"

j 2

)

u"

2

"

jru

"

j 2

+ u"

4 4

"

jrju

"

j 2

j 2

u

"

g(ju

"

j 2

)

!

=0

(7)

where

g(s)= 1

p

1+s

1;

"

= p

1+ju

"

j 2

;

A(u

"

)= 1

2 2

"

2+ju

"

j 2

u 2

"

u

"

2

(2+ju

"

j 2

)

:

We nowdrop for onveniene theparameter". We set u

0

=u

"

, u

1

=

1 u

"

, u

2

=

2 u

"

, u

3

=e q(ju0j

2

)

u

"

,

u

4

=

t u

"

and u

5

= 2

t u

"

. Here, u

0

doesnotdenote aninitial ondition. The funtion q will playthe

role ofagaugetransformand willbehosenlater(see [8℄ formoredetails). We alsosetU

=(u

j )

2

j=0 and

U =(u

j )

5

j=0

. Then,equation(7)anberewrittenas

2i

t u

0

t u

0

+A(u

0 )

u

0

u

0

"

2

1

u

5

1

u

5

+F

0 (U

)=0 (8)

where

F

0 (U

)= 0

B

B

B

B

u

0

2

(u

0 )

( 2

X

k =1 ju

k j

2

)+ u

0

4 4

(u

0 )

( 2

X

k =1 (u

0 u

k +u

0 u

k )

2

)+u

0 g(ju

0 j

2

)

u

0

2

(u

0 )

( 2

X

k =1 ju

k j

2

) u

0

4 4

(u

0 )

( 2

X

k =1 (u

0 u

k +u

0 u

k )

2

) u

0 g(ju

0 j

2

) 1

C

C

C

C

A :

Dierentiatingequation(7)withrespettox

j

forj=1;2,weobtain

2i

(u

j )

t

(u

j )

t

+A(u

0 )

u

j

u

j

+ 2

X

k =1 B(u

0

;u

k )

T

k j u

3

T

k j u

3

+C(u

0

;u

j )

(e q(ju

0 j

2

)

u

3 )

(e q(ju

0 j

2

)

u

3 )

!

"

2

j

1

u

5

j

1

u

5

+

F(U

;u

j )

F(U

;u

j )

=0

(6)

B(v;w)= v

4 4

(v) (2jvj

2

w+2v 2

w) v

2

(v) w

v

4 4

(v) (2jvj

2

w+2v 2

w) v

2

(v) w

v

4 4

(v) (2jvj

2

w+2v 2

w)+ v

2

(v) w

v

4 4

(v) (2jvj

2

w+2v 2

w)+ v

2

(v) w

!

C(v;w)= (

1

2 2

(v)

2+jvj 2

2 4

(v)

)(vw+vw) vw

2

(v) +

v 2

4 4

(v)

(vw+vw)

vw

2

(v) v

2

2 4

(v)

(vw+vw) ( 1

2 2

(v) +

2+jvj 2

2 4

(v)

)(vw+vw)

!

F(U

;w)=

u

0 w+u

0 w

4

(u

0 )

w

2

(u

0 )

N

X

k =1 ju

k j

2

!

+u

0 g

0

(ju

0 j

2

)(u

0 w+u

0 w)

+

w

4 4

(u

0 )

(u

0 w+u

0 w)

2 6

(u

0 )

N

X

k =1 (u

0 u

k +u

0 u

k )

2

+

!

+ u

0

4 4

(u

0 )

2(u

0 w+u

0 w)(

N

X

k =1 ju

k j

2

)+2u

0 w(

N

X

k =1 u

2

k )

+2u

0 w(

N

X

k =1 u

2

k )

!

+wg(ju

0 j

2

)

andfori;j=1;2,T

i;j

isthefollowingoperator oforder0

T

i;j v=

i

j

1

(e q(ju0j

2

)

v):

Furthermore,thisoperatorisontinuousfromL 2

toL 2

withanormindependentof". Weanrewritethese

equationsasfollows

2i

(u

j )

t

(u

j )

t

+A(u

0 )

u

j

u

j

"

2

j

1

u

5

j

1

u

5

+F

j (U

;u

3

;Tu

3

)=0 (9)

where

F

j (U

;u

3

;Tu

3 )=

F(U

;u

j )

F(U

;u

j )

+C(u

0

;u

j )

(e q(ju

0 j

2

)

u

3 )

(e q(ju

0 j

2

)

u

3 )

!

