Two-phase Stokes flow by capillarity in the plane: The case of different viscosities
aa r X i v : . [ m a t h . A P ] F e b TWO-PHASE STOKES FLOW BY CAPILLARITY IN THE PLANE:THE CASE OF DIFFERENT VISCOSITIES
BOGDAN–VASILE MATIOC AND GEORG PROKERT
Abstract.
We study the two-phase Stokes flow driven by surface tension for two fluids ofdifferent viscosities, separated by an asymptotically flat interface representable as graph ofa differentiable function. The flow is assumed to be two-dimensional with the fluids fillingthe entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces upto critical regularity. The main technical tools are an analysis of nonlinear singular integraloperators arising from the hydrodynamic single and double layer potential, spectral resultson the corresponding integral operators, and abstract results on nonlinear parabolic evolutionequations. Introduction
In the context of boundary value problems involving elliptic constant-coefficient PDE’slike the Laplace equation or the Stokes system, it is often natural to consider two-phaseproblems in unbounded domains, where the same equation has to be solved on both sidesof the boundary, and the boundary conditions typically are of “transmission” type, i.e. theyrelate limits of the solutions from both sides. The method of layer potentials is a classicaltechnique which is intrinsically suited to such settings. Typically, this method reduces theboundary 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 potentialsacross the boundary.The first applications of layer potentials in the analysis of moving boundary problems ofthe type described above are from the 1980s, for problems of Hele-Shaw or Muskat type [8](see also the recent surveys [11, 12] on further developments) as well as for Stokes flow prob-lems [5]. In these applications, the interfaces are represented as graphs of a time dependentfunction [ f f ( t )] , with f ( t ) ∈ C( R ) , for which an evolution equation can be derived. Thisequation involves singular integral operators originating from the layer potential, dependingnonlinearly and nonlocally on f ( t ) . However, in suitable geometries this nonlinearity can bedescribed rather explicitly, and technicalities resulting from transforming the problem to afixed reference domain can be avoided. More precisely, the operators determining the evo-lution belong to a class discussed in Section 3 below, and results are available concerningmapping properties, smoothness, localization etc. of the operators in this class.After reducing the moving boundary problem to an evolution equation for f , this equationhas to be analyzed. Initially, various approaches have been used that necessitated ratherrestrictive assumptions on the initial data. Recently, however, more general, in some sense Mathematics Subject Classification.
Key words and phrases.
Stokes problem; Two-phase; Singular integrals; Contour integral formulation. optimal existence, uniqueness, and smoothness results have been obtained. One of the crucialtools for this has been the meanwhile well-developed and versatile abstract theory of nonlinearparabolic evolution equations, cf. [2, 14, 19].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 betweenthe two phases. More precisely, we seek a moving interface [ t Γ( t )] between two liquidphases Ω ± ( t ) , and corresponding functions v ± ( t ) : Ω ± ( t ) −→ R and p ± ( t ) : Ω ± ( t ) −→ R , representing the velocity and pressure fields in Ω ± ( t ) , respectively, such that the followingequations are satisfied: µ ± ∆ v ± − ∇ p ± = 0 in Ω ± ( t ) , div v ± = 0 in Ω ± ( t ) , [ v ] = 0 on Γ( t ) , [ T µ ( v, p )]˜ ν = − σ ˜ κ ˜ ν on Γ( t ) , ( v ± , p ± )( x ) → for | x | → ∞ , x ∈ Ω ± ( t ) V n = v ± · ˜ ν on Γ( t ) . (1.1a)Here ˜ ν is the unit exterior normal to ∂ Ω − ( t ) and ˜ κ denotes the curvature of the interface.Moreover, T µ ( v, p ) = ( T µ,ij ( v, p )) ≤ i, j ≤ denotes the stress tensor that is given by T µ,ij ( v, p ) := − pδ ij + µ ( ∂ i v j + ∂ j v i ) , (1.1b)and [ v ] (respectively [ T µ ( v, p )] ) is the jump of the velocity (respectively stress tensor) acrossthe moving interface, see (2.2) below. The positive constants µ ± and σ denote the viscosity ofthe liquids in the two phases and the surface tension coefficient of the interface, respectively.We assume that Γ( t ) = ∂ Ω + ( t ) = ∂ Ω − ( t ) , Ω + ( t ) ∪ Ω − ( t ) ∪ Γ( t ) = R , Γ( t ) = graph f ( t ) so that Γ( t ) is a graph over the real line. Equation (1.1a) determines the motion of theinterface by prescribing its normal velocity V n as coinciding with the normal component ofthe velocity at Γ( t ) , i.e. the interface is transported by the liquid flow. The interface Γ( t ) isassumed to be known at time t = 0 : f (0) = f . (1.1c)In the previous paper [17], the authors considered Problem (1.1a) in the case of equalviscosities µ ± = µ . In that case, the solution to the fixed-time problem (1.1a) –(1.1a) canbe directly represented as a hydrodynamic single-layer potential [13] with density − σ ˜ κ ˜ ν , andthe resulting evolution equation represents the time derivative of f as a nonlinear singularintegral operator acting on f .If µ + = µ − this is not feasible. Instead, we first transform the unknowns such that thesame equation holds in both phases, introducing thereby a jump across the interface forthe transformed velocity field. In Proposition 5.1, we show that the corresponding fixed-timeStokes problem is uniquely solvable, and we represent the solution by a sum of a hydrodynamicsingle layer and a double layer potential. While the single layer potential is generated by thesame density as in the case of equal viscosities, the density β for the double layer potential isfound from solving a linear, singular integral equation of the second kind, cf. (5.8). As Γ( t ) is WO-PHASE STOKES FLOW 3 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 spectrumof the corresponding integral operator via a Rellich identity. Moreover, we also rely on afurther Rellich identity used in [15] in the study of the Muskat problem.The solution to the fixed-time problem is then used in the formulation of an evolutionequation for f , (cf. (5.9), (5.17), (5.18)) dfdt ( t ) = Φ( f ( t )) , t ≥ , f (0) = f , whose investigation will yield the following main result. Here and further, H s ( R ) := W s ( R ) denotes the usual Sobolev spaces of integer or noninteger order. Theorem 1.1.
