On well-posedness of incompressible two-phase flows with phase transitions: the case of equal densities
Jan Pruess, Senjo Shimizu, Yoshihiro Shibata, Gieri Simonett
aa r X i v : . [ m a t h . A P ] S e p ON WELL-POSEDNESS OF INCOMPRESSIBLETWO-PHASE FLOWS WITH PHASE TRANSITIONS:THE CASE OF EQUAL DENSITIES
J. PR ¨USS, Y. SHIBATA, S. SHIMIZU, AND G. SIMONETT
Abstract.
The basic model for incompressible two-phase flows with phasetransitions is derived from basic principles and shown to be thermodynami-cally consistent in the sense that the total energy is conserved and the totalentropy is nondecreasing. The local well-posedness of such problems is provedby means of the technique of maximal L p -regularity in the case of equal den-sities. This way we obtain a local semiflow on a well-defined nonlinear statemanifold. The equilibria of the system in absence of external forces are iden-tified and it is shown that the negative total entropy is a strict Ljapunovfunctional for the system. If a solution does not develop singularities, it isproved that it exists globally in time, its orbit is relatively compact, and itslimit set is nonempty and contained in the set of equilibria. Mathematics Subject Classification (2000):
Primary: 35R35, Secondary: 35Q30, 76D45, 76T05, 80A22.
Key words:
Two-phase Navier-Stokes equations, surface tension, phase transitions, en-tropy, well-posedness, time weights.Version of May 31, 2018 Introduction
Let Ω ⊂ R n be a bounded domain of class C − , n ≥
2. Ω contains two phases:at time t , phase i occupies the subdomain Ω i ( t ) of Ω, respectively, with i = 1 , . We assume that ∂ Ω ( t ) ∩ ∂ Ω = ∅ ; this means that no boundary contact can occur.The closed compact hypersurface Γ( t ) := ∂ Ω ( t ) ⊂ Ω forms the interface betweenthe phases.Let u denote the velocity field, π the pressure field, T the stress tensor, θ > ν Γ the outer normal of Ω ( t ), V Γ the normalvelocity of Γ( t ), H Γ = H (Γ( t )) = − div Γ ν Γ the sum of the principal curvatures ofΓ( t ), and [[ v ]] = v − v the jump of a quantity v across Γ( t ). Y.S., S.S., and G.S. express their thanks for hospitality to the Institute of Mathematics,Martin-Luther-Universit¨at Halle-Wittenberg, where important parts of this work originated. Theresearch of S.S. was partially supported by challenging Exploratory Research - 23654048, MEXT,Japan.
By the
Incompressible Two-Phase Flow with Phase Transition we mean the fol-lowing problem: find a family of closed compact hypersurfaces { Γ( t ) } t ≥ containedin Ω and appropriately smooth functions u : R + × ¯Ω → R n , π, θ : R + × ¯Ω → R such that ∂ t u + u · ∇ u − div T = 0 in Ω \ Γ( t ) ,T = µ ( θ )( ∇ u + [ ∇ u ] T ) − πI, div u = 0 in Ω \ Γ( t ) ,κ ( θ )( ∂ t θ + u · ∇ θ ) − div( d ( θ ) ∇ θ ) − T : ∇ u = 0 in Ω \ Γ( t ) ,u = ∂ ν θ = 0 on ∂ Ω , [[ u ]] = [[ θ ]] = 0 on Γ( t ) , [[ T ν Γ ]] + σH Γ ν Γ = 0 on Γ( t ) , [[ ψ ( θ )]] + σH Γ = 0 on Γ( t ) ,l ( θ ) j + [[ d ( θ ) ∂ ν θ ]] = 0 on Γ( t ) ,V Γ + j − u · ν Γ = 0 on Γ( t ) , Γ(0) = Γ , u (0 , x ) = u ( x ) , θ (0 , x ) = θ ( x ) in Ω . (1.1)The variable j , called phase flux , can in fact be eliminated from the system: solvingfor j yields j = − [[ d ( θ ) ∂ ν θ ]] /l ( θ ) , provided l ( θ ) = 0, a property which is assumed later on anyway. In fact, if l ( θ ) = 0then the interfacial velocity V is not uniquely determined which leads to non-well-posedness of the problem. The equation for V Γ then becomes V Γ − u · ν Γ − [[ d ( θ ) ∂ ν θ ]] /l ( θ ) = 0 . Note that the sign of the curvature H Γ is negative at a point x ∈ Γ if Ω ∩ B r ( x )is convex, for some sufficiently small r >
0. Thus if Ω is a ball, i.e. Γ = S R ( x ),then H Γ = − ( n − /R .Several quantities are derived from the specific Helmholtz free energies ψ i ( θ ) asfollows: • ǫ i ( θ ) = ψ i ( θ ) + θη i ( θ ) means the specific internal energy in phase i , • η i ( θ ) = − ψ ′ i ( θ ) the specific entropy, • κ i ( θ ) = ǫ ′ i ( θ ) > • l ( θ ) = θ [[ ψ ′ ( θ )]] = − θ [[ η ( θ )]] the latent heat.Further d i ( θ ) > µ i ( θ ) > ρ := ρ = ρ = 1 the constant density,and σ > i , as there is no danger of confusion; we just keep in mind that the coefficientsdepend on the phases.This model is explained in more detail in the next section. It has been recentlyproposed by Anderson et al. [2], see also the monographs by Ishii [9] and Ishiiand Takashi [10]. We will see below that it is thermodynamically consistent inthe sense that in the absence of exterior forces and of fluxes through the outer NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 3 boundary, the total energy is preserved and the total entropy is nondecreasing. Itis in some sense the simplest sharp interface model for incompressible Newtoniantwo-phase flows taking into account phase transitions driven by temperature.There is an extensive literature on isothermal incompressible Newtonian two-phase flows without phase transitions, and also on the two-phase Stefan problemwith surface tension modeling temperature driven phase transitions. On the otherhand, mathematical work on two-phase flow problems including phase transitionsis rare. In this direction, we only know of the papers by Hoffmann and Starovoitov[7, 8], dealing with a simplified two-phase flow model, and Kusaka and Tani [13,14] which are two-phase for the temperature but with only one phase moving.The papers of DiBenedetto and Friedman [3] and DiBenedetto and O’Leary[4]deal with weak solutions of conduction-convection problems with phase change.However, the models considered in these papers do not seem to be consistent withthermodynamics.It is the purpose of this paper to present a rigorous analysis of problem (1.1) inthe framework of L p -theory. We prove local well-posedness, and we identify theequilibria of the problem. In turns out that the equilibria are the same as those forthe thermodynamically consistent two-phase Stefan problem with surface tension.This heavily depends on the fact that the densities of the two phases are assumedto be equal; in this case the problem is temperature dominated .In a forthcoming paper we will consider the case where the densities are notequal; then the solution behavior is different, as the interfacial mass flux has adirect impact on the velocity field of the fluid. Physically this relates to the so-called Stefan currents which are induced by phase transitions. The velocity field isthen no longer continuous across the interface and this leads to different analyticproperties of the model. We call this case velocity dominated .The plan for this paper is as follows. In the next section we introduce and dis-cuss the model in some detail and show that it is thermodynamically consistent. InSection 3 we prove that the negative total entropy is a strict Ljapunov functionalfor the problem, i.e. is strictly decreasing along nonconstant solutions. We iden-tify the equilibria which consist of velocity zero, constant temperature, constantpressures in the phases, and the disperse phase Ω consists of finitely many nonin-tersecting balls of the same radius . This is in contrast to the isothermal two-phaseNavier-Stokes equation with surface tension where phase transitions are neglected:in that case the balls may have different sizes!In our approach we apply the well-known direct mapping method where theproblem with moving interface is transformed to a problem on a fixed domain. Thistransformation will be introduced in Section 4; as a result we obtain a quasilinearparabolic evolution problem with dynamic boundary condition. In Section 5 westudy maximal L p -regularity of the underlying linearized problem. It turns outthat the linear problem is equivalent to the linearized Stefan problem with surfacetension, followed by a Stokes problem where the curvature serves as an inputvariable on the interface. Here we rely on previous work Pr¨uss, Simonett andZacher [21]. Local well-posedness is discussed in Section 6, while the proof of the J. PR¨USS, Y. SHIBATA, S. SHIMIZU, AND G. SIMONETT main result is given in Section 7. It is based on maximal L p -regularity of the linearproblem and the contraction mapping principle. Finally, in Section 8 we considerglobal existence of solutions. It is shown that solutions exist globally in time,provided the interface satisfies a uniform ball condition, the temperature and themodulus of the latent heat are bounded from below, and certain a priori estimateson the solution hold. In a forthcoming paper we will show that in this situationthe solution converges to an equilibrium in the topology of the state manifold.2. The Model
