GLOBAL STABILITY OF STEADY STATES IN THE CLASSICAL STEFAN PROBLEM MAHIRHADZˇIC´ ANDSTEVESHKOLLER ABSTRACT. Theclassicalone-phaseStefanproblem(withoutsurfacetension)allowsforacontinuumofsteadystatesolu- tions,givenbyanarbitrary(butsufficientlysmooth)domaintogetherwithzerotemperature.Weproveglobal-in-timestability ofsuchsteadystates,assumingasufficientdegreeofsmoothnessontheinitialdomain,butwithoutanyapriorirestriction 5 ontheconvexitypropertiesoftheinitialshape. Thisisanextensionofourpreviousresult[28]inwhichwestudiednearly 1 sphericalshapes. 0 2 n a J 2 1. INTRODUCTION 1.1. The problem formulation. We consider the problem of global existence and asymptotic stability of classical ] P solutionstotheclassicalStefanproblem,whichmodelstheevolutionofthetime-dependentphaseboundarybetween A liquidandsolidphases. Thetemperaturep(t,x)oftheliquidandtheaprioriunknownmovingphaseboundaryΓ(t) . mustsatisfythefollowingsystemofequations: h t a p ∆p=0 in Ω(t); (1.1a) t m − (Γ(t))= ∂ p on Γ(t); (1.1b) n [ V − p=0 on Γ(t); (1.1c) 1 p(0, )=p ,Ω(0)=Ω. (1.1d) v 0 · 3 6 Foreachinstantoftimet [0,T],Ω(t)isatime-dependentopensubsetofRd withd 2,andΓ(t)d=ef∂Ω(t)denotes 4 ∈ ≥ themoving,time-dependentfree-boundary. 0 Theheatequation(1.1a)modelsthermaldiffusioninthebulkΩ(t)withthermaldiffusivitysetto1. Theboundary 0 . transportequation(1.1b)statesthateachpointonthemovingboundaryistransportedwithnormalvelocityequalto 1 ∂ p= p n,thenormalderivativeofponΓ(t). Here,n(t, )denotestheoutwardpointingunitnormaltoΓ(t), 0 − n −∇ · · and (Γ(t))denotesthespeedorthenormalvelocityofthehypersurfaceΓ(t). ThehomogeneousDirichletboundary 5 V 1 condition(1.1c) istermedthe classicalStefanconditionandproblem(1.1) iscalled theclassicalStefanproblem. It : impliesthatthefreezingoftheliquidoccursataconstanttemperaturep=0. Finally,in(1.1d)wespecifytheinitial v temperaturedistributionp :Ω R,aswellastheinitialgeometryΩ. BecausetheliquidphaseΩ(t)ischaracterized i 0 X by the set x Rd : p(x,t)>→0 , we shall consider initial data p >0 in Ω. Thanks to (1.1a), the parabolic Hopf 0 { ∈ } r lemma implies that ∂ p(t)<0 on Γ(t) for t>0, so we impose the non-degeneracy condition (also known as the a n Rayleigh-Taylorsignconditioninfluidmechanics[43,45,47,14,17,16]): ∂ p λ>0 on Γ(0) (1.2) n 0 − ≥ onourinitialtemperaturedistribution.Undertheaboveassumptions,weprovedin[27]that(1.1)islocallywell-posed. Steadystates(u¯,Γ¯)of(1.1)consistofarbitrarydomainswithΓ¯ C1 andwithtemperatureu¯ 0.Themaingoal ∈ ≡ ofthispaperistoproveglobal-in-timestabilityofsuchsteadystates,independentofanyconvexityassumptions. Our analysis employs high-order energy spaces, which are weighted by the normal derivative of the temperature along themovingboundary;wecreateahybridizedenergymethod,combiningintegratedquantitieswithpointwisemethods via the Pucci extremal operators, which allow us to track the time-decay propertiesof the normal derivative of the temperature. This hybrid approachappears to be new, and is a natural extension of our previouswork [28], which necessitatedperturbationsofsphericalinitialdomains. 1991MathematicsSubjectClassification. 35R35,35B65,35K05,80A22. Keywordsandphrases. free-boundaryproblems,Stefanproblem,regularity,stability,globalexistence. 1 2 MAHIRHADZˇIC´ANDSTEVESHKOLLER 1.2. Notation. Foranys 0andgivenfunctionsf:Ω R,ϕ:Γ Rweset ≥ → → f d=ef f and ϕ d=ef ϕ . s Hs(Ω) s Hs(Γ) k k k k | | k k Ifi=1,...,dthenf, =def∂ f isthepartialderivativeoff withrespecttoxi. Similarly,f, =def∂ ∂ f,etc. Fortime- i xi ij xi xj differentiation,f =def∂ f. Furthermore,fora functionf(t,x), we shalloftenwrite f(t)forf(t, ), andf(0)tomean t t · f(0,x). ThespaceofcontinuousfunctionsonΩisdenotedbyC0(Ω).Foranygivenmulti-indexα=(α ,...,α )we 1 d set ∂α~=∂α1...