Let s ∈ (3 / , be given. Then, the following statements hold true: (i) (Well-posedness) Given f ∈ H s ( R ) , there exists a unique maximal solution ( f, v ± , p ± ) to (1.1) such that • f = f ( · ; f ) ∈ C([0 , T + ) , H s ( R )) ∩ C ([0 , T + ) , H s − ( R )) , • v ± ( t ) ∈ C (Ω ± ( t )) ∩ C (Ω ± ( t )) , p ± ( t ) ∈ C (Ω ± ( t )) ∩ C(Ω ± ( t )) for all t ∈ (0 , T + ) , • v ( t ) ± | Γ( t ) ◦ Ξ( f ( t )) ∈ H ( R ) for all t ∈ (0 , T + ) ,where T + = T + ( f ) ∈ (0 , ∞ ] and Ξ( f ( t ))( ξ ) := ( ξ, f ( t )( ξ )) , ξ ∈ R . Moreover, the set M := { ( t, f ) | f ∈ H s ( R ) , < t < T + ( f ) } is open in (0 , ∞ ) × H s ( R ) , and [( t, f ) f ( t ; f )] is a semiflow on H s ( R ) which issmooth in M . (ii) (Parabolic smoothing)(iia) The map [( t, ξ ) f ( t )( ξ )] : (0 , T + ) × R −→ R is a C ∞ -function. (iib) For any k ∈ N , we have f ∈ C ∞ ((0 , T + ) , H k ( R )) . (iii) (Global existence) If sup [0 ,T ] ∩ [0 ,T + ( f )) k f ( t ) k H s < ∞ for each T > , then T + ( f ) = ∞ . Remark . 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 λ > , also [ t f λ ( t )] given by f λ ( t )( ξ ) := λ − f ( λt )( λξ ) , is a solution to (1.1). This identifies H / ( R ) as a critical space for the evolution problem (1.1).Hence, Theorem 1.1 covers all subcritical spaces.1.1. Outline.
The paper is structured as follows: In Section 2 we discuss a two-phase Stokesproblem with equal viscosities in both phases where the normal stresses are continuous acrossthe interface and the velocity has a prescribed jump there. In fact, the problem is solved bythe hydrodynamic double layer potential generated by that jump. Although the boundarybehavior of this potential is well-known, we prove the results on this in Appendix A as theydo not seem directly available in the literature for our unbounded geometry.
BOGDAN–VASILE MATIOC AND GEORG PROKERT
As we rely on the solvability of singular integral equations of the second kind arising fromthe hydrodynamic double-layer potential, the spectrum of the corresponding operator is inves-tigated in Sections 3 and 4, first in L ( R ) and then in H s ( R ) , with s ∈ (3 / , , and H ( R ) .The main technical tools in the latter cases are shift invariances and commutator properties forsingular integral operators of the type discussed here. In Section 5 we reformulate the movingboundary problem (1.1) as a nonlinear and nonlocal evolution equation problem, cf. (5.17).Finally, in Section 6 we carry out the linearization of (5.17) and locally approximate thelinearization by Fourier multipliers. This enables us to identify the parabolic character of theevolution equation and to prove our main result by invoking abstract results on equations ofthat type from [14].1.2. Notation.
Slightly deviating from the usual notation, if E , . . . , E k , F , k ∈ N , areBanach spaces, we write L k ( E , . . . , E k ; F ) for the Banach space of bounded k -linear mapsfrom Q i E i to F . Given Banach spaces X and Y , we let L k sym ( X, Y ) ⊂ L k ( X, . . . , X ; Y ) denote the space of k -linear, bounded symmetric maps A : X k → Y . Moreover, C − ( E, F ) will denote the space of locally Lipschitz continuous maps from a Banach space E to a Banachspace F . Given k ∈ N , we further let C k ( R ) denote the Banach space of functions withbounded and continuous derivatives up to order k and C k + α ( R ) , α ∈ (0 , , is its subspaceconsisting of functions with α -Hölder continuous k th derivative whose α -Hölder modulus isbounded. 2. An auxiliary fixed-time problem
As a preparation for solving the boundary value problem (1.1a) − (1.1a) for fixed time, inthis section we consider the related Stokes problem (2.3) with equal viscosities normed to .The unique solvability of (2.3) is established in Proposition 2.1 below and in Appendix A.In this section, f ∈ H ( R ) is fixed. We introduce the following notation: Ω ± := Ω ± f := { ( x , x ) ∈ R | x ≷ f ( x ) } , Γ := Γ f := ∂ Ω ± = { ( ξ, f ( ξ )) | ξ ∈ R } . Note that Γ is the image of R under the diffeomorphism Ξ := Ξ f := (id R , f ) . Further, let ν and τ be the componentwise pull-back under Ξ of the unit normal ˜ ν on Γ exterior to Ω − and of the unit tangent vector ˜ τ to Γ , that is ν := ω ( − f ′ , ⊤ , τ := ω (1 , f ′ ) ⊤ , ω := ω ( f ) := (1 + f ′ ) / . (2.1)For any function z defined on R \ Γ we set z ± := z | Ω ± and if z ± have limits at somepoint ( ξ, f ( ξ )) ∈ Γ we will write z ± ( ξ, f ( ξ )) for the limits, and we set [ z ]( ξ, f ( ξ )) := z + ( ξ, f ( ξ )) − z − ( ξ, f ( ξ )) . (2.2)For notational brevity we introduce the function space X := X f := ( ( w, q ) : R \ Γ −→ R × R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ± ∈ C (Ω ± , R ) ∩ C (Ω ± , R ) q ± ∈ C (Ω ± ) ∩ C(Ω ± ) ) . WO-PHASE STOKES FLOW 5
For given β = ( β , β ) ⊤ ∈ H ( R ) we seek solutions ( w, q ) ∈ X to the Stokes problem ∆ w ± − ∇ q ± = 0 in Ω ± , div w ± = 0 in Ω ± , [ w ] = β ◦ Ξ − on Γ , [ T ( w, q )]( ν ◦ Ξ − ) = 0 on Γ , ( w ± , q ± )( x ) → for | x | → ∞ . (2.3)For the construction of the solution to (2.3), let us first point out that for any smoothsolution ( U, P ) : E −→ R × R to the homogeneous Stokes system ∆ U − ∇ P = 0 , div U = 0 ) in E, (2.4)where E is a domain in R , the functions ( W i , Q i ) : E −→ R × R , i = 1 , , given by W ij := T ,ij ( U, P ) = − P δ ij + ∂ i U j + ∂ j U i , j = 1 , , and Q i = 2 ∂ i P are solutions to (2.4) as well. In particular, if E = R \ { } and ( U, P ) = ( U k , P k ) : R \ { } −→ R × R , k = 1 , , are the fundamental solutions to the Stokes equations (2.4), given by U kj ( y ) = − π (cid:18) δ jk ln 1 | y | + y j y k | y | (cid:19) , j = 1 , , and P k ( y ) = − π y k | y | (2.5)for y = ( y , y ) ∈ R \ { } , we obtain a system ( W i,k , Q i,k ) : R \ { } −→ R × R , i, k = 1 , ,of solutions to the homogeneous Stokes equations given by W i,kj ( y ) := ( −P k δ ij + ∂ i U kj + ∂ j U ki )( y ) = 1 π y i y j y k | y | , j = 1 , , Q i,k ( y ) := 2 ∂ i P k ( y ) = 1 π (cid:18) − δ ik | y | + 2 y i y k | y | (cid:19) , y ∈ R \ { } . We are going to show that ( w, q ) := ( w, q )[ β ] given by w j ( x ) := Z Γ W i,kj ( x − y )˜ ν i ( y ) β k ( y ) d Γ y := Z R W i,kj ( r ) ν i ( s ) β k ( s ) ω ( s ) ds, j = 1 , , (2.6) q ( x ) := Z Γ Q i,k ( x − y )˜ ν i ( y ) β k ( y ) d Γ y := Z R Q i,k ( r ) ν i ( s ) β k ( s ) ω ( s ) ds (2.7)for x ∈ R \ Γ and with r := r ( x, s ) := x − ( s, f ( s )) solves (2.3). Here and further, we sumover indices appearing twice in a product. We write this more explicitly as w ( x ) = 1 π Z R − f ′ r + r | r | (cid:18) r r r r r r (cid:19) β ds,q ( x ) = 1 π Z R | r | ( − f ′ (cid:18) r − r r r r r r − r (cid:19) β ds. (2.8)The solution ( w, q ) is the so-called hydrodynamic double-layer potential generated by thedensity β ◦ Ξ − on Γ , see [13]. BOGDAN–VASILE MATIOC AND GEORG PROKERT
Proposition 2.1.
