Two-phase Stokes flow by capillarity in the plane: The case of different viscosities

We study the two-phase Stokes flow driven by surface tension for two fluids of different viscosities, separated by an asymptotically flat interface representable as graph of a differentiable function. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single and double layer potential, spectral results on the corresponding integral operators, and abstract results on nonlinear parabolic evolution equations.


Introduction
In the context of boundary value problems involving elliptic constant-coefficient PDE's like the Laplace equation or the Stokes system, it is often natural to consider two-phase problems in unbounded domains, where the same equation has to be solved on both sides of the boundary, and the boundary conditions typically are of "transmission" type, i.e. they relate limits of the solutions from both sides.The method of layer potentials is a classical technique which is intrinsically suited to such settings.Typically, this method reduces the boundary value problem to a linear, singular integral equation (or system of such equations) on the boundary of the domain, on the basis of well-known jump relations for these potentials across the boundary.
The first applications of layer potentials in the analysis of moving boundary problems of the type described above are from the 1980s, for problems of Hele-Shaw or Muskat type [8] (see also the recent surveys [11,12] on further developments) as well as for Stokes flow problems [5].In these applications, the interfaces are represented as graphs of a time dependent function [f → f (t)], with f (t) ∈ C(R), for which an evolution equation can be derived.This equation involves singular integral operators originating from the layer potential, depending nonlinearly and nonlocally on f (t).However, in suitable geometries this nonlinearity can be described rather explicitly, and technicalities resulting from transforming the problem to a fixed reference domain can be avoided.More precisely, the operators determining the evolution belong to a class discussed in Section 3 below, and results are available concerning mapping properties, smoothness, localization etc. of the operators in this class.
After reducing the moving boundary problem to an evolution equation for f , this equation has to be analyzed.Initially, various approaches have been used that necessitated rather restrictive assumptions on the initial data.Recently, however, more general, in some sense optimal existence, uniqueness, and smoothness results have been obtained.One of the crucial tools for this has been the meanwhile well-developed and versatile abstract theory of nonlinear parabolic evolution equations, cf.[2,14,19].
This paper discusses, along the lines sketched above, the moving boundary problem of twophase Stokes flow in full 2D space driven by surface tension forces on the interface between the two phases.More precisely, we seek a moving interface [t → Γ(t)] between two liquid phases Ω ± (t), and corresponding functions v ± (t) : Ω ± (t) −→ R 2  and p ± (t) : Ω ± (t) −→ R, representing the velocity and pressure fields in Ω ± (t), respectively, such that the following equations are satisfied: Here ν is the unit exterior normal to ∂Ω − (t) and κ denotes the curvature of the interface.Moreover, T µ (v, p) = (T µ,ij (v, p)) 1≤i, j≤2 denotes the stress tensor that is given by and [v] (respectively [T µ (v, p)]) is the jump of the velocity (respectively stress tensor) across the moving interface, see (2.2) below.The positive constants µ ± and σ denote the viscosity of the liquids in the two phases and the surface tension coefficient of the interface, respectively.We assume that so that Γ(t) is a graph over the real line.Equation (1.1a) 6 determines the motion of the interface by prescribing its normal velocity V n as coinciding with the normal component of the velocity at Γ(t), i.e. the interface is transported by the liquid flow.The interface Γ(t) is assumed to be known at time t = 0: In the previous paper [17], the authors considered Problem (1.1a) in the case of equal viscosities µ ± = µ.In that case, the solution to the fixed-time problem (1.1a) 1 -(1.1a) 5 can be directly represented as a hydrodynamic single-layer potential [13] with density −σκν, and the resulting evolution equation represents the time derivative of f as a nonlinear singular integral operator acting on f .
If µ + = µ − this is not feasible.Instead, we first transform the unknowns such that the same equation holds in both phases, introducing thereby a jump across the interface for the transformed velocity field.In Proposition 5.1, we show that the corresponding fixed-time Stokes problem is uniquely solvable, and we represent the solution by a sum of a hydrodynamic single layer and a double layer potential.While the single layer potential is generated by the same density as in the case of equal viscosities, the density β for the double layer potential is found from solving a linear, singular integral equation of the second kind, cf.(5.8).As Γ(t) is unbounded we cannot rely on compactness arguments to show the solvability of this equation.Instead, we modify arguments from [7,10] to obtain the necessary information on the spectrum of the corresponding integral operator via a Rellich identity.Moreover, we also rely on a further Rellich identity used in [15] in the study of the Muskat problem.
