Global stability of steady states in the classical Stefan problem

0.22 MB·English

by  Mahir Hadžić

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
− ∈