+ 2

X

k =1 B(u

0

;u

k )

T

k j u

3

T

k j u

3

:

Inordertoderiveanequationforu

3

,wedierentiateequation(7) twiewithrespetto x

j

toobtain

2i

( 2

j u

0 )

t

( 2

j u

0 )

t

+A(u

0 )

2

j u

0

2

j u

0

+ 2

X

k =1 B(u

0

;u

k )

k

2

j u

0

k

2

j u

0

+2C(u

0

;u

j )

j u

0

j u

0

+

j C(u

0

;u

j )

u

0

u

0

"

2

( 2

j u

0 )

tt

( 2

j u

0 )

tt

+ 2

X

k =1

j B(u

0

;u

k )

k

j u

0

k

j u

0

+

j F(U

;u

j )

j F(U

;u

j )

=0:

(10)

(7)

Summing thetwoequationsandmultiplyingbyA (u

0

),weobtain

2iA 1

(u

0 )

(u

0 )

t

(u

0 )

t

+

u

0

u

0

+ 2

X

k =1 A

1

(u

0 )B(u

0

;u

k )

k u

0

k u

0

+2 2

X

j=1 A

1

(u

0 )C(u

0

;u

j )

j u

0

j u

0

+ 2

X

j=1 A

1

(u

0 )

j C(u

0

;u

j )

u

0

u

0

"

2

A 1

(u

0 )

(u

0 )

tt

(u

0 )

tt

+ 2

X

j=1 2

X

k =1 A

1

(u

0 )

j B(u

0

;u

k )

k

j u

0

k

j u

0

+ 2

X

j=1 A

1

(u

0 )

j F(U

;u

j )

j F(U

;u

j )

=0:

(11)

whihanberewritten,afterthemultipliationbye q(ju0j

2

)

,in thefollowingway

2i

(u

3 )

t

(u

3 )

t

+A(u

0 )

u

3

u

3

+ 2

X

k =1 E(u

0

;u

k )

k u

3

k u

3

"

2

u

5

u

5

+I(U;Tu

3

; 1

u

4 )=0

(12)

where

E(u

0

;u

k )=B(u

0

;u

k

)+2C(u

0

;u

k )

2A(u

0 )q

0

(ju

0 j

2

)

u

0 u

k +u

0 u

k

0

0 u

0 u

k +u

0 u

k

andI isamatrixdependingonU,T

ij u

3

fori;j =1;2and 1

u

4

inthefollowingway

I(U;Tu

3

; 1

u

4 )=

2

X

m=1 e

q(ju0j 2

)

m F(U

;u

j )

m F(U

;u

j )

2iq 0

(ju

0 j

2

)

u

0 (

1

u

4 )u

3 +u

0 (

1

u

4 )u

3

u

0 (

1

u

4 )u

3 +u

0 (

1

u

4 )u

3

2

X

k =1 (2C(u

0

;u

k

)+B(u

0

;u

k ))q

0

(ju

0 j

2

)

(u

0 u

k +u

0 u

k )u

3

(u

0 u

k +u

0 u

k )u

3

+ 2

X

k =1

k C(u

0

;u

k )

u

3

u

3

+ 2

X

j=1 2

X

k =1 e

q(ju

0 j

2

)

k B(u

0

;u

k )

T

k j u

3

T

k j u

3

+A(u

0 )

0

B

B

B

B

(q

0

2

(ju

0 j

2

) q 00

(ju

0 j

2

))(

2

X

k =1 (u

0 u

k +u

0 u

k )

2

)u

3

(q 0

2

(ju

0 j

2

) q 00

(ju

0 j

2

))(

2

X

k =1 (u

0 u

k +u

0 u

k )

2

)u

3 1

C

C

C

C

A

A(u

0 )

0

B

B

B

B

q

0

(ju

0 j

2

)(2 2

X

k =1 ju

k j

2

+u

0 (

2

X

k =1 T

k k u

3 )u

3

q 0

(ju

0 j

2

)(2 2

X

k =1 ju

k j

2

+u

0 (

2

X

k =1 T

k k u

3 ))u

3 1

C

C

C

C

A A(u

0 )

0

B

B

B

B

q

0

(ju

0 j

2

)u

0 (

2

X

k =1 T

k k u

3 )u

3

q 0

(ju

0 j

2

)u

0 (

2

X

k =1 T

k k u

3 )u

3 1

C

C

C

C

A

We have to notie here that the matrix I depends on 1

u

4

, whih is due to the gauge transformu

3

=

e q(ju0j

2

)

u

0

. Indeed,inequation(11),wehavetoreplaeu

0 bye

q(ju0j 2

)

u

3

intheterm(u

0 )

t

whih gives

theontributione q(ju0j

2

)

(q(ju

0 j

2

))

t u

3

. Then,reallingthat(u

0 )

t

= 1

u

4

,weinlude thislastterminI.

(8)

"

2

A 1

(u

0 )

(u

4 )

tt

(u

4 )

tt

"

2

t A

1

(u

0 )

u

5

u

5

+2iA 1

(u

0 )

(u

4 )

t

(u

4 )

t

+2i

t A

1

(u

0 )

u

4

u

4

+

u

4

u

4

+ 2

X

k =1 A

1

(u

0 )B(u

0

;u

k )

k u

4

k u

4

+ 2

X

k =1

t A

1

(u

0 )B(u

0

;u

k )

k (e

q(ju0j 2

)

u

3 )

k (e

q(ju0j 2

)

u

3 )

!

+2 2

X

k =1 A

1

(u

0 )C(u

0

;u

k )

k u

4

k u

4

+2 2

X

j=1

t A

1

(u

0 )C(u

0

;u

j )

j (e

q(ju0j 2

)

u

3 )

j (e

q(ju

0 j

2

)

u

3 )

!