The solution to the fixed-time problem is then used in the formulation of an evolution equation for f , (cf.(5.9), (5.17), (5.18)) whose investigation will yield the following main result.Here and further, H s (R) := W s 2 (R) denotes the usual Sobolev spaces of integer or noninteger order.Theorem 1.1.Let s ∈ (3/2, 2) be given.Then, the following statements hold true: where Moreover, the set is a solution to (1.1).This identifies H 3/2 (R) as a critical space for the evolution problem (1.1).Hence, Theorem 1.1 covers all subcritical spaces.1.1.Outline.The paper is structured as follows: In Section 2 we discuss a two-phase Stokes problem with equal viscosities in both phases where the normal stresses are continuous across the interface and the velocity has a prescribed jump there.In fact, the problem is solved by the hydrodynamic double layer potential generated by that jump.Although the boundary behavior of this potential is well-known, we prove the results on this in Appendix A as they do not seem directly available in the literature for our unbounded geometry.
As we rely on the solvability of singular integral equations of the second kind arising from the hydrodynamic double-layer potential, the spectrum of the corresponding operator is investigated in Sections 3 and 4, first in L 2 (R) 2 and then in H s (R) 2 , with s ∈ (3/2, 2), and H 2 (R) 2 .The main technical tools in the latter cases are shift invariances and commutator properties for singular integral operators of the type discussed here.In Section 5 we reformulate the moving boundary problem (1.1) as a nonlinear and nonlocal evolution equation problem, cf.(5.17).Finally, in Section 6 we carry out the linearization of (5.17) and locally approximate the linearization by Fourier multipliers.This enables us to identify the parabolic character of the evolution equation and to prove our main result by invoking abstract results on equations of that type from [14].
1.2.Notation.Slightly deviating from the usual notation, if E 1 , . . ., E k , F , k ∈ N, are Banach spaces, we write L k (E 1 , . . ., E k ; F ) for the Banach space of bounded k-linear maps from i E i to F .Given Banach spaces X and Y , we let L k sym (X, Y ) ⊂ L k (X, . . ., X; Y ) denote the space of k-linear, bounded symmetric maps A : X k → Y .Moreover, C −1 (E, F ) will denote the space of locally Lipschitz continuous maps from a Banach space E to a Banach space F .Given k ∈ N, we further let C k (R) denote the Banach space of functions with bounded and continuous derivatives up to order k and C k+α (R), α ∈ (0, 1), is its subspace consisting of functions with α-Hölder continuous kth derivative whose α-Hölder modulus is bounded.

An auxiliary fixed-time problem
As a preparation for solving the boundary value problem (1.1a) 1 − (1.1a) 5 for fixed time, in this section we consider the related Stokes problem (2.3) with equal viscosities normed to 1.The unique solvability of (2.3) is established in Proposition 2.1 below and in Appendix A.
In this section, f ∈ H 3 (R) is fixed.We introduce the following notation: Note that Γ is the image of R under the diffeomorphism Further, let ν and τ be the componentwise pull-back under Ξ of the unit normal ν on Γ exterior to Ω − and of the unit tangent vector τ to Γ, that is For any function z defined on R 2 \ Γ we set z ± := z| Ω ± and if z ± have limits at some point (ξ, f (ξ)) ∈ Γ we will write z ± (ξ, f (ξ)) for the limits, and we set For notational brevity we introduce the function space For given β = (β 1 , β 2 ) ⊤ ∈ H 2 (R) 2 we seek solutions (w, q) ∈ X to the Stokes problem For the construction of the solution to (2.3), let us first point out that for any smooth solution (U, P ) : where E is a domain in R 2 , the functions given by are solutions to (2.4) as well.In particular, if E = R 2 \ {0} and are the fundamental solutions to the Stokes equations (2.4), given by for y = (y 1 , y 2 ) ∈ R 2 \ {0}, we obtain a system of solutions to the homogeneous Stokes equations given by We are going to show that (w, q) := (w, q)[β] given by for x ∈ R 2 \ Γ and with r := r(x, s) := x − (s, f (s)) solves (2.3).Here and further, we sum over indices appearing twice in a product.We write this more explicitly as (2.8) The solution (w, q) is the so-called hydrodynamic double-layer potential generated by the density β • Ξ −1 on Γ, see [13].
It is given by (2.6), (2.7).Moreover, Proof.The uniqueness of the solution can be shown as in the proof of [17,Theorem 2.1].
Observe that w and q are defined by integrals of the form where for every α ∈ N 2 we have ∂ α x K(x, s) = O(s −1 ) for |s| → ∞ and locally uniformly in x ∈ R 2 \ Γ.This shows that w and q are well-defined by (2.6) and (2.7), and that integration and differentiation with respect to x may be interchanged.As (W i,k , Q i,k ) solve the homogeneous Stokes equations, this also holds for (w, q).