∂αd. 1 d Wealsodefinethetangentialgradient∂¯by∂¯f=def f ∂ fN,whereN standsfortheoutward-pointingunitnormal N ∇ − onto∂Ωand∂ f=N f isthenormalderivativeoff.ByextendingN smoothlyintoaneighborhoodofΓinsidethe N interiorofΩwecande·fi∇ne∂¯onthatneighborhoodinthesameway. Weemploythefollowingnotationalconvention: ∂¯f=(∂¯ f,...,∂¯ f), ∂¯α~f=def(∂¯α1f,...,∂¯αdf), 1 d 1 d whereα~=(α ,...,α )denotesamulti-index.TheidentitymaponΩisdenotedbye(x)=x,whiletheidentitymatrix 1 d isdenotedbyId. WeuseC todenoteauniversal(orgeneric)constantthatmaychangefrominequalitytoinequality. WewriteX.Y todenoteX CY. WeusethenotationP(s)todenoteagenericnon-zerorealpolynomialfunction ≤ ofs1/2withnon-negativecoefficientsoforderatleast3: m 3+i P(s)= cis 2 , ci 0, m N0. (1.3) ≥ ∈ Xi=0 TheEinsteinsummationconventionisemployed,indicatingsummationoverrepeatedindices. 1.3. TheinitialdomainΩandtheharmonicgauge. ForourinitialdomainΩwechooseasimplyconnecteddomain Ω Rd, wheretheboundary∂ΩwillbedenotedbyΓ.Wefurtherassume,withoutlossofgenerality,thattheorigin ⊂ iscontainedinΩ,i.e. 0 Ω.WetransformtheStefanproblem(1.1)setonthemovingdomainΩ(t),toanequivalent ∈ problemonthefixeddomainΩ;todoso,weuseasystemofharmoniccoordinates,alsoknownastheharmonicgauge orArbitraryLagrangianEulerian(ALE)coordinatesinfluidmechanics. ThemovingdomainΩ(t)willberepresentedastheimageofatime-dependentfamilyofdiffeomorphismsΨ(t): Ω Ω(t). LetN representtheoutwardpointingunitnormaltoΓandletΓ(t)begivenby 7→ Γ(t)= x x=x +h(t,x )N, x Γ . 0 0 0 { | ∈ } Assumingthatthesignedheightfunctionh(t, )issufficientlyregularandΓ(t)remainsasmallgraphoverΓ,wecan · defineadiffeomorphismΨ:Ω Ω(t)astheellipticextensionoftheboundarydiffeomorphismx x +h(t,x )N, 0 0 0 → 7→ bysolvingthefollowingDirichletproblem: ∆Ψ=0 in Ω, (1.4) Ψ(t,x)=x+h(t,x)N(x) x Γ. ∈ WeintroducethefollowingnewvariablessetonthefixeddomainΩ: q=p Ψ (temperature), ◦ v= p Ψ (“velocity”), −∇ ◦ A=[DΨ]−1 (inverseofthedeformationtensor), J=detDΨ (Jacobiandeterminant), We now pull-back the Stefan problem(1.1) from Ω(t) onto the fixed domain Ω. If we let g denote the Jacobian of the transformationΨ(t, ) :Γ Γ(t), and let n(t, ) denote the outward-pointingunitnormalvector to the moving Γ · | → · surfaceΓ(t),thenthefollowingrelationshipholds[15]: J−1√gn Ψ(t,x)=Ak(t,x)N (x). i◦ i k It thus follows that the outward-pointing unit normal vector n(t, ) to the moving surface Γ(t) can be written as · (n Ψ)(t,x)=ATN/ATN . We shall henceforth drop the explicit composition with the diffeomorphism Ψ, and ◦ | | simplywrite n(t,x)=ATN/ATN | | fortheunitnormaltothemovingboundaryatthepointΨ(t,x) Γ(t). ∈ GLOBALSTABILITYOFSTEADYSTATESINTHECLASSICALSTEFANPROBLEM 3 TheclassicalStefanproblemonthefixeddomainΩiswrittenas(see[27,28]) q Aj(Akq, ), = v Ψ in [0,T) Ω, (1.5a) t− i i k j − · t × vi+Akq, =0 in [0,T) Ω, (1.5b) i k × q=0 on [0,T) Γ, (1.5c) × v ATN h = · on [0,T) Γ, (1.5d) t N ATN × · ∆Ψ=0 on [0,T) Ω, (1.5e) × Ψ=e+hN on [0,T) Γ, (1.5f) × q=q >0 on t=0 Ω, (1.5g) 0 { }× h=0 on t=0 Γ, (1.5h) { }× Problem(1.5)isareformulationoftheproblem(1.1). Observethattheboundarycondition(1.5d)isequivalentto Ψ n(t)=v n(t) on [0,T) Γ sothat Ψ(t)(Γ)=Γ(t), (1.6) t · · × whichisbutarestatementoftheStefancondition(1.1b). SincethefactorN ATN willshowuprepeatedlyinvarious · calculations,itisusefultointroducetheabbreviation: Λd=efN ATN. (1.7) · Note thatinitially Λ=1andit willremainclose to 1, since forsmallh thetransitionmatrixA remainsclose to the identitymatrix. Since the identity map e:Ω Ω is harmonic in Ω and Ψ e=hN on Γ, standard elliptic regularity theory for → − solutionsto(1.4)showsthatfort [0,T), ∈ Ψ(t, ) e C h(t, ) , s>0.5, k · − kHs(Ω)≤ k · kHs−0.5(Γ) sothatforhsufficientlysmallandslargeenough,theSobolevembeddingtheoremshowsthat ΨisclosetoId,and ∇ bytheinversefunctiontheorem,Ψisadiffeomorphism. 