The boundary value problem (2.3) has precisely one solution ( w, q ) ∈ X .It is given by (2.6) , (2.7) . Moreover, w ± | Γ ◦ Ξ ∈ H ( R ) .Proof. The uniqueness of the solution can be shown as in the proof of [17, Theorem 2.1].Observe that w and q are defined by integrals of the form ( w, q )( x ) = Z R K ( x, s ) β ( s ) ds where for every α ∈ N we have ∂ αx K ( x, s ) = O ( s − ) for | s | → ∞ and locally uniformlyin x ∈ R \ Γ . This shows that w and q are well-defined by (2.6) and (2.7), and that inte-gration and differentiation with respect to x may be interchanged. As ( W i,k , Q i,k ) solve thehomogeneous Stokes equations, this also holds for ( w, q ) .To show the decay of q at infinity we obtain from the matrix equality π | r | ( − f ′ (cid:18) r − r r r r r r − r (cid:19) = − ∂ s ( P ( r ) − P ( r )) , via integration by parts q ( x ) = 2 Z R ( P − P )( r ) β ′ ds = 1 π Z R | r | ( − r r ) β ′ ds. In view of this representation, [15, Lemma 2.1] implies q ( x ) → as | x | → ∞ .In order to prove the decay of w we rewrite w ( x ) = 12 π Z R − f ′ r + r | r | (cid:18) I + 1 | r | (cid:18) r − r r r r r r − r (cid:19)(cid:19) β ds = 12 π Z R (cid:18) − f ′ r + r | r | I + ∂ s (cid:20) | r | (cid:18) r r r r − r r (cid:19)(cid:21)(cid:19) β ds = 12 π Z R (cid:18) − f ′ r + r | r | β − | r | (cid:18) r r r r − r r (cid:19) β ′ (cid:19) ds, where I ∈ R × is the identity matrix. Now [15, Lemma 2.1] and [17, Lemma B.2] imply thatindeed w ( x ) → for | x | → ∞ .The boundary conditions (2.3) and (2.3) together with the properties that ( w, q ) ∈ X and w ± | Γ ◦ Ξ ∈ H ( R ) are shown in Appendix A. (cid:3) The L -resolvent of the hydrodynamic double-layer potential operator In this section we study the resolvent set of the hydrodynamic double-layer potential op-erator D ( f ) , with f ∈ C ( R ) , introduced in (3.5) below, which we view in this section as anelement of L ( L ( R ) ) . The main result of this section is Theorem 3.3 below which providesin particular the invertibility of λ − D ( f ) for λ ∈ R with | λ | > / .To begin, we introduce a general class of singular integral operators suited to our ap-proach via layer potentials, cf. [16, 17]. Given n, m ∈ N and Lipschitz continuous func-tions a , . . . , a m , b , . . . , b n : R −→ R , we let B n,m denote the singular integral operator B n,m ( a , . . . , a m )[ b , . . . , b n , h ]( ξ ) := PV Z R h ( ξ − η ) η Q ni =1 (cid:0) δ [ ξ,η ] b i /η (cid:1)Q mi =1 (cid:2) (cid:0) δ [ ξ,η ] a i /η (cid:1) (cid:3) dη, (3.1) WO-PHASE STOKES FLOW 7 where PV R R denotes the principal value integral and δ [ ξ,η ] u := u ( ξ ) − u ( ξ − η ) . For brevitywe set B n,m ( f ) := B n,m ( f, . . . f )[ f, . . . , f, · ] . (3.2)In this section we several times use the following result. Lemma 3.1.
There exists a constant C depending only on n, m , and max i =1 ,...,m k a ′ i k ∞ with k B n,m ( a , . . . , a m )[ b , . . . , b n , · ] k L ( L ( R )) ≤ C n Y i =1 k b ′ i k ∞ . Moreover, B n,m ∈ C − ( W ∞ ( R ) m , L n sym ( W ∞ ( R ) , L ( L ( R )))) . Proof.
See [16, Remark 3.3]. (cid:3)
Given f ∈ C ( R ) , we introduce the linear operators D ( f ) and D ( f ) ∗ defined by D ( f )[ β ]( ξ ) := 1 π PV Z R r f ′ − r | r | (cid:18) r r r r r r (cid:19) β ds, D ( f ) ∗ [ β ]( ξ ) := 1 π PV Z R − r f ′ ( ξ ) + r | r | (cid:18) r r r r r r (cid:19) β ds, (3.3)where ξ ∈ R and β ∈ L ( R ) . Throughout this section r := ( r , r ) is given by r := r ( ξ, s ) := ( ξ − s, f ( ξ ) − f ( s )) . (3.4)We note that D ( f ) is related to the B n,m via D ( f )[ β ] = 1 π B , ( f ) B , ( f ) B , ( f ) B , ( f ) ! f ′ β f ′ β ! − π B , ( f ) B , ( f ) B , ( f ) B , ( f ) ! β β ! (3.5)for β = ( β , β ) ⊤ . Therefore, as a direct consequence of Lemma 3.1, D ( f ) is boundedon L ( R ) . Moreover, up to the sign and the push-forward via Ξ , D ( f )[ β ]( ξ ) is the “di-rect value” of the hydrodynamic double-layer potential w generated by β in ( ξ, f ( ξ )) ∈ Γ ,cf. (2.8) . One may also check that D ( f ) ∗ is the L -adjoint of D ( f ) .Using the same notation, we define the singular integral operators B ( f ) and B ( f ) by B ( f )[ θ ]( ξ ) := 1 π PV Z R − r f ′ + r | r | θ ds and B ( f )[ θ ]( ξ ) := 1 π PV Z R r + r f ′ | r | θ ds, where θ ∈ L ( R ) . The operators B i ( f ) , i = 1 , , play an important role also in the studyof the Muskat problem, cf. [15]. Lemma 3.1 implies that also B i ( f ) i = 1 , , is boundedon L ( R ) . Moreover, B ( f )[ θ ]( ξ ) is the direct value of the double layer potential for theLaplacian corresponding to the density θ in ( ξ, f ( ξ )) ∈ Γ .We are going to prove in Theorem 3.3 below that the resolvent sets of D ( f ) and D ( f ) ∗ contain all real λ with | λ | > / , with a bound on the resolvent that is uniform in λ awayfrom ± / , and in f as long as k f ′ k ∞ is bounded.Oriented at [7, 10], we obtain this property on the basis of a Rellich identity for the Stokesoperator. While eventually the result for D ( f ) is needed, it is helpful to consider D ( f ) ∗ , as thisoperator naturally arises from the jump relations for the single-layer hydrodynamic potentialgenerated by β , cf. (3.13) below. BOGDAN–VASILE MATIOC AND GEORG PROKERT
We next derive the Rellich identity (3.14), and based on it we establish an estimate thatrelates the operator D ( f ) ∗ to the operators B ( f ) and B ( f ) introduced above. Lemma 3.2.