+ 2

X

j=1 A

1

(u

0 )

j C(u

0

;u

j )

u

4

u

4

+ 2

X

j=1

t A

1

(u

0 )

j C(u

0

;u

j )

e q(ju0j

2

)

u

3

e q(ju

0 j

2

)

u

3

!

+ 2

X

j=1 2

X

k =1 A

1

(u

0 )

j B(u

0

;u

k )

k

j

1

u

4

k

j

1

u

4

+ 2

X

j=1 2

X

k =1

t A

1

(u

0 )

j B(u

0

;u

k )

T

k j u

3

T

k j u

3

+ 2

X

j=1

t

A 1

(u

0 )

j F(U

;u

j )

j F(U

;u

j )

=0:

(13)

Keepingin viewthat wehave

t u

0

= 1

u

4

,weanrewritethelastequationin thefollowingway

"

2

A 1

(u

0 )

(u

4 )

tt

(u

4 )

tt

+2iA 1

(u

0 )

(u

4 )

t

(u

4 )

t

+

u

4

u

4

+ 2

X

k =1 D

k (u

0

;u

k )

k u

4

k u

4

+G(DU;Tu

3 )

u

4

u

4

+ 2

X

j=0 2

X

k =0 H

jk

(DU;Tu

3 )

j

k

1

u

4

j

k

1

u

4

=0

(14)

where fork=1;2

D

k (u

0

;u

k )=A

1

(u

0 )(B(u

0

;u

k

)+2C(u

0

;u

k ));

andforj;k=1;2,H

jk

isamatrixdependingonU anditsrstderivativesandonT

mn u

3

(m;n=1;2). We

havetonotieherethatinequation(14),allthetermsofequation(13)whihontains

t u

0

anditsrstand

seond spaederivativesareinludedinthelastterms

2

X

j=0 2

X

k =0 H

jk

(DU;Tu

3 )

j

k

1

u

4

j

k

1

u

4

:

Indeed,thissuminludethevaluesj=0andk=0. Reallingthatwehaveadoptedthenotation

0 u=u,

thenitispossibleto writeforj=0;::;2andk=0;::;2

t

j

k u

0

=

j

k

1

u

4 :

Finally, in orderto obtainanequationsatisedbyu

5

,wedierentiatetwie withrespetto tequation

(9)

"

2

A 1

(u

0 )

(u

5 )

tt

(u

5 )

tt

2"

2

t A

1

(u

0 )

(u

5 )

t

(u

5 )

t

"

2

2

t A

1

(u

0 )

u

5

u

5

+2iA 1

(u

0 )

(u

5 )

t

(u

5 )

t

+4i

t A

1

(u

0 )

u

5

u

5

+2i 2

t A

1

(u

0 )

u

4

u

4

+

u

5

u

5

+ 2

X

k =1 A

1

(u

0 )B(u

0

;u

k )

k u

5

k u

5

+2 2

X

k =1

t A

1

(u

0 )B(u

0

;u

k )

k u

4

k u

4

+ 2

X

k =1

2

t A

1

(u

0 )B(u

0

;u

k )

k (e

q(ju0j 2

)

u

3 )

k (e

q(ju0j 2

)

u

3 )

!

+2 2

X

j=1 A

1

(u

0 )C(u

0

;u

j )

j u

5

j u

5

+4 2

X

j=1

t A

1

(u

0 )C(u

0

;u

j )

j u

4

j u

4

+2 2

X

j=1

2

t A

1

(u

0 )C(u

0

;u

j )

j (e

q(ju0j 2

)

u

3 )

j (e

q(ju0j 2

)

u

3 )

!

+ 2

X

j=1 A

1

(u

0 )

j C(u

0

;u

j )

u

5

u

5

+2 2

X

j=1

t A

1

(u

0 )

j C(u

0

;u

j )

u

4

u

4

+ 2

X

j=1

2

t A

1

(u

0 )

j C(u

0

;u

j )

(e q(ju0j

2

)

u

3 )

(e q(ju0j

2

)

u

3 )

!

+ 2

X

j=1 2

X

k =1 A

1

(u

0 )

j B(u

0

;u

k )

k

j

1

u

5

k

j

1

u

5

+2 2

X

j=1 2

X

k =1

t A

1

(u

0 )

j B(u

0

;u

k )

k

j

1

u

4

k

j

1

u

4

+ 2

X

j=1 2

X

k =1

2

t A

1

(u

0 )

j B(u

0

;u

k )

T

k j u

3

T

k j u

3

+ 2

t 0

2

X

j=1 A

1

(u

0 )

j F(U

;u

j )

j F(U

;u

j )

1

A

=0:

Thisequationtakestheform

"

2

A 1

(u

0 )

(u

5 )

tt

(u

5 )

tt

+C(u

0

; 1

u

4 )

(u

5 )

t

(u

5 )

t

+

u

5

u

5

+ 2

X

k =1 D

k (u

0

;u

k )

k u

5

k u

5

+I(DU;Tu

3 )

u

5

u

5

+ 2

X

j=0 2

X

k =0 J

jk

(DU;Tu

3 )

j

k

1

u

5

j

k

1

u

5

+ 2

X

j=0 2

X

k =0 K

jk

(DU;Tu

3

;D 2

1

u

4 )

j

k

1

u

4

j

k

1

u

4

=0:

(15)

where

C(u

0

; 1

u

4

)=2iA 1

(u

0 ) 2"

2

t A

1

(u

0 )

:

Forj;k=0;::;2,thematriesJ

jk

dependonj,k,onU anditsrstderivativesandonT

mn u

3

form;m=1;2.

