The Mullins–Sekerka problem via the method of potentials

It is shown that the two‐dimensional Mullins–Sekerka problem is well‐posed in all subcritical Sobolev spaces Hr(R)$H^r({\mathbb {R}})$ with r∈(3/2,2)$r\in (3/2,2)$ . This is the first result, where this issue is established in an unbounded geometry. The novelty of our approach is the use of the potential theory to formulate the model as an evolution problem with nonlinearities expressed by singular integral operators.


Introduction
The Mullins-Sekerka problem in a bounded geometry is a moving boundary problem which appears as the gradient flow of the area functional with respect to a metric which is formally equivalent to the H −1 -metric on the tangent space of all oriented hypersurfaces which enclose a fixed volume [23,36].It describes the evolution of two domains Ω + (t) and Ω − (t) together with the sharp interfaces Γ(t) that separates them in such a way that the volumes of Ω ± (t) are preserved and the area of Γ(t) is decreased [15,23,25].Known also as the two-phase Hele-Shaw problem, it may also be derived as a singular limit of the Cahn-Hilliard problem when the thickness of the transition layer between the phases vanishes [3,40].
In this paper we consider the situation when the two phases are both unbounded and we restrict to the two-dimensional case.To be more precise, we assume that at each time instant t ≥ 0 we have where f (t) : R → R, t ≥ 0, is an unknown function.The same setting has been also considered in [13] where the authors establish convergence rates to a planar interface for global solutions (assuming they exist).Our goal is to establish the well-posedness of the Mullins-Sekerka problem in this unbounded regime for initial data whose regularity is close of being optimal.To be more precise, the equations of motion are described by the following system of equations for t > 0. Above, ν Γ(t) , V (t), and κ Γ(t) are the unit normal which points into Ω + (t), the normal velocity, and the curvature of Γ(t).Moreover, represents the jump of ∇u(t) across Γ(t) in normal direction.The system (1.1a) is supplemented by the initial condition Before presenting our main result, we emphasize that, under suitable conditions, the interface f (t) identifies at each time instant t ≥ 0 the functions u ± (t) uniquely, see Proposition 2.4 below.Therefore, from now on, we shall only refer to f as being a solution to (1.1).A further observation is that if f is a solution to (1.1) then, given λ > 0, also the function is a solution to (1.1).Since where • Ḣ3/2 is the homogeneous Sobolev norm, we identify BUC 1 (R) and H 3/2 (R) as critical spaces for (1.1).In Theorem 1.1 we establish the well-posedness of (1.1) together with a parabolic smoothing property in all subcritical Sobolev spaces H r (R) with r ∈ (3/2, 2).With respect to this point we mention that all previous existence results in the setting of classical solutions [2,12,19,21,24,38] consider initial data with at least C 2 -regularity.
The main result of this paper is the following theorem.
Theorem 1.1.Let r ∈ (3/2, 2) and choose r ∈ (3/2, r).Then, given f 0 ∈ H r (R), there exists a unique maximal solution where In Theorem 1.1 and below we let (•) ′ denote the spatial derivative d/dx.The strategy to prove Theorem 1.1 consists in several steps.To begin with, we first prove that if f (t) is known and belongs to H 4 (R), then the first three equations of (1.1a) identify the functions u ± (t) uniquely, see Proposition 2.4.Furthermore, we can also represent the right side of (1.1a) 4 in terms of certain singular integral operators which involve only the function f (t, •).In this way we reformulate the problem as an evolution problem with only f as unknown, see (3.1).In the proof of Proposition 2.4 we rely on potential theory and some formulas, see Lemma 2.2 (iv), that relate the derivatives of certain singular integral operator evaluated at some density β to the L 2 -adjoints of these operators evaluated β ′ , which have been used already in the context of the Muskat problem in [14,29].Thanks to these formulas, we may formulate (1.1), see (3.1) in Section 3.1, as an evolution problem in H r−2 (R), r ∈ (3/2, 2), with nonlinearities which are expresses as a derivative.Then, using a direct localization argument, we show in Section 3.2 that the problem is of parabolic type by identifying the right side of (3.1) as the generator of an analytic semigroup.The proof of the main result is established in Section 3.3 and relies on the quasilinear parabolic theory presented in [5,34].
1.1.Notation.Given Banach spaces E 1 and E 0 , we define L(E 1 , E 0 ) as the space of bounded linear operators from E 1 to E 0 and L(E 0 ) := L(E 0 , E 0 ).Moreover, is the space of k-linear, bounded, and symmetric operators T : E k 1 → E 0 .The set of all locally Lipschitz continuous mappings from E 1 to E 0 is denoted by If E 1 is additionally densely embedded in E 0 , we set (following [6]) Given a Banach space E, an interval I ⊂ R, n ∈ N, and γ ∈ (0, 1), we define C n (I, E) as the set of all n-times continuously differentiable functions and C n+γ (I, E) is its subset consisting of those functions which posses a locally γ-Hölder continuous nth derivative.Moreover, BUC n (I, E) is the Banach space of functions with bounded and uniformly continuous derivatives up to order n and BUC n+γ (I, E) denotes its subspace which consists of those functions which have a uniformly γ-Hölder continuous nth derivative.We also set is the set of functions with uniformly continuous derivatives up to order n.