1.3.1. Thehigh-orderenergyandthehigh-ordernorm. Wewillspecializetothecased=2fortheremainderofthis paper. The case d=3 requiresonly our normsto contain one more degreeof differentiability,while the rest of the argumentisentirelyanalogous. Todefinethenaturalenergiesassociatedwiththemainproblem,wemustemploytangentialderivativesinaneigh- borhood which is sufficiently close to the boundary Γ. Near Γ=∂Ω, it is convenient to use tangential derivatives ∂¯α~, while away from the boundary, Cartesian partial derivatives ∂xi are natural. For this reason, we introduce a non-negativeC∞ cut-offfunctionµ:Ω¯ R withtheproperty + → µ(x) 0 if x ρ; µ(x) 1 if dist(x,Γ) σ. ≡ | |≤ ≡ ≤ Hereρ,σ R+ arechoseninsuchawaythatB (0)⋐Ωand x dist(x,Γ) σ Ω B (0). ρ ρ ∈ { | ≤ }∈ \ Definition1.1(Higher-orderenergies). Thefollowinghigh-orderenergyanddissipationfunctionalsarefundamental toouranalysis: def (t)= (q,h)(t)= E E 1 1 1 µ1/2∂¯α~∂bv 2 + ( ∂ q)1/2Λ∂¯α~∂bΨ2 + µ1/2(∂¯α~∂bq+∂¯α~∂bΨ v) 2 2 k t kL2x 2 | − N t |L2x 2 k t t · kL2x X X X |α~|+2b≤5 |α~|+2b≤6 |α~|+2b≤6 1 (1 µ)1/2∂α~∂bv 2 + (1 µ)1/2(∂α~∂bq+∂α~∂bΨ v) 2 k − t kL2x 2 k − t t · kL2x X X |α~|+2b≤5 |α~|+2b≤6 4 MAHIRHADZˇIC´ANDSTEVESHKOLLER and def (t)= (q,h)(t)= D D µ1/2∂¯α~∂bv 2 + ( ∂ q)1/2Λ∂¯α~∂bΨ 2 + µ1/2(∂¯α~∂bq +∂¯α~∂bΨ v) 2 k t kL2x | − N t t|L2x k t t t t· kL2x X X X |α~|+2b≤6 |α~|+2b≤5 |α~|+2b≤5 + (1 µ)1/2∂α~∂bv 2 + (1 µ)1/2(∂α~∂bq +∂α~∂bΨ v) 2 , k − t kL2x k − t t t t· kL2x X X |α~|+2b≤6 |α~|+2b≤5 wherewerecallthedefinitionofΛgivenin(1.7). Finally,weintroducethetotalenergyE(t): t E(t)d=ef sup (τ)+ (τ)dτ. (1.8) E Z D 0≤s≤t 0 Note that the boundary norms of the gauge function Ψ are weighted by √ ∂ q. We thus introduce the time- N − dependentfunction χ(t)d=efinf( ∂ q)(t,x)>0, N x∈Γ − whichwillbeusedtotracktheweightedbehaviorofh. Itisimportanttonote,thatduetothesmoothnessassumption onΓitiseasytoseethatforanylocalcoordinatechart(∂ ,...,∂ )forΓwehavetheequivalence s1 sd−1 ( ∂ q)1/2Λ∂¯α~∂bΨ2 ( ∂ q)1/2∂β1...∂βd−1h2 , (1.9) | − N t |L2x≈ | − N s1 sd−1 |L2(Γ) |α~|X+2b≤6 β=(β1X,...,βd−1) |β|+2b≤6 where X Y means that there exist positive constants C and C such that C Y X C Y. In our case, the two 1 2 1 2 ≈ ≤ ≤ constantsdependonthechoiceofthelocalchart. Definition1.2(High-ordernorm). Thefollowinghigh-ordernormisfundamentaltoouranalysis: 3 2 S(t)=def ∂lq 2 + q 2 + ∂lq 2 k t kL∞H6−2l k kL2H6.5 k t tkL2H5−2l Xl=0 Xl=0 + sup eβs q(s, ) 2 + ∂¯α~∂lv 2 k · kH5 k t kL2L2 0≤s≤t X |α~|+2l≤6 3 2 +χ(t) ∂lh2 +χ(t) ∂l+1h2 + h4 | t |L∞H6−2l | t |L2H5−2l | |L∞H4.5 Xl=0 Xl=0 (1.10) Here β=2λ η, where λ is the smallest eigenvalue of the Dirichlet-Laplacianon Ω and η>0 is a small but fixed − numbertobedeterminedlater. Remark1.3. Asubtlefeatureoftheabovedefinitionisthelossofa 1-derivative-phenomenonforthetemperatureq. 2 Bytheparabolicscaling(whereonetimederivativescalesliketwospatialderivatives),onemightexpectqtobelong to L2H7([0,T);Ω), since ∂l+1q L2H5−2l([0,T);Ω), for l=0,1,2. This is, however, not the case, as the height- t ∈ evolutionequation(1.5d)scalesina hyperbolicfashion,andthusplacesarestriction onthetop-orderregularityof theunknownq,allowingonlyforq L2H6.5([0,T);Ω). ∈ 1.4. Steady states. Note that any C1 simply connected domain represents a steady state of (1.1). In other words, for any simply connected domain Ω¯ C1, the pair (u¯ 0,Γ¯=∂Ω¯) forms a time-independentsolution to (1.1). In ∈ ≡ particular, it is challenging to determine which steady state a small perturbation will decay to. Thus the problem of asymptotic stability, rather than the optimalregularityof weak/viscositysolutions, is one of the main motivating questionsforthiswork. Inparticular,weworkwithclassicalsolutionswithahighdegreeofdifferentiabilityonthe initialdata. 1.5. Rayleigh-Taylorsign condition or non-degeneracycondition on q . With respect to q , condition (1.2) be- 0 0 comes inf[ ∂ q (x)] δ>0 onΓ. N 0 x∈Γ− ≥ GLOBALSTABILITYOFSTEADYSTATESINTHECLASSICALSTEFANPROBLEM 5 For initial temperature distributions that are not necessarily strictly positive in Ω, this condition was shown to be sufficientforlocalwell-posednessfor(1.1)(see[27,39,41]). Ontheotherhand,ifwerequirestrictpositivityofour initialtemperaturefunction1, q >0 inΩ, (1.11) 0 thentheparabolicHopflemma(see,forexample,[20])guaranteesthat ∂ q(t,x)>0for0<t<T onsomeapriori N − (possiblysmall)timeinterval,which,inturn,showsthat and arenormsfort>0,butuniformitymaybelostas E D t 0. To ensurea uniformlower-boundfor ∂ q(t) as t 0, we impose the Rayleigh-Taylorsign conditionwith N → − → thefollowinglower-bound: ∂ q C q ϕ dx, (1.12) N 0 ∗ 0 1 − ≥ Z Ω Here, ϕ is the positive first eigenfunction of the Dirichlet Laplacian ∆ on Ω, and C >0 denotes a universal 