fields in the equations of magnetohydrodynamics

-

Documents
21 pages
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

Niveau: Supérieur, Doctorat, Bac+8
On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics Lorenzo Brandolese ? and François Vigneron † 13th April 2006 Abstract We study the behavior at infinity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in Rd. We prove that if the initial magnetic field decays su?ciently fast, then the plasma flow behaves as a solution of the free nonstationnary NavierStokes equations when |x| ? +∞, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evo- lution. Our main tools are boundedness criteria for convolution operators in weighted spaces. Keywords: decay at infinity, instantaneous spreading, magnetohydrodynamics, MHD, spatial localisation, weighted spaces, convolution, asymptotic behavior. AMS 2000 Classification: 76W05, 35Q30, 76D05. 1 Introduction The magnetohydrodynamics equations are a well-known model in plasma physics, describing the interactions between a magnetic field and a fluid made of moving electrically charged particles. A common example of an application of this model is the design of tokamaks: the purpose of these machines is to confine a plasma in a region, with a density and a temperature large enough to entertain thermonuclear fusion reactions. This can be achieved, at least during a small time interval, by applying strong magnetic fields.

  • p1 ≥

  • ?1 ≤

  • equations when

  • let u0 ?

  • ?1 ?

  • rate ?1

  • any spatial

  • l2 decay


Sujets

Informations

Publié par
Nombre de lectures 16
Langue English
Signaler un problème


dR
|x|→+∞

∂u S 1 2 +(u∇)u S(B∇)B +∇ p+ |B| = u ∂t 2 Re∂B 1
+(u∇)B (B∇)u = B
∂t R m divu = divB = 0
u(0) =u B(0) =B .0 0
u p
dB R (d 2) R Re m
winggo,vtoernithefast,decaequationsymagneticoffothenonstationnaryplasma,WifWittisconstanpotherotheorlyequationslandovcalizedinitialattotheatbandeginningunknooprefthetheyevrefero-molution.toOursolarmainthetowrittenolseldareesbsolutionoundednessocriteriadecaforecondynamicsvvolution,opberators13thinydrowyeighvtedofspaces.theKeywinords:ThedecaandyngatWinnit[12]yof,ininstanultaneousstudyspreading,ofmagnetohInydromdynamydroibcs,theMHD,aspatialthelowhencalisation,vierStowtheeighastebdthespaces,suciencogneticnifvproolution,inasymptoticmagnetohbofehaevior.respAMSnit2000vClassication:study76W05,200635Q30,ran?ois76D05.Lorenzo1magnetohInintroHereductionareThecmagnetoheldydrouid,dynamicseloequationseldaredeneaandwcalizationell-knoositivwnlomobdelapplyiinstrongplasmaelds.pheysics,todescribingfortheapplicationsinthisteractionsdel,bpartetcwareentheaofmagneticdynamicsethelcorona.dnon-dimensionalandra,uidmagnetohmadedynamicsofcanmoevinginelectricallyfollocwhargedy:particles.magneticAthatcommon,exampleequationsofkanNaapplifreecofaatesiehaonwofplasmathisthenwilltlyareysspeldelyma1thedesignthatofvtokeamaks:.theequationspurpydroosetheofsolutionstheseariable,macspacehinesthisecttowithconneyainplasmaiorinehaatheregion,ewithAbstractaAprildensitVigneronyFandBrandoleseadynamicstempanderatureofltheargeeldsenough(MHD)totheenwnstertainthethermonelouclearifusionyreactions.citThisthecanthebssureeandacmagnetichiveallvded,theatmagneticleastofduring.apsmalletimetsinthetervOnal,moredelectivithesthe2S =M /(R R ) Me m
u B S = R = 1e
R = 1m
B 0
d 3
t → +∞
2L u B
2L
2 d 2f ∈L (R ) L |x|→ +∞ f
Z
2 2(f) = sup ∈R ; lim R |f(Rx)| dx = 0 .
R→+∞ 1 | x| 2
2L =(f) f |x| |x|→ +∞
2L f = O(|x| ) |x| → +∞ (f)
2L |f(x)| C(1+|x|) f = O(|x| )
|x|→ +∞
a da∈ [1,+∞] ∈R L (R )
Z 1/a
a a kfk a = |f(x)| (1+|x|) dx 1a< +∞L
dR
a = +∞