Solvability of some boundary value problems
Our strategy to solve (1.1) is to reformulate this model as an evolution problem for the function f only.To this end, we first solve via the method of potentials, for each given function f ∈ H 4 (R), the (decoupled) boundary value problems for u + and u − given by the systems where Below ν Γ is the outward unit normal at Γ which points into Ω + .The corresponding existence and uniqueness result is provided in Proposition 2.4 below.Before stating this result we first introduce some notation.Observe that Γ is the image of the diffeomorphism Ξ Γ : R → Γ defined by Ξ(x) := (x, f (x)) for x ∈ R.Then, the pulled-back curvature κ(f given by the relation Moreover, given functions w ± ∈ C(Ω ± ), we set (2.3) 2.1.Some singular integral operators.We now introduce some singular integral operators which are used when solving (2.1).Given f ∈ W 1 ∞ (R), we set where PV is the principal value and Lemma 2.1 (i) below ensures that these singular integral operators belong to L(L 2 (R)).
Their L 2 -adjoints are given by the relations (2.5) An important observation is that the operators defined in (2.4)-(2.5)can be represented in terms of a family of singular integral operators {B 0 n,m (f ) : n, m ∈ N} which we now introduce.Given n, m ∈ N and Lipschitz continuous mappings a 1 , . . ., a m , b 1 , . . ., b n : R → R, we set In particular, if f : R → R is Lipschitz continuous we use the short notation These operators have been defined in the context of the Muskat problem in [30].It is now a straight forward consequence of (2.4)-(2.7) to observe that (2.8) In view of the representation (2.8) several mapping properties for the operators introduced in (2.4)-(2.5)can be derived from the following result.The next lemma collects some important properties of the operators defined in (2.4)-(2.5).
Proof.The property (i) follows from [30,Theorem 3.5] Hence, ≥ C α 2 H 2 , the inequalities in the second last line of the formula (with a sufficiently small constant C independent of λ and α) being a straightforward consequence of (iii).The assertion (v) follows now from this estimate via the method of continuity [6, Proposition I.1.1.1].

2.2.
The solvability of the boundary value problems (2.1).As a preliminary result we provide in Proposition 2.3 the unique solvability of a transmission type boundary value problem which is used to establish the uniqueness claim in Proposition 2.4 below.
Proposition 2.3.Given f ∈ H 3 (R) and φ ∈ H 2 (R), the boundary value problem Moreover, the solution is, up to an additive constant, unique.
Proof.We first prove uniqueness of solutions in the class described above.Let therefore U be a solution to the homogeneous problem associated with (2.9) (that is with φ = 0).
Hence, U is the real part of a holomorphic function h : C → C. Since h ′ is also holomorphic and h ′ = ∇U is bounded and vanishes for In order to establish the existence of solutions, we set (2.10) and setting U ± := U | Ω ± , we next show that (U + , U − ) is a solution to (2.9) with the required properties.To start, we note that and, for every α ∈ N 2 , we have ∂ α (x,y) K(x, y, s) = O(s −1 ) for |s| → ∞ and locally uniformly in (x, y) ∈ R 2 \ Γ.This shows that U is well-defined and that integration and differentiation with respect to x and y may be commuted.