1 ∗ − constant. Theuniformlower-boundin(1.12)thusensuresthatoursolutionsarecontinuousintime;moreover,(1.12) allows usto establish a time-dependentoptimallower-boundfor the quantityχ(t)=inf ( ∂ q)(t,x)>0 for all x∈Γ N − timet 0,whichiscrucialforouranalysis. ≥ 1.6. Mainresult. Ourmainresultisaglobal-in-timestabilitytheoremforsolutionsoftheclassicalStefanproblem forsurfaceswhichareassumedtobeclosetoagivensufficientlysmoothdomainΩandfortemperaturefieldscloseto zero.Thenotionsofnearandclosearemeasuredbyourenergynormsaswellasthedimensionlessquantity q K=defk 0k4. (1.13) q 0 0 k k asexpressedinthefollowing Theorem 1.4. Let (q ,h ) satisfy the Rayleigh-Taylorsign condition(1.12), the strict positivity assumption (1.11), 0 0 andsuitablecompatibilityconditions. LetK bedefinedasin(1.13). Thenthereexistsanǫ >0andamonotonically 0 increasingfunctionF:(1, ) R ,suchthatif + ∞ → ǫ S(0)< 0 , (1.14) F(K) thenthereexistuniquesolutions(q,h)toproblem(1.5)satisfying S(t)<Cǫ , t [0, ), 0 ∈ ∞ forsomeuniversalconstantC>0.Moreover,thetemperatureq(t) 0ast withbound → →∞ q(t, ) 2 Ce−βt, k · kH4(Ω)≤ where β=2λ O(ǫ )andλis thesmallest eigenvalueofthe Dirichlet-LaplacianonΩ. The movingboundaryΓ(t) 0 settlesasympt−oticallytosomenearbysteadysurfaceΓ¯ andwehavetheuniform-in-timeestimate sup h(t, ) h .√ǫ 0 4.5 0 | · − | 0≤t<∞ Remark1.5. The increasing functionF(K)given in (1.14) hasan explicit form. ForuniversalconstantsC¯,C>1 choseninSection4, F(K)=defmax 8K2CC¯K2,C¯10(lnK)10K20C¯λ . (1.15) { } Remark1.6. The use of the constantK in our smallness assumption(1.14) allows us to determine a time T=T K whenthedynamicsoftheStefanproblembecomestronglydominatedbytheprojectionofqontothefirsteigenfunction ϕ oftheDirichlet-Laplacian. Explicitknowledgeofthe K-dependencein the smallnessassumption(1.14) permits 1 theuseofenergyestimatestoshowthatsolutionsexistinourenergyspaceonthetime-interval[0,T ]. Fort T , K K ≥ certainerrorterms(thatcannotbecontrolledbyournormsforlarget)becomesign-definitewithagoodsign. Remark1.7. Ananalogoustheoremwasstatedin[28],forperturbationsofsteadysurfacesinitiallyclosetoasphere. Therefore,thisworkgeneralizesthatresult. Moreover,ourmethodsaregeneralenoughtoapplytoothergeometries aswell. Anexampleisthatofafreeboundaryparametrizedasagraphoveraperiodicflatinterface. 1Condition(1.11)isnatural,sinceitdeterminesthephase:Ω(t)={q(t)>0}. 6 MAHIRHADZˇIC´ANDSTEVESHKOLLER Remark1.8(Oncompatibilityconditions). Thefirstcompatibilityconditionontheinitialtemperatureq is 0 q =0. 0 Γ | The secondconditionarises by restricting the parabolicequation(1.5a) to the boundaryΓ andusing the boundary conditions(1.5c)and(1.6). Itgives ∂ q +(d 1)κ ∂ q +(∂ q )2=0 on Γ. NN 0 Γ N 0 N 0 − Here κ standsforthe mean curvatureofΓ. Higherordercompatibilityconditionsarise by takingtime derivatives Γ of (1.5a),re-expressingthemintermsofpurelyspatialderivativesvia(1.5a)andrestrictingtheresultingequationto theboundaryΓattimet=0. Remark1.9. Aninterestingproblemistodeterminetheasymptoticattractor-thesteadystateΓ¯ justfromtheinitial data(u ,Γ ).Thisisstronglyconnectedtotheso-calledmomentumproblem,whichisaproblemofdeterminingthe 0 0 domainΩfromtheknowledgeofitsharmonicmomentac = φdx,φ:Rd R, ∆φ=0.Arelatedquestionarises φ Ω → intheHele-Shawproblem,see[26]. R 1.7. Localwell-posedness theories. In [27], we established the local-in-timeexistence, uniqueness, andregularity fortheclassicalStefanprobleminL2-basedSobolevspaces,withoutderivativeloss,usingthefunctionalframework givenbyDefinition1.1.Thisframeworkisnatural,andreliesonthegeometriccontrolofthefree-boundary,analogous tothatusedintheanalysisofthefree-boundaryincompressibleEulerequationsin[14,15];thesecond-fundamental formiscontrolledbyaanaturalcoercivequadraticform,generatedfromtheinner-productofthetangentialderivative ofthecofactormatrixJA,andthetangentialderivativeofthevelocityofthemovingboundary,andyieldscontrolof thenorm ( ∂ q(t))∂¯kh2dx′ foranyk 3. TheHopflemmaensurespositivityof ∂ q(t)andtheTaylorsign Γ − N | | ≥ − N conditionRonq0ensuresauniformlower-boundast 0. → The first local existence