Given
K > , there exists a positive constant C , that depends only on K , suchthat for all β ∈ L ( R ) , λ ∈ [ − K, K ] , and f ∈ C ( R ) with k f ′ k ∞ < K we have C k ( λ − D ( f ) ∗ )[ β ] k k β k ≥ k ( λ − B ( f ))[ ω − β · ν ] − B ( f )[ ω − β · τ ] k + m ( λ ) k ω − β · τ k , (3.6) where ω , ν , and τ are defined in (2.1) , and with m ( λ ) := max (cid:8)(cid:0) λ + (cid:1) (cid:0) λ − (cid:1) , (cid:0) λ − (cid:1) (cid:0) λ + (cid:1)(cid:9) . (3.7) Proof.
Let first f ∈ C ∞ ( R ) and β = ( β , β ) ⊤ with β k ∈ C ∞ ( R ) , k = 1 , . We define thehydrodynamic single-layer potential u with corresponding pressure Π by u ( x ) := − Z R U k ( x − ( s, f ( s ))) β k ( s ) ds and Π( x ) := − Z R P k ( x − ( s, f ( s ))) β k ( s ) ds for x ∈ R \ Γ , where and U k , P k defined by (2.5). Using the fact that β is compactlysupported, is is not difficult to see that the functions ( u, Π) are well-defined and smoothin Ω ± and satisfy ∆ u − ∇ Π = 0 , div u = 0 ) in Ω ± , (3.8)as well as Π , ∇ u = O ( | x | − ) for | x | → ∞ . (3.9)Moreover, [6, Lemma A.1] and the arguments in the proof of [17, Lemma A.1] show that thefunctions Π | Ω ± and u | Ω ± have extensions Π ± ∈ C(Ω ± ) and u ± ∈ C (Ω ± ) , and, given ξ ∈ R ,we have ∂ i u ± j ◦ Ξ( ξ ) = − PV Z R ∂ i U kj ( r ) β k ds ± − β j ν i + ν i ν j β · ν ω ( ξ ) , Π ± ◦ Ξ( ξ ) = − PV Z R P k ( r ) β k ds ± β · ν ω ( ξ )= 12 B ( f )[ ω − β · ν ]( ξ ) + 12 B ( f )[ ω − β · τ ]( ξ ) ± β · ν ω ( ξ ) , (3.10)where ν = ( ν , ν ) and r = r ( ξ, s ) are defined in (2.1) and (3.4). In particular, ∂ u ± ◦ Ξ( ξ ) = T ( f )[ β ]( ξ ) ∓ ( β · τ ) τ ω ( ξ ) , (3.11)where T ( f ) is the singular integral operator given by T ( f )[ β ]( ξ ) := 14 π PV Z R | r | (cid:18) − r − r r r − r r r − r r r r − r (cid:19) (cid:18) β β (cid:19) ds. Observe that T ( f ) is skew-adjoint on L ( R ) , i.e. T ( f ) ∗ = − T ( f ) , and therefore h T ( f )[ β ] | β i = 0 . (3.12)Here h· | ·i denotes the inner product of L ( R ) . WO-PHASE STOKES FLOW 9
Moreover, for the normal stress at the boundary we find ω ( T ( u, Π) ± ◦ Ξ) ν = (cid:16) ∓ − D ( f ) ∗ (cid:17) [ β ] . (3.13)For convenience we introduce the notation τ ij := ( T ( u, Π)) ij = − Π δ ij + ∂ i u j + ∂ j u i , i, j = 1 , , and observe that due to (3.8) ∂ i τ ij = 0 in Ω ± , j = 1 , , and δ ij ∂ i u j = 0 in Ω ± .The latter identities lead us to ∂ i ( τ ij ∂ u j ) = τ ij ∂ i ∂ u j = ( ∂ i u j + ∂ j u i ) ∂ ∂ i u j = 14 X i, j =1 ∂ ( ∂ i u j + ∂ j u i ) in Ω ± .In view of (3.9) we may integrate the latter relation over Ω ± and using Gauss’ theoremand (3.13) we get Z Γ ω X i, j =1 ( ∂ i u ± j + ∂ j u ± i ) d Γ = 4 Z Γ τ ± ij ˜ ν i ∂ u ± j d Γ = 4
D(cid:16) ∓ − D ( f ) ∗ (cid:17) [ β ] (cid:12)(cid:12)(cid:12) ∂ u ± ◦ Ξ E . (3.14)To estimate the term on the left we observe that the Cauchy-Schwarz inequality and | ˜ ν | = 1 yield X i, j =1 ( ∂ i u ± j + ∂ j u ± i ) ≥ X i =1 (( ∂ i u ± j + ∂ j u ± i )˜ ν j ) = X i =1 ( τ ± ij ˜ ν j + Π ± ˜ ν i ) on Γ .This inequality, the representations (3.10) and (3.13), and k B i ( f ) k L ( L ( R )) ≤ C ( K ) , i = 1 , ,cf. Lemma 3.1, now yield Z Γ ω X i,j =1 ( ∂ i u ± j + ∂ j u ± i ) d Γ ≥ (cid:13)(cid:13)(cid:13) ω (cid:16) ∓ − D ( f ) ∗ (cid:17) [ β ] + (Π ± ◦ Ξ) ν (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ω (cid:16) λ − D ( f ) ∗ (cid:17) [ β ] − ω (cid:16) λ ± (cid:17) β + (Π ± ◦ Ξ) ν (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) − ω (cid:16) λ ± (cid:17) β + (cid:16) B ( f )[ ω − β · ν ] + 12 B ( f )[ ω − β · τ ] ± β · ν ω (cid:17) ν (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) ω ( λ − D ( f ) ∗ )[ β ] (cid:13)(cid:13)(cid:13) − C k ( λ − D ( f ) ∗ )[ β ] k kk β k ≥ (cid:16) λ ± (cid:17) k ω − β · τ k + (cid:13)(cid:13)(cid:13)(cid:16) λ − B ( f ) (cid:17) [ ω − β · ν ] − B ( f )[ ω − β · τ ] (cid:13)(cid:13)(cid:13) − C k ( λ − D ( f ) ∗ )[ β ] k kk β k for any λ ∈ [ − K, K ] . We next consider the term on the right of (3.14). As a direct consequence of Lemma 3.1 wenote that k T ( f ) k L ( L ( R ) ) ≤ C = C ( K ) . This bound together with (3.11) and (3.12) implies D(cid:16) ∓ − D ( f ) ∗ (cid:17) [ β ] (cid:12)(cid:12)(cid:12) ∂ u ◦ Ξ E = 4 D(cid:16) λ − D ( f ) ∗ (cid:17) [ β ] − (cid:16) λ ± (cid:17) β (cid:12)(cid:12)(cid:12) T [ β ] ∓ ( β · τ ) τ ω E ≤ C k ( λ − D ( f ) ∗ )[ β ] k kk β k ± (cid:16) λ ± (cid:17) k ω − β · τ k . For f ∈ C ∞ ( R ) , the estimate (3.6) follows from (3.14) and the latter estimates upon rear-ranging terms and a standard density argument. For general f ∈ C ( R ) we additionally needto use the continuity of the mappings [ f D ( f ) ∗ ] : C ( R ) → L ( L ( R ) ) and [ f B i ( f )] : C ( R ) → L ( L ( R )) , i = 1 , , which are direct consequences of Lemma 3.1, together with the density of C ∞ ( R ) in C ( R ) . (cid:3) Based on Lemma 3.2 we now establish the following result.
