Abstract
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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 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 [13, 14] 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 \({[f\mapsto f(t)]}\), with \({f(t)\in \mathrm{C}(\mathbb {R})},\) 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 Sect. 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, 17, 22].
This paper discusses, along the lines sketched above, the moving boundary problem of two-phase 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 \([t\mapsto \Gamma (t)]\) between two liquid phases \(\Omega ^\pm (t)\), and corresponding functions
representing the velocity and pressure fields in \(\Omega ^\pm (t)\), respectively, such that the following equations are satisfied:
Here \(\tilde{\nu }\) is the unit exterior normal to \(\partial \Omega ^-(t)\) and \(\tilde{\kappa }\) denotes the curvature of the interface. Moreover, \(T_\mu (v,p)=(T_{\mu ,ij}(v,p))_{1\le i,\, j\le 2}\) denotes the stress tensor that is given by
and [v] (respectively \([T_\mu (v, p)]\)) is the jump of the velocity (respectively stress tensor) across the moving interface, see (2.3) below. The positive constants \(\mu ^\pm \) and \(\sigma \) denote the viscosity of the liquids in the two phases and the surface tension coefficient of the interface, respectively. We assume that
so that \(\Gamma (t)\) is a graph over the real line and \(\Omega ^+(t)\) (resp. \(\Omega ^-(t)\)) is the unbounded domain above (resp. beneath) the graph \(\Gamma (t)\), cf. (2.1). 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 \(\Gamma (t)\), i.e. the interface is transported by the liquid flow. The interface \(\Gamma (t)\) is assumed to be known at time \(t=0\):
In the previous paper [20], the authors considered Problem (1.1a) in the case of equal viscosities \(\mu ^\pm =\mu \). 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 [16] with density \(-\sigma \tilde{\kappa }\tilde{\nu }\), and the resulting evolution equation represents the time derivative of f as a nonlinear singular integral operator acting on f.
If \(\mu ^+\ne \mu ^-\) 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 \(\beta \) for the double layer potential is found from solving a linear, singular integral equation of the second kind, cf. (5.8). As \({\Gamma (t)}\) 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 [18] 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, \({H^s(\mathbb {R}):=W^{s}_2(\mathbb {R})}\) denotes the usual Sobolev spaces of integer or noninteger order.
Theorem 1.1
Let \(s\in (3/2,2) \) be given. Then, the following statements hold true:
-
(i)
(Well-posedness) Given \(f_0\in H^{s}(\mathbb {R})\), there exists a unique maximal solution \((f,v^\pm ,p^\pm )\) to (1.1) such that
-
\(f=f(\cdot ;f_0)\in \mathrm{C}([0,T_+), H^{s}(\mathbb {R}))\cap \mathrm{C}^1([0,T_+), H^{s-1}(\mathbb {R})),\)
-
\(v^\pm (t)\in \mathrm{C}^2(\Omega ^\pm (t))\cap \mathrm{C}^1(\overline{\Omega ^\pm (t)})\), \(p^\pm (t)\in \mathrm{C}^1(\Omega ^\pm (t))\cap \mathrm{C}(\overline{\Omega ^\pm (t)})\) for all \({t\in (0,T_+)}\),
-
\( v(t)^\pm |_{\Gamma (t)}\circ \Xi _{f(t)}\in H^2(\mathbb {R})^2\) for all \(t\in (0,T_+)\),
where \(T_+=T_+(f_0)\in (0,\infty ]\) and \(\Xi _{f(t)}(\xi ):=(\xi , f(t)(\xi ))\), \(\xi \in \mathbb {R}.\)
Moreover, the set
$$\mathcal {M}:=\{(t,f_0)\,|\,f_0\in H^s(\mathbb {R}),\,0< t<T_+(f_0)\}$$is open in \((0,\infty )\times H^s(\mathbb {R})\), and \([(t,f_0)\mapsto f(t;f_0)]\) is a semiflow on \(H^s(\mathbb {R})\) which is smooth in \(\mathcal {M}\).
-
-
(ii)
(Parabolic smoothing)
-
(iia)
The map \([(t,\xi )\mapsto f(t)(\xi )]:(0,T_+)\times \mathbb {R}\longrightarrow \mathbb {R}\) is a \(\mathrm{C}^\infty \)-function.
-
(iib)
For any \(k\in \mathbb {N}\), we have \(f\in \mathrm{C}^\infty ((0,T_+), H^k(\mathbb {R})).\)
-
(iia)
-
(iii)
(Global existence) If
$$\begin{aligned} \sup _{[0,T]\cap [0,T_+(f_0))} \Vert f(t)\Vert _{H^s}<\infty \end{aligned}$$for each \(T>0\), then \(T_+(f_0)=\infty .\)
Remark 1.2
Observe that the complete problem (1.1) is encoded in the time evolution of f. Besides, if f is a solution to (1.1), then, given \(\lambda >0\), also the function \([t\mapsto f_\lambda (t)]\) given by
is a solution to (1.1). This identifies \(H^{3/2}(\mathbb {R})\) as a critical space for the evolution problem (1.1). Hence, Theorem 1.1 covers all subcritical spaces. To our knowledge, this result is stronger than those found in the literature on the related problems with bounded liquid domain, e.g. [11, 12, 15, 23]. More generally, if the problem is treated using the general strategy described in [22], higher regularity demands on the initial interface are needed than in the approach used here. To be more precise, the authors of [15] establish the local well-posedness of the one phase problem for a bounded fluid domain in \(\mathbb {R}^d\) for \(H^{s+1}\)-data with \(s\ge s_1,\) where \(s_1\) is the smallest integer that satisfies \({s_1>3+(d-1)/2}\). Moreover, it is shown in [15] that balls are exponentially stable under \(H^{s+1}\)-perturbations. The exponential stability of balls for the one-phase problem has been also established in \(\mathbb {R}^2\) for \(H^5\)-initial data, see [12], and in \(\mathbb {R}^3\) for \(H^6\)-initial data, see [11]. The local well-posedness for \(C^{3+\alpha }\)-data, with \(\alpha >0\), in three space dimensions has been investigated in [23], and the same author has justified in [24] the quasistationary Stokes flow as a limit of the Stokes flow when the Reynolds number vanishes. Finally, the local well-posedness and stability properties for the two-phase Stokes flow (with or without phase transitions) in a bounded geometry in \(\mathbb {R}^d\), \(d\ge 2\), have been studied in [22] in a \(W^{2+\mu -2/p}_p\)-setting with \(1\ge \mu >(d+1)/p\).
1.1 Outline
The paper is structured as follows: In Sect. 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 Sects. 3 and 4, first in \(L_2(\mathbb {R})^2\) and then in \(H^{s-1}(\mathbb {R})^2,\) with \(s\in (3/2,2)\), and \(H^2(\mathbb {R})^2\). The main technical tools in the latter cases are shift invariances and commutator properties for singular integral operators of the type discussed here. In Sect. 5 we reformulate the moving boundary problem (1.1) as a nonlinear and nonlocal evolution equation problem, cf. (5.17). Finally, in Sect. 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 [17].
1.2 Notation
Slightly deviating from the usual notation, if \(E_1,\ldots ,E_k,\,F\), \(k\in \mathbb {N}\), are Banach spaces, we write \(\mathcal {L}^k(E_1,\ldots ,E_k;F)\) for the Banach space of k-linear bounded maps from \(\prod _i E_i\) to F. Given two Banach spaces X and Y, we let \({\mathcal {L}^k_\mathrm{sym}(X,Y)\subset \mathcal {L}^k(X,\ldots ,X;Y)}\) denote the space of bounded, k-linear, and symmetric maps \(A:\;X^k\rightarrow Y\). Moreover, \(\mathrm{C}^{-1}(E,F)\) will denote the space of locally Lipschitz continuous maps from a Banach space E to a Banach space F. Given \(k\in \mathbb {N}\), we further let \(\mathrm{C}^k(\mathbb {R})\) denote the Banach space of functions with bounded and continuous derivatives up to order k and \(\mathrm{C}^{k+\alpha }(\mathbb {R})\), with \(\alpha \in (0,1)\), is its subspace consisting of functions with \(\alpha \)-Hölder continuous kth derivative whose \(\alpha \)-Hölder modulus is bounded.