results of classical solutions for the classical Stefan problem were established by Meir- manov(see[39]andreferencestherein)andHanzawa[29]. Meirmanovregularizedtheproblembyaddingartificial viscosityto(1.1b)andfixedthemovingdomainbyswitchingtotheso-calledvonMisesvariables,obtainingsolutions withlessSobolev-regularitythantheinitialdata. Similarly,HanzawausedNash-Moseriterationtoconstructalocal- in-timesolution,butagain,withderivativeloss. Alocal-in-timeexistenceresultfortheone-phasemulti-dimensional Stefanproblemwasprovedin[24],usingLp-typeSobolevspaces. Forthetwo-phaseStefanproblem,alocal-in-time existenceresultforclassicalsolutionswasestablishedin[41]intheframeworkofLp-maximalregularitytheory. 1.8. Priorwork. Thereisalargeamountofliteratureontheclassicalone-phaseStefanproblem.Foranoverviewwe referthereaderto[22,39,46]aswellastheintroductionto[28].First,weaksolutionsweredefinedin[31,21,37].For theone-phaseproblemstudiedherein,avariationalformulationwasintroducedin[23],whereinadditionalregularity results for the free surface were obtained. In [6] it was shown that in some space-time neighborhoodof points x 0 onthefree-boundarythathaveLebesguedensity,theboundaryisC1 inbothspaceandtime,andsecondderivatives oftemperaturearecontinuousuptotheboundary. Undersomeregularityassumptionsonthetemperature,Lipschitz regularity of the free boundarywas shown in [7]. In related works [34, 35] it was shown that the free boundaryis analyticinspaceandofsecondGevreyclassintime,undertheaprioriassumptionthatthefreeboundaryisC1 with certainassumptionsonthetemperaturefunction.In[9]thecontinuityofthetemperaturewasprovedinddimensions. Asforthetwo-phaseclassicalStefanproblem,thecontinuityofthetemperatureinddimensionsforweaksolutions wasshownin[10]. SincetheStefanproblemsatisfiesamaximumprinciple,itsanalysisisideallysuitedtoanothertypeofweaksolu- tioncalledtheviscositysolution. Regularityofviscositysolutionsforthetwo-phaseStefanproblemwasestablished inaseriesofseminalpapers[3,4].Existenceofviscositysolutionsfortheone-phaseproblemwasestablishedin[32], andforthetwo-phaseproblemin[33]. Alocal-in-timeregularityresultwasestablishedin[12],whereitwasshown thatinitiallyLipschitzfree-boundariesbecomeC1 overapossiblysmallerspatialregion. Foranexhaustiveoverview andintroductiontotheregularitytheoryofviscositysolutionswereferthereaderto[11]. In[36]theauthorshowed bythe useofvonMises variablesandharmonicanalysis, thatanprioriC1 free-boundaryinthe two-phaseproblem becomessmooth. In order to understand the asymptotic behavior of the classical Stefan problem on external domains, in [42] the authorsprovedthatonacomplementofagivenboundeddomainG,withnon-zeroboundaryconditionsonthefixed boundary∂G, the solutiontothe classicalStefanproblemconverges,in a suitablesense, tothe correspondingsolu- tionoftheHele-Shawproblemandsharpglobal-in-timeexpansionratesfortheexpandingliquidblobareobtained. Moreover,theblobasymptoticallyhasthegeometryofaball. Notethatthenon-zeroboundaryconditionsactasan GLOBALSTABILITYOFSTEADYSTATESINTHECLASSICALSTEFANPROBLEM 7 effectiveforcing which is absentfrom our problemand the techniquesof [42] do not directly apply. Since the cor- respondingHele-Shawproblem(intheabsenceofsurfacetensionandforcing)isnotadynamicproblem,possessing only time-independent solutions, we are not able to use the Hele-Shaw solution as a comparison problem for our problem. A global stability result for the two-phase classical Stefan problem in a smooth functional framework was also establishedin[39]foraspecific(andsomewhatrestrictive)perturbationofaflatinterface,whereintheinitialgeometry is a strip with imposed Dirichlet temperatureconditions on the fixed top and bottom boundaries, allowing for only one equilibriumsolution. A globalexistence result for smooth solutions was given in [18] under the log-concavity assumption on the initial temperature function, which in light of the level-set reformulationof the Stefan problem, requiresconvexityoftheinitialdomain(apropertythatispreservedbythedynamics). Remark 1.10. We remark that global stability of solutions in the presence of surface tension does not require the use offunctionframework with a decayingweight, suchas ∂ q(t). Inthis regard, the surface tensionproblem is N − simpler for two importantreasons: first, the surface tension contributesa positive-definiteenergy-contributionthat is uniform-in-time, and provides better regularity of the free-boundary(by one spatial derivative), and second, the spaceofequilibriaisfinite-dimensionalandthusitiseasiertounderstandthedegrees-of-freedomthatdeterminethe asymptoticstateofthesystem. 