On well-posedness of incompressible two-phase flows with phase transitions: the case of non-equal densities

The basic model for incompressible two-phase flows with phase transitions consistent with thermodynamics is studied. The latter means that the total energy is conserved and the total entropy is nondecreasing. We consider the case of constant but non-equal densities of the phases, complementing our previous paper (Prüss et al. in Evol Equ Control Theory 1:171–194, 2012) where the case of equal densities is analyzed. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity, in a configuration where the interface is nearly flat and initial data are small.


Introduction
Let ⊂ R n+1 be a bounded domain of class C 3− , n ≥ 1. contains two phases: at time t, phase k, k = 1, 2, occupies subdomain k (t) of . Assume ∂ 1 (t) ∩ ∂ = ∅; this means no boundary intersection and no contact angles. The closed compact hypersurface (t) := ∂ 1 (t) ⊂ forms the interface between the phases.
Let u denote the velocity field, π the pressure field, T (u, π, θ) the stress tensor, D(u) = (∇u + [∇u] T )/2 the rate of deformation tensor, θ the (absolute) temperature field, ν the outer normal of 1 , u the interface velocity, V = u · ν the normal velocity of (t), H = H ( (t)) = −div ν the curvature of (t), j the phase flux, and the jump of a quantity v across (t).
Several quantities are derived from the specific free energy ψ k (θ ) in phase k as follows. Further, d k (θ ) > 0 denotes the coefficient of heat conduction in Fourier's law, μ k (θ ) > 0 the viscosity in Newton's law, and σ > 0 the constant coefficient of surface tension.
Concerning the second equation of (1.3), we remind that balance of mass across (t) requires [[ρ(u − u )]] · ν = 0, which implies and so Therefore, this equation is well-defined on (t). This model is explained in more detail in our previous paper [19], where we consider the case of equal densities. It has been recently proposed by Anderson et al. [1], see also the monographs by Ishii [12] and Ishii and Takashi [13], and it is thermodynamically consistent in the sense that in the absence of exterior forces and heat sources, the total energy is preserved and the total entropy is nondecreasing, see [19]. It is in some sense the simplest sharp interface model for incompressible Newtonian two-phase flows taking into account phase transitions driven by temperature.
Note that in the case of equal densities, the phase flux j does not enter (1.1), and so in this case, we obtain essentially a Stefan problem with surface tension, which is only weakly coupled to the standard two-phase Navier-Stokes problem via temperature-dependent viscosities. We call this case temperature dominated, and it has been studied in [19]. But in the case of different densities, the phase flux j causes a jump in the velocity field on the interface, which leads to so-called Stefan currents that are convections driven by phase transitions. In this situation, it turns out that the heat problem (1.2) is only weakly coupled to (1.1) and (1.3), we call this case velocity dominated. The resulting two-phase Navier-Stokes problem is non-standard, and therefore, it requires a new analysis.
The analytical properties of the problem appear to be different in these two cases. The spaces for well-posedness are not the same, and in the velocity-dominated case, the pressure is uniquely determined, while in the temperature-dominated case, it is only unique up to a constant. In the temperature-dominated case [[ρ]] = 0, the phase flux j can be eliminated by solving the second equation in (1.2) for j. This yields as long as l(θ ) = 0; this is the essential well-posedness condition in this case. Then, the equation describing the evolution of the interface becomes On the other hand, in the velocity-determined case [[ρ]] = 0, we can eliminate j by taking the inner product of the fourth equation in (1.1) with ν to the result In this case, the equation for V becomes which does not contain temperature, in contrast to the first case. Therefore, the analysis for these two cases necessarily is different, too.
There is a large literature on isothermal incompressible Newtonian two-phase flows without phase transitions [2,15,22,23,25,26], and also on the two-phase Stefan problem with surface tension modeling temperature driven phase transitions [3,8,18,21,24]. On the other hand, mathematical work on two-phase flow problems including phase transitions is rare. In this direction, we only know the papers by Hoffmann and Starovoitov [10,11] dealing with a simplified two-phase flow model, and Kusaka and Tani [16,17] which is two-phase for temperature but only one phase is moving. The papers of Di Benedetto and Friedman [4] and Di Benedetto and O'Leary [5] deal with weak solutions of conduction-convection problems with phase change. However, none of these papers considers models which are consistent with thermodynamics.
It is the purpose of this paper to present a rigorous analysis of problem (1.1), (1.2), (1.3) in the framework of L p -theory in the case of non-equal densities and an initial interface which is nearly flat. We consider the nonlinear problem (1.1)-(1.3) for = R n+1 and a nearly flat interface represented as a graph over R n , namely in the regions We let 0 = 1 (0) ∪ 2 (0) and ν 0 be the outer normal of 1 (0).
Then, given any finite interval J = [0, a], there exists η > 0 such that (1.1)-(1.3) admits a unique L p -solution on J provided the smallness conditions The notion L p -solution is explained in more detail in Section 5. For a proof of this result, we perform a detailed analysis of the linearized problem in an L p -setting, following the approach in [22] for the standard two-phase Navier-Stokes problem without phase transitions. This requires the detection and analysis of the underlying boundary symbol. We then show maximal regularity for the linear part of the problem and finally employ the contraction mapping principle to solve the nonlinear problem. In a forthcoming paper, we will consider problem ( 3) in general geometries without smallness assumptions.
The plan for this paper is as follows. In Sect. 2, we transform the problem to the configuration of a fixed flat interface. The principal part of the linearization is studied in Sect. 3, and the property of maximal L p -regularity is proved in Sect. 4. The last section contains the proof of well-posedness for the nonlinear problem.