2 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 (2.4) with equal viscosities normed to 1. The unique solvability of (2.4) is established in Proposition 2.1 below and in Appendix A. In this section, \(f\in H^3(\mathbb {R})\) is fixed. We introduce the following notation:
Note that \(\Gamma \) is the image of \(\mathbb {R}\) under the diffeomorphism
Further, let \(\nu \) and \(\tau \) be the componentwise pull-back under \(\Xi \) of the unit normal \(\tilde{\nu }\) on \(\Gamma \) exterior to \(\Omega ^-\) and of the unit tangent vector \(\tilde{\tau }\) to \(\Gamma \), that is
We indicate the dependence of the functions defined in (2.2) on f only where necessary. For any function z defined on \(\mathbb {R}^2\setminus \Gamma \) we set \(z^\pm :=z|_{\Omega ^\pm }\) and if \(z^\pm \) have limits at some point \((\xi ,f(\xi ))\in \Gamma \) we will write \(z^{\pm }(\xi ,f(\xi ))\) for the limits, and we set
For notational brevity we introduce the function space \(X:=X_f\) by setting
For given \(\beta =(\beta _1,\beta _2)^\top \in H^2(\mathbb {R})^2\) we seek solutions \((w,q)\in X\) to the Stokes problem
For the construction of the solution to (2.4), let us first point out that for any smooth solution \({(U,P):E\longrightarrow \mathbb {R}^2\times \mathbb {R}}\) to the homogeneous Stokes system
where E is a domain in \(\mathbb {R}^2\), the functions \((W^i,Q^i):E\longrightarrow \mathbb {R}^2\times \mathbb {R}\), \(i=1,2\), given by
are solutions to (2.5) as well. In particular, if \(E=\mathbb {R}^2\setminus \{0\}\) and
are the fundamental solutions to the Stokes equations (2.5), given by
we obtain a system \((\mathcal {W}^{i,k},\mathcal {Q}^{i,k}):\mathbb {R}^2\setminus \{0\} \longrightarrow \mathbb {R}^2\times \mathbb {R}\), \(i,k=1,\,2\), of solutions to the homogeneous Stokes equations given by
We are going to show that \((w,q):=(w,q)[\beta ]\) given by
for \(x\in \mathbb {R}^2\setminus \Gamma \) and with \(r:=r(x,s):=x-(s,f(s))\) solves (2.4). Here and further, we sum over indices appearing twice in a product. We write this more explicitly as
The solution (w, q) is the so-called hydrodynamic double-layer potential generated by the density \(\beta \circ \Xi ^{-1}\) on \(\Gamma \), see [16].
Proposition 2.1
The boundary value problem (2.4) has precisely one solution \({(w,q)\in X}\). It is given by (2.7), (2.8). Moreover, \(w^\pm |_{\Gamma }\circ \Xi \in H^2(\mathbb {R})^2\).
Proof
The uniqueness of the solution can be shown as in the proof of [20, Theorem 2.1]. Observe that w and q are defined by integrals of the form
where for every \(\alpha \in \mathbb {N}^2\) we have \(\partial ^\alpha _x K(x,s)=O(s^{-1})\) for \(|s|\rightarrow \infty \) and locally uniformly in \({x\in \mathbb {R}^2\setminus \Gamma }\). This shows that w and q are well-defined by (2.7) and (2.8), and that integration and differentiation with respect to x may be interchanged. As \((\mathcal {W}^{i,k},\mathcal {Q}^{i,k})\) 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
In view of this representation, [18, Lemma 2.1] implies \(q(x)\rightarrow 0\) as \(|x|\rightarrow \infty \).
In order to prove the decay of w we rewrite
where \(I\in \mathbb {R}^{2\times 2}\) is the identity matrix. In view of [18, Lemma 2.1] and [20, Lemma B.2] we conclude that indeed \(w(x)\rightarrow 0\) for \(|x|\rightarrow \infty \).
The boundary conditions (2.4)\(_3\) and (2.4)\(_4\) together with the properties that \((w,q)\in X\) and \({w^\pm |_{\Gamma }\circ \Xi \in H^2(\mathbb {R})^2}\) are shown in Appendix A. \(\square \)
3 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 \(\mathbb {D}(f),\) with \(f\in \mathrm{C}^1(\mathbb {R})\), introduced in (3.5) below, which we view in this section as an element of \(\mathcal {L}(L_2(\mathbb {R})^2)\). The main result of this section is Theorem 3.3 below which provides in particular the invertibility of \({\lambda -\mathbb {D}(f)}\) for \(\lambda \in \mathbb {R}\) with \(|\lambda |>1/2\).
To begin, we introduce a general class of singular integral operators suited to our approach via layer potentials, cf. [19, 20]. Given \({n,\,m\in \mathbb {N}}\) and Lipschitz continuous functions \({a_1,\ldots , a_{m},\, b_1, \ldots , b_n:\mathbb {R}\longrightarrow \mathbb {R}}\), we let \(B_{n,m}\) denote the singular integral operator
where \(\mathrm{PV}\int _\mathbb {R}\) denotes the principal value integral and \(\delta _{[\xi ,\eta ]}u:=u(\xi )-u(\xi -\eta )\). For brevity we set
In this section we several times use the following result.
Lemma 3.1
There exists a constant \(C=C(n,\, m, \,\max _{i=1,\ldots , m}\Vert a_i'\Vert _{\infty })\) with
Moreover, \(B_{n,m}\in \mathrm{C}^{1-}(W^1_\infty (\mathbb {R})^{m},\mathcal {L}_{\mathrm{sym}}^n(W^1_\infty (\mathbb {R}) ,\mathcal {L}(L_2(\mathbb {R})))).\)
Proof
See [19, Remark 3.3]. \(\square \)
As we are concerned exclusively with boundary integral operators in this section, it will be convenient to slightly change notation and write
Given \(f\in \mathrm{C^1}(\mathbb {R})\), we introduce the linear operators \(\mathbb {D}(f)\) and \(\mathbb {D}(f)^*\) defined by
where \(\xi \in \mathbb {R}\) and \(\beta \in L_2(\mathbb {R})^2\). We note that \(\mathbb {D}(f)\) is related to the \(B_{n,m}\) via
for \(\beta =(\beta _1,\,\beta _2)^\top \). Therefore, as a consequence of Lemma 3.1, \(\mathbb {D}(f)\) is bounded on \(L_2(\mathbb {R})^2\). Moreover, up to the sign and the push-forward via \(\Xi \), \(\mathbb {D}(f)[\beta ](\xi )\) is the “direct value” of the hydrodynamic double-layer potential w generated by \(\beta \) in \((\xi ,f(\xi ))\in \Gamma \), cf. (2.9)\(_1\). One may also check that \(\mathbb {D}(f)^*\) is the \(L_2\)-adjoint of \(\mathbb {D}(f)\).
Using the same notation, we define the singular integral operators \(\mathbb {B}_1(f)\) and \(\mathbb {B}_2(f)\) by
where \(\theta \in L_2(\mathbb {R})\). The operators \(\mathbb {B}_i(f)\), \(i=1,\, 2\), play an important role also in the study of the Muskat problem, cf. [18]. Lemma 3.1 implies in particular that also \(\mathbb {B}_i(f)\) \(i=1,\, 2\), is bounded on \(L_2(\mathbb {R})\). Moreover, \(\mathbb {B}_1(f)[\theta ](\xi )\) is the direct value of the double layer potential for the Laplacian corresponding to the density \(\theta \) in \((\xi ,f(\xi ))\in \Gamma \).
We are going to prove in Theorem 3.3 below that the resolvent sets of \(\mathbb {D}(f)\) and \(\mathbb {D}(f)^*\) contain all real \(\lambda \) with \(|\lambda |>1/2\), with a bound on the resolvent that is uniform in \(\lambda \) away from \(\pm 1/2\), and in f as long as \(\Vert f'\Vert _\infty \) 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 \(\mathbb {D}(f)\) is needed, it is helpful to consider \(\mathbb {D}(f)^*\), as this operator naturally arises from the jump relations for the single-layer hydrodynamic potential generated by \(\beta \), cf. (3.13) below.
We next derive the Rellich identity (3.14), and based on it we establish an estimate that relates the operator \(\mathbb {D}(f)^*\) to the operators \(\mathbb {B}_1(f)\) and \(\mathbb {B}_2(f)\) introduced above.