To show the decay of q at infinity we obtain from the matrix equality via integration by parts In view of this representation, [15, Lemma 2.1] implies q(x) → 0 as |x| → ∞.
In order to prove the decay of w we rewrite where I ∈ R 2×2 is the identity matrix.Now [15, Lemma 2.1] and [17,Lemma B.2] imply that indeed w(x) → 0 for |x| → ∞.
The boundary conditions (2.3) 3 and (2.3) 4 together with the properties that (w, q) ∈ X and

The L 2 -resolvent of the hydrodynamic double-layer potential operator
In this section we study the resolvent set of the hydrodynamic double-layer potential operator D(f ), with f ∈ C 1 (R), introduced in (3.5) below, which we view in this section as an element of L(L 2 (R) 2 ).The main result of this section is Theorem 3.3 below which provides in particular the invertibility of λ − D(f ) for λ ∈ R with |λ| > 1/2.
To begin, we introduce a general class of singular integral operators suited to our approach via layer potentials, cf.[16,17].Given n, m ∈ N and Lipschitz continuous functions a 1 , . . ., a m , b 1 , . . ., b n : R −→ R, we let B n,m denote the singular integral operator where PV R denotes the principal value integral and δ 2) In this section we several times use the following result.Lemma 3.1.There exists a constant C depending only on n, m, and max i=1,...,m a ′ i ∞ with We note that D(f ) is related to the B n,m via Moreover, up to the sign and the push-forward via Ξ, D(f Using the same notation, we define the singular integral operators B 1 (f ) and B 2 (f ) by , play an important role also in the study of the Muskat problem, cf.[15].Lemma 3.1 implies that also is the direct value of the double layer potential for the Laplacian corresponding to the density θ in (ξ, f (ξ)) ∈ Γ.
We are going to prove in Theorem 3.3 below that the resolvent sets of D(f ) and D(f ) * contain all real λ with |λ| > 1/2, with a bound on the resolvent that is uniform in λ away from ±1/2, and in f as long as f ′ ∞ is bounded.
Oriented at [7,10], we obtain this property on the basis of a Rellich identity for the Stokes operator.While eventually the result for D(f ) is needed, it is helpful to consider D(f ) * , as this operator naturally arises from the jump relations for the single-layer hydrodynamic potential generated by β, cf.(3.13) below.
We next derive the Rellich identity (3.14), and based on it we establish an estimate that relates the operator D(f ) * to the operators B 1 (f ) and B 2 (f ) introduced above.Lemma 3.2.Given K > 0, there exists a positive constant C, that depends only on K, such that for all where ω, ν, and τ are defined in (2.1), and with We define the hydrodynamic single-layer potential u with corresponding pressure Π by where and U k , P k defined by (2.5).Using the fact that β is compactly supported, is is not difficult to see that the functions (u, Π) are well-defined and smooth in Ω ± and satisfy ∆u − ∇Π = 0, as well as for |x| → ∞.
(3.9)Moreover, [6, Lemma A.1] and the arguments in the proof of [17,Lemma A.1] show that the functions Π| Ω ± and u| Ω ± have extensions Π ± ∈ C(Ω ± ) and u ± ∈ C 1 (Ω ± ), and, given ξ ∈ R, we have where ν = (ν 1 , ν 2 ) and r = r(ξ, s) are defined in (2.1) and (3.4).In particular, where T(f ) is the singular integral operator given by Observe that T(f ) is skew-adjoint on L 2 (R) 2 , i.e.T(f ) * = −T(f ), and therefore Moreover, for the normal stress at the boundary we find For convenience we introduce the notation and observe that due to (3.8) The latter identities lead us to In view of (3.9) we may integrate the latter relation over Ω ± and using Gauss' theorem and (3.13) we get To estimate the term on the left we observe that the Cauchy-Schwarz inequality and |ν| = 1 yield 2 i, j=1 This inequality, the representations (3.10) and (3.13), and We next consider the term on the right of (3.14).As a direct consequence of Lemma 3.1 we note that T(f This bound together with (3.11) and (3.12) implies For f ∈ C ∞ (R), the estimate (3.6) follows from (3.14) and the latter estimates upon rearranging terms and a standard density argument.For general f ∈ C 1 (R) we additionally need to use the continuity of the mappings which are direct consequences of Lemma 3.1, together with the density of Based on Lemma 3.2 we now establish the following result.
Theorem 3.3 (Spectral properties of D(f ) and D(f ) * ).Given δ ∈ (0, 1), there exists a constant Proof.In order to prove (3.15) we assume the opposite.Then we may find sequences As the operators This contradicts the property that β k 2 = 1 for all k ∈ N and (3.15) follows.