Theorem 3.3 (Spectral properties of D ( f ) and D ( f ) ∗ ) . Given δ ∈ (0 , , there exists a con-stant C = C ( δ ) > such that for all λ ∈ R with | λ | ≥ / δ and f ∈ C ( R ) with k f ′ k ∞ ≤ /δ we have k ( λ − D ( f ) ∗ )[ β ] k ≥ C k β k for all β ∈ L ( R ) . (3.15) Moreover, λ − D ( f ) ∗ , λ − D ( f ) ∈ L ( L ( R ) ) are isomorphisms for all λ ∈ R with | λ | > / and f ∈ C ( R ) .Proof. In order to prove (3.15) we assume the opposite. Then we may find sequences ( λ k ) in R , ( f k ) in C ( R ) , and ( β k ) in L ( R ) such that | λ k | ≥ / δ, k f ′ k k ∞ ≤ /δ , and k β k k = 1 for all k ∈ N , and ( λ k − D ( f k ) ∗ )[ β k ] → in L ( R ) .Given k ∈ N , we set ω k := ω ( f k ) , (2.1). As the operators D ( f k ) ∗ are bounded, uniformlyin k ∈ N , in L ( L ( R ) ) , cf. Lemma 3.1, the sequence ( λ k ) is bounded. Observing that forthe constant m = m ( λ ) from (3.7) we have m ( λ k ) ≥ δ (2 + δ ) > for all k ∈ N , we get fromLemma 3.2 that ω − k β k · τ → , (cid:0) λ k − B ( f k ) (cid:1) [ ω − k β k · ν ] − B ( f k )[ ω − k β k · τ ] → in L ( R ) .As the operators B ( f k ) are bounded, uniformly in k ∈ N , in L ( L ( R ) ) , cf. Lemma 3.1, thisimplies (cid:0) λ k − B ( f k ) (cid:1) [ ω − k β k · ν ] → in L ( R ) .Let A ( f ) := B ( f ) ∗ . Since | λ k | ≥ , it follows from the proof of [15, Theorem 3.5] that theoperator λ k − A ( f k ) ∈ L ( L ( R )) , k ∈ N , is an isomorphism with k (2 λ k − A ( f k )) − k L ( L ( R )) ≤ C ( δ ) . This implies that also λ k − B ( f k ) ∈ L ( L ( R )) , k ∈ N , is an isomorphism and (cid:13)(cid:13) ( λ k − B ( f k )) − (cid:13)(cid:13) L ( L ( R )) ≤ C ( δ ) . Thus ω − k β k · ν → in L ( R ) , so that β k = ω k (cid:0) ω − k ( β k · ν ) ν + ω − k ( β k · τ ) τ (cid:1) → in L ( R ) . WO-PHASE STOKES FLOW 11
This contradicts the property that k β k k = 1 for all k ∈ N and (3.15) follows.To complete the proof we fix f ∈ C ( R ) and λ ∈ R with | λ | > / and we choose δ ∈ (0 , such that | λ | ≥ / δ and k f ′ k ∞ ≤ /δ . As D ( f ) ∗ is bounded, λ − D ( f ) ∗ ∈ L ( L ( R ) ) is anisomorphism if | λ | is sufficiently large. The estimate (3.15) together with a standard continuityargument, cf. e.g. [3, Proposition I.1.1.1], now implies that λ − D ( f ) ∗ is an isomorphism aswell. The result for D ( f ) is an immediate consequence of this property. (cid:3) The resolvent of the hydrodynamic double-layer potential operator inhigher order Sobolev spaces
The main goal of this section is to establish spectral properties for D ( f ) , parallel to thosein Theorem 3.3, in the spaces H s − ( R ) , s ∈ (3 / / , and in H ( R ) . The latter are neededwhen solving the fixed-time problem (5.1), see Proposition 5.1, and the former are used toderive 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 ≥ , s ∈ (3 / , , and a , . . . , a m ∈ H s ( R ) be given. Then, there exists aconstant C , depending only on n, m , s , and max ≤ i ≤ m k a i k H s , such that k B n,m ( a , . . . , a m )[ b , . . . , b n , h ] k ≤ C k b k H k h k H s − n Y i =2 k b i k H s (4.1) for all b , . . . , b n ∈ H s ( R ) and h ∈ H s − ( R ) . Moreover, B n,m ∈ C − ( H s ( R ) m , L n +1 ( H ( R ) , H s ( R ) , . . . , H s ( R ) , H s − ( R ); L ( R ))) . (ii) Given s ∈ (3 / , and a , . . . , a m ∈ H s ( R ) , there exists a constant C, depending onlyon n, m, s , and max ≤ i ≤ m k a i k H s , such that k B n,m ( a , . . . , a m )[ b , . . . , b n , h ] k H s − ≤ C k h k H s − n Y i =1 k b i k H s for all b , . . . , b n ∈ H s ( R ) and h ∈ H s − ( R ) . Moreover, B n,m ∈ C − ( H s ( R ) m , L n sym ( H s ( R ) , L ( H s − ( R )))) . (iii) Let n ≥ , / < s ′ < s < , and a , . . . , a m ∈ H s ( R ) be given. Then, there exists aconstant C , depending only on n, m , s , s ′ , and max ≤ i ≤ m k a i k H s , such that k B n,m ( a , . . . , a m )[ b , . . . , b n , h ] − hB n − ,m ( a , . . . , a m )[ b , . . . , b n , b ′ ] k H s − ≤ C k b k H s ′ k h k H s − n Y i =2 k b i k H s for all b , . . . , b n ∈ H s ( R ) and h ∈ H s − ( R ) . Proof.
