Two-phase Stokes flow by capillarity in full 2D space: an approach via hydrodynamic potentials

We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. 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-layer potential and abstract results on nonlinear parabolic evolution equations.


Introduction
One of the standard methods in the analysis of moving boundary problems is the reformulation of these problems as evolution equations in function spaces to which methods of Functional Analysis can be applied, depending on the character of the problem under investigation.The difficulty of this typically consists in the fact that the resulting evolution equations are nonlocal and strongly nonlinear.For moving boundary problems with domains of general shape this approach typically involves the transformation of the moving domain to a fixed reference domain by an unknown, time dependent diffeomorphism, and the (explicit or implicit) use of solution operators for boundary value problems with variable coefficients on this reference domain.This approach often implies restrictions to results of perturbation type, i.e. either short-time solutions, or solutions for (in some sense) small data.However, this can be avoided in special situations where • the geometry is simpler (e.g.full space, with the moving boundary being a graph), and • the underlying PDE is elliptic and has constant coefficients.
In such situations, one can use the classical methods of potential theory to solve the PDEs directly, i.e. without transformations of the domain, and reformulate the moving boundary problem as an evolution equation that involves nonlinear, singular integral operators.
This strategy proved to be successful for various versions of the Muskat (or two-phase Hele-Shaw) problem, see e.g. the survey articles [6,7].While the constant coefficient elliptic operator underlying the Muskat problem is simply the Laplacian, the related moving boundary problems of quasistationary Stokes flow are based on the Stokes operator (which is also elliptic in a sense that can be made precise).Concretely, in this paper we are interested in the following moving boundary problem of Stokes flow driven by the capillarity of the moving interface t → Γ(t) between two fluid phases Ω ± (t) in R 2 : on Γ(t), [T (v, q)]ν = −σκν on Γ(t), (v ± , q ± ) → 0 for |x| → ∞, Here, v ± : Ω ± (t) −→ R 2 is a vector field representing the velocity of the liquid located in Ω ± (t) and q ± : Ω ± (t) −→ R its pressure.Furthermore, ν is the unit exterior normal to Ω − (t) and κ denotes the curvature of the interface.Moreover, [T (v, q)] denotes the jump of the stress tensor across Γ(t), see (2.4), (2.6) below.The positive constants µ and σ denote the viscosity of the liquids and the surface tension coefficient of the interface, respectively.We assume that and that Γ(t) is a graph over a suitable straight line.Equation (1.1) 6 determines the motion of the interface by prescribing its normal velocity 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.
For the Stokes operator, is is possible to set up a treatment of boundary value problems based on so-called hydrodynamic potentials [8] in strict analogy to the potentials for the Laplacian.It is this analogy that enables us to study the moving boundary problem of two-phase Stokes flow driven by capillarity (at least in 2D and with equal viscosity in both phases) along the same lines as for the Muskat problem.This has first been exploited in [4] to obtain an existence result for all positive times, with initial data that are small in a space of Fourier transforms of bounded measures.To the best of our knowledge, this is the only result available on two-phase Stokes flow in the unbounded geometry considered here.
It is the aim of the present paper to analyze Problem (1.1) in Sobolev spaces (up to critical regularity) in an L 2 -based setting.The use of these spaces also implies that the interface is asymptotically flat.
We will establish • existence and uniqueness of maximal solutions with initial data that are arbitrary within our phase space; • a corresponding semiflow property; • parabolic smoothing up to C ∞ of solutions in time and space (away from the initial time); • a criterion for global existence of solutions, or equivalently, a necessary condition for blow-up.