In this section we explain the model. We begin with
Balance of Mass: ∂ t ρ + div ( ρu ) = 0 in Ω \ Γ( t ) , [[ ρ ( u − u Γ )]] · ν Γ = 0 on Γ( t ) , where u Γ is the interfacial velocity. Hence the normal velocity V Γ can be expressedby V Γ = u Γ · ν Γ . We define the phase flux , more precisely, the interfacial massflux , by means of j := ρ ( u − u Γ ) · ν Γ , i.e. [[ 1 ρ ]] j = [[ u · ν Γ ]] , (2.1)and we note that j is independent of the phases, hence well-defined. Therefore, a phase transition takes place if j = 0. If j ≡ u · ν Γ = u Γ · ν Γ = V Γ and inthis case the interface is advected with the velocity field u .In this paper we consider the completely incompressible case, i.e. we assume thatthe densities are constant in the phases Ω i . Then conservation of mass reduces todiv u = 0 in Ω \ Γ( t ) . (2.2)If only the latter property holds, we say that the material is incompressible .Integrating over the finite domain Ω ( t ) we obtain by the Reynolds transporttheorem ddt Z Ω ( t ) ρ dx = Z Γ( t ) ρV Γ ds + Z Ω ( t ) ∂ t ρ dx = Z Γ( t ) ρV Γ ds − Z Ω ( t ) div( ρu ) dx = Z Γ( t ) ( ρu Γ · ν Γ − ρu · ν Γ ) ds = − Z Γ( t ) j ds. In case | Ω ( t ) | is finite as well, we obtain in the same way ddt Z Ω ( t ) ρ dx = Z Γ( t ) j ds, proving conservation of total mass.Therefore, if both phases are completely incompressible we have ρ | Ω ( t ) | + ρ | Ω ( t ) | ≡ ρ | Ω (0) | + ρ | Ω (0) | =: c . NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 5
This implies [[ ρ ]] | Ω ( t ) | = ρ | Ω | − c , hence | Ω ( t ) | = constant in the case of nonequal densities, i.e. the phase volumesare preserved. If the densities are equal, there is no restriction on the phasevolumes due to conservation of mass. This constitutes a big difference betweeenthe two cases.In later sections of this paper we are concerned with the completely incom-pressible case where, in addition, the densities are equal, and hence conservationof mass does then not imply conservation of phase volumes.Next we consider Balance of Momentum: ∂ t ( ρu ) + div ( ρu ⊗ u ) − div T = ρf in Ω \ Γ( t ) , [[ ρu ⊗ ( u − u Γ ) − T ]] ν Γ = div Γ T Γ on Γ( t ) . Using balance of mass and the definition of the phase flux j we may rewrite thisconservation law as follows: ρ ( ∂ t u + u · ∇ u ) − div T = ρf in Ω \ Γ( t ) , [[ u ]] j − [[ T ν Γ ]] = div Γ T Γ on Γ( t ) . (2.3)Here T Γ denotes surface stress, which in this paper will be assumed to be of theform T Γ = σP Γ , P Γ = I − ν Γ ⊗ ν Γ , where σ > Γ T Γ = σH Γ ν Γ (2.4)which is the usual assumption for surface stress.We now turn our attention to Balance of Energy:
Let q denote the heat flux and r an external heat source. Then balance of energybecomes ∂ t ( ρ | u | + ρǫ ) + div { ( ρ | u | + ρǫ ) u } − div( T u − q ) = ρf · u + ρr in Ω \ Γ( t ) , [[( ρ | u | + ρǫ )( u − u Γ ) − ( T u − q )]] · ν Γ = div Γ T Γ · u Γ on Γ( t ) . Note that the second line implies that the only surface energy taken into accounthere is that induced by surface tension.Using balance of mass, balance of momentum, and the definition of the phaseflux j we may rewrite this conservation law as follows: ρ ( ∂ t ǫ + u · ∇ ǫ ) + div q − T : ∇ u = ρr in Ω \ Γ( t ) , ([[ ǫ ]] + [[ 12 | u − u Γ | ]]) j − [[ T ν Γ · ( u − u Γ )]] + [[ q · ν Γ ]] = 0 on Γ( t ) . (2.5) J. PR¨USS, Y. SHIBATA, S. SHIMIZU, AND G. SIMONETT
The total energy is given by E := E ( u, θ, Γ) := 12 Z Ω ρ | u | dx + Z Ω ρǫ ( θ ) dx + σ | Γ | , where | Γ | denotes the surface area of Γ. Note that here the assumption of σ beingconstant is important! Otherwise, one has to take into account surface energy, locally , which means that we would have to include a balance of surface energy.In the absence of external forces f and heat sources r , for the time derivativeof E we obtain by the transport theorem ddt E = Z Ω { u · ρ∂ t u + ρ∂ t ǫ ( θ ) } dx − Z Γ { [[ ρ | u | + ρǫ ( θ )]] + σH Γ } V Γ ds = − Z Ω { ρ ( u · ∇ ) u · u − div T · u + ρ ( u · ∇ ) ǫ ( θ ) + div q − T : ∇ u } dx − Z Γ (cid:8) [[ ρ | u | + ρǫ ( θ )]] + σH Γ (cid:9) u Γ · ν Γ ds = Z Γ (cid:8) [[( ρ | u | + ρǫ ( θ ))( u − u Γ ) · ν Γ ]] − [[ T ν Γ · u ]] + [[ q · ν Γ ]] − σH Γ V Γ (cid:9) ds = Z Γ (cid:8) [[ 12 | u | + ǫ ( θ )]] j − [[ T u · ν Γ ]] + [[ q · ν Γ ]] − σH Γ V Γ (cid:9) ds, provided that energy transport through the outer boundary is zero, which means (cid:8) ρ | u | ρǫ ( θ ) (cid:9) u · ν − T ν · u + q · ν = 0 on ∂ Ω . Hence ddt E ( u, θ, Γ) = 0 , which implies that the total energy is preserved. Entropy Production:
We introduce now the fundamental thermodynamic relations which read ǫ ( θ ) = ψ ( θ ) + θη ( θ ) , η ( θ ) = − ψ ′ ( θ ) ,κ ( θ ) = ǫ ′ ( θ ) , l ( θ ) = θ [[ ψ ′ ( θ )]] = − θ [[ η ( θ )]] , where ψ ( θ ) means the Helmholtz free energy which should be considered as given,but depends on the phases. The quantities η and κ , l are called entropy , heatcapacity , and latent heat , respectively.The total entropy is defined byΦ( θ, Γ) = Z Ω ρη ( θ ) dx. The main idea of the modeling approach presented here is that there should be noentropy production on the interface, as it is considered to be ideal; in particular
NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 7 it is assumed to carry no mass and no energy except for surface tension. We havewith (2.2)-(2.8) for f = r = 0, and with ǫ ′ ( θ ) = θη ′ ( θ ), ddt Φ( u, θ, Γ) = Z Ω ρ∂ t η ( θ ) dx − Z Γ [[ ρη ( θ )]] V Γ ds = Z Ω ρη ′ ( θ ) ∂ t θ dx − Z Γ [[ ρη ( θ )]] u Γ · ν Γ ds = Z Ω n η ′ ( θ ) ǫ ′ ( θ ) { T : D − div q } − ρu · ∇ η ( θ ) o dx − Z Γ [[ ρη ( θ )]] u Γ · ν Γ ds = Z Ω n T : Dθ − q · ∇ θθ o dx + Z Γ n [[ ρη ( θ )( u − u Γ ) · ν Γ ]] + 1 θ [[ q · ν Γ ]] o ds = Z Ω n T : Dθ − q · ∇ θθ o dx + Z Γ θ n [[ − l ( θ ) j + [[ q · ν Γ ]] o ds, provided there is no entropy flux through the outer boundary ∂ Ω, which means q · ν + ρθη ( θ ) u · ν = 0 on ∂ Ω . To ensure conservation of energy as well as entropy through the outer boundarywe impose in this paper for simplicity the following
Constitutive Laws on the Outer Boundary ∂ Ω : q · ν = 0 , u = 0 . (2.6)Actually, we may consider more general conditions at the outer boundary, like q · ν = 0 and the partial slip condition u · ν = 0, P ∂ Ω T ν = 0, but we refrain fromdoing this here.As constitutive laws in the phases we employ Newton’s law for the stress tensorand Fourier’s law for the heat flux; these ensure nonnegative entropy productionin the bulk. Recall that we consider the completely incompressible case.