The resolvent of the hydrodynamic double-layer potential operator in higher order Sobolev spaces
The main goal of this section is to establish spectral properties for D(f ), parallel to those in Theorem 3.3, in the spaces H s−1 (R) 2 , s ∈ (3/2/2), and in H 2 (R) 2 .The latter are needed when solving the fixed-time problem (5.1), see Proposition 5.1, and the former are used to derive and study the contour integral formulation (5.17) of the evolution problem (1.1).
For this purpose, we first recall some further results on the singular integral operators B n,m introduced in (3.1).Lemma 4.1.
(i) Let n ≥ 1, s ∈ (3/2, 2), and a 1 , . . ., a m ∈ H s (R) be given.Then, there exists a constant C, depending only on n, m, s, and max 1≤i≤m a i H s , such that (ii) Given s ∈ (3/2, 2) and a 1 , . . ., a m ∈ H s (R), there exists a constant C, depending only on n, m, s, and max 1≤i≤m a i H s , such that and a 1 , . . ., a m ∈ H s (R) be given.Then, there exists a constant C, depending only on n, m, s, s ′ , and max 1≤i≤m a i H s , such that For ξ ∈ R we define the left shift operator τ ξ on L 2 (R) by τ ξ u(x) := u(x + ξ) and observe the invariance property Differences of B n,m with respect to the nonlinear arguments a i can be represented by the identity We will also use the interpolation property where [•, •] θ denotes the complex interpolation functor of exponent θ.
where 0 ≤ n ≤ 3 and i ∈ {1, 2}.Let s ′ ∈ (3/2, s) be fixed.We first consider terms of the second type and estimate in view of Lemma 3.1 Furthermore, using (4.3), we have and together with Lemma 4.1 (i) (with s ′ instead of s), we conclude that Combining this estimate with (4.7) we get and by (4.6) and the interpolation property (4.4) we arrive at Finally, using Theorem 3.3 again, we obtain the estimate (4.5).The isomorphism property of λ − D(f ), with λ ∈ R with |λ| > 1/2 and f ∈ H s (R), follows by the same continuity argument as in the L 2 result.
For the H 2 result we need an additional estimate for the operators B n,m with higher regularity of the arguments.Lemma 4.3.Let n, m ∈ N and a 1 , . . ., a m ∈ H 2 (R) be given.Then, there exists a constant C, depending only on n, m, and max 1≤i≤m a i H 2 , such that Proof.We first show that ϕ it suffices to show that D ξ ϕ := (τ ξ ϕ − ϕ)/ξ converges in L 2 (R) when letting ξ → 0. In view of (4.3) we write Lemma 3.1 and Lemma 4.1 (i) enable us to pass to the limit ξ → 0 in L 2 (R) in this equality.Hence, ϕ ∈ H 1 (R) and (4.9) The estimate (4.8) is a consequence of Lemma 3.1 and Lemma 4.1 (i).The local Lipschitz continuity property follows from an repeated application of (4.3) and (4.8).
As a consequence of Lemma 4.3 and (4.9) we obtain the following result.

The contour integral formulation
In this section we formulate the Stokes evolution problem (1.1) as an nonlinear evolution problem having only f as unknown, cf.(5.17).
Based on the results established in Section 2, Section 4, and Appendix A we start by proving that for each f ∈ H 3 (R), the boundary value problem has a unique solution (v, p) ∈ X f with the property that 2 .This is established in Proposition 5.1 below, where we also provide an implicit formula for v ± | Γ in terms of contour integrals on Γ.This representation allows to recast the kinematic boundary condition (1.1a) 6 in the form (5.17).
Summarizing, we have shown the following result: Proposition 5.1.Given f ∈ H 3 (R), the boundary value problem (5.1) has a unique solution where 2 is the unique solution to (5.8).