Γ
It is now easy to infer from (2.13) that also (2.9) 1 holds true, and therewith we established the existence of a solution.
We are now in a position to solve the boundary value problems (2.1) for u + and u − .
and with density functions α ± ∈ H 1 (R) given by the relation Proof.(i) Existence.According to Lemma 2.2 (iii) we have ∓1 + A(f ) ∈ Isom(H 1 (R)) and, since (κ(f )) ′ ∈ H 1 (R), the density functions α ± defined in (2.15) are well-defined and belong to H 1 (R).We next infer from [9, Lemma A.1] that the vector fields v ± defined in (2.14) belong to C ∞ (Ω ± ) ∩ UC(Ω ± ) and (2.16) Moreover, v ± satisfies the asymptotic boundary condition and, since v ± are divergence free, (2.1) 1 is satisfied.It is clear that also the asymptotic boundary conditions (2.1) 2 hold.Combining (2.4), (2.16), and the relation ∇u ± = v ± on Γ, we further have In order to show that B(f )[α ± ] are derivatives of functions in H 2 (R) we define β ± ∈ H 2 (R) by the relations (2.18) see Lemma 2.2 (v).We next differentiate (2.18) with respect to x and infer then from Lemma 2.2 (iii)-(iv) that (β ± ) ′ = α ± and As a final step we show that the additive constants c ± can be chosen such that also (2.1) 2 are satisfied.Indeed, in view of (2.15) and (2.16), we have Therewith, we established the existence of a solution to (2.1).
(ii) Uniqueness.It suffices to show that the homogeneous problems have unique solutions u ± with the required properties.We establish only the uniqueness of u + (that of u − follows by similar arguments).Let thus φ + ∈ H 2 (R) be the function which satisfies the relation Setting U − := 0 and U + := u + , we note that (U + , U − ) solves the boundary value problem (2.9) (with φ = φ + ) and it is thus given by the formula (2.11).In particular, it follows from (2.11) and [9, Lemma A.1] that and together with Lemma 2.2 (iv) we get However, as shown in [30,Eqs. (3.22) and (3.25)], there exits a positive constant C such that B(f . Therefore φ ′′ = 0, hence also φ = 0. We now infer from (2.11) that U + = u + = 0, and the uniqueness claim is proven.

The evolution problem and the proof of the main result
In this section we first formulate the original problem (1.1) as an evolution problem for f , see (3.1).Subsequently, we prove that the linearization of the right side of (3.1) is the generator of an analytic semigroup, see Theorem 3.1 below, and we conclude the section with the proof of the main result stated in Theorem 1.1.