1.9. Methodology. Broadly speaking, our methods combine high-orderenergy estimates with maximum principle techniques.Oncetheproblemisformulatedonthefixeddomainwiththehelpoftheharmonicgaugeexplainedabove, wenoticethatthenaturalquadraticenergyquantitiesthattracktheregularitybehaviorofthemovingboundary,come weighted with the normalderivativeof the temperature. This weightis a time-dependentquantityand its evolution is tied to the free boundary itself. This coupling is nonlinear and it is one of the central difficulties in closing our estimates. Ourstrategyisbasedon[28]anditcontainsthreebasicsteps. Wefirstshowthatundertheassumptionofsmallness onthenormS(t)oversometimeinterval[0,T],theenergyE andthenormS areequivalent,i.e. S(t).E(t).S(t), t [0,T]. (1.16) ∈ Oursecondstepistoestablishthekeyenergyinequalityintheform 1 t E(t) C + (∂ q )∂¯α~∂lh2dS(Γ)+P(S(t)), (1.17) ≤ 0 2 Z Z N t | t | |α~|X+2l≤6 0 Γ where P is a cubic polynomial(see (1.3)) and C is a small quantity depending only on the initial data. Combin- 0 ing(1.16)and(1.17),weinferthat t S(t) C˜ +C (∂ q )∂¯α~∂lh2dS(Γ)+P(S(t)) (1.18) ≤ 0 Z Z N t | t | X 0 Γ |α~|+2l≤6 dangerousterm onthetimeintervalofexistence.Ifitwere|notforthesumo{nztheright-handsid}eabove,asimplecontinuityargument wouldyieldaglobalexistenceresultforsmallinitialdata.However,thesumappearingontheright-handsideof(1.18), whileseeminglycubic,cannotbeboundedbyP(S(t)). Instead,inthethirdstepweshowthatafteracertain,precisely quantifiedamountoftime,this“dangerousterm”becomesnegativeandcanthusbetriviallyboundedfromaboveby zero. Thekeynoveltywithrespectto[28]isanewquantitativelowerboundontheweight ∂ q whichappearsinour N − definitionoftheenergyE(t).Notethatthisquantityisexpectedtoconvergeexponentiallyfastto0astheunknowns settle to anasymptoticequilibrium. We employthe theoryof “halfeigenvalues”associatedwith the Bellman-Pucci- typeoperatorstogenerateacomparisonfunction,whichthenallowsustousethemaximumprincipleandgetanearly sharplowerbound: ∂ q&e(−λ+O(ǫ))t, N − whereλdenotesthefirstDirichleteigenvalueassociatedwiththedomainΩ.Inourpreviouswork[28],wereliedon aratherexplicitBessel-typecomparisonfunctionsusedbyOddsonin[40],whichinparticular,requiredthatwework inanearlysphericaldomain.TheabovelowerboundismuchmoreflexibleanditisexplainedcarefullyinSection3. Thepresentationinthepaperisconsiderablysimplifiedwithrespectto[28]andwebelievethatourenergymethod in conjunctionwith maximumprinciplescan be useful for the stability analysis in other free boundaryproblemsin absenceofsurfacetension. 8 MAHIRHADZˇIC´ANDSTEVESHKOLLER 1.10. Planof the paper. In Section 2, we introducethe bootstrapassumptionsand formulatethe equivalencerela- tionshipbetweentheenergyandthenorm. InSection3weprovideadynamiclowerboundestimateonχ(t). Thisis themainnewingredientwithrespectto[28]andweusethetheoryofhalf-eigenvaluesforthePuccioperators.Finally, in Section 4, we give the proof of Theorem1.4, thereby explainingour continuitymethod as well as a comparison argumentusedtoshowthesign-definitenessofthe“dangerouslinearterms”describedabove. 2. BOOTSTRAP ASSUMPTIONS AND NORM-ENERGY EQUIVALENCE 2.1. The bootstrapassumptions. Let[0,T)be agiventime-intervalofexistenceofsolutionsto(1.5). We assume thatthefollowingtwoassumptionshold: S(t) ǫ, t [0,T), (2.19) ≤ ∈ χ(t)&c1e−(λ+η2)t, t [0,T), (2.20) ∈ whereǫandη aretobechosensufficientlysmalllaterandλstandsforthefirstDirichleteigenvalueassociatedwith thedomainΩ. 