∞kfk = ess sup |f(x)|(1+|x|) .L
dx∈R
a d b dL (R ) L (R )
d d
+ = + .
a b
thater,antheeakasymptoticybofehabvior.oferthwhereeIfsolutionsthefor.vtheirMoreothediscussed.thealsoalsoisloquiteeredwFenitelthelrunderstoregularitoyd:offorbutexample,case,[l13]theproralvidesthethe,optimalAfterdecatyumratesandofrelatingthe[7].isvsolutionsenormsingularofmenselfsimilarwandheardhformeana.largehclasstofio,ws.)OnsolutionstheinothervierStokhand,hnothingreducesseemsthetoehaalidvresultse,boneenlossdonewto(1.2b)studyointthendecavyustoft,solutionsnofregularit(MHD)theorywithtinrespobtainedebctlutions,tonthewritespofacpeonvariable..hand,Inwritethisusdpapofer,whenmotivresults,atedwbartialyproblemrecourse,cenfunctionteresults,obtainedothbtheyothnesssevweralunicitauthorsexist,for(MHtheFNawvierStokthisesJustequationsequations.(see,bratede.g.the,normed[1],the[2],(M[6],case[11]ts.anddications[14]),withwtheeifwwouldand,lik.eytoedescribnoegeneralitinWithwhiccanhandwbaromyalizationtheviewpresenceoofthetheermagneticer;eldReynoldsaectsetheequivspatialmagneticlobcalizationtheofythethev[5]eloeerman'skwandcConstaneld.inDenitions(1.1)andeennotations.eWhaeisstartthebwywillinsotrowducingsettheossiblenotiwhenonofofsidecadiyOnrateotheratwheninniteyinoaHawteakoundssense,videwhicprohwhicgeneralizesythe,usualenotionthatofPpforoinntOfwiseandecameasurableysucratethatinoptheremainsframewaorkdaofsmolocasecallynsquasatisesrsmoeellinastegrableyfunctions.theirAwhensimpledomotivDationtois2.thatortheeakbaglobalfparticularloandcasregularit,yspaceisestheNaminimaleoisneBanacforspacewhicbhcethetosystemHD)(MHD)systemmak,esparticularsense.In1.constanLetinomononexistencesimplthecasewheregeneloincv[9],remaininevenhogiv(1.2a).ifWAllethatdenebtheassumeisshallsolutionswdewcfromayyrofateminoras.larassumeselfsimieard,forwrescalingofer.constructionumofFAthe,casp8].of[theinwdynamicsspacesydroHartmanmagnetohisto,ndedandextemoreoeenbbnhasmyborticitconsid-vasthealenofwhendirectionsthetheertoumwReynoldsocit2y2 La d (+d/a)f ∈ L (R ) a 2 f = O |x| |x| → +∞
a bL L
+d/a> +d/b ab

d a(f) = sup + ; a 2 f ∈La
2 df ∈L (R )
A B
AB ε ,
AB = 0 A<B = 0
A B ε a1/a
a = +∞
+() = max{ ,0}
p d p d0 1(u ,B )∈L (R )L (R ).0 0 ϑ ϑ0 1
2 2 L L(ϑ +d/p ) (ϑ +d/p )0 0 1 1u(t) =O |x| B(t) =O |x| |x|→ +∞.
p p0 d 1 d du ∈ L (R ) B ∈ L (R ) R0 0ϑ ϑ0 1
d 2
( (
ϑ 0 ϑ 00 1
d<p +∞ d<p +∞.0 1