3.1.
The evolution problem.In order to formulate the system (1.1) as an evolution problem for f we first infer from Proposition 2.4 that if (f, u ± ) is a solution to (1.1) as stated in Theorem 1.1, then, for each t > 0, we have Together with (1.1) 4 we arrive at the following evolution equation As we want to solve the latter equation in the phase space H r (R) with r ∈ (3/2, 2) we encounter the problem that the curvature κ(f ) is in general not a function, but a distribution.However, taking full advantage of the quasilinear character of the curvature operator we can formulate the system (1.1) as the following quasilinear evolution problem where ) is defined by the following formula Moreover, arguing as in [32, Appendix C], it is not difficult to prove that Recalling (2.8), it follows from Lemma 2.1 (iii) and Lemma 2.2 (ii), by also using the smoothness of the map which associate to an isomorphism its inverse, that 3.2.The parabolicity property.Our next goal is to prove that the problem (3.1) is of parabolic type in the sense that, for each f ∈ H r (R), r ∈ (3/2, 2), the operator Φ(f ) is the generator of an analytic semigroup in L(H r−2 (R)).This is the content of the next result.
In the proof of Theorem 3.1 we exploit of the fact that, given h ∈ H r+1 (R), the action Φ(f )[h] is the derivative of a function which lies in H r−1 (R).The proof of Theorem 3.1 is postponed to the end of this subsection and it relies on a strategy inspired by [16,17,20].
As a first step we associate to Φ(f ) the continuous path and we note that , where H is the Hilbert transform.In particular, Φ(0) is the Fourier multiplier defined by the symbol [ξ → 2|ξ| 3 ].As a second step we locally approximate in Proposition 3.2 the operator Φ(τ f ) by Fourier multipliers which coincide, up to some positive multiplicative constants, with Φ(0).As a final third step we establish for these Fourier multipliers suitable (uniform) resolvent estimates, see (3.14)- (3.15).The proof of Theorem 3.1 follows then by combining the results established in these three steps.
Before presenting Proposition 3.2, we choose for each ε ∈ (0, 1), a finite ε-localization family, that is a family To each finite ε-localization family we associate a second family with the following properties j is an interval of length 3ε and with the same midpoint as supp π ε j , |j| ≤ N − 1.
It is not difficult to prove that, given r ∈ R and ε ∈ (0, 1), there exists c = c(ε, r) ∈ (0, 1) such that for all h ∈ H r (R) we have We are now in a position to establish the aforementioned localization result.
, and ν > 0 be given.Then, there exist ε ∈ (0, 1), a ε-localization family {(π ε j , x ε j ) : −N +1 ≤ j ≤ N }, and a constant Proof.In the following C and C 0 are constants that do not depend on ε, while constants denoted by K may depend on ε.
, and h ∈ H r+1 (R) we have where, in view of (2.8), (3.3), Lemma 2.1 (i), and Lemma 2.2 (i) we have ) is a contraction, we have shown that (3.9)In remains to estimated the first term on the right of (3.9).To this end several steps are needed. Step ) see (3.4) and Lemma 2.1 (ii).In this step we prove there exists a constant C 0 > 0 such that for all ε ∈ (0, 1), τ ∈ [0, 1], −N + 1 ≤ j ≤ N , and h ∈ H r+1 (R) we have Indeed, after multiplying (3.10) by π ε j , we arrive at ), and it can bee easily shown that . Moreover, since r − 1 < 1, the commutator estimate in Lemma A.1 together with (2.8) yields The estimates (3.11) follow now from Lemma 2.2 (ii).

3.3.
The proof of the main result.We complete this section with the proof of the main result which exploits the abstract quasilinear parabolic theory presented in [5] (see also [34,Theorem 1.1]).
Proof of Theorem 1.1. where We next prove that the uniqueness claim holds in the class of classical solutions; that is of solutions which satisfy merely (3.19).To this end prove that each such solution with the property (3.19) satisfies (3.20) for some small ζ.Let therefore T ∈ (0, T + ) be arbitrary but fixed.Then, there exists a positive constant C such that for all t ∈ [0, T ] we have and together with (3.16) and the observation that 0 < 2 − r < r − 1 we get that ±1 − A(f (t)) ∈ Isom(H 2−r (R)).Since by Lemma 2.1 (i)-(ii) and (2.8) the mapping is in particular continuous, we may chose C > 0 sufficiently large to guarantee that for all t ∈ [0, T ] it holds that Therefore, setting ϑ ± (t) := (±1 − A(f (t) * ) −1 [κ(f (t))] ∈ H r−1 (R), t ∈ (0, T ], we infer from (3.21) and (3.22) that there exists a constant C > 0 such that for all t ∈ (0, T ] we have 2) may be shown by using a parameter trick employed also in other settings, see [7,18,31,37].
Since the arguments are more or less identical to those used in [31,Theorem 1.3], we refrain to present them here.
Appendix A. Some properties of the singular integral operators B 0 n,m (f ) We recall some recent results that are available for the singular integrals operators B 0 n,m (f ) introduced in (2.7) and which are used in the analysis in Section 3. We begin with a commutator type estimate.
The next results describe how to localize the singular integrals operators B 0 n,m (f ).They may be viewed as generalizations of the method of freezing the coefficients of elliptic differential operators.