Forj;k=0;::;2,thematriesK

jk

dependonj, k,onU anditsrstderivatives,onT

mn u

3

form;m=1;2

andon 1

u

4

anditsrstandseond derivatives. Inorder tosimplifythenotations,wenowintroduefor

j;k=0;::;2theoperatorR

jk

R

jk

=

j

k

1

:

Oneagain,wehaveinludedthetermswhihontain

t u

0

anditsrstandseondspaederivativesin

2

X

j=0 2

X

k =0 K

jk

(DU;Tu

3

;D 2

1

u

4 )

j

k

1

u

4

j

k

1

u

4

(10)

andthetermswhihontain

t u

0

anditsrstandseondspaederivativesin

2

X

j=0 2

X

k =0 J

jk

(DU;Tu

3 )

j

k

1

u

5

j

k

1

u

5

:

Inonlusion,wehavetransformedequation(7)intothefollowingsystem

2i

t u

0

t u

0

+A(u

0 )

u

0

u

0

"

2

1

u

5

1

u

5

+F

0 (U

)=0 (16)

forj=1;2,

2i

(u

j )

t

(u

j )

t

+A(u

0 )

u

j

u

j

"

2

j

1

u

5

j

1

u

5

+F

j (U

;u

3

;Tu

3

)=0 (17)

2i

(u

3 )

t

(u

3 )

t

+A(u

0 )

u

3

u

3

+ 2

X

k =1 E(u

0

;u

k )

k u

3

k u

3

"

2

u

5

u

5

+I(U;Tu

3

; 1

u

4 )=0

(18)

"

2

A 1

(u

0 )

(u

4 )

tt

(u

4 )

tt

+2iA 1

(u

0 )

(u

4 )

t

(u

4 )

t

+

u

4

u

4

+ 2

X

k =1 D

k (u

0

;u

k )

k u

4

k u

4

+ 2

X

j=0 2

X

k =0 H

jk

(DU;Tu

3 )

R

jk u

4

R

jk u

4

+G(DU;Tu

3 )

u

4

u

4

=0

(19)

"

2

A 1

(u

0 )

(u

5 )

tt

(u

5 )

tt

+C(u

0

; 1

u

4 )

(u

5 )

t

(u

5 )

t

+

u

5

u

5

+ 2

X

k =1 D

k (u

0

;u

k )

k u

5

k u

5

+ 2

X

j=0 2

X

k =0 J

jk

(DU;Tu

3 )

R

jk u

5

R

jk u

5

+ 2

X

j=0 2

X

k =0 K

jk

(DU;Tu

3

;D 2

1

u

4 )

R

jk u

4

R

jk u

4

+I(DU;Tu

3 )

u

5

u

5

=0:

(20)

3 Proof of Theorem 1:1

Inthis setion,wegiveaproofofTheorem 1:1. Westudythesystem(16),(17), (18), (19)and(20)inthe

followingfuntion spae

X

T

= 8

>

<

>

:

U =(u

j )

5

j=0 :u

j

2C([0;T℄;L 2

(R 2

))\L 1

(0;T;H 4

(R 2

));

jjUjj

XT

= 5

X

j=0 sup

0tT jju

j (t)jj

H 4

(R 2

)

<1

9

>

=

>

;

ForM =(m

j )

5

j=0 2R

6

+

andr2R

+

,wedenote

X

T

(M;r)=

U =(u

j )

5

j=0 2X

T

:8j=0;::;5 jju

j jj

L 1

(0;T;H 4

(R 2

))

m

j

jj(u

0 )

t jj

L 1

(0;T;H 2

(R 2

))

r andu

0

(;0)=u

"

0 ()

(11)

andweletV =(v

j )

j=0 2X

T

(M;r). WealsosetV =(v

j )

j=0

. Considerthelinearizedversionofequations

(16),(17),(18),(19)and(20)

2i

t u

0

t u

0

+A(v

0 )

u

0

u

0

"

2

1

v

5

1

v

5

+F

0 (V

)=0 (21)

forj=1;2,

2i

(u

j )

t

(u

j )

t

+A(v

0 )

u

j

u

j

"

2

j

1

v

5

j

1

v

5

+F

j (V

;v

3

;Tv

3

)=0 (22)

2i

(u

3 )

t

(u

3 )

t

+A(v

0 )

u

3

u

3

+ 2

X

k =1 E(v

0

;v

k )

k u

3

k u

3

"

2

v

5

v

5

+I(V;Tv

3

; 1

v

4 )=0

(23)

"

2

A 1

(v

0 )

(u

4 )

tt

(u

4 )

tt

+2iA 1

(v

0 )

(u

4 )

t

(u

4 )

t

+

u

4

u

4

+ 2

X

k =1 D

k (v

0

;v

k )

k u

4

k u

4

+ 2

X

j=0 2

X

k =0 H

jk

(DV;Tv

3 )