Essentially, these results are obtained by applying the theory of maximal regularity for nonlinear parabolic equations in weighted Hölder spaces of vector-valued functions presented in [9].Since the Stokes flow (1.1) is driven by capillarity, it turns out that the problem is parabolic "everywhere", i.e. the parabolicity condition is just positivity of the surface energy.We emphasize that while for the discussion of the boundary value problem (1.1) 1−5 at a fixed time t we need to assume H 3 -smoothness of the interface (cf.Section 2), the corresponding nonlinear evolution equation (3.4) is shown to be well-posed in all subcritical spaces H s with s > 3/2.Hence, we obtain a "weak" solution concept allowing for less regular initial data.Nevertheless, for positive times all solutions are classical due to parabolic smoothing.The structure of the paper is as follows: In Section 2 we consider the underlying twophase boundary value problem for the Stokes equations (1.1) 1−5 with fixed interface, and show that it is solved by the so-called hydrodynamic single-layer potential.We prove this by investigating its behavior near and on the interface (recovering results from [8] in our slightly different setting) and show that it vanishes in the far-field limit.This asymptotic result can be interpreted as nonoccurrence of the 2D Stokes paradoxon in our setting, which is essentially due to the fact that the curvature vector of the interface is a derivative (by arclength) of a vector that approaches a constant at infinity (see Eqn. (2.1)).In Section 3 we first rewrite our moving boundary problem as an evolution equation for the function that parametrizes the interface between the fluids and announce our main result.The remainder of the section is dedicated to its proof.We linearize the evolution equation, and then establish its parabolic character (see Proposition 3.7).The localization procedure by which this is accomplished demands the main technical effort.Once parabolicity is established, the results follow from general facts on (fully) nonlinear problems of this type as given in [9].
Throughout the paper, some longer proofs are deferred to appendices.

The fixed time problem
In this section we study the two-phase boundary value problem for the Stokes equations (1.1) 1−5 with fixed domains Ω ± and boundary ∂Ω ± := Γ as defined by The function f ∈ H 3 (R) is fixed.Note that Γ is the image of the first coordinate axis under the diffeomorphism Ξ := Ξ f := (id R , f ).Further, let ν be the componentwise pull-back under Ξ of the unit normal on Γ exterior to Ω − , i.e.
Let κ := ω −3 f ′′ ∈ H 1 (R) be the pull-back under Ξ of the curvature of Γ.In view of we will use the relation where For any functions z ± defined on Ω ± , respectively, and having limits at some (ξ, f (ξ)) ∈ Γ we will write [z](ξ, f (ξ)) := lim We fix a common viscosity µ > 0 as well as a surface tension coefficient σ > 0 and seek solutions to the two-phase boundary value problem with Ω ± and Γ as defined above.Here T (v, q) = (T ij (v, q)) 1≤i, j≤2 denotes the stress tensor that is given by (2.6) The structure of the problem allows us to represent the solution as a hydrodynamic singlelayer potential [8].For this, we introduce the fundamental solutions to the Stokes equations in R 2 by These functions solve (in distributional sense) the Stokes equations with e 1 = (1, 0) and e 2 = (0, 1).Moreover, differentiating the fundamental solutions with respect to y 1 and y 2 we get the following solutions to the homogeneous Stokes system in R 2 \ {0}: (2.7) We are going to prove that the functions (v ± , q ± ) : and g 1 , g 2 as defined in (2.2) constitute the unique solution to (2.5).First we check that the integrals exist.
Observe that the kernels of the integral operators in (2.8) are smooth with respect to x ∈ R 2 \ Γ.Moreover, g ′ k ∈ H 1 (R) and so that the integrand in (2.8) 2 belongs to L 1 (R).Furthermore, g k → 0 for |s| → ∞, so we can use integration by parts to obtain Recalling (2.7), we get and since g k ∈ H 2 (R), it follows that also v is well-defined.Altogether, we obtain the following representation for the velocity field and the pressure: for x ∈ R 2 \ Γ, where Theorem 2.1.Given f ∈ H 3 (R), Problem (2.5) has the unique solution (v ± , q ± ) given by (2.8) or, equivalently, (2.9).
Proof. 1. (v ± , q ± ) solves the Stokes equations: Denote the integrand in (2.9) by J = [(x, s) → J(x, s)], Any partial derivative ∂ α x J can be dominated by an absolutely integrable function with respect to s, locally uniformly in x, so that differentiation with respect to x and integration with respect to s can be interchanged.In particular, it follows that As the columns of ∂ 1 M , ∂ 2 M represent solutions to the homogeneous Stokes equations (cf.(2.7)), (v ± , q ± ) is also a solution to these equations on R 2 \ Γ.

Uniqueness:
We have to show that any solution be the unit tangential vector field along Γ, oriented to the right.Observe first that We now define Taking distributional derivatives and using the continuity of u across Γ yields where, given a ∈ L 1,loc (Γ), the distribution aδ Γ is defined by So, from this and (2.11) we get In particular, taking the divergence of this equation yields ∆Q = 0, i.e.Q is a harmonic function on the full space R 2 , and the asymptotic condition implies Q = 0 via Liouville's theorem.This implies in turn that V 1 and V 2 are harmonic, and are therefore zero by the same argument.