Transformation to a flat interface
In the situation of a nearly flat interface, the nonlinear problem (1.1)-(1.3) can be transformed to a problem onṘ n+1 := R n+1 \ [R n × {0}] by means of the transformations v(t, x, y) := (u 1 , . . . , u n ) T (t, x, y + h(t, x)), where t ∈ J = [0, a], x ∈ R n−1 , y ∈ R, y = 0. Here, θ ∞ > 0 denotes the (equilibrium) temperature at infinity and π ∞ the corresponding (equilibrium) pressure at infinity defined by the relations With a slight abuse of notation, we will denote in the sequel the transformed velocity again by u, that is, u = (v, w) T , the transformed temperature by θ , and the transformed pressure by π . For given initial data u 0 (x) and θ 0 (x), we set again u 0 (x, y) : With this notation, we have the transformed problem where the phase flux j already has been eliminated, according to Sect. 1. Here, it reads The nonlinear right-hand sides are defined by The curvature of (t) is given by where ∇ 2 h denotes the Hessian of h.

The linear problem
The principal part of the linearized problem in the case of a nearly flat initial interface reads as follows 2) decouples from the remaining problem. Since it is well-known that this problem has maximal L p -regularity, we concentrate on the remaining one. It reduces to two separate problems. With u = (v, w) T , the first one is the following non-standard Stokes problem.
Having solved the first one, the second one results from replacing g w by g w +σ x h and solving Before stating maximal regularity results of linear problems, let us introduce the relevant function spaces. Let ⊂ R m be open and X be an arbitrary Banach space. By L p ( ; X ) and H s p ( ; X ), for 1 ≤ p ≤ ∞, s ∈ R, we denote the X -valued Lebesgue and the X -valued Bessel potential spaces of order s, respectively. We will also make use of the fractional Sobolev-Slobodeckij spaces W s The spaces 0 H s p (J ; X ) are defined analogously. We remind that H k p = W k p for k ∈ N and 1 < p < ∞ and that W s p = B s pp for s > 0, s ∈ N. For s ∈ R and 1 < p < ∞, we consider the homogeneous Bessel-potentianl spacė where S (R n ) denotes the space of all tempered distributions, andİ s is the Riesz potential given bẏ For s ∈ R \ Z, the homogeneous Sobolev-Slobodeckij spacesẆ s p (R n ) of fractional order can be obtained by real interpolation aṡ where (·, ·) θ, p is the real interpolation functor. For problem (3.4), we have the following maximal regularity result.
if and only if the data ( f u , f d , g v , g w , g j , g π , u 0 ) satisfy the following regularity and compatibility conditions: In addition, for the pressure traces π k on the interface we have if and only if is continuous between the corresponding spaces.
For the problem (3.1) and (3.3), we also have maximal regularity result in the L p -setting.
if and only if the data ( f u , f d , g u , g j , g π , g h , u 0 , h 0 ) satisfy the following regularity and compatibility conditions: The solution map [( f u , f d , g u , g j , g π , g h , u 0 , h 0 ) → (u, π, h)] is continuous between the corresponding spaces. Equation (3.2) is a two-phase heat problem with Neumann condition, which has the property of the maximal L p -regularity (cf. [8]). Therefore, the linearized problem (3.1)-(3.3) has the property of maximal L p -regularity as well. To state this result, we introduce appropriate function spaces. We set and define the solution space for (3.1)-(3.3) as We denote by γ π the two one-sided traces of π on R n . The generic elements of E(a) are functions (u, π, γ π, θ, h).E(a) is a Banach space with norm Moreover we set and define the regularity of the data space for (3.1)-(3.3) as F(a) is a Banach space with its natural norm, and the generic elements of F(a) are functions ( f u , f d , g u , g j , f θ , g θ , g π , g h ). Finally, we define the time trace space X γ of E(a) as The main result on the linearized problem (3.1)-(3.3) now can be stated as THEOREM 3.3. Let 1 < p < ∞ be fixed, p = 3/2, 3, and assume that ρ k and μ k are positive constants for k = 1, 2, ρ 2 = ρ 1 , κ, d > 0. (u 0 , θ 0 , h 0 ) and ( f u , f d , g u , g j , f θ , g θ , g π , g h ) satisfy the regularity conditions