2.2. NormS andtotalenergyE areequivalent. Recallthenotation“ ”introducedin(1.9). ≈ Proposition2.1. Thereexistsasufficientlysmallǫ′suchthatifS(t) ǫ′onatimeinterval[0,T]then ≤ S(t) E(t), t [0,T]. ≈ ∀ ∈ Proof. Theproofofthisfactisoneofthepillarsofourstrategy. IthasbeenpresentedindetailinSections2.1-2.5 andSection4.2of[28]and,therefore,weomitithere. WenotethatthedirectionS(t).E(t)isobviouslyharderto prove,as the energyfunctionE(t) a-prioricontrolsonly tangentialderivativesof the temperatureq. In [28] we use aversionoftheellipticregularitystatementforequationswithSobolev-classcoefficientstoobtaincontrolofnormal derivatives(see[13]). (cid:3) 3. LOWERBOUND ONχ(t)AND IMPROVEMENT OF THESECOND BOOTSTRAP ASSUMPTION Theheatequation(1.5a)forqcanbewritteninnon-divergenceformas q a q, b q, =0 in Ω, (3.21a) t kj kj k k − − q=0 on Γ, (3.21b) q(0, )= q >0 in Ω (3.21c) 0 · wherethecoefficientmatrixa=(a ) ,andthevectorb=(b ,b )areexplicitlygivenby kj k,j=1,2 1 2 a =defAkAj; b d=efAk Aj+AkΨi. (3.22) kj i i k i,j i i t Bythebootstrapassumption(2.19)andthedefinition(1.10)ofS(t),wehavethat h .√ǫon[0,T),andthere- 4.5 | | forebytheSobolevembeddingH1(Γ)֒ L∞(Γ),weinferthat h .√ǫ.Fromthisobservation,(3.22),andthe W3,∞ → | | definitionofthetransitionmatrixA,weinferthat a δ .√ǫ, (k,j=1,2), kj kj | − | b .√ǫ, (i=1,2). i | | Therefore,thereexistsaconstantK>0suchthattheellipticityconstantsassociatedwiththematrix(a ) are ij i,j=1,2 betweenthevaluesµ′ =1 K√ǫandµ′ =1+K√ǫuniformlyover[0,T). 1 − 2 2 2 Before we proceed with calculating a lower bound for χ(t), we briefly explain the Bellman operators[2, 5, 19, 25, 38] whichareclosely connectedto the well-knownextremalPuccioperators. Theywillallowusto formulatea nonlinearanalogueofthe“first”eigenvaluefortheellipticpartoftheoperatordefinedin(3.21a). LetΩ bean arbitrarysimply connectedC1-domain. We definethe extremalPuccioperator − [25, 5] with Mµ1,µ2 parameters0<µ µ by 1 2 ≤ − ϕ(x)d=ef inf ϕ(x). (3.23) Mµ1,µ2 L∈Kµ1,µ2L GLOBALSTABILITYOFSTEADYSTATESINTHECLASSICALSTEFANPROBLEM 9 Here denotesthesetofalllinearsecond-orderellipticoperators,whoseellipticityconstantisbetweenµ and Kµ1,µ2 1 µ ,i.e., 2 d=ef L L=a ∂ +b ∂ +c, a ,b ,c C0(Ω), (3.24) Kµ1,µ2 | ij ij i i ij i ∈ (cid:8) µ ξ 2 a ξ ξ µ ξ 2, ξ Rd . 1 ij i j 2 | | ≤ ≤ | | ∈ (cid:9) Itiswellknownthattheoperators − are,ingeneral,fullynonlinearsecond-orderellipticoperators,positive, Mµ1,µ2 andhomogenousoforderone.Thelatterpropertyallowsustoformulateanassociated“eigenvalue”problem,looking forthesolutionsof − u=λu in Ω, (3.25) −Mµ1,µ2 u=0 on ∂Ω. Wenextstatesomeoftheresultsfrom[38]thatthatwillplayanimportantroleinthispaper(forfurtherreferenceson theso-calledhalf-eigenvaluesassociatedwithpositivehomogenousfullynonlinearoperatorswereferthereader,for example,to[5,2,19]): There exist two positive constants λ and λ called the first half-eigenvalues and two functions ̺ ,̺ 1 2 1 2 • C2(Ω) C(Ω¯)suchthat(λ ,̺ )and(λ ,̺ )solve(3.25),and̺ >0,̺ <0inΩ. ∈ 1 1 2 2 1 2 ∩ The first two half-eigenvaluesare simple, i.e. all positivesolutionsto (3.25) are ofthe form(λ ,α̺ )with 1 1 • α>0andanalogously,allnegativesolutionsareoftheform(λ ,α̺ ),α>0. 2 2 Finally,thefirsttwohalf-eigenvaluesarecharacterizedinthefollowingmanner: • λ = sup µ(A), λ = inf µ(A), (3.26) 1 2 A∈Kµ1,µ2 A∈Kµ1,µ2 whereµ(A)standsforthesmallestDirichleteigenvalueassociatedwiththesecondorderlinearellipticoper- atorA. 3.1. Lower bound on χ(t) and the improvement of (2.20). The key ingredientto the proofs of Propositions 2.1 and 4.1 is a quantitative lower bound on the weight χ(t). This is achieved by using the maximum principle and constructinganappropriatecomparisonfunction. Lemma3.1. Underthebootstrapassumptions(2.19)-(2.20)withǫsufficientlysmall,thefollowinginequalityholds: χ(t)&c e−(λ+λ˜(t))t, 1 wherec = q ϕ dxisthefirstcoefficientintheeigenfunctionexpansionoftheinitialdatumq withrespecttothe 1 Ω 0 1 0 L2orthonorRmalbasis ϕ ,ϕ ,... oftheeigenvectorsoftheoperator ∆onΩ,i.eq =c ϕ +c ϕ +.... Moreover, 1 2 0 1 1 2 2 { } − λstandsforthesmallestDirichleteigenvalueassociatedwiththedomainΩandλ˜(t)satisfiestheestimate: λ˜(t) C√ǫ. | |≤ Inparticular,withǫ>0sufficientlysmallsothatC√ǫ<η/4,weobtaintheimprovementofthebootstrapbound(2.20) givenbyχ(t)&c1e−(λ1+η/4)t. Proof. Let us chooseµ d=ef1 K√ǫ and µ d=ef1+K√ǫ. Recall thatK was definedin the paragraphafter (3.22). It 1 2 − followsthatL . Welet̺ bethefirsthalf-eigenvectorassociatedto − asabove.Considerthefollowing ∈Kµ1,µ2 1 Mµ1,µ2 comparisonfunction v(t,x)d=efe−λ1t̺ . 