Lemma 3.2
Given \(K>0\), there exists a positive constant C, that depends only on K, such that for all \({\beta \in L_2(\mathbb {R})^2}\), \(\lambda \in [-K,K]\), and \(f\in \mathrm{C}^1(\mathbb {R})\) which satisfy \(\Vert f'\Vert _\infty <K\) we have
where \(\omega \), \(\nu \), and \(\tau \) are defined in (2.2), and with
Proof
Let first \(f\in \mathrm{C}^\infty (\mathbb {R})\) and \(\beta =(\beta _1,\beta _2)^\top \) with \(\beta _k\in \mathrm{C}_0^\infty (\mathbb {R})\), \({k=1,\, 2}\). We define the hydrodynamic single-layer potential u with corresponding pressure \(\Pi \) by
for \(x\in \mathbb {R}^2\setminus \Gamma \), where and \(\mathcal {U}^k\), \(\mathcal {P}^k\) defined by (2.6). Using the fact that \(\beta \) is compactly supported, is is not difficult to see that the functions \((u,\Pi )\) are well-defined and smooth in \(\Omega ^\pm \) and satisfy
as well as
Moreover, [6, Lemma A.1] and the arguments in the proof of [20, Lemma A.1] show that \(\Pi |_{\Omega ^\pm }\) and \(u|_{\Omega ^\pm }\) have extensions \(\Pi ^\pm \in \mathrm{C}(\overline{\Omega ^\pm })\) and \({u^\pm \in \mathrm{C}^1(\overline{\Omega ^\pm })}\), and, given \(\xi \in \mathbb {R}\), we have
where \(\nu =(\nu ^1,\nu ^2)\) and \(r=r(\xi ,s)\) are defined in (2.2) and (3.3). In particular,
where \(\mathbb {T}(f)\) is the singular integral operator given by
Observe that \(\mathbb {T}(f)\) is skew-adjoint on \(L_2(\mathbb {R})^2\), i.e. \(\mathbb {T}(f)^*=-\mathbb {T}(f)\), and therefore
Here \(\langle \cdot \,|\,\cdot \rangle _2\) denotes the inner product of \(L_2(\mathbb {R})^2\).
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 the following identities in \(\Omega ^\pm \):
In view of (3.9) we may integrate the latter relation over \(\Omega ^\pm \) and using Gauss’ theorem and (3.13) we get
where \(\tilde{\omega }:=\omega \circ \Xi ^{-1}.\)
To estimate the term on the left we observe that the Cauchy-Schwarz inequality and \({|\tilde{\nu }|=1}\) yield
This inequality, the estimate \(\Vert \mathbb {B}_i(f)\Vert _{\mathcal {L}(L_2(\mathbb {R}))}\le C(K)\), \(i=1, 2\), cf. Lemma 3.1, and the representations (3.10) and (3.13), now yield
for any \(\lambda \in [-K,K]\).
We next consider the term on the right of (3.14). As a direct consequence of Lemma 3.1 we note that \(\Vert \mathbb {T}(f)\Vert _{\mathcal {L}(L_2(\mathbb {R})^2)}\le C=C(K)\). This bound together with (3.11) and (3.12) implies
For \(f\in \mathrm{C}^\infty (\mathbb {R})\), the estimate (3.6) follows from (3.14) and the latter estimates upon rearranging terms and a standard density argument. For general functions \({f\in \mathrm{C^1}(\mathbb {R})}\) we additionally need to use the continuity of the mappings
which is a straightforward consequence of Lemma 3.1, together with the density of \({\mathrm{C}^\infty (\mathbb {R})}\) in \({\mathrm{C^1}(\mathbb {R})}\). \(\square \)
Based on Lemma 3.2 we now establish the following result.
Theorem 3.3
(Spectral properties of \(\mathbb {D}(f)\) and \(\mathbb {D}(f)^*\)) Given \(\delta \in (0,1)\), there exists a constant \(C=C(\delta )>0\) such that for all \(\lambda \in \mathbb {R}\) with \(|\lambda |\ge 1/2+\delta \) and \(f\in \mathrm{C}^1(\mathbb {R})\) with \({\Vert f'\Vert _\infty \le 1/\delta }\) we have
Moreover, \(\lambda -\mathbb {D}(f)^*\), \(\lambda -\mathbb {D}(f)\in \mathcal {L}(L_2(\mathbb {R})^2)\) are isomorphisms for all \(\lambda \in \mathbb {R}\) with \(|\lambda |>1/2\) and \(f\in \mathrm{C}^1(\mathbb {R})\).
Proof
In order to prove (3.15) we assume the opposite. Then we may find sequences \((\lambda _k)\) in \(\mathbb {R}\), \((f_k)\) in \(\mathrm{C}^1(\mathbb {R})\), and \((\beta _k)\) in \(L_2(\mathbb {R})^2\) with the property that \(|\lambda _k|\ge 1/2+\delta ,\) \(\Vert f_k'\Vert _\infty \le 1/\delta \), and \(\Vert \beta _k\Vert _2=1\) for all \(k\in \mathbb {N}\), and
Given \(k\in \mathbb {N},\) we set \(\nu _k:=\nu _{f_k}\), \(\tau _k:=\tau _{f_k}\), and \(\omega _k:=\omega _{f_k}\), cf. (2.2). As the operators \(\mathbb {D}(f_k)^*\) are bounded, uniformly in \({k\in \mathbb {N}}\), in \(\mathcal {L}(L_2(\mathbb {R})^2)\), cf. Lemma 3.1, the sequence \((\lambda _k)\) is bounded. Observing that for the constant \({m=m(\lambda )}\) from (3.7) we have \(m(\lambda _k)\ge \delta (2+\delta )>0\) for all \(k\in \mathbb {N}\), we get from Lemma 3.2 that
in \(L_2(\mathbb {R})\). As the operators \(\mathbb {B}_2(f_k)\) are bounded, uniformly with respect to \({k\in \mathbb {N}}\), in \(\mathcal {L}(L_2(\mathbb {R})^2)\), cf. Lemma 3.1, this implies
Let \(\mathbb {A}(f):=\mathbb {B}_1(f)^*\). Since \(|2\lambda _k|\ge 1\), it follows from the proof of [18, Theorem 3.5] that the operator \(2\lambda _k-\mathbb {A}(f_k)\in \mathcal {L}(L_2(\mathbb {R}))\), \(k\in \mathbb {N}\), is an isomorphism with
This implies that also \(2\lambda _k- \mathbb {B}_1(f_k)\in \mathcal {L}(L_2(\mathbb {R}))\), \(k\in \mathbb {N}\), is an isomorphism and
Thus \(\omega _k^{-1}\beta _k\cdot \nu _k\rightarrow 0\) in \(L_2(\mathbb {R})\), so that
This contradicts the property that \(\Vert \beta _k\Vert _2=1\) for all \(k\in \mathbb {N}\) and (3.15) follows.