R

jk u

4

R

jk u

4

+G(DV;Tv

3 )

u

4

u

4

=0

(24)

"

2

A 1

(v

0 )

(u

5 )

tt

(u

5 )

tt

+C(v

0

; 1

v

4 )

(u

5 )

t

(u

5 )

t

+

u

5

u

5

+ 2

X

k =1 D

k (v

0

;v

k )

k u

5

k u

5

+ 2

X

j=0 2

X

k =0 J

jk

(DV;Tv

3 )

R

jk u

5

R

jk u

5

+ 2

X

j=0 2

X

k =0 K

jk

(DV;Tv

3

;D 2

1

v

4 )

R

jk u

4

R

jk u

4

+I(DV;Tv

3 )

u

5

u

5

=0:

(25)

LetZ =

L 1

(0;T;H 4

(R 2

))\C([0;T℄;L 2

(R 2

))

6

. Then,it isnotdiÆult tosee that thelinearinhomoge-

neousCauhyproblem(21),(22),(23),(24)and(25)withinitialondition

u

0

(;0)=u

"

0 (); u

1

(;0)=

1 u

"

0 (); u

2

(;0)=

2 u

"

0 ();

u

3

(;0)=e q(ju

"

0 j

2

)

u

"

0 (); u

4

(;0)=u

"

1 (); u

5

(;0)=

t u

4 (;0);

t u

4 (;0)=

1

"

2

u

"

0 +2iu

"

1 (

1

p

1+ju

"

0 j

2 1)u

"

0

1

p

1+ju

"

0 j

2 (

p

1+ju

"

0 j

2

)u

"

0

;

t u

5 (;0)=

1

"

2

u

"

1 (

1

p

1+ju

"

0 j

2 1)u

"

1 +

u

"

0 u

"

1 +u

"

0 u

"

1

2(1+ju

"

0 j

2

) 3

2 u

"

0

1

p

1+ju

"

0 j

2 (

p

1+ju

"

0 j

2

)u

"

1 +

u

"

0 u

"

1 +u

"

0 u

"

1

2(1+ju

"

0 j

2

) 3

2 (

p

1+ju

"

0 j

2

)u

"

0

+ u

"

0

p

1+ju

"

0 j

2

u

"

0 u

"

1 +u

"

0 u

"

1

2(1+ju

"

0 j

2

) 3

2

+ 2i

"

2 (u

"

0 +2iu

"

1

( 1

p

1+ju

"

0 j

2 1)u

"

0 1

p

1+ju

"

0 j

2 (

p

1+ju

"

0 j

2

)u

"

0

:

(26)

denesamappingS

S : Z !Z

V 7 !U:

(12)

wehavetondatime T andonstantsM andrsuh thatS mapsthelosedballX

T

(M;r)intoitselfand

isaontratingmappinginthenorm

sup

t2[0;T℄

5

X

j=0 jjv

j jj

2 :

Therstthingto doisto estimateu

4

byusingequation (24). Aswehavealreadysaidin theintrodution

of thisartile,wewill treat(24)asawaveequation. Multiplying equation(24)byA(v

0

)andapplyingthe

operator(1 ),weobtaindenoting

4

=(1 )u

4

"

2

(

4 )

tt

(

4 )

tt

+2i

(

4 )

t

(

4 )

t

+A(v

0 )

4

4

+ 2

X

k =1 L(v

0

;v

k

;

k v

0 )

k

4

k

4

X

jj2 M

(D

h1 1

V;D h2 1

Tv

3 )

1

1

2

2 u

4

1

1

2

2 u

4

+ X

jj2 2

X

j;k =0 N

(D

h

2

V;D h

2 1

Tv

3 )

1

1

2

2 R

jk u

4

1

1

2

2 R

jk u

4

=0:

(27)

where thematriesM

dependonthemulti-index2N 2

,N

on,j andk

L(v

0

;v

k

;

k v

0

)=A(v

0 )D

k (v

0

;v

k )+2

k A(v

0 );

h

1

=3 ifjj=1;2; h

1

=4 ifjj=0;

h

2

=3 jj:

Finally,dierentiating(27)withrespettox

m

form=1;2anddenoting m

4

=

m

4

,weobtain

"

2

( m

4 )

tt

( m

4 )

tt

+2i

( m

4 )

t

( m

4 )

t

+A(v

0 )

m

4

m

4

+ 2

X

k =1 L(v

0

;v

k

;

k v

0 )

k

m

4

k

m

4

+ 2

X

k =1

m A(v

0 )

k

k

4

k

k

4

!

+ X

jj3

M

(D

h1

V;D h2

Tv

3 )

1

1

2

2 u

4

1

1

2

2 u

4

+ X

jj3 2

X

j;k =0

N

(D

h

2 +1

V;D h

2

Tv

3 )

1

1

2

2 R

jk u

4

1

1

2

2 R

jk u

4

=0:

(28)

where

h

1

=2; h

2

=0 ifjj=3:

Thematries

M

and

N

depend onforjj3. TheyalsodependonrespetivelyD h

1

V, D h

2

Tv

3 and

D h2+1

V,D h2

Tv

3

. Wehavetonotiethat in(27),thedierentiationof

A(v

0 )

4

4

withrespetto x

m

foraxedmgivestheterm

m (A(v

0 ))

4

4

whihisrewrittenin(28)as

2

X

k =1

m A(v

0 )

k

k

4

k

k

4

!