The claims (i) is established in [15, Lemmas 3.2], while (ii) and (iii) are proven in [1,Lemma 5 and Lemma 6]. (cid:3)
For ξ ∈ R we define the left shift operator τ ξ on L ( R ) by τ ξ u ( x ) := u ( x + ξ ) and observethe invariance property τ ξ B n,m ( a , . . . , a m )[ b , . . . , b n , h ] = B n,m ( τ ξ a , . . . , τ ξ a m )[ τ ξ b , . . . , τ ξ b n , τ ξ h ] . (4.2)Differences of B n,m with respect to the nonlinear arguments a i can be represented by theidentity B n,m ( a , a . . . , a m )[ b , . . . , b n , · ] − B n,m (˜ a , a . . . , a m )[ b , . . . , b n , · ]= B n +2 ,m +1 (˜ a , a , a . . . , a m )[ b , . . . , b n , ˜ a + a , ˜ a − a , · ] . (4.3)We will also use the interpolation property [ H s ( R ) , H s ( R )] θ = H (1 − θ ) s + θs ( R ) , θ ∈ (0 , , −∞ < s ≤ s < ∞ , (4.4)where [ · , · ] θ denotes the complex interpolation functor of exponent θ . Theorem 4.2.
Given δ ∈ (0 , and s ∈ (3 / , , there exists a constant C = C ( δ, s ) > such that k ( λ − D ( f ))[ β ] k H s − ≥ C k β k H s − (4.5) for all λ ∈ R with | λ | ≥ / δ , f ∈ H s ( R ) with k f k H s ≤ /δ, and β ∈ H s − ( R ) .Moreover, λ − D ( f ) ∈ L ( H s − ( R ) ) is an isomorphism for all λ ∈ R with | λ | > / and f ∈ H s ( R ) .Proof. Given f ∈ H s ( R ) , the relation (3.5) and Lemma 4.1 (ii) imply D ( f ) ∈ L ( H s − ( R ) ) .In order to prove (4.5), let λ ∈ R with | λ | ≥ / δ and f ∈ H s ( R ) with k f k H s ≤ /δ be fixed.Theorem 3.3 together with the embedding H s ( R ) ֒ → L ∞ ( R ) implies there exists C = C ( δ ) > such that k ( λ − D ( τ ξ f )) − k L ( L ( R ) ) ≤ C for all ξ ∈ R . It is well-known there exists aconstant C > such that [ β ] H s − := k [ ξ
7→ | ξ | s − F [ β ]( ξ )] k = C (cid:16) Z R k β − τ ξ β k | ξ | s − dξ (cid:17) / =: [ β ] W s − , where F [ β ] is the Fourier transform of β . Together with (4.2) we then get [ β ] H s − ≤ C Z R k ( λ − D ( τ ξ f ))[ β − τ ξ β ] k | ξ | s − dξ ≤ C (cid:16) Z R k ( λ − D ( f ))[ β ] − τ ξ (( λ − D ( f ))[ β ]) k | ξ | s − dξ + Z R k ( D ( f ) − D ( τ ξ f ))[ β ] k | ξ | s − dξ (cid:17) = C [( λ − D ( f ))[ β ]] H s − + C Z R k ( D ( f ) − D ( τ ξ f ))[ β ] k | ξ | s − dξ. (4.6)The term k ( D ( f ) − D ( τ ξ f ))[ β ] k can be estimated by a finite sum of terms of the form k ( B n, ( f ) − B n, ( τ ξ f ))[ β i ] k and k B n, ( f )[ f ′ β i ] − B n, ( τ ξ f )[( τ ξ f ′ ) β i ] k , WO-PHASE STOKES FLOW 13 where ≤ n ≤ and i ∈ { , } . Let s ′ ∈ (3 / , s ) be fixed. We first consider terms of thesecond type and estimate in view of Lemma 3.1 k B n, ( f )[ f ′ β i ] − B n, ( τ ξ f )[( τ ξ f ′ ) β i ] k ≤ k ( B n, ( f ) − B n, ( τ ξ f ))[ f ′ β i ] k + k B n, ( τ ξ f )[( τ ξ f ′ − f ′ ) β i ] k ≤ k ( B n, ( f ) − B n, ( τ ξ f ))[ f ′ β i ] k + C k τ ξ f ′ − f ′ k k β k H s ′− . (4.7)Furthermore, using (4.3), we have B n, ( f ) − B n, ( τ ξ f ) = n X ℓ =1 B n, ( f, f )[ τ ξ f, . . . , τ ξ f | {z } ℓ − , f − τ ξ f, f, . . . , f, · ]+ B n +2 , ( τ ξ f, f, f )[ τ ξ f, . . . , τ ξ f, τ ξ − f, τ ξ f + f, · ]+ B n +2 , ( τ ξ f, τ ξ f, f )[ τ ξ f, . . . , τ ξ f, τ ξ f − f, τ ξ f + f, · ] , and together with Lemma 4.1 (i) (with s ′ instead of s ), we conclude that B n, ( f ) − B n, ( τ ξ f ) belongs to L ( H s ′ − ( R ) , L ( R )) and satisfies k B n, ( f ) − B n, ( τ ξ f ) k L ( H s ′− ( R ) ,L ( R )) ≤ C k f − τ ξ f k H ( R ) . Combining this estimate with (4.7) we get Z R k ( D ( f ) − D ( τ ξ f ))[ β ] k | ξ | s − dξ ≤ C k f k H s k β k H s ′− , and by (4.6) and the interpolation property (4.4) we arrive at k β k H s − ≤ C (cid:0) [ λ − D ( f )[ β ]] H s − + k β k (cid:1) + 12 k β k H s − . Finally, using Theorem 3.3 again, we obtain the estimate (4.5). The isomorphism propertyof λ − D ( f ) , with λ ∈ R with | λ | > / and f ∈ H s ( R ) , follows by the same continuityargument as in the L result. (cid:3) For the H result we need an additional estimate for the operators B n,m with higherregularity of the arguments. Lemma 4.3.