Constitutive Laws in the Phases: T = S − πI, S = 2 µ ( θ ) D, D = 12 ( ∇ u + [ ∇ u ] T ) ,q = − d ( θ ) ∇ θ. (2.7)We assume κ ( θ ) = ǫ ′ ( θ ) = − θψ ′′ ( θ ) >
0, as well µ ( θ ) > d ( θ ) > Constitutive Laws on the Interface Γ( t ):[[ θ ]] = [[ P Γ u ]] = 0 , − l ( θ ) j + [[ q · ν Γ ]] = 0 . (2.8)Hence in this model the temperature and the tangential part of the velocity arecontinuous across the interface; the latter means that there is no tangential slip J. PR¨USS, Y. SHIBATA, S. SHIMIZU, AND G. SIMONETT at the interface, as it is considered ideal. The third equation in (2.8) means thatno entropy is generated on the interface. This implies the relation ddt Φ( θ, Γ) = Z Ω n µ ( θ ) θ | D | + d ( θ ) θ |∇ θ | o dx ≥ P Γ u ]] = 0 with P Γ [[ T ν Γ ]] = 0 yields[[ | u − u Γ | ]] = [[ 1 ρ ]] j , [[ T ν Γ · ( u − u Γ )]] = [[ T ν Γ · P Γ ( u − u Γ )]] + [[ 1 ρ T ν Γ · ν Γ ]] j = [[ 1 ρ T ν Γ · ν Γ ]] j. Combining these conditions with the energy balance across the interface and theconstitutive law for q results into the following jump conditions, assuming that j may take arbitrary values: l ( θ ) j + [[ d ( θ ) ∂ ν θ ]] = 0 on Γ( t ) , { [[ ψ ( θ )]] + [[ 12 ρ ]] j − [[ 1 ρ T ν Γ · ν Γ ]] } j = 0 on Γ( t ) . (2.10)The first of these two equations is the Stefan law and the second the (generalized)
Gibbs-Thomson law .Note that if j ≡
0, i.e. in the absence of a phase transition, the Stefan lawbecomes [[ d ( θ ) ∂ ν θ ]] = 0, and the Gibbs-Thomson relation trivializes. On the otherhand, if phase transitions occur then j must be considered as arbitrary, and so theGibbs-Thomson law becomes[[ ψ ( θ )]] + [[ 12 ρ ]] j − [[ 1 ρ T ν Γ · ν Γ ]] = 0 . (2.11)We summarize to obtain the following model for incompressible two-phase flowswith phase transitions. The Complete Model
In the bulk Ω \ Γ( t ): ρ ( ∂ t u + u · ∇ u ) − div T = 0 ,T = µ ( θ )( ∇ u + [ ∇ u ] T ) − πI, div u = 0 ,ρκ ( θ )( ∂ t θ + u · ∇ θ ) − div( d ( θ ) ∇ θ ) − T : ∇ u = 0 . (2.12) On the interface Γ( t ):[[ 1 ρ ]] j ν Γ − [[ T ν Γ ]] − σH Γ ν Γ = 0 , [[ u ]] = [[ 1 ρ ]] jν Γ ,l ( θ ) j + [[ d ( θ ) ∂ ν θ ]] = 0 , [[ θ ]] = 0 , [[ ψ ( θ )]] + [[ 12 ρ ]] j − [[ 1 ρ T ν Γ · ν Γ ]] = 0 , V Γ = u · ν Γ − ρ j. (2.13) NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 9
On the outer boundary ∂ Ω: u = 0 , ∂ ν θ = 0 . (2.14) Initial conditions : Γ(0) = Γ , u (0) = u , θ (0) = θ . (2.15)This results in the model (1.1) if we assume in addition that the densities areequal, i.e. ρ = ρ = 1. 3. Equilibria
As we have seen in Section 2, the negative total entropy is a Ljapunov functionalfor the problem, and it is even a strict one. To see this, assume that Φ is constanton some interval ( t , t ). Then ddt Φ( θ, Γ) = 0 in ( t , t ), hence D = 0 and ∇ θ = 0in ( t , t ) × Ω. Therefore, θ is constant which implies [[ d ( θ ) ∂ ν θ ]] = 0, and thenfrom the interfacial boundary condition we obtain j = 0, provided l ( θ ) = 0. Then[[ u ]] = 0, hence by Korn’s inequality we have ∇ u = 0 and then u = 0 by the no-slipcondition on ∂ Ω. This implies further ∂ t θ = ∂ t u = 0 and V Γ = 0, i.e. we areat equilibrium. Further, ∇ π = 0, i.e. the pressure is constant in the componentsof the phases, and σH Γ = [[ π ]], [[ ψ ( θ )]] = − [[ π ]] are constant as well. Since θ iscontinuous across the interface the last relation shows that π is constant in all ofΩ , even if it is not connected. From this we finally deduce that Ω is a ball ifit is connected, or a finite union of balls of equal radii, as Ω is bounded. Let ussummarize. Theorem 3.1.
Let σ > , ψ i ∈ C (0 , ∞ ) , µ i , d i ∈ C (0 , ∞ ) , − ψ ′′ i ( s ) , µ i ( s ) , d i ( s ) > , [[ ψ ′ ( s )]] = 0 , for s > , i = 1 , . Then the following assertions hold. (i) The total energy E is conserved along smooth solutions. (ii) The negative total entropy − Φ is a strict Ljapunov functional, which meansthat − Φ is strictly decreasing along nonconstant smooth solutions. (iii) The non-degenerate equilibria (i.e. no boundary contact) are zero velocity,constant temperature, constant pressure in each phase, and Ω consists ofa finite number of nonintersecting balls of equal size. Therefore the equilibria are the same as those for the thermodynamically con-sistent Stefan problem with surface tension which has been discussed recently inPr¨uss, Simonett and Zacher [21]. We now discuss some basic considerations fromthat paper. Suppose ( u, π, θ,
Γ) is an equilibrium withΓ = n m [ j =1 S ( x j , R ) : B ( x j , R ) ∩ B ( x k , R ) = ∅ , j = k, B ( x j , R ) ⊂ Ω o , where B ( x, R ) denotes the ball with center x and radius R , and S ( x, R ) its bound-ary. To determine ([[ π ]] , θ, R ) we have to solve the system ϕ ( θ, R ) := | Ω | ǫ ( θ ) + | Ω | ǫ ( θ ) + σ | Γ | = E , [[ π ]] = σH Γ , (3.1)[[ ψ ( θ )]] = − [[ π ]] , where E = E ( u , θ , Γ ) is the prescribed initial total energy, and ϕ ( θ, R ) meansthe total energy at a given equilibrium to be studied. We have[[ π ]] = σH Γ = − ( n − σ/R, R = R ( θ ) = ( n − σ/ [[ ψ ( θ )]] . Hence there remains a single equation for the equilibrium temperature θ , namely E = ϕ ( θ ) := | Ω | ǫ ( θ ) − m ( ω n /n ) R n ( θ )[[ ǫ ( θ )]] + σmω n R n − ( θ ) , where ω n denotes the area of the unit sphere in R n . Note that only the temperaturerange [[ ψ ( θ )]] > R >
0, and with R ∗ m = sup { R > m disjoint balls of radius R } we must also have R < R ∗ m , i.e. with h ( θ ) = [[ ψ ( θ )]] h ( θ ) > σ ( n − R ∗ m . With ǫ ( θ ) = ψ ( θ ) − θψ ′ ( θ ) we may rewrite ϕ ( θ ) as ϕ ( θ ) = | Ω | ǫ ( θ ) + c n (cid:16) h ( θ ) n − + ( n − θ h ′ ( θ ) h ( θ ) n (cid:17) , where we have set c n = m ω n n ( n −
1) (( n − σ ) n . Next, with R ′ ( θ ) = − σ ( n − h ′ ( θ ) h ( θ ) = − h ′ ( θ ) R ( θ ) σ ( n − ϕ ′ ( θ ) = | Ω | ǫ ′ ( θ ) − [[ ǫ ′ ( θ )]] | Ω | + mω n (cid:16) σ ( n − R ( θ ) − [[ e ( θ )]] (cid:17) R n − ( θ ) R ′ ( θ )= | Ω | κ ( θ ) − [[ κ ( θ )]] | Ω | + mω n θh ′ ( θ ) R n − ( θ ) R ′ ( θ )= ( κ | Ω − θh ′ ( θ ) | Γ | h ′ ( θ ) R ( θ ) σ ( n − κ | Ω R ( θ ) σ ( n − n σ ( n − R ( θ ) − l ( θ ) | Γ | θ ( κ | Ω o , with l ( θ ) = θh ′ ( θ ). It will turn out that the term in the parentheses determineswhether an equilibrium is stable: it is stable if and only if m = 1 and ϕ ′ ( θ ) <
0. Inparticular, if Ω is not connected this equilibrium is unstable; this fact is relatedto the physical phenomenon called Ostwald ripening . NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 11