To complete the proof we fix \(f\in \mathrm{C}^1(\mathbb {R})\) and \(\lambda _0\in \mathbb {R}\) with \(|\lambda _0|>1/2\) and we choose \({\delta \in (0,1)}\) such that \(|\lambda _0|\ge 1/2+\delta \) and \(\Vert f'\Vert _\infty \le 1/\delta \). As \(\mathbb {D}(f)^*\) is bounded, \(\lambda -\mathbb {D}(f)^*\in \mathcal {L}(L_2(\mathbb {R})^2)\) is an isomorphism if \(|\lambda |\) is sufficiently large. The estimate (3.15) together with a standard continuity argument, cf. e.g. [3, Proposition I.1.1.1], now implies that \(\lambda _0-\mathbb {D}(f)^*\) is an isomorphism as well. The result for \(\mathbb {D}(f)\) is an immediate consequence of this property. \(\square \)
4 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 \(\mathbb {D}(f)\), parallel to those in Theorem 3.3, in the spaces \(H^{s-1}(\mathbb {R})^2\), \(s\in (3/2,2)\), and in \(H^2(\mathbb {R})^2\). 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 \(n\ge 1,\) \(s\in (3/2,2),\) and \(a_1,\ldots , a_m\in H^s(\mathbb {R})\) be given. Then, there exists a constant C, depending only on \(n,\, m\), s, and \(\max _{1\le i\le m}\Vert a_i\Vert _{H^s}\), such that
$$\begin{aligned}&\Vert B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,h]\Vert _2\le C\Vert b_1\Vert _{H^1}\Vert h\Vert _{H^{s-1}}\prod _{i=2}^{n}\Vert b_i\Vert _{H^s} \end{aligned}$$(4.1)for all \(b_1,\ldots , b_n\in H^s(\mathbb {R})\) and \(h\in H^{s-1}(\mathbb {R}).\)
Moreover, \([ (a_1,\ldots , a_{m})\mapsto B_{n,m}(a_1,\ldots , a_{m})]\) is locally Lipschitz continuous as a mapping from \(H^s(\mathbb {R})^m\) to
$$\begin{aligned} \mathcal {L}^{n+1}( H^1(\mathbb {R}), H^{s}(\mathbb {R}),\ldots ,H^s(\mathbb {R}), H^{s-1}(\mathbb {R}); L_2(\mathbb {R})). \end{aligned}$$ -
(ii)
Given \(s\in (3/2 ,2)\) and \(a_1,\ldots , a_m \in H^s(\mathbb {R})\), there exists a constant C, depending only on \(n,\, m,\, s\), and \(\max _{1\le i\le m}\Vert a_i\Vert _{H^s},\) such that
$$\begin{aligned} \Vert B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,h]\Vert _{H^{s-1}}\le C \Vert h\Vert _{H^{s-1}}\prod _{i=1}^{n}\Vert b_i\Vert _{H^{s}} \end{aligned}$$for all \( b_1,\ldots , b_n\in H^s(\mathbb {R})\) and \(h\in H^{s-1}(\mathbb {R}). \)
Moreover, \( B_{n,m}\in \mathrm{C}^{1-}(H^s(\mathbb {R})^m,\mathcal {L}^{n}_\mathrm{sym}( H^s(\mathbb {R}) , \mathcal {L}(H^{s-1}(\mathbb {R})))).\)
-
(iii)
Let \(n\ge 1\), \(3/2<s'<s<2\), and \(a_1,\ldots , a_m \in H^s(\mathbb {R})\) be given. Then, there exists a constant C, which depends only on \(n,\, m\), s, \(s'\), and \(\max _{1\le i\le m}\Vert a_i\Vert _{H^s}\), such that
$$\begin{aligned} \begin{aligned}&\Vert B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,h] -h B_{n-1,m}(a_1,\ldots , a_{m})[b_2,\ldots , b_n,b_1']\Vert _{H^{s-1}}\\&\quad \le C \Vert b_1\Vert _{H^{s'}}\Vert h\Vert _{H^{s-1}}\prod _{i=2}^{n}\Vert b_i\Vert _{H^s} \end{aligned} \end{aligned}$$for all \(b_1,\ldots , b_n\in H^s(\mathbb {R})\) and \(h\in H^{s-1}(\mathbb {R}).\)
Proof
The claims (i) is established in [18, Lemmas 3.2], while (ii) and (iii) are proven in [1, Lemma 5 and Lemma 6]. \(\square \)
For \(\xi \in \mathbb {R}\) we define the left shift operator \(\tau _\xi \) on \(L_2(\mathbb {R})\) by the relation \({\tau _\xi u(x):=u(x+\xi )}\) 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 \([\cdot ,\cdot ]_\theta \) denotes the complex interpolation functor of exponent \(\theta \).
Theorem 4.2
Given \(\delta \in (0,1)\) and \(s\in (3/2,2)\), there exists a positive constant \(C=C(\delta ,s)\) such that
for all \(\lambda \in \mathbb {R}\) which satisfy \({|\lambda |\ge 1/2+\delta }\), \(f\in H^s(\mathbb {R})\) with \(\Vert f\Vert _{H^s}\le 1/\delta ,\) and all \({\beta \in H^{s-1}(\mathbb {R})^2}\).
Moreover, \(\lambda -\mathbb {D}(f)\in \mathcal {L}(H^{s-1}(\mathbb {R})^2)\) is an isomorphism for all \(\lambda \in \mathbb {R}\) with \(|\lambda |> 1/2\) and \(f\in H^s(\mathbb {R})\).
Proof
Given \(f\in H^s(\mathbb {R})\), the relation (3.5) and Lemma 4.1 (ii) combined imply that \({\mathbb {D}(f)\in \mathcal {L}(H^{s-1}(\mathbb {R})^2)}\). In order to prove the estimate (4.5), let \({\lambda \in \mathbb {R}}\) with \(|\lambda |\ge 1/2+\delta \) and \(f\in H^s(\mathbb {R})\) with \(\Vert f\Vert _{H^s}\le 1/\delta \) be fixed. Theorem 3.3 together with the embedding \(H^s(\mathbb {R})\hookrightarrow L_\infty (\mathbb {R})\) implies there exists \({C=C(\delta )>0}\) such that \(\Vert (\lambda -\mathbb {D}(\tau _\xi f))^{-1}\Vert _{\mathcal {L}(L_2(\mathbb {R})^2)}\le C\) for all \(\xi \in \mathbb {R}\). It is well-known there exists a constant \({C>0}\) such that
where \(\mathcal {F}[\beta ] \) is the Fourier transform of \(\beta \). Together with (4.2) we then get
The term \(\Vert (\mathbb {D}(f) -\mathbb {D}(\tau _{\xi }f))[\beta ]\Vert _2\) can be estimated by a finite sum of terms of the form
where \(0\le n\le 3\) and \(i\in \{1,2\}\). Let \(s'\in (3/2,s)\) 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 the operator \(B_{n,2}^0(f)-B_{n,2}^0(\tau _\xi f)\) belongs to \(\mathcal {L}(H^{s'-1}(\mathbb {R}),L_2(\mathbb {R}))\) and satisfies
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 \({\lambda -\mathbb {D}(f)}\), with \(\lambda \in \mathbb {R}\) with \(|\lambda |> 1/2\) and \(f\in H^s(\mathbb {R})\), follows by the same continuity argument as in the \(L_2\) result. \(\square \)
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 \( n,\, m\in \mathbb {N}\) and \(a_1,\ldots , a_m\in H^2(\mathbb {R})\) be given. Then, there exists a constant C, depending only on \(n,\, m\), and \(\max _{1\le i\le m}\Vert a_i\Vert _{H^2}\), such that
for all \(b_1,\ldots , b_n\in H^2(\mathbb {R})\) and \(h\in H^1(\mathbb {R}).\)
Moreover, \(B_{n,m}\in \mathrm{C}^{1-}(H^2(\mathbb {R})^m,\mathcal {L}^{n}_\mathrm{sym}(H^2(\mathbb {R}),\mathcal {L}(H^1(\mathbb {R})))).\)
Proof
We first show that the function \(\varphi :=B_{n,m}(a_1,\ldots , a_{m})[b_1,\ldots , b_n,h]\) belongs to \({H^1(\mathbb {R})}\). Recalling that the group \(\{\tau _\xi \}_{\xi \in \mathbb {R}}\subset \mathcal {L}(H^r(\mathbb {R})),\) \(r\ge 0\), has generator \([f\mapsto f']\in \mathcal {L}(H^{r+1}(\mathbb {R}),H^r(\mathbb {R})),\) it suffices to prove that the quotient \({D_\xi \varphi :=(\tau _\xi \varphi -\varphi )/\xi }\) converges in \(L_2(\mathbb {R})\) when letting \(\xi \rightarrow 0\). In view of (4.3) we write
where \(B_{n+2,m+1}^i(\xi ):=B_{n+2,m+1}(\tau _\xi a_1,\ldots ,\tau _\xi a_i,a_i,\ldots ,a_m)\) for \(1\le i\le m\). Lemma 3.1 and Lemma 4.1 (i) enable us to pass to the limit \(\xi \rightarrow 0\) in \(L_2(\mathbb {R})\) in this equality. Hence, \(\varphi \in H^1(\mathbb {R})\) and
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). \(\square \)
As a consequence of Lemma 4.3 and (4.9) we obtain the following result.
Corollary 4.4
\(B_{n,m}\in \mathrm{C}^{1-}(H^3(\mathbb {R})^{m},\mathcal {L}^{n}_\mathrm{sym}(H^3(\mathbb {R}),\mathcal {L}( H^2(\mathbb {R})))) \) for \({n,\, m\in \mathbb {N}}\).
Theorem 4.5
The operator \({\lambda -\mathbb {D}(f)\in \mathcal {L}(H^2(\mathbb {R})^2)}\) is an isomorphism for all \(f\in H^3(\mathbb {R})\) and \(\lambda \in \mathbb {R}\) with \({|\lambda |>1/2}\).