From this result and (1.1) we infer, under the assumption that Γ(t) is at each time instant t ≥ 0 the graph of a function f (t) ∈ H 3 (R) and that (v(t), p(t)) belongs to X f (t) and satisfies v(t 2 , that (1.1a) can be recast as (5.9) Here • | • denotes the scalar product on R 2 .Using the results in Section 4 and [17] we can formulate the latter equation as an evolution equation in H s−1 (R) 2 , where s ∈ (3/2, 2) is fixed in the remaining.To this end we first infer from [17,Corollary C.5] that, given n, m ∈ N, we have (5.10) Further, [17,Lemma 3.5] ensures for the mappings defined in (5.7) that (5.11) Additionally, for any f 0 ∈ H s (R), the Fréchet derivative ∂φ i (f 0 ) is given by (5.12) It is easy to check, by arguing as in [17, Lemma C.1], that φ i , i = 1, 2, maps bounded sets in H s (R) to bounded sets in H s−1 (R).This observation, the relations (5.6), (5.10), (5.11), and Lemma 4.1 combined enable us to conclude that the map defined in (5.5)-(5.6)satisfies and also that G maps bounded sets in H s (R) to bounded sets in H s−1 (R) 2 .Moreover, recalling (3.5), we infer from (5.10) that In view of (5.13) and of Theorem 4.2 we can solve, for given f ∈ H s (R), the equation (5.8) in H s−1 (R) 2 .Its unique solution is given by and, since the mapping which associates to an isomorphism its inverse is smooth, we obtain from Theorem 4.2, (5.13), and (5.14) that (5.16) Furthermore, (5.15) and the estimate (4.5) imply that β inherits from G the property to map bounded sets in H s (R) to bounded sets in H s−1 (R) 2 .Summarizing, in a compact form, the Stokes flow problem (1.1) can be recast as the evolution problem where Φ : (5.9), by Observe that, due to (5.16), and that Φ maps bounded sets in H s (R) to bounded sets in H s−1 (R).

Linearization, localization, and proof of the main result
We are going to prove that the nonlinear and nonlocal problem (5.17) is parabolic in H s (R) in the sense that the Fréchet derivative ∂Φ(f 0 ), generates an analytic semigroup in L(H s−1 (R)) for each f 0 ∈ H s (R).This property then enables us to use the abstract existence results from [14] in the proof of our main result Theorem 1.1.Theorem 6.1.For any f 0 ∈ H s (R), the Fréchet derivative ∂Φ(f 0 ), as an unbounded operator in H s−1 (R) with dense domain H s (R), generates an analytic semigroup in L(H s−1 (R)).
is well-defined, and with C independent of f and τ .We also note that both paths B and Ψ are continuous and Ψ(1) = ∂Φ(f 0 ).Besides, since where H = π −1 B 0,0 is the Hilbert transform, we observe that Ψ(0) is the Fourier multiplier We next locally approximate the operator Ψ(τ ), τ ∈ [0, 1], by certain Fourier multipliers A j,τ , cf.Theorem 6.2 below.For this purpose, given ε ∈ (0, 1), we choose N = N (ε) ∈ N and a so-called finite ε-localization family, that is a set The real number ξ ε N plays no role in the analysis below.To each ε-localization family we associate a norm on H r (R), r ≥ 0, which is equivalent to the standard norm.Indeed, given r ≥ 0 and ε ∈ (0, 1) , there exists a constant c = c(ε, r) ∈ (0, 1) such that To introduce the aforementioned Fourier multipliers A j,τ , we first define the coefficient functions α τ , β τ : R −→ R, τ ∈ [0, 1], by the relations We now set We obviously have The following estimate of the localization error is the main step in the proof of Theorem 6.1.
Before proving Theorem 6.2 we first present some auxiliary lemmas which are used in the proof.We start with an estimate for the commutator [B 0 n,m (f ), ϕ] (we will apply this estimate in the particular case ϕ = π ε j , −N + 1 ≤ j ≤ N ). Lemma Proof.This result is a particular case of [1, Lemma 12].
We now provide a similar result as in Lemma 6.4, the difference to the latter being that the linear argument of B n,m is now multiplied by a function a that also needs to be frozen at ξ ε j .
We are now in a position to prove Theorem 6.2.
Below we denote by C constants that do not depend on ε and by K constants that may depend on ε.We need to approximate the linear operators f , see (6.8)-(6.9),where we set B(τ ) =: (B 1 (τ ), B 2 (τ )) ⊤ .The proof is divided in several steps.
Step 1.We consider the operator [f → β 1 0 f ′ ].Since χ ε j π ε j = π ε j , (6.17) yields From (5.16) we have The approximation procedure for f Step 2. We prove there exists a constant C B such that To start, we infer from (6.9) that To estimate the terms on the right, we use the representations and estimates (6.3)-(6.7)together with the commutator estimate from Lemma 6.3 and the H s−1 -estimate for the operators B m,n provided in Lemma 4.1 (ii).So we get and similarly, using (3.5) and (6.10) with s replaced by s ′ , The estimate (6.19) follows now from (6.20)-(6.22)and Theorem 4.2.
We are now in a position to prove the main result, for which we can exploit abstract theory for fully nonlinear parabolic problems from [14].

(4. 6 )
The term (D(f ) − D(τ ξ f ))[β] 2 can be estimated by a finite sum of terms of the form