In general it is not a simple task to analyze the equation for the temperature ϕ ( θ ) = | Ω | ǫ ( θ ) + c n (cid:16) h ( θ ) n − + ( n − θ h ′ ( θ ) h ( θ ) n (cid:17) = E , unless more properties of the functions ǫ ( θ ) and, in particular, of [[ ψ ( θ )]] areknown; cf. Pr¨uss, Simonett and Zacher [21] for further discussion and results.4. Transformation to a Fixed Domain
Let Ω ⊂ R n be a bounded domain with boundary ∂ Ω of class C , and supposeΓ ⊂ Ω is a closed hypersurface of class C , i.e. a C -manifold which is the boundaryof a bounded domain Ω ⊂ Ω; we then set Ω = Ω \ ¯Ω . Note that Ω is connected,but Ω maybe disconnected, however, it consists of finitely many components only,since ∂ Ω = Γ by assumption is a manifold, at least of class C . Recall that the second order bundle of Γ is given by N Γ := { ( p, ν Γ ( p ) , ∇ Γ ν Γ ( p )) : p ∈ Γ } . Here ν Γ ( p ) denotes the outer normal of Ω at p ∈ Γ and ∇ Γ the surface gradienton Γ. The Weingarten tensor L Γ on Γ is defined by L Γ ( p ) = −∇ Γ ν Γ ( p ) , p ∈ Γ . The eigenvalues κ j ( p ) of L Γ ( p ) are the principal curvatures of Γ at p ∈ Γ, and wehave | L Γ ( p ) | = max j | κ j ( p ) | . The curvature H Γ ( p ) (more precisely ( n −
1) timesmean curvature) is defined as the trace of L Γ ( p ), i.e. H Γ ( p ) = n − X j =1 κ j ( p ) = − div Γ ν Γ ( p ) , where div Γ means surface divergence. Recall also the Hausdorff distance d H be-tween the two closed subsets A, B ⊂ R m , defined by d H ( A, B ) := max { sup a ∈ A dist( a, B ) , sup b ∈ B dist( b, A ) } . Then we may approximate Γ by a real analytic hypersurface Σ (or merely Σ ∈ C ),in the sense that the Hausdorff distance of the second order bundles of Γ and Σ isas small as we want. More precisely, for each η > d H ( N Σ , N Γ) ≤ η . If η > Σ1 with Ω Σ1 ⊂ Ω, and we set Ω Σ2 = Ω \ Ω Σ1 .It is well known that a hypersurface Σ of class C admits a tubular neighbor-hood, which means that there is a > × ( − a, a ) → R n Λ( p, r ) := p + rν Σ ( p )is a diffeomorphism from Σ × ( − a, a ) onto R (Λ). The inverseΛ − : R (Λ) Σ × ( − a, a ) of this map is conveniently decomposed asΛ − ( x ) = (Π Σ ( x ) , d Σ ( x )) , x ∈ R (Λ) . Here Π Σ ( x ) means the orthogonal projection of x to Σ and d Σ ( x ) the signeddistance from x to Σ; so | d Σ ( x ) | = dist( x, Σ) and d Σ ( x ) < x ∈ Ω Σ1 .In particular we have R (Λ) = { x ∈ R n : dist( x, Σ) < a } .Note that on the one hand, a is determined by the curvatures of Σ, i.e. we musthave 0 < a < min { / | κ j ( p ) | : j = 1 , . . . , n − , p ∈ Σ } , where κ j ( p ) mean the principal curvatures of Σ at p ∈ Σ. But on the other hand, a is also connected to the topology of Σ, which can be expressed as follows. SinceΣ is a compact C manifold of dimension n − r Σ > p ∈ Σ there are points x j ∈ Ω Σ i , i = 1 ,
2, such that B r Σ ( x j ) ⊂ Ω Σ i , and ¯ B r Σ ( x i ) ∩ Σ = { p } . Choosing r Σ maximal, we then must also have a < r Σ . In the sequel we fix a = 12 min { r Σ , | κ j ( p ) | : j = 1 , . . . , n − , p ∈ Σ } . For later use we note that the derivatives of Π Σ ( x ) and d Σ ( x ) are given by ∇ d Σ ( x ) = ν Σ (Π Σ ( x )) , D Π Σ ( x ) = M ( d Σ ( x ))(Π( x )) P Σ (Π Σ ( x ))for | d Σ ( x ) | < a , where P Σ ( p ) = I − ν Σ ( p ) ⊗ ν Σ ( p ) denotes the orthogonal projectiononto the tangent space T p Σ of Σ at p ∈ Σ, and M ( r )( p ) = ( I − rL Σ ( p )) − , ( r, p ) ∈ ( − a, a ) × Σ . Note that | M ( r )( p ) | ≤ / (1 − r | L Σ ( p ) | ) ≤ , for all ( r, p ) ∈ ( − a, a ) × Σ . Setting Γ = Γ( t ), we may use the map Λ to parameterize the unknown free bound-ary Γ( t ) over Σ by means of a height function h ( t, p ) viaΓ( t ) = { p + h ( t, p ) ν Σ ( p ) : p ∈ Σ , t ≥ } , at least for small | h | ∞ . Extend this diffeomorphism to all of ¯Ω by means ofΞ h ( t, x ) = x + χ ( d Σ ( x ) /a ) h ( t, Π Σ ( x )) ν Σ (Π Σ ( x )) =: x + ξ h ( t, x ) . Here χ denotes a suitable cut-off function. More precisely, χ ∈ D ( R ), 0 ≤ χ ≤ χ ( r ) = 1 for | r | < /
3, and χ ( r ) = 0 for | r | > /
3. Note that Ξ h ( t, x ) = x for | d ( x ) | > a/
3, and Ξ − h ( t, x ) = x − h ( t, x ) ν Σ ( x ) , x ∈ Σ , for | h | ∞ sufficiently small. Now we define the transformed quantities¯ u ( t, x ) = u ( t, Ξ h ( t, x )) , ¯ π ( t, x ) = π ( t, Ξ h ( t, x )) , t > , x ∈ Ω \ Σ , ¯ θ ( t, x ) = θ ( t, Ξ h ( t, x )) , t > , x ∈ Ω \ Σ , ¯ j ( t, x ) = j ( t, Ξ h ( t, x )) , t > , x ∈ Σ , NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 13 the pull back of ( u, π, θ ) and j . This way we have transformed the time varyingregions Ω \ Γ( t ) to the fixed domain Ω \ Σ.This transformation gives the following problem for (¯ u, ¯ π, ¯ θ, h ): ∂ t ¯ u − G ( h ) · µ (¯ θ )( G ( h )¯ u + [ G ( h )¯ u ] T ) + G ( h )¯ π = R u (¯ u, ¯ θ, h ) in Ω \ Σ , G ( h ) · ¯ u = 0 in Ω \ Σ ,κ (¯ θ ) ∂ t ¯ θ − G ( h ) · d (¯ θ ) G ( h )¯ θ = R θ (¯ u, ¯ θ, h ) in Ω \ Σ , ¯ u = ∂ ν ¯ θ = 0 on ∂ Ω , − [[ µ (¯ θ )( G ( h )¯ u + [ G ( h )¯ u ] T ) ν Γ ( h ) − ¯ πν Γ ( h )]] = σH Γ ( h ) ν Γ ( h ) on Σ , [[¯ u ]] = [[¯ θ ]] = 0 on Σ , [[ ψ (¯ θ )]] + σH Γ ( h ) = 0 on Σ ,β ( h ) ∂ t h − ¯ u · ν Γ − [[ d (¯ θ ) G ( h )¯ θ · ν Γ ]] /l (¯ θ ) = 0 on Σ , ¯ u (0) = ¯ u , ¯ θ (0) = ¯ θ , h (0) = h . (4.1)Here G ( h ) and H Γ ( h ) denote the transformed gradient and curvature, respectively.More precisely we have D Ξ h = I + Dξ h , [ D Ξ h ] − = I − [ I + Dξ h ] − Dξ h =: I − M ( h ) T , with Dξ h ( t, x ) = 1 a χ ′ ( d Σ ( x ) /a ) h ( t, Π Σ ( x )) ν Σ (Π Σ ( x )) ⊗ ν Σ (Π Σ ( x ))+ χ ( d Σ ( x ) /a ) ν Σ (Π Σ ( x )) ⊗ M ( d Σ ( x )) ∇ Σ h ( t, Π Σ ( x )) − χ ( d Σ ( x ) /a ) h ( t, Π Σ ( x )) L Σ (Π( x )) M ( d Σ ( x )) P Σ (Π Σ ( x )) − h ( t, Π( x )) L Σ (Π( x )) M ( d Σ ( x )) P Σ (Π Σ ( x )) for | d Σ ( x ) | < a/ ,Dξ h ( t, x ) = 0 for | d Σ ( x ) | > a/ . In particular, Dξ h ( t, x ) = ν Σ (Π( x )) ⊗ M ( d Σ ( x )) ∇ Σ h ( t, Π( x )) for | d Σ ( x ) | < a/ . Thus, [ I + Dξ h ] is boundedly invertible if h and ∇ Σ h are sufficiently small, e.g. if | h | ∞ <
