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Publié par | mijec |
Nombre de lectures | 16 |
Langue | English |
Extrait
†
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