1 Notethatvvanisheson∂Ω=Γ.Astraightforwardcalculationtogetherwiththedefinitionof − showsthat Mµ1,µ2 (∂t L)v= λ1v e−λ−1tL̺1 − − − λ v e−λ1t − ̺ ≤− 1 − Mµ1,µ2 1 = λ v+e−λ1tλ ̺ 1 1 1 − =0. Thereforevisasubsolutiontotheparabolicproblem(3.21). Thenextkeyobservationisthattheeigenfunction̺ (x) 1 behaveslikeaconstantmultipleofthedistancefunctiondist(x,Γ)asxapproachestheboundaryΓ. Namely,sincethe operator −isconcave,thesolutionisC2,α [44,8]andtheHopflemma ∂ ̺ >0holds(seeforinstanceLemma N 1 M − 2.1in[5]). Therefore,functionvbehaveslikecdist(x,Γ)e−λ−1tasxapproachestheboundaryΓforsomeconstantc. 10 MAHIRHADZˇIC´ANDSTEVESHKOLLER Heredist(x,Γ) denotesthe distancefunctionto the boundaryΓ. We first wantto showthatforanyarbitrarilysmall timeσ>0thereexistsastrictlypositiveconstantδ(σ)>0suchthatq δvisapositivesupersolutiontotheparabolic − problem(3.21)onthetimeinterval[σ,T). Sincevisasubsolutionandqisasolution,itfollowsthatforanyδ>0,q δvisasupersolution.Thepositivityof − q δvatt=σfollowsfromtheparabolicHopflemma,fromwhichweinfertheexistenceofaconstantδ(σ)suchthat q−>δ(σ)uniformlyoverΩ¯. Notethatwehaveusedthefactthatv(σ,x)behaveslikec dist(x)neartheboundaryΓ v × forsomepositiveconstantc. Thusbythemaximumprinciple,q δ(σ)v 0on[σ,T).Thisimplies − ≥ q(t,x) δ(σ)v(t,x) Cδ(σ)dist(x,Γ)e−λ1t, t [σ,T), ≥ ≥ ∈ whichyields ∂q(t,x) Cδ(σ)e−λ1t, t [σ,T). − ∂N ≥ ∈ Theaboveestimateishowevernotyetsatisfactory,astheconstantδ(σ)maydegenerateasσgoestozero. WenowrevisitourusageoftheparabolicHopflemmaabove.Forsmallt>0let Ω = x Ω dist(x,Γ) t , t>0. t { ∈ ≥ } (cid:12) NotethatΩtisacompactpropersubsetofΩ.Fromth(cid:12)eproofoftheparabolicHopflemma(seeforinstanceTheorem 3.14in[20]),thevalue ∂q/∂N isproportionaltotheminimalvalueofthetemperatureqonaspace-timeregion t=σ − | strictlycontainedinthespace-timeslabK :=Ω [t/2,3t/2] Ω [0,2t]dividedbyt(whichisproportionaltothe t t × × distanceofK fromtheparabolicboundaryofΩ [0,2t]). Notethat,astapproaches0wemaylooseuniformity-in- t × timeinourconstants. Thisishowevernotthecasesince∂ qiscontinuousatt=0andbytheassumption(1.12) N ∂ q ∂ q =− N 0c C c . (3.27) N 0 1 ∗ 1 − c ≥ 1 Assumption(1.12)isusedonlyin(3.27)toinsurethatthereexistsauniversalconstantC independentofc suchthat ∗ 1 L=( ∂ q )/c >C .ThequantityLisdimensionless,andtheassumptionL>C isnotarestrictionontheinitial N 0 1 ∗ ∗ − data. Inotherwords,ifwehadnotassumed(1.12),theonlymodificationinthestatementofthemaintheoremwould bethatthesmallnessassumptiononinitialdata(1.14)isadditionallyexpressedintermsofLaswell. Astotheboundonλ˜,notethatby(3.26),theexponentλ ischaracterizedbythecondition 1 λ = sup µ(A). 1 A∈Kµ1,µ2 Since µ 1 .√ǫ, i=1,2, it follows that for any matrix A the estimate A Id .√ǫ holds. Since the | i− | ∈Kµ1,µ2 | − | functionµ()isacontinuousfunctionfromthespaceof2 2matricesintoR,itthusfollowsthat · × λ˜ = λ µ(Id) = sup µ(A) µ(Id) .√ǫ. 1 | | | − | | − | A∈Kµ1,µ2 (cid:3) 4. ENERGY ESTIMATESAND IMPROVEMENT OF THEFIRST BOOTSTRAP ASSUMPTION Proposition4.1. Assumingthebootstrapassumption(2.19)andwithǫ>0chosensufficientlysmall, 1 t E(t) C + ( ∂ q )∂¯α~∂lΨ2dS(Γ)+CP(S(t)), (4.28) ≤ 0 2 Z Z − N t | t | X 0 Γ |α~|+2l≤6 where C dependsonlyonthe initialdata,C>0isa genericpositive constantdependingonlyonthe dimensiond, 0 andP denotesanorder-rpolynomialwithr 3oftheform(1.3). ≥ Proof. TheproofofthepropositionisentirelyanalogoustotheproofofProposition3.4from[28]. (cid:3) Proposition 4.2. Let the solution (q,h) to the Stefan problem (1.5) exist on a given maximal interval of existence [0, )onwhichthebootstrapassumptions(2.19)and(2.20)aresatisfied. T (a) ThereexistsauniversalconstantC¯ suchthatifthesmallnessassumption(1.14)fortheinitialdataholdsand if T =defC¯lnK,then K T ≥ q (T ,x)>Cc e−λ1TKϕ (x), x B (0), t K 1 1 1 − ∈