13 min { a/ | χ ′| ∞ , / | L Σ | ∞ } and |∇ Σ h | ∞ < . (4.2)With these properties we derive ∇ π ◦ Ξ h = G ( h )¯ π = [ ∇ Ξ − h ◦ Ξ h ] T ∇ ¯ π = [ ∇ Ξ h ] − , T ∇ ¯ π = ( I − M ( h )) ∇ ¯ π div u ◦ Ξ h = G ( h ) · ¯ u = ( I − M ( h )) ∇ · ¯ u. Next we note that ∂ t u ◦ Ξ h = ∂ t ¯ u − [ Du ◦ Ξ h ] ∂ t Ξ h = ∂ t ¯ u − D ¯ u [ D Ξ h ] − ∂ t ξ h = ∂ t ¯ u − ([ I + Dξ h ] − ∂ t ξ h · ∇ ) u =: ∂ t ¯ u + ( R ( h ) · ∇ )¯ u. with R ( h ) = − [ I + Dξ h ] − ∂ t ξ h . Hence R u (¯ u, ¯ θ, h ) = − ¯ u · G ( h )¯ u − ( R ( h ) · ∇ )¯ u. Observe that the function R ( h ) contains a time derivative of h linearly. Similarlywe get ∇ θ ◦ Ξ h = G ( h )¯ θ = ( I − M ( h )) ∇ ¯ θ∂ t θ ◦ Ξ h = ∂ t ¯ θ − ∇ ¯ θ · [ ∇ Ξ h ] − ∂ t ξ h =: ∂ t ¯ θ + ( R ( h ) · ∇ )¯ θ, and so R θ (¯ u, ¯ θ, h )) = − κ (¯ θ )¯ u · G ( h )¯ θ − κ ( θ )( R ( h ) · ∇ )¯ θ + µ (¯ θ ) (cid:0) G ( h )¯ u + [ G ( h )¯ u ] T (cid:1) : G ( h )¯ u. With the Weingarten tensor L Σ and the surface gradient ∇ Σ we further have ν Γ ( h ) = β ( h )( ν Σ − α ( h )) , α ( h ) = M ( h ) ∇ Σ h,M ( h ) = ( I − hL Σ ) − , β ( h ) = (1 + | α ( h ) | ) − / , and V Γ = ∂ t Ξ · ν Γ = ∂ t hν Γ · ν Σ = β ( h ) ∂ t h. The curvature H Γ ( h ) becomes H Γ ( h ) = β ( h ) { tr[ M ( h )( L Σ + ∇ Σ α ( h ))] − β ( h ) M ( h ) α ( h ) · [ ∇ Σ α ( h )] α ( h ) } , a differential expression involving second order derivatives of h only linearly. Itslinearization is given by H ′ Γ (0) = tr L + ∆ Σ . Here ∆ Σ denotes the Laplace-Beltrami operator on Σ.It is convenient to decompose the stress boundary condition into tangential andnormal parts. For this purpose let as before P Σ = I − ν Σ ⊗ ν Σ denote the projectiononto the tangent space of Σ. Multiplying the stress interface condition with ν Σ /β we obtain[[¯ π ]] − σH Γ ( h ) = [[ µ ( θ )( G ( h )¯ u + [ G ( h )¯ u ] T )( ν Σ − M ( h ) ∇ Σ h ) · ν Σ ]] , for the normal part of the stress interface condition. Substituting this expressionfor [[¯ π ]] − σH Γ ( h ) in (4.1) and applying the tangential projection yields − P Σ [[ µ ( θ )( G ( h )¯ u + [ G ( h )¯ u ] T )( ν Σ − M ( h ) ∇ Σ h )]]= [[ µ ( θ )( G ( h )¯ u + [ G ( h )¯ u ] T )( ν Σ − M ( h ) ∇ Σ h ) · ν Σ ]] M ( h ) ∇ Σ h for the tangential part. Note that the latter neither contains the pressure jumpnor the curvature which is the advantage of this decomposition.The idea of our approach can be described as follows. We consider the trans-formed problem (4.1). Based on maximal L p -regularity of the linear problem givenby the left hand side of (4.1), we employ the contraction mapping principle to ob-tain local well-posedness of the nonlinear problem. This program will be carriedout in the next sections. NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 15 The Linear Problem
The principal part of the linearization of (4.1) reads as follows. ∂ t u − µ ( x )∆ u + ∇ π = f u in Ω \ Σ , div u = f d in Ω \ Σ , [[ u ]] = 0 on Σ , − [[ µ ( x )( ∇ u + [ ∇ u ] T ) ν Σ − πν Σ ]] = σ ∆ Σ hν Σ + g u on Σ ,u = 0 on ∂ Ω ,u (0) = u in Ω , (5.1) κ ( x ) ∂ t θ − d ( x )∆ θ = f θ in Ω \ Σ , [[ θ ]] = 0 on Σ ,∂ ν θ = 0 on ∂ Ω ,θ (0) = θ in Ω , (5.2) l ( t, x ) θ + σ ∆ Σ h = g θ on Σ ,∂ t h − [[ d ( x ) ∂ ν θ ]] /l ( x ) = g h on Σ ,h (0) = h on Σ . (5.3)Here µ ( x ) = µ ( θ ( x )) , κ ( x ) = κ ( θ ( x )) , d ( x ) = d ( θ ( x )) ,l ( x ) = l ( θ ( x )) , l ( t, x ) = [[ ψ ′ (( e ∆ Σ t γθ )( x ))]] , where γθ means the restriction of θ to Σ. Observe that the term u · ν Γ in theequation for h is of lower order as it enjoys more regularity than the trace of θ onΣ. Therefore this system is triangular, since u does neither appear in (5.2) norin (5.3). The latter system comprises the linearized Stefan problem with surfacetension which has been studied in [21, Theorem 3.3]. To state this result we definethe solution spaces E θ ( J ) = { θ ∈ H p ( J ; L p (Ω)) ∩ L p ( J ; H p (Ω \ Σ) ∩ C ( ¯Ω)) : ∂ ν θ = 0 on ∂ Ω } , E h ( J ) = W / − / pp ( J ; L p (Σ)) ∩ W − / pp ( J ; H p (Σ)) ∩ L p ( J ; W − /pp (Σ)) , E ( J ) = E θ ( J ) × E h ( J )and the spaces of data F θ ( J ) = L p ( J × Ω) , F H ( J ) = W − / pp ( J ; L p (Σ)) ∩ L p ( J ; W − /pp (Σ)) , F h ( J ) = W / − / pp ( J ; L p (Σ)) ∩ L p ( J ; W − /pp (Σ)) , F ( J ) = F θ ( J ) × F H ( J ) × F h ( J ) . Then we have
Theorem 5.1.
Let p > n + 2 and σ > , suppose κ ∈ C ( ¯Ω i ) , d ∈ C ( ¯Ω i ) , i = 1 , , κ , d > on ¯Ω , l ∈ C (Σ) , and let l ∈ F H ( J ) such that l l > on J × Σ , where J = [0 , t ] is a finite time interval.Then there is a unique solution ( θ, h ) ∈ E ( J ) of (5.2) - (5.3) if and only if the data ( f θ , g θ , g h , θ , h ) satisfy ( f θ , g θ , g h ) ∈ F ( J ) , ( θ , h ) ∈ [ W − /pp (Ω \ Σ) ∩ C ( ¯Ω)] × W − /pp (Σ) , and the compatibility conditions ∂ ν θ = 0 on ∂ Ω , l (0) θ + σ ∆ Σ h = g θ (0) , g h (0) + [[ d ∂ ν θ ]] /l ∈ W − /pp (Σ) . The solution map [( f θ , g θ , g h , z ) ( θ, h )] is continuous between the correspondingspaces. Having solved this problem, ∆ Σ h ∈ F h ( J ), hence we may now solve the remain-ing Stokes problem to obtain a unique solution ( u, π ) in the class u ∈ H p ( J ; L p (Ω)) n ∩ L p ( J ; H p (Ω \ Σ) ∩ C ( ¯Ω)) n , ∇ π ∈ L p ( J, L p (Ω)) n , provided the data satisfy f u ∈ F θ ( J ) n , g u ∈ F h ( J ) n , f d ∈ H p ( J ; ˙ H − p (Ω)) ∩ L p ( J ; H p (Ω \ Σ)) , as well as u ∈ [ W − /pp (Ω \ Σ) ∩ C ( ¯Ω)] n , and the compatibility conditions u = 0 on ∂ Ω , div u = f d (0) , − P Σ [[ µ ( ∇ u + [ ∇ u ] T ) ν Σ ]] = P Σ g u (0) , are satisfied. Concerning the two-phase Stokes problem we refer to Shibata andShimizu [22] and to K¨ohne, Pr¨uss and Wilke [11], where the case µ = const has been treated. This result can be extended to nonconstant µ via the methodof localization. For the one-phase case this has been carried out for much moregeneral Stokes problems in Bothe and Pr¨uss [1]Therefore the linearized problem (5.1)-(5.3) has the property of maximal L p -regularity. To state this result we set E u ( J ) := { u ∈ [ H p ( J ; L p (Ω)) ∩ L p ( J ; H p (Ω \ Σ) ∩ C ( ¯Ω))] n : u = 0 on ∂ Ω } , E π ( J ) := L p ( J ; ˙ H p (Ω \ Σ)) , and define the solution space for (5.1)-(5.3) as E ( J ) = E u ( J ) × E π ( J ) × F h ( J ) × E θ ( J ) × E h ( J ) . Then the main result on the linearized problem reads
Theorem 5.2.
Let p > n + 2 and σ > . Suppose µ , κ ∈ C ( ¯Ω i ) , d ∈ C ( ¯Ω i ) , i = 1 , , κ , d > on ¯Ω , l ∈ C (Σ) , and l ∈ F h ( J ) such that l l > on J × Σ ,where J = [0 , t ] is a finite time interval. NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 17
Then the linear problem (5.1) - (5.3) admits a unique solution ( u, π, [[ π ]] , θ, h ) ∈ E ( J ) if and only if the data ( u , θ , h ) and ( f u , f d , g u , f θ , g θ , g h ) satisfy the regu-larity conditions: ( u , θ , h ) ∈ W − /pp (Ω \ Σ) n × W − /pp (Ω \ Σ) × W − /pp (Σ) , ( f u , f θ ) ∈ F θ ( J ) n +1 , f d ∈ H p ( J ; ˙ H − p (Ω)) ∩ L p ( J ; H p (Ω \ Σ)) , ( g u , g h ) ∈ F h ( J ) n +1 , g θ ∈ F H ( J ) , and the compatibility conditions: div u = f d (0) in Ω \ Σ ,u = ∂ ν θ = 0 on ∂ Ω , [[ u ]] = [[ θ ]] = 0 on Σ , − P Σ [[ µ ( ∇ u + [ ∇ u ] T ) ν Σ ]] = P Σ g u (0) on Σ ,l (0) θ + σ ∆ Σ h = g θ (0) on Σ g h (0) + [[ d ∂ ν θ ]] /l ∈ W − /pp (Σ) . The solution map [( u , θ , h , f u , f d , g u , f θ , g θ , g h ) ( u, π, [[ π ]] , θ, h )] is continuousbetween the corresponding spaces. Local Well-Posedness
The basic result for local well-posedness of problem (1.1) in an L p -setting is thefollowing. Theorem 6.1.
