On a linearized Mullins-Sekerka/Stokes system for two-phase flows

We study a linearized Mullins-Sekerka/Stokes system in a bounded domain with various boundary conditions. This system plays an important role to prove the convergence of a Stokes/Cahn-Hilliard systemto its sharp interface limit, which is a Stokes/Mullins-Sekerka system, and to prove solvability of the latter system locally in time. We prove solvability of the linearized system in suitable $L^2$-Sobolev spaces with the aid of a maximal regularity result for non-autonomous abstract linear evolution equations.

This system arises in the construction of approximate solutions in the proof of convergence of a Stokes/Cahn-Hilliard system to its sharp interface limit, which is a Stokes/Mullins-Sekerka system, cf. [2]- [3]. Here v ± : t∈[0,T ] Ω ± (t) × {t} → R d and p ± : t∈[0,T ] Ω ± (t) × {t} → R are the velocity and pressure incompressible viscous Newtonian fluids filling the domains Ω ± (t) at time t, which are separated by the (fluid) interface Γ t . Furthermore, h : Γ 0 × [0, T ] → R is a linearized height function that describes the evolution of the interface at a certain order and µ ± : t∈[0,T ] Ω ± (t)× {t} → R is a linearized chemical potential related to the fluids in Ω ± (t). If one neglects the terms related to v ± , p ± , a similar linearized system arises in the study of the sharp interface limit of the Cahn-Hilliard equation, cf. [6]. Moreover, similar systems arise in the construction of strong solutions for a Navier-Stokes/Mullins-Sekerka system locally in time, cf. [4].
We consider different kinds of boundary conditions for v . and µ − simultaneously. More precisely, we assume that where Γ µ,1 , Γ µ,2 and Γ S,1 , Γ S,2 , Γ S,3 are disjoint and closed. Moreover, we have where n ∂Ω denotes the exterior normal on ∂Ω. To avoid a non-trivial kernel in the following we assume that one of the following cases holds true: for all v ∈ H 1 (Ω) d with v| Γ S,1 = 0, n ∂Ω · v| Γ S,2 = 0, cf. [5,Corollary 5.9]. The structure of this contribution is as follows: In Section 2 we summarize some preliminaries on the parametrization of the interface Γ t and non-autonomuous evolution equations. In Section 3 we present and prove our main results on existence and smoothness of solutions to the linearized Mullins-Sekerka system. Finally, in the appendix we prove an auxilliary result on the existence of a pressure.
The results of this paper are extensions of results in the second author's PhD Thesis.

Notation
Throughout this manuscript we denote by ξ ∈ C ∞ (R) a cut-off function such that

Coordinates
We will parametrize (Γ t ) t∈[0,T 0 ] with the aid of a family of smooth diffeomorphisms X 0 : Γ 0 × [0, T 0 ] → Ω such that d s X 0 (s, t) has full rank for all s ∈ Γ 0 , t ∈ [0, T 0 ]. Here d s X 0 (s, t) is the differential of the mapping and n(s, t) normal vectors on Γ t at X 0 (s, t) such that τ 1 (s, t), . . . , τ d−1 (s, t), n(s, t) is a positively oriented orthonormal basis of R d . Furthermore, P τ = P τ (s, t) denotes the orthogonal projection onto the tangent space of Γ t at X 0 (s, t) and can be represented as P τ = I − n(s, t) ⊗ n(s, t).
We choose the orientation of Γ t (induced by X 0 (·, t)) such that n(s, t) is the exterior normal with respect to Ω − (t). Moreover, we denote n Γt (x) := n(s, t) for all x = X 0 (s, t) ∈ Γ t .
Furthermore, V Γt and H Γt should be the normal velocity and (mean) curvature of Γ t (with respect to n Γt ) and we define In the following we will need a tubular neighborhood of Γ t : For δ > 0 sufficiently small, the orthogonal projection P Γt (x) of all is well-defined and smooth. Moreover, we choose δ so small that dist(∂Ω, Γ t ) > 3δ for every t ∈ [0, T 0 ]. Every x ∈ Γ t (3δ) has a unique representation For the following we define for δ ′ ∈ (0, 3δ] We introduce new coordinates in Γ(3δ) which we denote by for all (x, t) ∈ Γ(3δ), (q, t) ∈ Γ, resp., cf. Chen et al. [8,Section 4.1]. Moreover, we define for all (x, t) ∈ Γ(3δ).
In the case that h is twice continuously differentiable with respect to s and continuously differentiable with respect to t, we introduce the notations