Proof
Fix \(f\in H^3(\mathbb {R})\). We then infer from (3.5) and Corollary 4.4 that we have \({\mathbb {D}(f)\in \mathcal {L}(H^2(\mathbb {R})^2)}\). Recalling (4.9), we further compute
where each component of \(T_\mathrm{lot}[\beta ]\) is a linear combination of terms
where \(n,\,m\in \mathbb {N}\) satisfy \(0\le n,\,m\le 7\) and \(k\in \{0,\, 1\}\). From Lemma 3.1 and Lemma 4.1 (i) (with \(s=7/4\)) we conclude that
Given \(\lambda \in \mathbb {R}\) with \(|\lambda |>1/2\), we pick \(\delta \in (0,1)\) such that \({|\lambda |\ge 1/2+\delta }\) and additionally \({\Vert f'\Vert _{\infty }\le 1/\delta .}\) Since \(\Vert (\mu -\mathbb {D}(f))^{-1}\Vert _{\mathcal {L}(L_2(\mathbb {R})^2)}\le C\) for all \(\mu \in \mathbb {R}\) with \(|\mu |\ge 1/2+\delta \), cf. Theorem 3.3, we deduce from (4.10), (4.11), and (4.4) that
hence
for all \(\beta \in H^2(\mathbb {R})^2\) and \(\mu \in \mathbb {R}\) with \(|\mu |\ge 1/2+\delta \). The result follows now by the same continuity argument as in the proof of Theorem 4.2. \(\square \)
5 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 Sect. 2, Sect. 4, and Appendix A we start by proving that for each \(f\in H^3(\mathbb {R})\), the boundary value problem
has a unique solution \((v,p)\in X_f\) with the property that \( v^\pm |_\Gamma \circ \Xi _f\in H^2(\mathbb {R})^2\). This is established in Proposition 5.1 below, where we also provide an implicit formula for \(v^\pm |_\Gamma \) in terms of contour integrals on \(\Gamma \). This representation allows to recast the kinematic boundary condition (1.1a)\(_6\) in the form (5.17).
With the substitution \(\tilde{v}^\pm :=\mu _\pm v^\pm \), Problem (5.1) is equivalent to
We construct the solution to (5.2) by splitting
where \((w_s,q_s),\,(w_d,q_d)\in X_f\) satisfy
and
The system (5.3) has been studied in [20]. According to [20, Theorem 2.1 and Remark A.2], there exists exactly one solution \({(w_{s} ,q_{s}):=(w_{s}(f),q_{s}(f))\in X_f}\) to (5.3). It satisfies
Moreover, recalling (3.2) and [20, Eqns. (2.2), (2.3), (A.2)], the trace \(w_{s}(f)|_\Gamma \) can be expressed via
with
where \( \phi _i(f)\in H^2(\mathbb {R}),\) \(i\in \{1,\,2\}\), are given by
We point out that Corollary 4.4 yields \(G_i(f)\in H^2(\mathbb {R}),\) \(i\in \{1,\,2\}\).
It remains to show that the boundary value problem (5.4) has a unique solution \({(w_d,q_d)\in X_f}\) with \(w_d^\pm |_\Gamma \circ \Xi \in H^2(\mathbb {R})^2.\) To construct a solution, we use the ansatz \((w_d,q_d)=(w,q)[\beta ]\), where \(\beta \in H^2(\mathbb {R})^2\) and \((w,q)[\beta ]\) is defined by (2.7), (2.8). We recall from Proposition 2.1 that \((w,q)[\beta ]\) is the unique solution to (2.4) in \(X_f\). In view of Lemma A we have
Therefore \((w_d,q_d)\) solves (5.4) if and only if
where
Theorem 4.5 implies that (5.8) has a unique solution \(\beta =:\beta (f)\in H^2(\mathbb {R})^2\). This establishes not only the existence but also the uniqueness of the solution to (5.4).
Summarizing, we have shown the following result:
Proposition 5.1
Given \(f\in H^3(\mathbb {R})\), the boundary value problem (5.1) has a unique solution \({(v,p)\in X_f}\) such that \( v^\pm |_\Gamma \circ \Xi \in H^2(\mathbb {R})^2.\) Moreover,
where \(G(f)\in H^2(\mathbb {R})^2\) is defined in (5.5)-(5.6) and \(\beta (f)\in H^2(\mathbb {R})^2\) is the unique solution to (5.8).
From this result and (1.1) we infer, under the assumption that \(\Gamma (t)\) is at each time instant \({t\ge 0}\) the graph of a function \(f(t)\in H^3(\mathbb {R})\) and that the pair (v(t), p(t)) belongs to \(X_{f(t)}\) and satisfies \(v(t)^\pm |_{\Gamma (t)}\circ \Xi _{f(t)}\in H^2(\mathbb {R})^2,\) that (1.1a) can be recast as
Here \(\langle \cdot \,|\,\cdot \rangle \) denotes the scalar product on \(\mathbb {R}^2\).
Using the results in Sect. 4 and [20] we can formulate the latter equation as an evolution equation in \(H^{s-1}(\mathbb {R})^2\), where \(s\in (3/2,2)\) is fixed in the remaining. To this end we first infer from [20, Corollary C.5] that, given \(n,\, m\in \mathbb {N}\), we have
Further, [20, Lemma 3.5] ensures for the mappings defined in (5.7) that
Additionally, for any \({f_0\in H^s(\mathbb {R}),}\) the Fréchet derivative \(\partial \phi _i(f_0)\) is given by
with
It is easy to check, by arguing as in [20, Lemma C.1], that \(\phi _i\), \(i=1,\, 2\), maps bounded sets in \(H^s(\mathbb {R})\) to bounded sets in \(H^{s-1}(\mathbb {R})\). 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 \(H^s(\mathbb {R})\) to bounded sets in \(H^{s-1}(\mathbb {R})^2\).
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 \(f\in H^s(\mathbb {R})\), the equation (5.8) in \(H^{s-1}(\mathbb {R})^2\). 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
Furthermore, (5.15) and the estimate (4.5) imply that \(\beta \) inherits from G the property to map bounded sets in \(H^s(\mathbb {R})\) to bounded sets in \(H^{s-1}(\mathbb {R})^2\). Summarizing, in a compact form, the Stokes flow problem (1.1) can be recast as the evolution problem
where \(\Phi :H^{s}(\mathbb {R})\rightarrow H^{s-1}(\mathbb {R})\) is defined, cf. (5.9), by
Observe that, due to (5.16),
and that \(\Phi \) maps bounded sets in \(H^s(\mathbb {R})\) to bounded sets in \(H^{s-1}(\mathbb {R}).\)
6 Linearization, localization, and proof of the main result
We are going to prove that the nonlinear and nonlocal problem (5.17) is parabolic in \(H^s(\mathbb {R})\) in the sense that the Fréchet derivative \(\partial \Phi (f_0)\), generates an analytic semigroup in \(\mathcal {L}(H^{s-1}(\mathbb {R}))\) for each \({f_0\in H^s(\mathbb {R})}\). This property then enables us to use the abstract existence results from [17] in the proof of our main result Theorem 1.1.
Theorem 6.1
For any \(f_0\in H^s(\mathbb {R})\), the Fréchet derivative \(\partial \Phi (f_0)\), considered as an unbounded operator in \(H^{s-1}(\mathbb {R})\) with dense domain \(H^{s}(\mathbb {R})\), generates an analytic semigroup in \(\mathcal {L}(H^{s-1}(\mathbb {R}))\).