and the compatibility conditions
is continuous between the corresponding spaces.

Proofs for Theorems 3.1 and 3.2
For necessity, we employ trace arguments as in [15,22]. The more difficult part of sufficiency also follows the lines of these papers, but has to be modified as it is more involved.
1. In order to remove u 0 which has a jump at the interface, we first solve the parabolic problem where E R n+1 g j := (g j , 0) T , to meet the necessary compatibility conditions, and set π 1 = 0. Then, Next, to remove the divergence data we solve the problem According to [22], this problem has a unique solution in the maximal L p -regularity class. Note that the compatibility conditions do not involve the normal part of the stress boundary condition. Therefore, it remains to study the problem This way the problem for h is decoupled from the Stokes problem where we have set g 1 = σ x h and u = (v, w). Having solved this problem for given h we insert into and solve the remaining equation for h, which reads.
Here, the remaining data satisfy

2.
Assume for a moment that we have a solution of (4.5) in the proper regularity class even on the half-line J = R + . Then, we may employ the Laplace transform in t and the Fourier transform in the tangential variables x ∈ R n , to obtain the following boundary value problem for a system of ordinary differential equations onṘ : This system of equations is easily solved to the result for y > 0, and for y < 0. Here a k ∈ C n and α k ∈ C have to be determined by the interface conditions which in frequency domain readv Inserting the representation of the transformed solution into the first two of these equations, we obtain the following system.
Using the formulas for β k and solving the resulting system in terms of α k , we arrive after some elementary algebra at the expressions where we have set Here, we observe that the surface pressure π k have transforms λα k . Since the entries in the matrix defining λα k are bounded and holomorphic, we may conclude that π k have the same regularity as g k , and that the pressure π belongs to L p (J ;Ḣ 1 p (Ṙ n+1 )). Next, let us compute the boundary velocities v b w k (x, 0). Their transforms are given bŷ Some algebra yield for w b k the representation This representation shows thatŵ b k is bounded by |ξ |ĝ i /ω 1 ω 2 . As in [22], Section 4, we obtain that the operator with symbol |ξ |/ω 1 ω 2 maps L p (J ;Ẇ 1−1/ p p (R n )) into the right space for the boundary values of w, i.e., we have To keep this paper self-contained, we prove the mapping properties stated above. We set G : . It is well-known that G is closed, invertible and sectorial with angle π/2, and −G is the generator of a C 0 -semigroup of contractions in L p (R n ). Moreover, G admits an H ∞ -calculus in X with H ∞ -angle π/2 as well; see e.g. [9]. The symbol of G is λ, the time covariable.
Next, we set D n := − , the Laplacian in L p (R n ) with domain D(D n ) = H 2 p (R n ). It is well-known that D n is closed and sectorial with angle 0, and it admits a bounded H ∞ -calculus which is even R-bounded with RH ∞ -angle 0; see e.g. [6]. These results also hold for the canonical extension of D n to X , and also for the fractional power is |ξ |, where ξ is the covariable of x. By the Dore-Venni theorem for sums of commutating sectorial operators (cf. [7]), we see that L k := ρ k G + μ k D n with natural domain are closed, invertible and sectorial