Maximal Regularity for Non-autonomous Equations
In order to prove our main result we use of the theory of maximal regularity for non-autonomous abstract evolution equations. Therefore, we give a short overview of the basic definitions and results which we will use. These are taken from [7] and all the proofs of the statements can be found in that article.
In this subsection let X and D be two Banach spaces such that D is continuously and densely embedded in X.
2. Let T > 0 and A : [0, T ] → L (D, X) be a bounded and strongly measurable function. Then A has L p -maximal regularity and we write It can be shown that if A ∈ MR p for some p ∈ (1, ∞) then A ∈ MR p for all p ∈ (1, ∞). Hence, we often simply write A ∈ MR.
A very important tool for proving maximal regularity properties of different operators are perturbation techniques. Employing these can often help to show maximal regularity for a variety of operators by separating them into a main part (for which maximal regularity can be readily shown) and a perturbation.
In the following we give a perturbation result which is key to many results in the next chapter. 3 Main Results

Parabolic Equations on Evolving Surfaces
We introduce the space for T ∈ (0, ∞), where we equip X T with the norm hold for some constant C > 0 independent of µ and h.
Proof. We may write (15) in abstract form as . Now we fix t 0 ∈ [0, T ] and analyze the operator A(t 0 ), where we replace t with the fixed t 0 in all time dependent coefficients.
In order to understand this operator we define In the literature the concatenation B t 0 • S N t 0 is often referred to as the Dirichletto-Neumann operator and A 0 (t 0 ) := B t 0 • S N t 0 • D t 0 is called the Mullins-Sekerka operator. It can be shown that has L p -maximal regularity, i.e., A 0 ∈ MR p (0, T ). We will not prove this in detail but just give a short sketch describing the essential ideas: first, a reference surface Σ ⊂⊂ Ω is fixed such that Γ t can be expressed as a graph over Σ for t in some time interval t ,t + ǫ ⊂ [0, T ]. e.g. one may choose Σ := Γ 0 and then determine ǫ 0 > 0 such that Γ t may be written as graph over Γ 0 for all t ∈ [0, ǫ 0 ], which is possible since Γ is a smoothly evolving hypersurface. Next, a Hanzawa transformation is applied, enabling us to consider (18c) as a system on fixed domains Ω ± and Σ, but with time dependent coefficients (see e.g. [4,Chapter 2.2] or and [12,Chapter 4]). Here, Ω + , Ω − and Σ denote disjoint sets such that ∂Ω + = Σ and Ω = Ω + ∪ Ω − ∪ Σ holds and we assume in the following that t 0 ∈ [0, ǫ 0 ]. To be more specific, the Hanzawa transformation results in a system of the form where a is the transformed Laplacian, depending smoothly on t andf is the transformation of f . Applying the Hanzawa transformation (and the diffeomorphism X 0 ) also to the operators D t 0 and B t 0 , we end up with a transformed operator A 0 (t 0 ) ∈ L H 7 2 (Σ), H 1 2 (Σ) and [11, Corollary 6.6.5] implies thatÃ 0 (t 0 ) has L pmaximal regularity. As all involved differential operators and coefficients depend smoothly on t, it is possible to show thatÃ 0 : [0, ǫ 0 ] → L H 7 2 (Σ), H 1 2 (Σ) is relatively continuous. Therefore Theorem 2.3 impliesÃ 0 ∈ MR p (0, ǫ 0 ) and, transforming back, also A 0 ∈ MR p (0, ǫ 0 ). Repeating this procedure with a new reference surface Σ := Γ ǫ 0 and iteratively continuing the argumentation, we end up with A 0 ∈ MR p (0, T ).
We proceed by showing that A(t 0 ) = A 0 (t 0 ) + B(t 0 ) holds for some lower order perturbation B. We introduce and elliptic regularity theory implies andμ + ≡ 0 in Ω + (t 0 ). For the further argumentation, we show To this end let γ(x) := ξ(4d B (x)) for all x ∈ Ω, where ξ is a cut-off function satisfying (11). In particular suppγ ∩ Γ t = ∅ for all t ∈ [0, T 0 ] by our assumptions and γ ≡ 1 in ∂Ω( δ 4 ). Denotingμ := γµ − N ∈ H 2 (Ω − (t 0 )), we compute using ∆µ − N = 0 in Ω − (t 0 ) thatμ is a solution to which, again regarding elliptic regularity theory, implies μ ) . This is essential in view of (21) as it leads to where we used the continuity of the trace operator tr : H 2 (Ω − (t 0 )) → H where we employed the continuity of the trace in the first line, (20) in the second, (21) in the third and the definition of µ − N in the fourth. As H which shows in regard to (19) that we may view B t 0 • S D t 0 • D t 0 as a lower order perturbation of A 0 (t 0 ).
Next we take care of the term involving b 2 in (16b). For this we consider the operator We estimate where C > 0 can be chosen independent of h and t 0 ∈ [0, T ]. Here we again employed the continuity of the trace operator and elliptic theory. Defining withb := b · ∇S, and using (23) and (22), we find that By elliptic theory for almost all t ∈ [0, T ] and thus µ ± L 6 (0,T ;H 1 (Ω ± (t))) ≤ C h L 6 0,T ;H holds.
Proof. We can assume for simplicity that g = 0 on Γ S,1 and n ∂Ω · g = 0 on Γ S,2 .
Otherwise we substract a suitable extension of g. As a first step, we reduce the system (25)-(29) to the case s = 0. Elliptic theory implies that the equation n ∂Ω · ∇q = n ∂Ω · g 2 on Γ S,1 ∪ Γ S,2 , q = 0 on Γ S,3 has a unique solution q ∈ H 3 (Ω − (t)) with´Ω − (t) q dx = 0 if Γ S,3 = ∅ since s ∈ H 3 2 (Γ t ) d and n ∂Ω ·g 2 ∈ H 3 2 (Γ S,1 ∪Γ S,2 ). Here, if Γ S,3 = ∅, the necessary compatibility condition is satisfied because of (24). Moreover, we have the estimate Regarding the tangential part of s, we may solve the stationary Stokes system We may find a solution (w,p) ∈ H 2 (Ω − (t)) d × H 1 (Ω − (t)) (made unique by the normalization´Ω − (t)p dx = 0) and also get the estimate Thus, definingw := w + ∇q, the couple (w,p) solves and may be estimated by s in strong norms. Next, let where the regularity is due to the properties of the trace operator. Then, for every strong solution (v ± ,p ± ) of (25)-(29), with s ≡0 and g, a substituted byg,ã, the functions v + , p + := v + ,p + and v − , p − := v − +w,p − +p are solutions to the original system (25)-(29). So, we will consider s ≡ 0 in the following and show existence of strong solutions in that case.
Proof. We show this by a perturbation argument. First of all note that we may without loss of generality assume that a 1 , a 3 , a 4 , a 1 , a 2 , a 5 , a 6 = 0 on their respective domains. The above system may be reduced to this case by solving with the help of standard elliptic theory and on Γ S,j , j = 1, 2, 3, with the help of Theorem 3.2 and settinĝ be the solution to (44)-(48). Multiplying (44) by v ± h and integrating in Ω ± (t) together with integration by parts and the consideration of the boundary values (47) and (48) allows us to deducê Hence, by [5,Corollary 5.8] and the continuity of the trace we find for C independent of h and t. [5, Corollary 5.8], also implieŝ due to v + h = v − h on Γ t , (50) and (49). Defining · n Γt , we may use (50) and (51) to confirm for C > 0 independent of h and t. As H 7 2 (Γ 0 ) is dense in H 2 (Γ 0 ) we can extend B(t) to an operator B(t) : H 2 (Γ 0 ) → H 1 2 (Γ 0 ) and H 2 (Γ 0 ) is close to H 1 2 (Γ 0 ) compared with H 7 2 (Γ 0 ). The existence of a unique solution h ∈ X T with the properties stated in the theorem is now a consequence of Theorem 2.5. Higher regularity may be shown by localization and e.g. the usage of difference quotients. F (ψ) = 0 for all ψ ∈ V (Ω) = {ψ ∈ H 1 (Ω) d : div ψ = 0, ψ| Γ S,1 = 0, n ∂Ω ·ψ| Γ S,2 = 0}.
Then T is onto, which can be seen as follows: Let g ∈ Y .
This proves the statement of the lemma.