:

(13)

WealsoreallheretheexpressionofA (v

0 )

A 1

(v

0 )=

1

2

2+jv

0 j

2

v 2

0

v 2

0

(2+jv

0 j

2

)

:

Wemultiplyequation(28)byA 1

(v

0

)andweonsidertherstlineoftheresultingequation. Furthermore,

weaddtobothsideoftheequationtheterm m

4

. Weannowperformtheusualenergyestimateforwave

equations,namelywemultiplytheequationby( m

4 )

t

andweintegrateoverR 2

.

"

2 Z

R 2

(2+jv

0 j

2

)

2 (

m

4 )

tt (

m

4 )

t dx "

2 Z

R 2

v 2

0

2 (

m

4 )

tt (

m

4 )

t dx

i Z

R 2

(2+jv

0 j

2

)j(

m

4 )

t j

2

dx+i Z

R 2

v 2

0 (

m

4 )

t (

m

4 )

t dx

Z

R 2

m

4 (

m

4 )

t dx

2

X

k =1 Z

R 2

O 11

(v

0

;v

k

;

k v

0 )

k

m

4 (

m

4 )

t dx

2

X

k =1 Z

R 2

O 12

(v

0

;v

k

;

k v

0 )

k

m

4 (

m

4 )

t dx+

Z

R 2

m

4 (

m

4 )

t dx

2

X

k =1 Z

R 2

A 1

(v

0 )

m A(v

0 )

11

k

k

4 (

m

4 )

t dx

Z

R 2

m

4 (

m

4 )

t dx

2

X

k =1 Z

R 2

A 1

(v

0 )

m A(v

0 )

12

k

k

4 (

m

4 )

t dx

X

jj3 Z

R 2

P 11

(D

h1

V;D h2

Tv

3 )(

1

1

2

2 u

4 )(

m

4 )

t dx

X

jj3 Z

R 2

P 12

(D

h1

V;D h2

Tv

3 )(

1

1

2

2 u

4 )(

m

4 )

t dx

X

jj3 2

X

j;k =0 Z

R 2

Q 11

(D

h2+1

V;D h2

Tv

3 )

1

1

2

2 R

jk u

4 (

m

4 )

t dx

X

jj3 2

X

j;k =0 Z

R 2

Q 12

(D

h

2 +1

V;D h

2

Tv

3 )

1

1

2

2 R

jk u

4 (

m

4 )

t dx=0:

(29)

where O=A 1

(v

0 )L, P

=A 1

(v

0 )

M

and Q

=A 1

(v

0 )

N

.

Wenowtaketherealpartofequation(29). Therstlineoftheresultingexpressiongives

Re

"

2 Z

R 2

(2+jv

0 j

2

)

2 (

m

4 )

tt (

m

4 )

t dx "

2 Z

R 2

v 2

0

2 (

m

4 )

tt (

m

4 )

t dx

="

2 Z

R 2

(2+jv

0 j

2

)

4 j(

m

4 )

t j

2

t dx "

2 Z

R 2

v 2

0

8 d

dt (

m

4 )

2

t +

v 2

0

8 d

dt (

m

4 )

2

t

dx

= d

dt

"

2 Z

R 2

(2+jv

0 j

2

)

4 j(

m

4 )

t j

2

dx "

2 Z

R 2

v 2

0

8 (

m

4 )

2

t +

v 2

0

8 (

m

4 )

2

t

dx

"

2 Z

R 2

(2+jv

0 j

2

)

t

4

j(

m

4 )

t j

2

dx+"

2 Z

R 2

(v 2

0 )

t

8 (

m

4 )

2

t +

(v 2

0 )

t

8 (

m

4 )

2

t

dx:

Fortheseondline,sineV 2X

T

(M;r)wehave,usingtheontinuousembeddingofH 2

(R 2

)intoL 1

(R 2

)

i

Z

R 2

(2+jv

0 j

2

)(

m

4 )

t (

m

4 )

t dx+i

Z

R 2

v 2

0 (

m

4 )

t (

m

4 )

t dx

C(M) Z

R 2

j(

m

4 )

t j

2

dx:

(14)

Re Z

R 2

m

4 (

m

4 )

t dx

= 1

2 d

dt Z

R 2

jr m

4 j

2

dx:

UsingagainthefatthatV 2X

T

(M;r)andthatthematrixO

m

dependsonlyonV and

k v

0

,weobtain

byCauhy-Shwarzinequality

2

X

k =1 Z

R 2

O 11

(v

0

;v

k

;

k v

0 )

k

m

4 (

m

4 )

t dx+

Z

R 2

O 12

(v

0

;v

k

;

k v

0 )

k

m

4 (

m

4 )

t dx

C(M) 2

X

k =1 jj

k

m

4 jj

2 jj(

m

4 )

t jj

2

C(M) 2

X

k =1 jj

k

m

4 jj

2

2 +jj(

m

4 )

t jj

2

2

!