Let p > n + 2 , σ > , suppose ψ i ∈ C (0 , ∞ ) , µ i , d i ∈ C (0 , ∞ ) such that κ i ( s ) = − sψ ′′ i ( s ) > , µ i ( s ) > , d i ( s ) > , s ∈ (0 , ∞ ) , i = 1 , . Let Ω ⊂ R n be a bounded domain with boundary ∂ Ω ∈ C − and suppose Γ ⊂ Ω is a closed hypersurface. Assume the regularity conditions u ∈ W − /pp (Ω \ Γ ) n , θ ∈ W − /pp (Ω \ Γ ) , Γ ∈ W − /pp , the compatibility conditions div u = 0 in Ω \ Γ ,u = ∂ ν θ = 0 on ∂ Ω , [[ u ]] = [[ θ ]] = 0 on Γ ,P Γ [[ µ ( ∇ u + [ ∇ u ] T ) ν Γ ]] = 0 on Γ , [[ ψ ( θ )]] + σH Γ = 0 on Γ , [[ d ∂ ν θ ]] ∈ W − /pp (Γ ) , and the well-posedness condition l ( θ ) = 0 on Γ and θ > on ¯Ω . Then there exists a unique L p -solution of problem (1.1) on some possibly small butnontrivial time interval J = [0 , a ] . Here the notation Γ ∈ W − /pp means that Γ is a C -manifold, such that its(outer) normal field ν Γ is of class W − /pp (Γ ). Therefore the Weingarten tensor L Γ = −∇ Γ ν Γ of Γ belongs to W − /pp (Γ ) which embeds into C α (Γ ), with α = 1 − ( n + 2) /p > p > n + 2 by assumption. For the same reason we alsohave u ∈ C α ( ¯Ω i (0))) n , and θ ∈ C α ( ¯Ω i (0))), i = 1 ,
2, and V ∈ C α (Γ ).The notion L p -solution means that ( u, π, θ, Γ) is obtained as the push-forward ofan L p -solution of the transformed problem (4.1). The proof of Theorem 6.1 isgiven in the next section.For later use we discuss an extension of the local existence results to spaces withtime weights. For this purpose, given a UMD -Banach space Y and µ ∈ (1 /p, J = (0 , t ) K sp,µ ( J ; Y ) := { u ∈ L p,loc ( J ; Y ) : t − µ u ∈ K sp ( J ; Y ) } , where s ≥ K ∈ { H, W } . It has been shown in Pr¨uss and Simonett [17] thatthe operator d/dt in L p,µ ( J ; Y ) with domain D ( d/dt ) = H p,µ ( J ; Y ) = { u ∈ H p,µ ( J ; Y ) : u (0) = 0 } is sectorial and admits an H ∞ -calculus with angle π/
2. This is the main tool toextend the results for the linear problem, i.e. Theorem 5.2, to the time weightedsetting, where the solution space E ( J ) is replaced by E µ ( J ) and F ( J ) by F µ ( J )respectively, where z ∈ E µ ( J ) ⇔ t − µ z ∈ E ( J ) , f ∈ F µ ( J ) ⇔ t − µ f ∈ F ( J ) . The trace spaces for ( u, θ ) and h for p > u , θ ) ∈ [ W µ − /pp (Ω \ Σ) ∩ C ( ¯Ω)] n +1 , h ∈ W µ − /pp (Σ) ,h := ( ∂ t h )(0) ∈ W µ − − /pp (Σ) , (6.1)where for the last trace we need in addition µ > / / p . Note that theembeddings E µ,u ( J ) × E µ,θ ( J ) ֒ → [ C ( J × ¯Ω) ∩ C ( J ; C ( ¯Ω i )] n +1 , E µ,h ( J ) ֒ → C ( J ; C (Σ))require µ > / n + 2) / p , which is feasible since p > n + 2 by assumption.This restriction is needed for the estimation of the nonlinearities.The assertions for the linear problem remain valid for such µ , replacing E ( J )by E µ ( J ), F ( J ) by F µ ( J ), for initial data subject to (6.1). This relies on the factmentioned above that d/dt admits a bounded H ∞ -calculus with angle π/ L p,µ ( J ; Y ). Concerning such time weights, we refer to Pr¨uss, Simonettand Zacher [21] for the Stefan problem, and to K¨ohne, Pr¨uss and Wilke [11]for the isothermal incompressible two-phase Navier-Stokes problem, and also to NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 19
Meyries and Schnaubelt [15] for a general theory. Thus as a consequence of theseconsiderations we have the following result.
Corollary 6.2.
Let p > n + 2 , µ ∈ (1 / n + 2) / p, , σ > , and suppose ψ ∈ C (0 , ∞ ) , µ, d ∈ C (0 , ∞ ) are such that κ i ( s ) = − sψ ′′ i ( s ) > , µ i ( s ) > , d i ( s ) > , s ∈ (0 , ∞ ) , i = 1 , . Let Ω ⊂ R n be a bounded domain with boundary ∂ Ω ∈ C − and suppose Γ ⊂ Ω is a closed hypersurface. Assume the regularity conditions ( u , θ ) ∈ [ W µ − /pp (Ω \ Γ ) ∩ C ( ¯Ω)] n +1 , Γ ∈ W µ − /pp , the compatibility conditions div u = 0 in Ω \ Γ ,u = ∂ ν θ = 0 in ∂ Ω , [[ u ]] = [[ θ ]] = 0 , P Γ [[ µ ( ∇ u + [ ∇ u ] T ) ν Γ ]] = 0 on Γ , [[ ψ ( θ )]] + σH Γ = 0 on Γ , [[ d ∂ ν θ ]] ∈ W µ − − /pp (Γ ) , as well as the well-posedness condition l ( θ ) = 0 on Γ and θ > on ¯Ω . Then the transformed problem (4.1) admits a unique solution ( u, π, [[ π ]] , θ, h ) ∈ E µ ((0 , a )) for some a > . The solution depends continuously on the data. Foreach δ ∈ (0 , a ) , the solution belongs to E ( δ, a ) , i.e. regularizes instantly. Proof of the Main Result
In this section we prove Theorem 6.1, for given initial data Γ ∈ W − /pp , u ∈ W − /pp (Ω \ Γ ) n and θ ∈ W − /pp (Ω \ Γ ) satisfying the compatibilityconditions and the well-posedness condition stated in Theorem 6.1. We supposethat ψ i ∈ C (0 , ∞ ), µ i , d i ∈ C (0 , ∞ ) satisfy κ i ( s ) = − sψ ′′ i ( s ) > , µ i ( s ) > , d i ( s ) > , s ∈ (0 , ∞ ) , i = 1 , . According to the considerations in Section 4, Γ can be approximated by a realanalytic hypersurface for any prescribed η > d H ( N Σ , N Γ ) <η , and is parameterized by h ∈ W − /pp (Σ). It is sufficient to prove the localwell-posedness of the nonlinear problem (4.1), because the notion of L p -solutionmeans that the solution of (1.1) is obtained as the push-forward of an L p -solutionof the transformed problem (4.1). In order to facilitate this task, we rewrite(4.1) in quasilinear form, dropping the bars and collecting its principal linear parton the left hand side. We set as before µ ( x ) = µ ( θ ( x )), κ ( x ) = κ ( θ ( x )), d ( x ) = d ( θ ( x )), l ( x ) = l ( θ ( x )), and l ( t, x ) = [[ ψ ′ ( e ∆ Σ t γθ ( x ))]]. We then have ∂ t u − µ ( x )∆ u + ∇ π = F u ( u, π, θ, h ) in Ω \ Σ , div u = F d ( u, h ) in Ω \ Σ , [[ u ]] = 0 on Σ , − P Σ [[ µ ( x )( ∇ u + [ ∇ u ] T ) ν Σ ]] = G tanu ( u, θ, h ) on Σ , − µ ( x ) ∇ u ]] ν Σ · ν Σ + [[ π ]] − σ ∆ Σ h = G noru ( u, θ, h ) on Σ ,u = ∂ ν θ = 0 on ∂ Ω ,κ ( x ) ∂ t θ − d ( x )∆ θ = F θ ( u, θ, h ) on Ω \ Σ , [[ θ ]] = 0 on Σ ,l ( t, x ) θ + σ ∆ Σ h = G θ ( θ, h ) on Σ ,∂ t h − [[ d ( x ) ∂ ν θ ]] /l ( x ) = G h ( u, θ, h ) on Σ ,u (0) = u , θ (0) = θ in Ω ,h (0) = h in Σ . (7.1)The nonlinearities are defined by F u ( u, θ, π, h ) = ( µ ( θ ) − µ ( θ ))∆ u + M ( h ) ∇ π − ( u · ( I − M ( h )) ∇ ) u − ( R ( h ) · ∇ ) u + µ ′ ( θ ) (cid:0) ( I − M ( h )) ∇ θ · (( I − M ( h )) ∇ u + [( I − M ( h )) ∇ u ] T ) (cid:1) − µ ( θ )( M ( h ) : ∇ ) u − µ ( θ )( M ( h ) · ∇ ) u + µ ( θ ) M ( h ) : ∇ u,F d ( u, h ) = M ( h ) : ∇ u,G tanu ( u, h ) = P Σ [[( µ ( θ ) − µ ( θ ))( ∇ u + [ ∇ u ] T ) ν Σ ]] − P Σ [[ µ ( θ )( ∇ u + [ ∇ u ] T ) M ( h ) ∇ Σ h ]] − P Σ [[ µ ( θ )( M ( h ) ∇ u + [ M ( h ) ∇ u ] T )( ν Σ − M ( h ) ∇ Σ h )]]+ [[ µ (( I − M ) ∇ u + [( I − M ) ∇ u ] T )( ν Σ − M ∇ Σ h ) · ν Σ ]] M ( h ) ∇ Σ h,G noru ( u, h ) = [[( µ ( θ ) − µ ( θ ))( ∇ u + [ ∇ u ] T ) ν Σ · ν Σ ]] − [[ µ ( θ )( ∇ u + [ ∇ u ] T ) M ( h ) ∇ Σ h · ν Σ ]] − [[ µ ( θ )( M ( h ) ∇ u + [ M ( h ) ∇ u ] T )( ν Σ − M ( h ) ∇ Σ h ) · ν Σ ]]+ σ ( H Γ ( h ) − ∆ Σ h ) ,F θ ( u, θ, h ) = ( κ ( θ ) − κ ( θ )) ∂ t θ − ( d ( θ ) − d ( θ ))∆ θ − d ( θ ) M ( h ) : ∇ θ + d ′ ( θ ) | ( I − M ( h )) ∇ θ | − d ( θ ) M ( h ) · ∇ θ − κ ( θ )( R ( h ) · ∇ ) θ − κ ( θ ) u · ( I − M ( h )) ∇ θ + µ ( θ )(( I − M ( h )) ∇ u + [( I − M ( h )) ∇ u ] T ) : ( I − M ( h )) ∇ u,G θ ( θ, h ) = l θ − [[ ψ ( θ )]] − σ ( H Γ ( h ) − ∆ Σ h ) ,G h ( u, θ, h ) = [[ (cid:0) d ( θ ) /l ( θ ) − d ( θ ) /l ( θ ) (cid:1) ∂ ν θ ]] + u · ( ν Σ − M ( h ) ∇ Σ h ) − [[( d/l ) M ( h ) ∇ θ · ( ν Σ − M ( h ) ∇ Σ h )]] − [[( d/l ) ∇ θ · M ( h ) ∇ Σ h ]] . NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 21
Here we employed the abbreviations M ( h ) = M ( h ) + M T ( h ) − M ( h ) M T ( h ) ,M ( h ) = ( I − M ( h )) : ∇ M ( h ) ,M ( h ) = (( I − M ( h )) ∇ ) M ( h ) − [(( I − M ( h )) ∇ ) M ( h )] T . We prove the local well-posedness of (7.1) by means of maximal L p -regularity ofthe linear problem (Theorem 5.2) and the contraction mapping principle. Theright hand side of problem (7.1) consist of either lower order terms, or terms ofthe same order as those appearing on the left hand side but carry factors whichcan be made small by construction. Indeed, we have smallness of h , ∇ Σ h andeven of ∇ h uniformly on Σ, because Γ is approximated by Σ in the secondorder bundle. θ appears nonlinearly in ψ, κ, µ, d , but only to order zero; hence e.g.the difference µ ( θ ( t )) − µ ( θ ) will be uniformly small for small times.We introduce appropriate function spaces. Let J = [0 , a ]. The solution spacesare defined by E ( a ) := { u ∈ H p ( J ; L p (Ω)) n ∩ L p ( J ; H p (Ω \ Σ)) n : u = 0 on ∂ Ω , [[ u ]] = 0 } , E ( a ) := L p ( J ; ˙ H p (Ω \ Σ)) , E ( a ) := W / − / pp ( J ; L p (Σ)) ∩ L p ( J ; W − /pp (Σ)) , E ( a ) := { θ ∈ H p ( J ; L p (Ω)) ∩ L p ( J ; H p (Ω \ Σ)) : ∂ ν θ = 0 on ∂ Ω , [[ θ ]] = 0 } , E ( a ) := W / − / pp ( J ; L p (Σ)) ∩ W − / pp ( J ; H p (Σ)) ∩ L p ( J ; W − /pp (Σ)) . We abbreviate E ( a ) := { ( u, π, [[ π ]] , θ, h ) ∈ E ( a ) × E ( a ) × E ( a ) × E ( a ) × E ( a ) } , and equip E j ( a ) ( j = 1 , . . . ,
5) with their natural norms, which turn E ( a ) into aBanach space. A left subscript 0 always means that the time trace at t = 0 of thefunction in question is zero whenever it exists.The data spaces are defined by F ( a ) := L p ( J ; L p (Ω)) n , F ( a ) := H p ( J ; ˙ H − p (Ω)) ∩ L p ( J ; H p (Ω \ Σ)) , F ( a ) := W / − / pp ( J ; L p (Σ)) n ∩ L p ( J ; W − /pp (Σ)) n , F ( a ) := L p ( J ; L p (Ω)) , F ( a ) := W − / pp ( J ; L p (Σ)) ∩ L p ( J ; W − /pp (Σ)) , F ( a ) := W / − / pp ( J ; L p (Σ)) ∩ L p ( J ; W − /pp (Σ)) . We abbreviate F ( a ) := { ( f u , f d , g u , g θ , g θ , g h ) ∈ Y j =1 F j ( a ) } , and equip F j ( a ) ( j = 1 , . . . ,
6) with their natural norms, which turn F ( a ) into aBanach space. Step 1.
In order to economize our notation, we set z = ( u, π, [[ π ]] , θ, h ) ∈ E ( a )and reformulate the quasilinear problem (7.1) as Lz = N ( z ) ( u (0) , θ (0) , h (0)) = ( u , θ , h ) , (7.2)where L denotes the linear operator on the left hand side of (7.1), and N denotesthe nonlinear mapping on the right-hand side of (7.1). From Section 5 we knowthat L : E ( a ) → F ( a ) is bounded and linear, and that L : E ( a ) → F ( a ) is anisomorphism for each a >
0, with norm independent of 0 < a ≤ a < ∞ .Concerning the nonlinearity N , we have the following result. Proposition 7.1.
Suppose p > n + 2 , σ > , and let ψ i ∈ C (0 , ∞ ) , µ i , d i ∈ C (0 , ∞ ) such that κ i ( s ) = − sψ ′′ i ( s ) > , µ i ( s ) > , d i ( s ) > , s ∈ (0 , ∞ ) , i = 1 , . Then for each a > the nonlinearity satisfies N ∈ C ( E ( a ) , F ( a )) and its Fr´echetderivative N ′ satisfies in addition N ′ ( u, π, [[ π ]] , θ, h ) ∈ L ( E ( a ) , F ( a )) . Moreover,there is η > such that for a given z ∗ ∈ E ( a ) with | h | C (Σ) ≤ η , there arecontinuous functions α ( r ) > and β ( a ) > with α (0) = β (0) = 0 , such that || N ′ (¯ z + z ∗ ) || B ( E ( a ) , F ( a )) ≤ α ( r ) + β ( a ) , ¯ z ∈ B r ⊂ E ( a ) . Proof.
The nonlinear terms F u ( u, θ, π, h ), F d ( u, h ), G tanu ( u, θ, h ) and G noru ( u, θ, h )are essentially the same as those for the isothermal two-phase Navier-Stokes prob-lem (cf. Proposition 4.2 in [11]). The nonlinearities F θ ( u, θ, h ), G θ ( θ, h ), and G h ( u, θ, h ) are similar to those for u ≡ (cid:3) Step 2.
We reduce the problem to initial values 0 and resolve the compatibilitiesas follows. Thanks to Proposition 4.1 in [11], we find extensions f ∗ d ∈ F ( a ), g ∗ u ∈ F ( a ) which satisfy f ∗ d (0) = div u , P Σ g ∗ u = − P Σ [[ µ ( ∇ u + [ ∇ u ] T ) ν Σ ]] . Then we define g ∗ θ := e ∆ Σ t [ G θ ( θ, h )(0)] , g ∗ h := e ∆ Σ t G h ( u , θ , h ) , and set f ∗ u = f ∗ θ = 0. With these extensions, by Theorem 5.2 we may solvethe linear problem (5.1)-(5.3) with initial data ( u , θ , h ) and inhomogeneities( f ∗ u , f ∗ d , g ∗ u , f ∗ θ , g ∗ θ , g ∗ h ), which satisfy the required regularity conditions and, by con-struction, the compatibility conditions, to obtain a unique solution z ∗ = ( u ∗ , π ∗ , [[ π ∗ ]] , θ ∗ , h ∗ ) ∈ E ( J )with u ∗ (0) = u , θ ∗ (0) = θ , and h ∗ (0) = h . Step 3.