with angle π/2.L k also admits a bounded H ∞calculus in X with H ∞ -angle π/2 (cf. [20]). The same results are valid for operators k , their H ∞ -angle is π/4, and their domains are The symbol of L k is ρ k λ + μ k |ξ | 2 and that of F k is ρ k λ + μ k |ξ | 2 .F −1 k have the following mapping properties: hence, inserting the expressions for α k and β k , we obtain after some more algebra Here g 4 is determined by the data alone and has the same regularity as g 3 . We set τ = |ξ |. The boundary symbol s(λ, τ ) is defined by (4.14) where we employed the scaling z = λ/τ 2 . The holomorphic function m(z) in turn is given by , (4.15) with the abbreviations .
We derive the formula in the "Appendix". Note that ω k (z) is holomorphic in the sliced plane C \ (−∞, −μ k /ρ k ]; hence, the function ϕ k (z) has this property as well. This function has exactly one zero z k in this set, it is real and satisfies −μ k /ρ k < z k < −8μ k /9ρ k < 0. It is easy to see that ϕ k mapsC + into C + , and as ϕ k (0) = 2 and ϕ k (z) ∼ √ ρ k z/μ k as z → ∞, we see that ϕ k (C + ) ⊂ φ k , for some angle φ k < π/2.
On the other hand, choosing |λ| ≥ C|τ | we obtain If λ 0 is chosen large enough, this implies the lower bound In order to economize our notation, we set z = (u, π, γ π, θ, h) for (u, π, γ π, θ, h) ∈ E(a). and set z 0 = (u 0 , θ 0 , h 0 ) for (u 0 , θ 0 , h 0 ) ∈ X γ . With this notation, the nonlinear problem (2.1) can be recast as where L denotes the linear operator on the left hand side of (2.1), and N denotes the nonlinear mapping on its right-hand side.
In the following, we say that a function space is a multiplication algebra if it is a Banach algebra under the operator of multiplication. LEMMA 5.2. (Lemma 6.1 in [22]) Suppose p > n + 3. Then, γ π (a), G j (a), G u (a), G θ (a), G π (a), and G h (a) are multiplication algebras.
Concerning the nonlinearity N , we have the following result. Proof. This result is proved in a similar way as Proposition 6.2 in [22]. Now we prove Theorem 5.1, where a > 0 is a fixed life time.
Step 1. First, we reduce the problem to initial values 0 and resolve the compatibility conditions. Thanks to Theorem 6.3 in [22], we find an extension f * d ∈ F d (a) which satisfies f * d (0) = div u 0 . We define and set f * u = f * θ = 0. With these extensions, by Theorem 3.3, we may solve the linear problem (3.1)-(3.3) with initial data (u 0 , θ 0 , h 0 ) and inhomogeneities ( f * u , f * d , g * u , g * j , f * θ , g * θ , g * π , g * h ) which satisfy the required regularity conditions, and by construction, the required compatibility conditions, to obtain a unique solution z * = (u * , π * , γ π * , θ * , h * ) ∈ E(a) with u * (0) = u 0 , θ * (0) = θ 0 , and h * (0) = h 0 . As the solution map of Theorem 3.3 is continuous, we know that for any r > 0, there exists η > 0 such that z 0 X γ ≤ η ⇒ z * E(a) ≤ r. (5.4) Step 2. We rewrite problem ( We may assume that M ≥ 1. Thanks to Proposition 5.3 and due to K (0) = N (z * ) − Lz * , we may choose first r > 0 and then η > 0 sufficiently small such that for all z ∈ 0 E(a) with z E(a) ≤ r , hence which ensures that L −1 K : B 0 E(a) (0, r ) → B 0 E(a) (0, r ) is a contraction. Thus, we may employ the contraction mapping principle to obtain a unique solution on the fixed time interval [0, a]. As the map z 0 → z * is continuous, the solution map z 0 → z is continuous as well. This completes the proof of Theorem 5.1.