:

Inthesameway,wehave

2

X

k =1 Z

R 2

A 1

(v

0 )

m A(v

0 )

11

k

k

4 (

m

4 )

t dx

C(M) 2

X

k =1 jj

k

k

4 jj

2

2 +jj(

m

4 )

t jj

2

2

!

;

2

X

k =1 Z

R 2

A 1

(v

0 )

m A(v

0 )

12

k

k

4 (

m

4 )

t dx

C(M) 2

X

k =1 jj

k

k

4 jj

2

2 +jj(

m

4 )

t jj

2

2

!

:

Wenowtreatthetwotermsontaining m

4 (

m

4 )

t

bytwodierentmethods. Ononehand,oneanwrite

Re Z

R 2

m

4 (

m

4 )

t dx

= 1

2 d

dt Z

R 2

j m

4 j

2

dx:

Ontheotherhand,wederivefromCauhy-Shwarzinequality

Z

R 2

m

4 (

m

4 )

t dx

C(M) jj m

4 jj

2

2 +jj(

m

4 )

t jj

2

2

:

Forthenextterm,wehaveto bemoreareful. If jj=3,P

depends onderivativesof V oforder less

thanorequalto2andonTv

3

anditsrstorderderivatives. Thus,itanbeestimatedinL 1

(R 2

). Thetwo

other termsareestimated byCauhy-Shwarzinequality. Asaonsequene,one anndapositiveC(M)

suhthat

X

jj=3 Z

R 2

P 11

(D

h

1

V;D h

2

Tv

3 )(

1

1

2

2 u

4 )(

m

4 )

t dx

C(M) X

jj=3 jj

1

1

2

2 u

4 jj

2 jj(

m

4 )

t jj

2

C(M) 2

X

k =1 jj

k

4 jj

2

2 +jj(

m

4 )

t jj

2

2

!

:

Ifjj=0;::;2, P

depends onderivativesof V of order lessorequalto 4and onderivativesof Tv

3 of

order lessor equal to 3. Thus it has to be estimated in L 2

(R 2

) together with ( m

4 )

t

by Cauhy-Shwarz

(15)

inequalitywhereas

1

2 u

4

isestimated inL 1

(R 2

).

2

X

jj=0 Z

R 2

P 11

(D

h

1

V;D h

2

Tv

3 )(

1

1

2

2 u

4 )(

m

4 )

t dx

C(M) 2

X

jj=0 jj

1

1

2

2 u

4 jj

L 1

(R 2

) jj(

m

4 )

t jj

2

C(M) jj

4 jj

2

2 +

2

X

k =1 (jj

k

k

4 jj

2

2 +jj

k

4 jj

2

2 )+jj(

m

4 )

t jj

2

2

!

:

Indenitive,wehaveprovedthat

X

jj3 Z

R 2

P 11

(D

h1

V;D h2

Tv

3 )(

1

1

2

2 u

4 )(

m

4 )

t dx

C(M) jj

4 jj

2

2 +

2

X

k =1 (jj

k

4 jj

2

2 +jj

k

k

4 jj

2

2 )+jj(

m

4 )

t jj

2

2

!

:

Inthesameway,oneanalsoobtain

X

jj3 Z

R 2

P 12

(D

h1

V;D h2

Tv

3 )(

1

1

2

2 u

4 )(

m

4 )

t dx

C(M) jj

4 jj

2

2 +

2

X

k =1 (jj

k

4 jj

2

2 +jj

k

k

4 jj

2

2 )+jj(

m

4 )

t jj

2

2

!

:

Thelasttwotermsaretreatedexatlyin thesameway,theonlythingtonotieisthattheoperatorR

jk

isontinuousfromL 2

(R 2

)to L 2

(R 2

). Thusweanwrite

X

jj3 2

X

j;k =0 Z

R 2

Q 11

(D

h

2 +1

V;D h

2

Tv

3 )

1

1

2

2 R

jk u

4 (

m

4 )

t dx

jj

4 jj

2

2 +

2

X

k =1 (jj

k

4 jj

2

2 +jj

k

k

4 jj

2

2 )+jj(

m

4 )

t jj

2

2

!

;

X

jj3 2

X

j;k =0 Z

R 2

Q 12

(D

h2+1

V;D h2

Tv

3 )

1

1

2

2 R

jk u

4 (

m

4 )

t dx

jj

4 jj

2

2 +

2

X

k =1 (jj

k

4 jj

2

2 +jj

k

k

4 jj

2

2 )+jj(

m

4 )

t jj

2

2

!

:

(16)

d

dt

"

2 Z

R 2

(2+jv

0 j

2

)

4 j(

m

4 )

t j

2

dx "

2 Z

R 2

v 2

0

8 (

m

4 )

2

t +

v 2

0

8 (

m

4 )

2

t

dx+ 1

2 Z

R 2

jr m

4 j

2

dx

+ 1

2 d

dt Z

R 2

j m

4 j

2

dx

"

2 Z

R 2

jv

0 j

2

t

4 j(

m

4 )

t j

2

dx+"

2 Z

R 2

(v 2

0 )

t

8 (

m

4 )

2

t +

(v 2

0 )

t

8 (

m

4 )

2

t

dx

C(M) jj

4 jj

2

2 +

2

X

k =1 jj

k

k

4 jj

2

2 +jj

k

4 jj

2

2

+jj(

m

4 )

t jj

2

2

!