The proof of Theorem 6.1 requires some preparation. To start, we fix a function \({f_0\in H^s(\mathbb {R})}\), \(s'\in (3/2,s)\), and we set \(\beta _0:=\beta (f_0):=(\beta _{0}^1,\beta _{0}^2)^\top \). We have \(\beta _0\in H^{s-1}(\mathbb {R})^2.\)
Differentiating the relations (5.18) and (5.15), we get
and
For the computation of \(\partial \mathbb {D}(f_0)[f][\beta _0]\) and \(\partial G(f_0)[f]\) we use the relation
see [20, Lemma C.4]. Additionally we use Lemma 4.1 (iii) to rewrite this expression as
where \(n B^0_{n-1,3}(f_0):=0\) for \(n=0\) and
with a constant C independent of \(f\in H^{s}(\mathbb {R})\) and \(h\in H^{s-1}(\mathbb {R}).\) Using these relations, we infer from (3.5) that
for \(i=1,\,2\), where we used the shorthand notation \(B_{n,m}^0:=B_{n,m}^0(f_0)\) and
Taking the derivative of (5.6), the same arguments yield
where
cf. [20, Eq. (3.7)-(3.9)]. Here we have used the shortened notation \(a_i:=a_i(f_0)\) and \({\phi _i:=\phi _i(f_0)}\) for \(i=1,\,2\) and
In order to prove Theorem 6.1 we consider the path
defined by
for \(\tau \in [0,1]\) and \(f\in H^s(\mathbb {R}),\) where \(\mathcal {B}(\tau )[f]\) is defined by
Theorem 4.2, (6.3)–(6.7), and Lemma 4.1 (ii) combined ensure that the mapping \({\mathcal {B}:[0,1]\longrightarrow \mathcal {L}\big (H^s(\mathbb {R}),H^{s-1}(\mathbb {R})^2\big )}\) is well-defined, and
with C independent of f and \(\tau \). We also note that both paths \(\mathcal {B}\) and \(\Psi \) are continuous and \({\Psi (1)=\partial \Phi (f_0)}\). Besides, since
where \(H =\pi ^{-1}B_{0,0}\) is the Hilbert transform, we observe that \(\Psi (0)\) is the Fourier multiplier
We next locally approximate the operator \(\Psi (\tau )\), \(\tau \in [0,1]\), by certain Fourier multipliers \(\mathbb {A}_{j,\tau }\), cf. Theorem 6.2. For this purpose, given \({\varepsilon \in (0,1)}\), we choose \(N=N(\varepsilon )\in \mathbb {N}\) and a so-called finite \(\varepsilon \)-localization family, that is a set
such that
The real number \(\xi _N^\varepsilon \) plays no role in the analysis below. To each \(\varepsilon \)-localization family we associate a norm on \(H^r(\mathbb {R}),\) \(r\ge 0\), which is equivalent to the standard norm on \(H^r(\mathbb {R})\). Indeed, given \({r\ge 0}\) and \(\varepsilon \in (0,1)\) , there exists a constant \({c=c(\varepsilon ,r)\in (0,1)}\) such that
To introduce the aforementioned Fourier multipliers \(\mathbb {A}_{j,\tau }\), we first define the coefficient functions \(\alpha _\tau ,\, \beta _\tau :\mathbb {R}\longrightarrow \mathbb {R}\), \(\tau \in [0,1]\), 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.
Theorem 6.2
Let \(\mu >0\) be given and fix \(s'\in (3/2,s)\). Then there exist \({\varepsilon \in (0,1)}\) and a constant \(K=K(\varepsilon )\) such that
for all \(-N+1\le j\le N\), \(\tau \in [0,1],\) and \(f\in H^s(\mathbb {R})\).
Before proving Theorem 6.2 we first present some auxiliary lemmas which are used in the proof (which is presented below). We start with an estimate for the commutator \([B_{n,m}^0(f),\varphi ]\) (we will apply this estimate in the particular case \(\varphi =\pi _j^\varepsilon \), \(-N+1\le j\le N\)).
Lemma 6.3
Let \(n,\, m \in \mathbb {N}\), \(s\in (3/2, 2)\), \(f\in H^s(\mathbb {R})\), and \({\varphi \in \mathrm{C}^1(\mathbb {R})}\) with uniformly continuous derivative \(\varphi '\) be given. Then, there exists a constant K that depends only on n, m, \(\Vert \varphi '\Vert _\infty , \) and \(\Vert f\Vert _{H^s}\) such that
for all \(h\in L_2(\mathbb {R})\).
Proof
This result is a particular case of [1, Lemma 12]. \(\square \)
The results in Lemma 6.4-Lemma 6.8 below describe how to “freeze the coefficients” of the multilinear operators \(B_{n,m}^0.\) For these operators, this technique has been first developed in [19] in the study of the Muskat problem.
Lemma 6.4
Let \(n,\, m \in \mathbb {N}\), \(3/2<s'<s<2\), and \(\nu \in (0,\infty )\) be given. Let further \({f\in H^s(\mathbb {R})}\) and \( \overline{\omega }\in \{1\}\cup H^{s-1}(\mathbb {R})\). For any sufficiently small \({\varepsilon \in (0,1)}\), there exists a constant K depending only on \(\varepsilon ,\, n,\, m,\, \Vert f\Vert _{H^s},\) and \(\Vert \overline{\omega }\Vert _{H^{s-1}}\) (if \({\overline{\omega }\ne 1}\)) such that
for all \(|j|\le N-1\) and \(h\in H^{s-1}(\mathbb {R})\).
Proof
See [1, Lemma 13]. \(\square \)
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 \(\xi _j^\varepsilon \).
Lemma 6.5
Let \(n,\, m \in \mathbb {N}\), \(3/2<s'<s<2\), and \(\nu \in (0,\infty )\) be given. Let further \({f\in H^s(\mathbb {R})}\), \(a\in H^{s-1}(\mathbb {R})\), and \(\overline{\omega }\in \{1\}\cup H^{s-1}(\mathbb {R})\). For any sufficiently small \(\varepsilon \in (0,1)\), there is a constant K depending on \(\varepsilon ,\) n, m, \( \Vert f\Vert _{H^s},\) \({\Vert a\Vert _{H^{s-1}},}\) and \( \Vert \overline{\omega }\Vert _{H^{s-1}}\) (if \(\overline{\omega }\ne 1\)) such that
for all \(|j|\le N-1\) and \(h\in H^{s-1}(\mathbb {R})\).
Proof
See [20, Lemma D.5]. \(\square \)
Lemma 6.6 and Lemma 6.7 are the analogues of Lemma 6.4 corresponding to the case \(j=N\).
Lemma 6.6
Let \(n,\, m \in \mathbb {N}\), \(3/2<s'<s<2\), and \(\nu \in (0,\infty )\) be given. Let further \({f\in H^s(\mathbb {R})}\) and \(\overline{\omega }\in H^{s-1}(\mathbb {R})\). For any sufficiently small \(\varepsilon \in (0,1)\), there is a constant K depending only on \(\varepsilon ,\, n,\, m,\, \Vert f\Vert _{H^s},\) and \(\Vert \overline{\omega }\Vert _{H^{s-1}}\) such that
for all \(h\in H^{s-1}(\mathbb {R})\).
Proof
See [1, Lemma 14]. \(\square \)
Lemma 6.7 is the counterpart of Lemma 6.6 in the case when \(\overline{\omega }=1\).
Lemma 6.7
Let \(n,\, m \in \mathbb {N}\), \(3/2<s'<s<2\), and \(\nu \in (0,\infty )\) be given. Let further \({f\in H^s(\mathbb {R})}\). For any sufficiently small \(\varepsilon \in (0,1)\), there is a constant K depending only on \(\varepsilon ,\, n,\, m,\) and \( \Vert f\Vert _{H^s}\) such that
and
for all \(h\in H^{s-1}(\mathbb {R})\).
Proof
See [1, Lemma 15]. \(\square \)
Finally, Lemma 6.8 below is the analogue of Lemma 6.5 corresponding to the case \({j=N}\).
Lemma 6.8
Let \(n,\, m \in \mathbb {N}\), \(3/2<s'<s<2\), and \(\nu \in (0,\infty )\) be given. Let further \({f\in H^s(\mathbb {R})}\), \(a\in H^{s-1}(\mathbb {R})\), and \(\overline{\omega }\in \{1\}\cup H^{s-1}(\mathbb {R})\). For any sufficiently small \(\varepsilon \in (0,1)\), there is a constant K depending on \(\varepsilon ,\) n, m, \( \Vert f\Vert _{H^s},\) \(\Vert a\Vert _{H^{s-1}},\) and \( \Vert \overline{\omega }\Vert _{H^{s-1}}\) (if \(\overline{\omega }\ne 1\)) such that
for all \(h\in H^{s-1}(\mathbb {R})\).
Proof
See [20, Lemma D.6]. \(\square \)
We are now in a position to prove Theorem 6.2.
Proof of Theorem 6.2
Fix \(\mu >0\) and let \(\varepsilon \in (0,1)\). We next choose a finite \(\varepsilon \)-localization family \(\{(\pi _j^\varepsilon ,\xi _j^\varepsilon )\,|\, -N+1\le j\le N\}\) and, associated to it, a second family \(\{\chi _j^\varepsilon \,|\, -N+1\le j\le N\}\) with the following properties:
In the arguments that follow we repeatedly use the estimate
which holds for \(g,\, h\in H^{s-1}(\mathbb {R})\) and \(s\in (3/2,2)\), with a constant C independent of g and h.