3. The behavior of (v ± , q ± ) near Γ is addressed in Appendix A. In particular, it is shown that 4. The far-field boundary condition (2.5) 5 is established in Appendix B.

The evolution problem
In the first part of this section we introduce some notation which is then used to recast the Stokes problem (1.1) as an evolution problem for f only, see (3.4) below.In the second part we establish our main result stated in Theorem 3.2. where , and for brevity (with the appropriate number of identical arguments f filled in).Here PV denotes the principle value.Below we write C 1− (X, Y ) for the space of locally Lipschitz maps from X to Y .Furthermore, given Banach spaces X and Y , we let L k sym (X, Y ) denote the space of k-linear, bounded symmetric maps A : X k −→ Y .The following properties are extensively used in our analysis.
, there exists a constant C, depending only on n, m, s, and ). (iii) Let n ≥ 1 and 3/2 < s ′ < s < 2 be given.There exists a constant C, depending only on n, m, s, s ′ , and max 1≤i≤m a i H s , such that Proof.
given by (2.8).As shown in Lemma A.1, the extension of v to Γ exists and is given by with g j from (2.2).Therefore (3.3) can be written as an evolution equation for f in the form with where the nonlinear operators Ψ j , j = 1, 2, are defined by . We recall from (2.3) the shorthand notations The following theorem contains the main results of this paper.
Remark 3.4.We expect the solutions to be even analytic in space and time away from t = 0.However, we prefer to formulate and prove our result in the C ∞ -class, refraining from the considerable technicalities needed for a proof of the analytic counterpart of Lemma 3.6 below (see [11,Proposition 5.1] for a related analyticity result).
In order to study the mapping properties of the operator Ψ we need the following lemmas.
with a i defined by For the smoothness result we refer to Lemma C.3 in Appendix C. The representations for the derivatives ∂φ i (f 0 ) follow from straightforward calculations.Lemma 3.6.Given s ∈ (3/2, 2), we have Proof.The claim follows from Lemma 3.5 and Corollary C.5.
For two Banach spaces X 0 , X 1 with dense embedding X 1 ֒→ X 0 , let H(X 1 , X 0 ) denote the set of operators A ∈ L(X 1 , X 0 ) such that −A generates a strongly continuous and analytic semigroup of operators on X 0 .
In order to establish our main result in Theorem 3.2 we next prove a generator property for the Fréchet derivative ∂Ψ(f 0 ) ∈ L(H s (R), H s−1 (R)) which identifies (3.4) as a nonlinear evolution problem of parabolic type.
The subsequent analysis is devoted to the proof of Proposition 3.7.To start, we fix a function f 0 ∈ H s (R) and s ′ ∈ (3/2, s), and we note that To calculate the derivatives of Ψ i we use Lemma C.4 to get and Lemma 3.1 (iii) to rewrite this for n ≥ 0 as The constant C is independent of f ∈ H s (R) and h ∈ H s−1 (R).From this and the definition of Ψ i , i = 1, 2, we get where with shortened notation a i = a i (f 0 ), Having computed the derivative ∂Ψ(f 0 ), it remains to establish (3.6), which is achieved via a localization procedure.To proceed, we fix for each ε ∈ (0, 1) 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 finite ε-localization family we associate a second family ) with the following properties • supp χ ε j is an interval of length 3ε and with the same midpoint as supp π ε j , |j| ≤ N − 1; Using the ε-localization family we define norms on H s (R), s ≥ 0 that are equivalent to the standard norm.Indeed, it is not difficult to prove that, given ε ∈ (0, 1) and s ≥ 0, there exists a constant c = c(ε, s) ∈ (0, 1) such that To show (3.6) we use a homotopy argument.For this we consider the continuous path defined by We next locally approximate the operator Φ(τ ), τ ∈ [0, 1], by certain Fourier multipliers A j,τ .It is worth emphasizing that Φ(1) = ∂Ψ(f 0 ), while Φ(0) is the Fourier multiplier given by , with H denoting the Hilbert transform.The homotopy Φ will be used to conclude invertibility of λ − Φ(1) from λ − Φ(0) for sufficiently large λ.We also point out the estimate which is used several times in the arguments that follow.From a technical point of view, the following proposition is at the core of the proof of Proposition 3.7.It provides estimates for the errors introduced by replacing the operator Φ(τ ) by the localizations A j,τ .In fact, this amounts to the well-known "freezing of coefficients" in the context of our nonlocal operators.Proposition 3.8.Let γ > 0 be given and fix s ′ ∈ (3/2, s).Then, there exist ε ∈ (0, 1), a constant K = K(ε), and bounded operators 12) for all j ∈ {−N + 1, . . ., N }, τ ∈ [0, 1], and f ∈ H s (R).The operators A j,τ are defined by , with functions α τ , β τ given by Proof.Let ε ∈ (0, 1).In the following we denote by K constants that may depend on ε.