:

(30)

Sine

4

=(1 )u

4

,itisnotdiÆultto derivefrom(24)

jj

4 jj

2

2

C(M) 2

X

k =1 jj

k

k

4 jj

2

2 +jj

k

4 jj

2

2

+jj(

m

4 )

t jj

2

2

!

:

Then,integratinginequality(30)from 0tot,oneobtains

"

2

2 Z

R 2

j(

m

4 (t))

t j

2

dx+ 1

2 Z

R 2

jr m

4 (t)j

2

dx+

"

2

2 Z

R 2

jv

0 (t)j

2

2 j(

m

4 (t))

t j

2

dx

+ 1

2 Z

R 2

j m

4 j

2

dx

"

2

2 Z

R 2

v 2

0 (t)

4 (

m

4 (t))

2

t v

2

0 (t)

4 (

m

4 (t))

2

t

dx

"

2 Z

R 2

(2+jv

0 (0)j

2

)

4

j(

m

4 )

t (0)j

2

dx "

2 Z

R 2

v 2

0 (0)

8 (

m

4 )

2

t (0)+

v 2

0 (0)

8 (

m

4 )

2

t (0)

dx

+ 1

2 Z

R 2

jr m

4 (0)j

2

dx+ 1

2 Z

R 2

j m

4 (0)j

2

dx+"

2 Z

t

0 Z

R 2

(jv

0 (s)j

2

)

s

4 j(

m

4 (s))

s j

2

dxds

+"

2 Z

t

0 Z

R 2

j(v 2

0 (s))

s j

8 j(

m

4 (s))

2

s j+

j(v 2

0 (s))

s j

8 j(

m

4 (s))

2

s j

dxds

+C(M) Z

t

0 2

X

k =1 jj

k

k

4 (s)jj

2

2 +jj

k

4 (s)jj

2

2

+jj(

m

4 )

s (s)jj

2

2

!

ds:

(31)

Usingthefatforalls2[0;t℄, jv0(s)j

2

2 j(

4 (s))

s j

2 v

2

0 (s)

4 (

m

4 (s))

2

s v

2

0 (s)

4 (

m

4 (s))

2

s

0,thatjj(u

0 )

t jj

L 1

(R 2

)

r, anddenoting

f m

(0)= 1

2 Z

R 2

jr

"

m

4 (0)j

2

dx+ 1

2 Z

R 2

j m

4 (0)j

2

dx

+"

2 Z

R 2

(2+jv

0 (0)j

2

)

4

j(

m

4 )

t (0)j

2

dx

"

2 Z

R 2

v 2

0 (0)

8 (

m

4 )

2

t (0)+

v 2

0 (0)

8 (

m

4 )

2

t (0)

dx;

(17)

2

X

m=1

"

2

2 Z

R 2

j(

m

4 (t))

t j

2

dx+ 1

2 Z

R 2

jr m

4 (t)j

2

dx+ 1

2 Z

R 2

j m

4 (t)j

2

dx

2

X

m=1 f

m

(0)+C(M;r) 2

X

m=1 Z

t

0 2

X

k =1 jj

k

m

4 (s)jj

2

2 +jj

k

4 (s)jj

2

2

+jj(

m

4 (s))

s jj

2

2

!

ds:

(32)

Thisimplies,

2

X

m=1

"

2

2 Z

R 2

j(

m

4 (t))

t j

2

dx+ 2

X

k =1 Z

R 2

j

k

m

4 (t)j

2

dx+ Z

R 2

j m

4 (t)j

2

dx

!

2

X

m=1 f

m

(0)+C(M;r) 2

X

m=1 Z

t

0 2

X

k =1 jj

k

m

4 (s)jj

2

2 +jj

k

4 (s)jj

2

2

+jj(

m

4 (s))

s jj

2

2

!

ds:

(33)

ApplyingGronwalllemmato(33), weget

sup

t2[0;T℄

2

X

m=1

"

2

2 Z

R 2

j(

m

4 (t))

t j

2

dx+ 2

X

k =1 Z

R 2

j

k

m

4 (t)j

2

dx+ Z

R 2

j m

4 (t)j

2

dx

!

e C(M;r)

T

"

2 2

X

m=1 f

m

(0):

(34)

Then,hoosingC(M;r)T 1andu

4

(0)suhthat

2

X

m=1 f

m

(0)e T

"

2

m

4

(35)

gives

sup

t2[0;T℄

jju

4 (t)jj

H 4

(R 2

)

m

4

: (36)

It isimportanttonotieherethat(35)impliesthat

jju

4 (;0)jj

H 4

(R 2

) e

T

"

2

m

4 :

Furthermore,wealsohave

sup

t2[0;T℄

"jj

t u

4 (t)jj

H 3

(R 2

)

m

4

: (37)

Wenow studyequation (25)satisedbyu

5

. Thisequation isof thesametypeasequation (24)andit

willbetreatedwiththesamemethod; wemultiply(25)byA(v

0

)andweapplytheoperator

m

(1 )for

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