Below we denote by C constants that do not depend on \(\varepsilon \) and by K constants that may depend on \(\varepsilon \). We need to approximate the linear operators \({\big [f\mapsto \mathcal {B}_2(\tau )[f]-\tau f_0'\mathcal {B}_1(\tau )[f]\big ]}\) and \({[f\mapsto \beta _0^1f']}\), see (6.8)-(6.9), where we set \(\mathcal {B}(\tau )=:(\mathcal {B}_1(\tau ),\mathcal {B}_2(\tau ))^\top \). The proof is divided in several steps.
Step 1. We consider the operator \([f\mapsto \beta _0^1f']\). Since \(\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon \), (6.17) yields
for \(|j|\le N-1\) and
From (5.16) we have \(\beta _0^1\in \mathrm{C}^{s-3/2}(\mathbb {R})\) and \(\beta _0^1(\xi )\rightarrow 0\) for \(|\xi |\rightarrow \infty \). Hence, if \(\varepsilon \) is sufficiently small, then
for \( |j|\le N-1\).
The approximation procedure for \(\big [f\mapsto \mathcal {B}_2(\tau )[f]-\tau f_0'\mathcal {B}_1(\tau )[f]\big ]\) is more involved.
Step 2. We prove there exists a constant \(C_\mathcal {B}\) such that
for all \(-N+1\le j\le N\), \(\tau \in [0,1],\) and \(f\in H^s(\mathbb {R})\). To start, we infer from (6.9) that
In order 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.
Step 3. Given \(\tau \in [0,1]\) and \( -N+1\le j\le N\), let \(\mathbb {B}_{j,\tau }\in \mathcal {L}(H^{s}(\mathbb {R})^2, H^{s-1}(\mathbb {R})^2)\) denote the Fourier multipliers
We next prove that given \(\nu >0\), we have
for all \( -N+1\le j\le N\), \(\tau \in [0,1],\) \(f\in H^s(\mathbb {R})\) and all sufficiently small \(\varepsilon \). To start, we multiply (6.9) by \(\pi _j^\varepsilon \) and get
We consider the terms on the right hand side of (6.24) one by one. To deal with the first term we recall (6.5)–(6.7). Repeated use of Lemma 6.4 and Lemma 6.5 then shows that
for \(|j|\le N-1\), while Lemma 6.6, Lemma 6.7, and Lemma 6.8 yield
provided that \(\varepsilon \) is sufficiently small.
We estimate the second term on the right of (6.24) and let \(|j|\le N-1\) first. Combining (3.5), Lemma 6.4, Lemma 6.5, (6.10) with s replaced by \(s'\), and (6.19) we obtain
provided that \(\varepsilon \) is sufficiently small. Similarly, if \(j=N\), then Lemma 6.7, Lemma 6.8, (6.10) with s replaced by \(s'\), and (6.19) imply that
provided that \(\varepsilon \) is sufficiently small.
It remains to consider the term \(\pi _j^\varepsilon \partial \mathbb {D}(f_0)[f][\beta _0]\) on the right of (6.24). To this end we argue similarly as in the proof of (6.27) by adding and subtracting suitable localization operators. Recalling (6.3)-(6.4), we get from Lemma 6.4 and Lemma 6.5 if \(|j|\le N-1\), respectively from Lemma 6.6 and Lemma 6.8 if \({j=N}\), that
provided that \(\varepsilon \) is sufficiently small. The estimate (6.23) follows now from (6.24)–(6.29).
Step 4. We now localize the operators \(\big [f\mapsto \mathcal {B}_2(\tau )[f]-\tau f_0'\mathcal {B}_1(\tau )[f]\big ]\). The estimate (6.23) shows that, choosing \(\varepsilon \) sufficiently small, we have
for \(|j|\le N-1\) and
Moreover, for \(|j|\le N-1\), we write in view of \(\chi _j^\varepsilon \pi _j^\varepsilon =\pi _j^\varepsilon \)
The first term on the right hand side may be estimated by using (6.10) (with s replaced by \(s'\)), (6.17), (6.19), and the fact that \(f_0'\in \mathrm{C}^{s-3/2}(\mathbb {R})\). For the second term we rely on (6.23). Hence, if \(\varepsilon \) is sufficiently small then
For \(j=N\), it follows from (6.10) (with s replaced by \(s'\)), (6.17), (6.19), and the fact that \(f_0' \) vanishes at infinity that
The desired claim (6.15) follows now from (6.8), (6.18), (6.30), and (6.32) if \({|j|\le N-1}\), respectively from (6.8), (6.18), (6.31), and (6.33) if \(j=N.\) \(\square \)
We now investigate the Fourier multipliers \(\mathbb {A}_{j,\tau } \) found in Theorem 6.2. We recall the definitions (5.12), (6.13), and (6.14) and observe that as the functions \(f_0', \, \beta _0^1,\) and \( a_i(\tau f_0)\) belong to \(H^{s-1}(\mathbb {R})\), \(i=1,\,2\) and \(\tau \in [0,1]\), there is a constant \(\eta \in (0,1)\) such that
Based on this, it can be shown as in [19, Proposition 4.3], that there is a constant \({\kappa _0\ge 1}\) such that for all \(\varepsilon \in (0,1)\), \( -N+1\le j\le N\), and \(\tau \in [0,1]\) we have
The properties (6.34)-(6.35) together with Theorem 6.2 enable us to prove Theorem 6.1.
Proof of Theorem 6.1
Let \(s'\in (3/2,s)\) and let \(\kappa _0\ge 1\) be the constant in (6.35). Theorem 6.2 with \(\mu :=1/2\kappa _0 \) implies that there are \(\varepsilon \in (0,1)\), a constant \(K=K(\varepsilon )>0\) and bounded operators \(\mathbb {A}_{j,\tau }\in \mathcal {L}(H^s(\mathbb {R}), H^{s-1}(\mathbb {R}))\), for \({-N+1\le j\le N}\) and \(\tau \in [0,1],\) satisfying
Moreover, (6.35) yields
for all \(-N+1\le j\le N\), \(\tau \in [0,1],\) \(\mathrm{Re\,}\lambda \ge 1\), and \(f\in H^s(\mathbb {R})\). The latter estimates combined lead us to
Summing over j, we deduce from (6.12), Young’s inequality, and the interpolation property (4.4) that there exist constants \(\kappa \ge 1\) and \(\omega >1 \) such that
for all \(\tau \in [0,1],\) \(\mathrm{Re\,}\lambda \ge \omega \), and \(f\in H^s(\mathbb {R})\).
From (6.11) we also deduce that \(\omega -\Psi (0) \in \mathcal {L}(H^s(\mathbb {R}), H^{s-1}(\mathbb {R}))\) is an isomorphism. This together with method of continuity [3, Proposition I.1.1.1] and (6.36) implies that also
is an isomorphism. The estimate (6.36) (with \(\tau =1\)) and (6.37) finally imply that \(\partial \Phi (f_0)\) generates an analytic semigroup in \(\mathcal {L}(H^{s-1}(\mathbb {R}))\), cf. [3, Chapter I], and the proof is complete. \(\square \)
We are now in a position to prove the main result, for which we can exploit abstract theory for fully nonlinear parabolic problems from [17].
Proof of Theorem 1.1
Well-posedness: Given \(\alpha \in (0,1)\), \(T>0\), and a Banach space X we set
where
The property (5.19) together with Theorem 6.1 shows that the assumptions of [17, Theorem 8.1.1] are satisfied for the evolution problem (5.17). According to this theorem, (5.17) has, for each \({f_0\in H^{s}(\mathbb {R})}\), a local solution \(f(\cdot ;f_0)\) such that
where \(T=T(f_0)>0\) and \(\alpha \in (0,1)\) is fixed (but arbitrary). This solution is unique within the set
We improve the uniqueness property by showing that the solution is unique within
Indeed, let f now be any solution to (5.17) in that space, let \(s'\in (3/2,s)\) be fixed and set \({\alpha := s-s'\in (0,1)}\). Using (4.4), we find a constant \(C>0\) such that
which shows in particular that \(f \in \mathrm{C}^{\alpha }_{\alpha }((0,T], H^{s'}(\mathbb {R}))\). The uniqueness statement of[17, Theorem 8.1.1] applied in the context of the evolution problem (5.17) with \(\Phi \in \mathrm{C}^{\infty }(H^{s'}(\mathbb {R}), H^{s'-1}(\mathbb {R}))\) establishes the uniqueness claim. This unique solution can be extended up to a maximal existence time \({T_+(f_0)}\), see [17, Section 8.2]. Finally, [17, Proposition 8.2.3] shows that the solution map defines a semiflow on \(H^s(\mathbb {R})\) which, according to [17, Corollary 8.3.8], is smooth in the open set \(\{(t,f_0)\,|\, 0<t<T_+(f_0)\}\). This proves (i).