Step 1.We first consider the term R) and the property χ ε j π ε j = π ε j together with (3.11), we get for |j| ≤ N − 1, provided that ε is sufficiently small.Furthermore, since Ψ 1 (f 0 ) vanishes at infinity, we have provided that ε is sufficiently small.
Making use of Ψ 1 (f 0 ) ∈ H s−1 (R) and recalling the definition of the functions a i in Lemma 3.5, we conclude there exists η ∈ (0, 1) such that Given α ∈ [η, 1/η] and |β| ≤ 1/η, we now introduce the Fourier multipliers For two Banach spaces X, Y , let Isom(X, Y ) denote the set of (linear and topological) isomorphism from X to Y .
Proof of Theorem 3.2.Well-posedness: For α ∈ (0, 1), T > 0, and a Banach space X, we first introduce the weighted Hölders spaces C α α ((0, T ], X) which are essential for the theory in [9,Chapter 8].They are defined by Lemma 3.6 and Proposition 3.7 show that the assumptions of [9, Theorem 8.1.1]are satisfied for the evolution problem (3.4).This theorem ensures that for each f 0 ∈ H s (R) there exists a positive time T > 0 and a solution f for some α ∈ (0, 1).Furthermore, it states that the solution is unique within the set α∈(0,1) We improve this statement by showing that the solution is actually unique within Indeed, suppose f : [0, T ] −→ H s (R) is another solution to (3.4) satisfying the same initial condition f 0 .Since (3.4) is an autonomous problem, we may assume This unique solution can be extended up to a maximal existence time T + (f 0 ), see [9,Section 8.2].Finally, [9,Proposition 8.2.3] shows that the solution map defines a semiflow on H s (R).This proves (i).
Parabolic smoothing: The uniqueness statement in (i) enables us to use a parameter trick which was successfully applied also to other problems, cf., e.g., [3,5,11,12], in order to establish (iia) and (iib).In our setting the proof details are similar to those in [10, Theorem 1.2 (v)] or [1, Theorem 1.2 (ii)] and therefore we omit them.
Global existence: We prove the statement by contradiction.Assume there exists a maximal solution (3.26) The bound (3.26) together with Lemma 3.1 (ii) implies that Choosing s ′ ∈ (3/2, s), we may argue as above, see (3.25), to conclude from (3.26) and (3.27) , we may extend the solution f to an interval [0, T ′ + ) with ).The parabolic smoothing property established in (iib) (with s replaced by s ′ ) implies in particular that f ∈ C 1 ((0, T ′ + ), H s (R)), in contradiction to the maximality of f .This completes our arguments.
Appendix A. The hydrodynamic potential near Γ This appendix is devoted to the study of the properties of the functions (v ± , q ± ) defined in (2.8) near the boundary Γ.Lemma A.1 below establishes the corresponding part of Theorem 2.1.
Lemma A.1.Given f ∈ H 3 (R), the functions (v ± , q ± ) given by (2.8) satisfy and solve the equations Moreover, the velocity field v on Γ has an explicit representation in terms of nonlinear singular integral operators, given by with B 0 k,2 and g j defined in (3.1), (3.2) and (2.2), (2.3), respectively.Before establishing Lemma A.1 we make the following observation.
Remark A.2.It is shown in the proof of Lemma A.1 that not only v is continuous in R 2 , but also the first order partial derivatives of v, that is In particular, we get on Γ.
For k = 0, . . ., 3, replace φ by a ki φ in the i-th equation of (A.11) and sum over i.This gives or equivalently 2 ) ⊤ |r| 4  φ ds In view of this, we get directly from (2.7) and (2.8) 1 that [v] = 0 and the representation (A.2) is valid.Moreover, first differentiating with respect to x under the integral in (2.8) 1 and then using integration by parts and (2.2) we find It is now straightforward to check that v ± ∈ C 1 (Ω ± ) and [∇v] = 0. Together with (A.5), this implies (A.1) 2 , and the proof is complete.