+ε min d+1; 2 , 0 1
+
2d d =ϑ +d/p =ϑ +d/p = 1 p = min{p ; ε }0 0 0 1 1 1 0 0p 1
additional.oifTheoremH?lder.Indeed,.inequalitMainyb(1.5)andWillWtheeuniquethesolutionyofb(MHD)ofpreservbemsucofhwaeconditioneinofsomeefutureandtimebinectortervealum?andDepuseendingusonstrictthethatparameters,(1.6b)thealsoansw6erexpressionswhencan,bwepropfolloositivLeteersistenceorconcernednegativresults.e.impliesInforcasecoftaergence-freenegativinenotedanswAssumeer,ercanrealweeThestilleensurelargeth(1.6a)aassumetnitethebspatialinequalitlofollocalization:offormthewritesolutioneisIfcifonservoimpliestainingthata3,wethenakthensenseW?willInvotherthewwing:ords,1.1wtheeproblemwpouldwithlikareeWto,kno.wthatwhethe(1.4)ran(1.3)lowhenev.ereandw.divItvalsoeldseylik(conditiondecalization).lothatawillConsiderbquestions.nwingafolloparttheositiverpand4.answWtoshallaimwhenw)eelocanbLet1.3alsoandthat1.1the(TheoremforresultsemainustOuryelds.theymeaningcitwingelonotationsv3.thewithwhentheandexpressionsmagneticoftentheshallof,calization.loandspatialareAgain,tthisandconditionandmacony,bifeparameterconservFinallyed,deneorthatinstanmeantaneouslywbreakwhenandewritedown.ed.inthep p10T > 0 (u,B) C([0,T];L L )
2 2L L 0 1u(t) =O |x| B(t) =O |x| |x|→ +∞.
d = 2 T
pe pe0 1(u ,B ) L L0 0 e eϑ ϑ0 1
p p pe pe1 0 10L L L L
0
d+10
20 1
2 d 2 d(u,B)∈C([0,T];L (R )L (R ))
2L (d+1+ε)sup |u(t,x)| =O |x|
t∈[0,T]
2L (d+1+ε)/2sup |B(t,x)| =O |x|
t∈[0,T]
ε > 0 t ∈ [0,T] C(t) 0
u(t) B(t)
Z
j k j k(u u B B )(t,x)dx = C(t), (j,k = 1,...,d)j,k
dR
= 1 j =k = 0j,k j,k
u B0 0
p ddL (R ) p > d ϑ+ = (d+1+ε)/2 > 0ϑ p
(u ,B ) t = 00 0
u
2L 1
ifweNexteone.sucsamealsotheinactuallyonareeaksolutionsatmaximaltheothclass,bsandbagreeelongandableinsomeifetimeslol(1.9)abraksolutionviouslytosolutions(MHD)restrictionsucofhedthatcalizationthew.thenand,Theorem(1.6)vassumptionsesatisfyingtheindices.ondingwithcorrespethebutwithnot,conditiontoOnelongstitiesbinalsoaifeser,solutionsvucMoreoPhlarge.theoremarbitrarilythateThreebAncanthetimebthethe(1.8a)theandof,LetwillaewithwandHere,eresolution.ButIfstable(1.7)fowhenoandmeanssatisesesolutionwThisinitial.thatnoih(MHD)etheforof,izationmofdosolutionothermildtegraluniqueareaNevand5existsseethereofThen(1.9)(1.8b)Insideforcansomehalyscfasterlogeneric.inThen,1.1.foriallsomespatialbthethetocanrelatedwn:sspatialion,eldthereconservexiststheatheconstanytashoptimalitwhicy(1.6b))bintoare1.2ofresult.suchh,thatetheprocomptoonennottswofwforsolutions.oundwbforer,andruppabthev,This(namelythatsatisfywthestartfolloawingellincalizedtegraldatumidenrestrictions.titleastyr:,conditionsucSthatucdoasandholdsystem,sharp,theisofthenosedness(1.8a)ell-pustwethewn.totherelatedhand,conditionsinfewidena(1.9)areobthereunstable.herthless,restricsectionkinds:woshallwthattclasstiexceptionalonssatisfyingrestrictiondotheexist.thatthisiesoneimplexhibitemsucrthatodecathemwinghfollothanThethe(1.9)case.withysicalcondition.terpretationthatTheoremectThisthisreonnforceifmathematicallycusfactsdiscusscaneeobservbinarbitrarilyapplications.Thconclusionsspatiablodraof1.eldyalsoloareassumptilimitationsonthismagneticertwillw,econditioned(1.8b)ywillobIndeed,eotherfullleddecaasratesofocanonlye.large.Weexplecalizationvthecitelosharp.yisisconservbuted,otherwise.thereBysomeTheoremto1.3propbyelo4andθ θ
d+1d+1
2η1
2η −δ1
2η1
2θ1
2θ1
δ
1 1
p p0 2/p 1/d1 0 δ/d 1/d 2/p1
( (
η ≤ (d+1)/2 η ≤ (d+1+δ)/21 1
p ≥ 2d d < p < 2d1 1
(p ,ϑ )0 0
(p ,ϑ )1 1
θ
2η1
+d+1 2dB = 1
p1
u0
1
p p0∞0 1/d u ∈ L ([0,T];L )ϑ0(
η ≥ (d+1+δ)/21
p > d1
(d + 1 +)/2 u1
|x|→ +∞ 0 < 1
2L u 2 11
p 2d = 0 B1 0
p0 1
2L =ϑ +d/p1 1 1
:yrequiringasonethedecasolutiontheofwiceNaratevierStokyingescanequa-.esisitlodisapptheaeld)bthatmagnetic5ininitialratedatumev).ehaen(seeto[14]).edThewingdarkoundgraRoughlybregionsadmissiblecorrespguresondthtoofini-AstialwlydatathefordecawicAbohewbewwillyproallviseonceinthisaddi-protiontothatofbTheelderyallocittoelospvvThelargereld.ivmagneticdecayingatdecaernedastyFe:y.theTheeld.dash-magneticdotteddecalinesSloillustrate(1.6b),themaximalbarriersveusedyinofthethatprobofconservofd?4.3.y2.oFexceedsorregionspgraoWhenorly(logivcalized,magnetichaseldshold,(namelyandwn-LeftrateDoim.vthroughupoftythatlarit(1.7)regu-.thepathologicalonwtlybslighon),fortheearsbo.ehayvioreaking,ofaluesendstowhenedep(forresultsgTheend.thelyewisshoinn-ThegoFig.1vtionsofwimagnetictmeanshthethesame1B |x|0
u B
2 (d+1+)/2 L1
d+10
p0 dL (R )ϑ0
ϑ +d/p d+1 ε .0 0 1/p0
p < +∞0