Parabolic smoothing: The uniqueness result established in (i) enables us to use a parameter trick applied also to other problems, cf., e.g., [4, 9, 19, 21], in order to establish (iia) and (iib). The proof details are similar to those in [18, Theorem 1.2 (v)] or [1, Theorem 2 (ii)] and therefore we omit them.
Global existence: We prove the statement by contradiction. Assume there exists a maximal solution \(f\in \mathrm{C}([0,T_+),H^{s}(\mathbb {R}))\cap \mathrm{C}^1([0,T_+), H^{s-1}(\mathbb {R}))\) to (5.17) with \(T_+<\infty \) and such that
Recalling that \(\Phi \) maps bounded sets in \(H^s(\mathbb {R})\) to bounded sets in \(H^{s-1}(\mathbb {R}) \), we get
Let \(s'\in (3/2,s)\) be fixed. Arguing as above, see (6.38), from the bounds (6.39) and (6.40) we get that \({f:[0,T_+)\longrightarrow H^{s'}(\mathbb {R})}\) is uniformly continuous. Applying [17, Theorem 8.1.1] to (5.17) with \(\Phi \in \mathrm{C}^\infty (H^{s'}(\mathbb {R}), H^{s'-1}(\mathbb {R}))\), we may extend the solution f to a time interval \([0,T_+')\) with \({T_+<T_+'}\) and such that
Since by (iib) (with s replaced by \(s'\)) we have \(f\in \mathrm{C}^1((0,T_+'),H^{s}(\mathbb {R}))\), this contradicts the maximality property of f and the proof is complete. \(\square \)
References
Abels, H., Matioc, B.V.: Well-posedness of the Muskat problem in subcritical \(L_p\)-Sobolev spaces, European J. Appl. Math. 33(2), 224–266 (2022)
Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992) vol. 133 of Teubner-Texte Math., Teubner, Stuttgart, pp. 9–126 (1993)
Amann, H.: Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1995).. (Abstract linear theory)
Angenent, S.B.: Nonlinear analytic semiflows. Proc. Roy. Soc. Edinburgh Sect. A 115, 91–107 (1990)
Badea, A., Duchon, J.: Capillary driven evolution of an interface between viscous fluids. Nonlinear Anal. 31, 385–403 (1998)
Bierler, J., Matioc, B.V.: The multiphase Muskat problem with equal viscosities in two dimensions. Interfaces Free Bound. 24(2), 163–196 (2022)
Chang, T.K., Pahk, D.H.: Spectral properties for layer potentials associated to the Stokes equation in Lipschitz domains. Manuscripta Math. 130, 359–373 (2009)
Duchon, J., Robert, R.: Estimation d’opérateurs intégraux du type de Cauchy dans les échelles d’Ovsjannikov et application. C. R. Acad. Sci. Paris Sér. I Math. 299, 595–598 (1984)
Escher, J., Simonett, G.: Analyticity of the interface in a free boundary problem. Math. Ann. 305, 439–459 (1996)
Fabes, E.B., Kenig, C.E., Verchota, G.C.: The Dirichlet problem for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)
Friedman, A., Reitich, F.: Quasi-static motion of a capillary drop. II. The three-dimensional case. J. Differential Equations 186, 509–557 (2002)
Friedman, A., Reitich, F.: Quasistatic motion of a capillary drop. I. The two-dimensional case. J. Differential Equations 178, 212–263 (2002)
Gancedo, F.: A survey for the Muskat problem and a new estimate. SeMA J. 74, 21–35 (2017)
Granero-Belinchón, R., Lazar, O.: Growth in the Muskat problem. Math. Model. Nat. Phenom. 15, Paper No. 7, 23 (2020)
Günther, M., Prokert, G.: Existence results for the quasistationary motion of a free capillary liquid drop. Z. Anal. Anwendungen 16, 311–348 (1997)
Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York-London (1963)
Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel (1995)
Matioc, B.-V.: Viscous displacement in porous media: the Muskat problem in 2D. Trans. Amer. Math. Soc. 370, 7511–7556 (2018)
Matioc, B.-V.: The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12, 281–332 (2019)
Matioc, B.-V., Prokert, G.: Two-phase Stokes flow by capillarity in full 2d space: an approach via hydrodynamic potentials. Proc. Roy. Soc. Edinburgh Sect. A 151, 1815–1845 (2021)
Prüss, J., Shao, Y., Simonett, G.: On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension. Interfaces Free Bound. 17, 555–600 (2015)
Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations. vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham] (2016)
Solonnikov, V.A.: On quasistationary approximation in the problem of motion of a capillary drop. In: Topics in nonlinear analysis, vol. 35 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, pp. 643–671 (1999)
Solonnikov, V.A.: On the justification of the quasistationary approximation in the problem of motion of a viscous capillary drop. Interfaces Free Bound. 1, 125–173 (1999)
Acknowledgements
The research leading to this paper was carried out while the second author enjoyed the hospitality of DFG Research Training Group 2339 “Interfaces, Complex Structures, and Singular Limits in Continuum Mechanics - Analysis and Numerics” at the Faculty of Mathematics of Regensburg University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A. The hydrodynamic double-layer potential near \(\Gamma \)
Appendix A. The hydrodynamic double-layer potential near \(\Gamma \)
Given \(f\in H^3(\mathbb {R})\) and \(\beta \in H^2(\mathbb {R})\), we let (w, q) be given by (2.7) and (2.8). We recall the definitions (3.4) of \(\mathbb {D}(f)\) and (3.2) of the operators \(B^0_{n,m}\).
Lemma A.1
We have \(w^\pm \in \mathrm{C}^1(\overline{\Omega ^\pm },\mathbb {R}^2)\), \(q^\pm \in \mathrm{C}(\overline{\Omega ^\pm })\), \(w^\pm |_\Gamma \circ \Xi \in H^2(\mathbb {R})^2\), and
Proof
For \(j=0,\ldots ,3\), let \(\mathcal {Z}_j\in \mathrm{C}^1(\mathbb {R}^2\setminus \{0\})\) be given by
Given \(\phi \in H^1(\mathbb {R})\), we define the function \(Z_j[\phi ]:\mathbb {R}^2\setminus \Gamma \longrightarrow \mathbb {R}\), \(j=0,\ldots ,3,\) by
Recalling (2.9)\(_1\), we have
It is shown in [20, Lemma A.1] that \(Z_j[\phi ]^\pm \in \mathrm{C}(\overline{\Omega ^\pm })\), with
Consequently, \(w^\pm \in \mathrm{C}(\overline{\Omega ^\pm },\mathbb {R}^2)\), and the jump relations (A.2) imply (A.1)\(_1\). Moreover, recalling Corollary 4.4, we get \(w^\pm |_{\Gamma }\circ \Xi \in H^2(\mathbb {R})^2\). Further, the property \(q^\pm \in \mathrm{C}(\overline{\Omega ^\pm })\) follows from [18, Lemma 2.1].
Exchanging integration with respect to s and differentiation with respect to x by dominated convergence we find from (1.1b), (2.7), and (2.8) that
for \(x\in \mathbb {R}^2\setminus \Gamma \) and \( l,\, j=1,\, 2\).
For \(E\subset \mathbb {R}^2\) open, \(\mathcal {Z}\in \mathrm{C}^1(E)\), \(i=1,2\), we let \(\mathrm{rot\,} \mathcal {Z}:= (\mathrm{rot}^1 \mathcal {Z}, \mathrm{rot}^2 \mathcal {Z})\), with \(\mathrm{rot\,} \mathcal {Z}\in \mathrm{C}(E,\mathbb {R}^2)\), be defined by
With this notation, we find from integration by parts
Together with (A.3), (A.4), and the identities
and
this yields \(w^\pm \in \mathrm{C}^1(\overline{\Omega ^\pm },\mathbb {R}^2)\) and (A.1)\(_2\).
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Matioc, B., Prokert, G. Two-phase Stokes flow by capillarity in the plane: The case of different viscosities. Nonlinear Differ. Equ. Appl. 29, 54 (2022). https://doi.org/10.1007/s00030-022-00785-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-022-00785-0