We rewrite problem (7.2) as Lz = N ( z + z ∗ ) − Lz ∗ =: K ( z ) , z ∈ E ( a ) . NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 23
The solution is given by the fixed point problem z = L − K ( z ), since Theorem 5.2implies that L : E ( a ) → F ( a ) is an isomorphism with | L − | L ( F ( a ) , E ( a )) ≤ M, a ∈ (0 , a ] , where M is independent of a ≤ a . We may assume that M ≥
1. Thanks toProposition 7.1 and due to K (0) = N ( z ∗ ) − Lz ∗ , we may choose a ∈ (0 , a ] and r > | K (0) | F ( a ) ≤ r M , | K ′ ( z ) | L ( E ( a ) , F ( a )) ≤ M , z ∈ E ( a ) , | z | E ( a ) ≤ r hence | K ( z ) | F ( a ) ≤ rM , which ensures that L − K ( z ) : B E ( a ) (0 , r ) → B E ( a ) (0 , r ) is a contraction. Thuswe may employ the contraction mapping principle to obtain a unique solution ona time interval [0 , a ], which completes the proof of Theorem 6.1.8. The Local Semiflow
We follow here the approach introduced in K¨ohne, Pr¨uss and Wilke [11] for theisothermal incompressible two-phase Navier-Stokes problem without phase transi-tions and in Pr¨uss, Simonett and Zacher [21] for the Stefan problem with surfacetension.Recall that the closed C -hypersurfaces contained in Ω form a C -manifold,which we denote by MH (Ω). The charts are the parameterizations over a givenhypersurface Σ according to Section 4, and the tangent space T Σ MH (Ω) consistsof the normal vector fields on Σ. We define a metric on MH (Ω) by means of d MH (Σ , Σ ) := d H ( N Σ , N Σ ) , where d H denotes the Hausdorff metric on the compact subsets of R n . This way MH (Ω) becomes a Banach manifold of class C .Let d Σ ( x ) denote the signed distance for Σ as in Section 4. We may then definethe level function ϕ Σ by means of ϕ Σ ( x ) = φ ( d Σ ( x )) , x ∈ R n , where φ ( s ) = sχ ( s/a ) + (sgn s )(1 − χ ( s/a )) , s ∈ R . It is easy to see that Σ = ϕ − (0), and ∇ ϕ Σ ( x ) = ν Σ ( x ), for each x ∈ Σ. Moreover, κ = 0 is an eigenvalue of ∇ ϕ Σ ( x ), the remaining eigenvalues of ∇ ϕ Σ ( x ) are theprincipal curvatures κ j of Σ at x ∈ Σ.If we consider the subset MH (Ω , r ) of MH (Ω) which consists of all closedhypersurfaces Γ ∈ MH (Ω) such that Γ ⊂ Ω satisfies the ball condition with fixedradius r > MH (Ω , r ) → C ( ¯Ω)defined by Φ(Γ) = ϕ Γ is an isomorphism of the metric space MH (Ω , r ) ontoΦ( MH (Ω , r )) ⊂ C ( ¯Ω). Let s − ( n − /p >
2; for Γ ∈ MH (Ω , r ), we define Γ ∈ W sp (Ω , r ) if ϕ Γ ∈ W sp (Ω).In this case the local charts for Γ can be chosen of class W sp as well. A subset A ⊂ W sp (Ω , r ) is (relatively) compact, if and only if Φ( A ) ⊂ W sp (Ω) is (relatively)compact.As an ambient space for the state manifold SM of problem (1.1) we considerthe product space C ( ¯Ω) n +1 × MH (Ω), due to continuity of velocity, temperatureand curvature.We define the state manifold SM as follows. SM := n ( u, θ, Γ) ∈ C ( ¯Ω) n +1 × MH : ( u, θ ) ∈ W − /pp (Ω \ Γ) n +1 , Γ ∈ W − /pp , div u = 0 in Ω \ Γ , θ > , u = ∂ ν θ = 0 on ∂ Ω , [[ ψ ( θ )]] + σH Γ = P Γ [[ µ ( θ )( ∇ u + [ ∇ u ] T ) ν Γ ]] = 0 on Γ , l ( θ ) = 0 on Γ , [[ d ( θ ) ∂ ν θ ]] ∈ W − /pp (Γ) o , Charts for these manifolds are obtained by the charts induced by MH (Ω), fol-lowed by a Hanzawa transformation as defined in Section 4.Applying Theorem 6.1 and re-parameterizing the interface repeatedly, we seethat (1.1) yields a local semiflow on SM . Theorem 8.1.
Let p > n +2 , σ > , and suppose ψ i ∈ C (0 , ∞ ) , µ i , d i ∈ C (0 , ∞ ) such that κ i ( s ) = − sψ ′′ i ( s ) > , µ i ( s ) > , d i ( s ) > , s ∈ (0 , ∞ ) , i = 1 , . Then problem (1.1) generates a local semiflow on the state manifold SM . Eachsolution ( u, θ, Γ) exists on a maximal time interval [0 , t ∗ ) , where t ∗ = t ∗ ( u , θ , Γ ) . Note that the pressure does not occur explicitly as a variable in the local semiflow,as the latter is only formulated in terms of the temperature θ , the velocity field u , and the free boundary Γ. The pressure π is determined for each t from ( u, θ, Γ)by means of the weak transmission problem( ∇ π |∇ φ ) L (Ω) = 2(div( µ ( θ ) D ) − u · ∇ u |∇ φ ) L (Ω) , φ ∈ H p ′ (Ω) , [[ π ]] = 2[[ µ ( θ ) D ]] on Γ . Concerning such transmission problems we refer to Section 8 in [11].Let ( u, θ,
Γ) be a solution in the state manifold SM with maximal interval ofexistence [0 , t ∗ ). By the uniform ball condition we mean the existence of a radius r > t ∈ [0 , t ∗ ), at each point x ∈ Γ( t ) there exists centers x i ∈ Ω i ( t ) such that B r ( x i ) ⊂ Ω i and Γ( t ) ∩ ¯ B r ( x i ) = { x } , i = 1 ,
2. Note thatthis condition bounds the curvature of Γ( t ), prevents parts of it to shrink to points,to touch the outer boundary ∂ Ω, and to undergo topological changes.With this property, combining the local semiflow for (1.1) with the Ljapunovfunctional and compactness we obtain the following result.
NCOMPRESSIBLE TWO-PHASE FLOWS WITH PHASE TRANSITIONS 25
Theorem 8.2.
Let p > n +2 , σ > , and suppose ψ i ∈ C (0 , ∞ ) , µ i , d i ∈ C (0 , ∞ ) such that κ i ( s ) = − sψ ′′ i ( s ) > , µ i ( s ) > , d i ( s ) > , s ∈ (0 , ∞ ) , i = 1 , . Suppose that ( u, θ, Γ) is a solution of (1.1) in the state manifold SM on its maxi-mal time interval [0 , t ∗ ) . Assume there is constant M > such that the followingconditions hold on [0 , t ∗ ) : (i) | u ( t ) | W − /pp n , | θ ( t ) | W − /pp , | Γ( t ) | W − /pp , | [[ d ( θ ( t )) ∂ ν u ( t )]] | W − /pp ≤ M < ∞ ; (ii) | l ( θ ( t )) | , θ ( t ) ≥ /M ; (iii) Γ( t ) satisfies the uniform ball condition.Then t ∗ = ∞ , i.e. the solution exists globally, and ω + ( u, θ, Γ) ⊂ E is non-empty.Proof. Assume that (i), (ii), and (iii) are valid. Then Γ([0 , t ∗ )) ⊂ W − /pp (Ω , r )is bounded, hence relatively compact in W − /p − εp (Ω , r ). Thus we may coverthis set by finitely many balls with centers Σ k real analytic in such a way thatdist W − /p − εp (Γ( t ) , Σ j ) ≤ δ for some j = j ( t ), t ∈ [0 , t ∗ ). Let J k = { t ∈ [0 , t ∗ ) : j ( t ) = k } ; using for each k a Hanzawa-transformation Ξ k , we see that the pullbacks { ( u ( t, · ) , θ ( t, · )) ◦ Ξ k : t ∈ J k } are bounded in W − /pp (Ω \ Σ k ) n +1 , hencerelatively compact in W − /p − εp (Ω \ Σ k ) n +1 . Employing now Corollary 6.2 weobtain solutions ( u , θ , Γ ) with initial configurations ( u ( t ) , θ ( t ) , Γ( t )) in the statemanifold on a common time interval, say (0 , τ ], and by uniqueness we have( u ( τ ) , θ ( τ ) , Γ ( a )) = ( u ( t + τ ) , θ ( t + τ ) , Γ( t + τ )) . Continuous dependence implies that the orbit of the solution ( u ( · ) , θ ( · ) , Γ( · )) isrelative compact in SM , in particular t ∗ = ∞ and ( u, θ, Γ)( R + ) ⊂ SM is relativelycompact. The negative total entropy is a strict Ljapunov functional, hence thelimit set ω + ( u, θ, Γ) ⊂ SM of a solution is contained in the set E of equilibria. Bycompactness ω + ( u, θ, Γ) ⊂ SM is non-empty, hence the solution comes close to E ,and stays there. (cid:3) Remark.
We can prove that each equilibrium is normally hyperbolic. Therefore,taking into account the results in [21], one may expect that each orbit satisfyingthe assumptions of Theorem 8.2 actually converges to an equilibrium. This topicwill be considered elsewhere.
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Preprint
Japan J. Mech. (1995), 1–41. Institut f¨ur Mathematik, Martin-Luther-Universit¨at Halle-Wittenberg, D-60120Halle, Germany
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