p p∞ 0 1(u,B)∈L [0,T]; L L .
ϑ ϑ0 1
p0u ∈L (p ,ϑ )0 0 0ϑ0
(p ,ϑ )0 0
p p0 d 1 d du ∈ L (R ) B ∈ L (R ) R0 0ϑ ϑ0 1
d 2 ϑ ,ϑ 0 d<p +∞0 1 0
2 1 1
< + .
p p d1 0
T > 0 d = 2 T = +∞
p p0 1(u,B)∈C([0,T];L L ).
u B = ϑ +d/p = ϑ +d/p0 0 0 0 0 1 1 1

d 2d
min d+1 ε ; 2 ε ; 2 + ,0 1/p 1 2ϑ ϑ 10 1 0 p p0 1
caseifergence-freehoursucatoer,elongsLetbfundamen,rateprorestrictionvidedsucthatelds,stabilitinnitywing:is(ifassertedwiseasofinOne(1.7).yAthemoreThisphtoysicaltheexplanationyfolirhthedynamics.ab,otakvdecaeasconclusionsdenedisofthethat,folloconditionwingin1.1..ThisThedark-grainductionyequationthemeansrthatftheimagneticseldoillinesthearedecatransptortedinbAssumeyithewsovidedwexistwhile,simtheultaneouslyofundergoingorresistivusualeaddition,diusioantio.andThisviortransptheort-diusionhoprotocessandguarantheteeslighthat,ofwherewtheexactlyvrestrictioneloincitregion,ypropvesanishes,essenthetmagnetictheeldItwillcasenotanishingspatiallyresultspread[14]outvierStokduringTheoremsmallwntime(1.6b)inelotervofals,ratesinceonthebmoeectorcthathanismeldsofmagneticdiusiontheisreasonquitehslolaw.yAsisforThenthebuidisomaw,ythe)magneticmildeldmagneticactstlyup3.onoinitinonlIf,ydecathroughofthedecLorenytzfforce:towhenevclosereinnitrsatisfythisqualdisapptoearsevtyheevknoeloevcitthatysameactsisintheat-grapurelyregionsNaFigvierStokHoesevwifaisy;.thonlyus,istheayyossiblethen1hexplanationertasdotohold.bisthetially6statemenyofisnexteossenem.tiallyextendsgothevoernednon-vbmagneticythetheestablishedcompnetitionforbNaetewequations.een1.3diusion,towhosedoeectbiseldimpcitortanvt,onlyyforatlargeytime,theandmitationsincompressibilitourye,wthatdivimmediatelyvpreveldsen,ts(thesuco).wthatfromforremainingfollotoso,loThecalized.ydroStabilitofyandintalwtheeighbtedprospaces.butConclusion(1.10a)(1.7)theredoyesaectednotnotmeanofthatonecityeloevdecainitialelds,theandifuniqueinsolutionremains(MHD)equationsyingesfastvierStoksucienNaFthesense.oftsolutionpmildthethe,that(1.10b)[14]inresulttheVigneron'syF.sfromyswandknothatAbctuallyn,uncwaethatdoclosernotandknoiswyifatthisehapisenyitbyeholds(1.4),whenthankseerwwIndeed,noticeequations.maessharp.vierStokbNawntheinandThisforspaceearsapppropertspatia(1.11a)lThisspreadingwofsuggestedtheusinitialyvReferee.elocit