Appendix B. The hydrodynamic potential in the far-field limit In this appendix we prove that the functions (v, q) defined in (2.8) satisfy the far-field boundary condition (2.5) 5 .While the claim for q follows directly from [10, Lemma 2.1], proving that the velocity vanishes at infinity is more elaborate and necessitates some preparation.
Recall that f ∈ H 3 (R) is fixed.Using the notation (2.10) once again, we define functions φ → (F, G)[φ] according to (cf.(A.7)) We recall from Appendix A that F, G ∈ C(R 2 ).Moreover, from (A.9) and Lemma 3.1 (ii), we get for some β ∈ (0, 1), and We first prove a bound for F and G at moderate distances from the interface, in terms of their values at the interface.
Lemma B.1 (Vertical differences).Given f ∈ H 3 (R) and φ ∈ H 2 (R), there exist constants α ∈ (0, 1) and C 0 > 0 such that for Proof.We show the estimate for F only, the arguments for G are analogous with some obvious modifications.Given x ∈ S, we choose x := (ξ 0 , f After a change of variables we split We estimate the terms on the right separately.For the first one we use φ ′ ∈ C 2α (R) for some α ∈ (0, min{β, 1/2}] to obtain To estimate the second term we write for brevity We split the integral on the right.For |s| < 1 we use the minimality property of x to obtain and thus For |s| > 1 we estimate directly Summarizing and using the boundedness of F on S (which follows by applying Hölder's inequality to (B.1)) we get and consequently, using (B.2), where the minimality property of x has been used again in the last step.
We next prove that the functions F, G defined in (B.1) vanish at infinity.
Proof.We will show the result for F , the proof for G is essentially analogous.The result is proved in the following three steps: (i) For any ε > 0 there are ξ 0 > 0, δ ∈ (0, 1) To show (i), fix ε > 0, choose first δ > 0 small enough to ensure C 0 (2δ) α < ε/2 with C 0 and α from Lemma B.1, and then ξ 0 large enough to guarantee that |f (x 1 )| < δ and For (ii) we have to prove that F (x) → 0 for |x 2 | → ∞, uniformly in x 1 ∈ R. From (A.8) we immediately have and using the Cauchy-Schwarz inequality we get where we changed variables according to t := (x 1 − s)/|x 2 | in the last step.This proves (ii).
We are now in a position to prove the desired decay behavior.
Lemma B.3.Given f ∈ H 3 (R), the functions (v ± , q ± ) given by (2.8) satisfy Appendix C. Smoothness of some nonlinear operators In this appendix we establish the smoothness of certain nonlinear operators we are confronted with in Section 3. In the following r ∈ (1/2, 1) is fixed.Proof of Lemma C.1.Recall that a norm in H r (R), which is equivalent to the standard norm, is given by Proof.Let first ω ∈ H s−1 (R) (the case ω = 1 is similar).It is suitable to decompose where T 2 := B 0 n,m (f )[π ε j (a − a(ξ ε j ))h], Lemma D.1 yields ωT 1 H s−1 ≤ K (a − a(ξ ε j ))h 2 ≤ K h 2 .Besides, using Lemma 3.1 (ii), (3.11), the identity χ ε j π ε j = π ε j , and the fact that a ∈ C s−3/2 (R), we see that provided that ε is sufficiently small (where C = C(n, m, ω H s−1 , f H s )).Since T 3 can be estimated by using Lemma D.2, we have established the desired claim.

3. 1 .
A class of singular integral operators.We first introduce a class of multilinear singular integral operators which are needed in the second part of this section.Given n, m ∈ N and Lipschitz continuous functions a 1 , . . ., a m , b 1 , . . ., b n : R −→ R, denote by B n,m the singular integral operator

2 |η|
1+2r dη, where τ η := [z → z(• − η)] denotes the right shift operator.Let K = z ∞ and observe Formulation of the evolution equation and the main result.In view of Theorem 2.1 we may recast the two-phase Stokes moving boundary problem (1.1) as a nonlinear and nonlocal evolution problem of the form The claim (i) is established in [10, Lemma 3.1], while the properties (ii) and (iii) are proven in [1, Lemmas 2.5 and 2.6].3.2.