Let n, m ∈ N and a , . . . , a m ∈ H ( R ) be given. Then, there exists a constant C ,depending only on n, m , and max ≤ i ≤ m k a i k H , such that k B n,m ( a , . . . , a m )[ b , . . . , b n , h ] k H ≤ C k h k H n Y i =1 k b i k H (4.8) for all b , . . . , b n ∈ H ( R ) and h ∈ H ( R ) . Moreover, B n,m ∈ C − ( H ( R ) m , L n sym ( H ( R ) , L ( H ( R )))) . Proof.
We first show that ϕ := B n,m ( a , . . . , a m )[ b , . . . , b n , h ] belongs to H ( R ) . Recallingthat the group { τ ξ } ξ ∈ R ⊂ L ( H r ( R )) , r ≥ , has generator [ f f ′ ] ∈ L ( H r +1 ( R ) , H r ( R )) , it suffices to show that D ξ ϕ := ( τ ξ ϕ − ϕ ) /ξ converges in L ( R ) when letting ξ → . In viewof (4.3) we write D ξ ϕ = n X i =1 B n,m ( τ ξ a , . . . , τ ξ a m ) (cid:2) b , . . . , b i − , D ξ b i , τ ξ b i +1 , . . . , τ ξ b n , τ ξ h (cid:3) + B n,m ( τ ξ a , . . . , τ ξ a m ) (cid:2) b , . . . , , b n , D ξ h (cid:3) − m X i =1 B n +2 ,m +1 ( τ ξ a , . . . , τ ξ a i , a i , . . . , a m ) (cid:2) b , . . . , b n , D ξ a i , τ ξ a i + a i , h (cid:3) . Lemma 3.1 and Lemma 4.1 (i) enable us to pass to the limit ξ → in L ( R ) in this equality.Hence, ϕ ∈ H ( R ) and ϕ ′ = B n,m ( a , . . . , a m )[ b , . . . , b n , h ′ ]+ n X i =1 B n,m ( a , . . . , a m )[ b , . . . , b i − , b ′ i , b i +1 , . . . b n , h ] − m X i =1 B n +2 ,m +1 ( a , . . . , a i , a i , . . . , a m )[ b , . . . , b n , a ′ i , a i , h ] . (4.9)The estimate (4.8) is a consequence of Lemma 3.1 and Lemma 4.1 (i). The local Lipschitzcontinuity property follows from an repeated application of (4.3) and (4.8). (cid:3) As a consequence of Lemma 4.3 and (4.9) we obtain the following result.
Corollary 4.4. B n,m ∈ C − ( H ( R ) m , L n sym ( H ( R ) , L ( H ( R )))) for all n, m ∈ N . Theorem 4.5.
The operator λ − D ( f ) ∈ L ( H ( R ) ) is an isomorphism for all f ∈ H ( R ) and λ ∈ R with | λ | > / .Proof. Fix f ∈ H ( R ) . From (3.5) and Corollary 4.4 we get D ( f ) ∈ L ( H ( R ) ) . Recall-ing (4.9), we further have ( D ( f )[ β ]) ′′ − D ( f )[ β ′′ ] = T lot [ β ] , β ∈ H ( R ) , (4.10)where each component of T lot [ β ] is a linear combination of terms B n,m ( f, . . . , f )[ f ′′ , f, . . . , f, ( f ′ ) k β i ] , B n,m ( f, . . . , f )[ f ′ , f ′ , f, . . . , f, ( f ′ ) k β i ] ,B n,m ( f, . . . , f )[ f ′ , f, . . . , f, (( f ′ ) k β i ) ′ ] , B n,m ( f, . . . , f )[ f, . . . , f, f ′′′ β i ] , where n, m ∈ N satisfy ≤ n, m ≤ and k ∈ { , } . From Lemma 3.1 and Lemma 4.1 (i)(with s = 7 / ) we conclude that k T lot [ β ] k ≤ C k β k H , β ∈ H ( R ) . (4.11)Given λ ∈ R with | λ | > / , we pick δ ∈ (0 , with | λ | ≥ / δ and k f ′ k ∞ ≤ /δ. Sinceby Theorem 3.3 we have k ( µ − D ( f )) − k L ( L ( R ) ) ≤ C for all µ ∈ R with | µ | ≥ / δ , we WO-PHASE STOKES FLOW 15 deduce from (4.10), (4.11), and (4.4) that k β k H ≤ C ( k β ′′ k + k β k ) ≤ C ( k ( µ − D ( f ))[ β ′′ ] k + k β k ) ≤ C (cid:0) k ( µ − D ( f ))[ β ] ′′ k + k T lot [ β ] k + k β k (cid:1) ≤ C (cid:0) k ( µ − D ( f ))[ β ] ′′ k + k β k H (cid:1) ≤ k β k H + C (cid:0) k ( µ − D ( f ))[ β ] ′′ k + k β k (cid:1) ≤ k β k H + C (cid:0) k ( µ − D ( f ))[ β ] ′′ k + k ( µ − D ( f ))[ β ] k (cid:1) , hence k β k H ≤ C k ( µ − D ( f ))[ β ] k H for all β ∈ H ( R ) and µ ∈ R with | µ | ≥ / δ . The result follows now by the samecontinuity argument as in the proof of Theorem 4.2. (cid:3) The contour integral formulation
In this section we formulate the Stokes evolution problem (1.1) as an nonlinear evolutionproblem having only f as unknown, cf. (5.17).Based on the results established in Section 2, Section 4, and Appendix A we start byproving that for each f ∈ H ( R ) , the boundary value problem µ ± ∆ v ± − ∇ p ± = 0 in Ω ± , div v ± = 0 in Ω ± , v + = v − on Γ , [ T µ ( v, p )]˜ ν = − σ ˜ κ ˜ ν on Γ , ( v ± , p ± )( x ) → for | x | → ∞ (5.1)has a unique solution ( v, p ) ∈ X f with the property that v ± | Γ ◦ Ξ f ∈ H ( R ) . This isestablished in Proposition 5.1 below, where we also provide an implicit formula for v ± | Γ interms of contour integrals on Γ . This representation allows to recast the kinematic boundarycondition (1.1a) in the form (5.17).With the substitution ˜ v ± := µ ± v ± , Problem (5.1) is equivalent to ∆˜ v ± − ∇ p ± = 0 in Ω ± , div ˜ v ± = 0 in Ω ± , µ − ˜ v + − µ + ˜ v − = 0 on Γ , [ T (˜ v, p )]˜ ν = − σ ˜ κ ˜ ν on Γ , (˜ v ± , p ± ) → for | x | → ∞ . (5.2)We construct the solution to (5.2) by splitting (˜ v, p ) = ( w s , q s ) + ( w d , q d ) where ( w s , q s ) , ( w d , q d ) ∈ X f satisfy ∆ w ± s − ∇ q ± s = 0 in Ω ± , div w ± s = 0 in Ω ± , w + s − w − s = 0 on Γ , [ T ( w s , q s )]˜ ν = − σ ˜ κ ˜ ν on Γ , ( w ± s , q ± s ) → for | x | → ∞ (5.3) and ∆ w ± d − ∇ q ± d = 0 in Ω ± , div w ± d = 0 in Ω ± , µ − w + d − µ + w − d = ( µ + − µ − ) w s on Γ , [ T ( w d , q d )]˜ ν = 0 on Γ , ( w ± d , q ± d ) → for | x | → ∞ . (5.4)The system (5.3) has been studied in [17]. According to [17, Theorem 2.1 and Remark A.2],there exists precisely one solution ( w s , q s ) := ( w s ( f ) , q s ( f )) ∈ X f to (5.3). It satisfies w s ∈ C ∞ ( R \ Γ) ∩ C ( R ) and q ± s ∈ C ∞ (Ω ± ) ∩ C(Ω ± ) . Moreover, recalling (3.2) and [17, Eqns. (2.2), (2.3), (A.2)], the trace w s ( f ) | Γ can be expressedvia w s ( f ) | Γ ◦ Ξ =: G ( f ) := ( G ( f ) , G ( f )) , (5.5)with πσ − G ( f ) := ( B , ( f ) − B , ( f ))[ φ ( f ) + f ′ φ ( f )]+ B , ( f )[3 f ′ φ ( f ) − φ ( f )] + B , ( f )[ f ′ φ ( f ) + φ ( f )] , πσ − G ( f ) := ( B , ( f ) − B , ( f ))[ φ ( f ) + f ′ φ ( f )] − B , ( f )[ f ′ φ ( f ) + φ ( f )] + B , ( f )[ f ′ φ ( f ) − φ ( f )] , (5.6)where φ i ( f ) ∈ H ( R ) , i ∈ { , } , are given by φ ( f ) := f ′ ω + ω and φ ( f ) := f ′ ω . (5.7)We point out that Corollary 4.4 yields G i ( f ) ∈ H ( R ) , i ∈ { , } .It remains to show that the boundary value problem (5.4) has a unique solution ( w d , q d ) ∈ X f with w ± d | Γ ◦ Ξ ∈ H ( R ) . To prove the existence, we solve in X f , for given β ∈ H ( R ) , theauxiliary problem (2.3) and denote its solution by ( w, q ) = ( w, q )[ β ] . In view of Lemma A.1 (i)we have ( µ − w + − µ + w − ) | Γ ◦ Ξ = ( µ + + µ − ) (cid:16)
12 + a µ D ( f ) (cid:17) [ β ] . Therefore ( w d , q d ) := ( w, q )[ β ] solves (5.4) if and only if (cid:16)
12 + a µ D ( f ) (cid:17) [ β ] = a µ G ( f ) , (5.8)where a µ := µ + − µ − µ + + µ − ∈ ( − , . Theorem 4.5 implies that equation (5.8) has a unique solution β =: β ( f ) ∈ H ( R ) . Thisestablishes not only the existence but also the uniqueness of the solution to (5.4). WO-PHASE STOKES FLOW 17
Summarizing, we have shown the following result:
Proposition 5.1.
Given f ∈ H ( R ) , the boundary value problem (5.1) has a unique solu-tion ( v, p ) ∈ X f such that v ± | Γ ◦ Ξ ∈ H ( R ) . Moreover, v ± | Γ ◦ Ξ = G ( f ) µ ± + 1 µ ± (cid:18) − D ( f ) ± (cid:19) [ β ( f )] , where G ( f ) ∈ H ( R ) is defined in (5.5) - (5.6) and β ( f ) ∈ H ( R ) is the unique solutionto (5.8) . From this result and (1.1) we infer, under the assumption that Γ( t ) is at each time in-stant t ≥ the graph of a function f ( t ) ∈ H ( R ) and that ( v ( t ) , p ( t )) belongs to X f ( t ) andsatisfies v ( t ) ± | Γ( t ) ◦ Ξ( f ( t )) ∈ H ( R ) , that (1.1a) can be recast as ∂ t f = 1 µ + D G ( f ) − D ( f )[ β ( f )] + 12 β ( f ) (cid:12)(cid:12)(cid:12) ( − f ′ , ⊤ E = 1 µ + − µ − (cid:10) β ( f ) | ( − f ′ , ⊤ (cid:11) . (5.9)Here h· | ·i denotes the scalar product on R .Using the results in Section 4 and [17] we can formulate the latter equation as an evolutionequation in H s − ( R ) , where s ∈ (3 / , is fixed in the remaining. To this end we first inferfrom [17, Corollary C.5] that, given n, m ∈ N , we have [ f B n,m ( f )] ∈ C ∞ ( H s ( R ) , L ( H s − ( R ))) . (5.10)Further, [17, Lemma 3.5] ensures for the mappings defined in (5.7) that [ f φ i ( f )] ∈ C ∞ ( H s ( R ) , H s − ( R )) , i = 1 , . (5.11)Additionally, for any f ∈ H s ( R ) , the Fréchet derivative ∂φ i ( f ) is given by ∂φ i ( f ) = a i ( f ) ddx , i = 1 , , with a ( f ) := f ′ (2 + f ′ + 2 p f ′ ) p f ′ ( p f ′ + 1 + f ′ ) and a ( f ) := 1(1 + f ′ ) / . (5.12)It is easy to check, by arguing as in [17, Lemma C.1], that φ i , i = 1 , , maps bounded setsin H s ( R ) to bounded sets in H s − ( 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 [ f G ( f )] ∈ C ∞ ( H s ( R ) , H s − ( R ) ) , (5.13)and also that G maps bounded sets in H s ( R ) to bounded sets in H s − ( R ) .Moreover, recalling (3.5), we infer from (5.10) that D ∈ C ∞ ( H s ( R ) , L ( H s − ( R ) )) . (5.14)In view of (5.13) and of Theorem 4.2 we can solve, for given f ∈ H s ( R ) , the equation (5.8)in H s − ( R ) . Its unique solution is given by β ( f ) := 2 a µ (1 + 2 a µ D ( f )) − [ G ( f )] ∈ H s − ( R ) , (5.15) and, since the mapping which associates to an isomorphism its inverse is smooth, we obtainfrom Theorem 4.2, (5.13), and (5.14) that (cid:2) f β ( f )] (cid:3) ∈ C ∞ ( H s ( R ) , H s − ( R ) ) . (5.16)Furthermore, (5.15) and the estimate (4.5) imply that β inherits from G the property to mapbounded sets in H s ( R ) to bounded sets in H s − ( R ) . Summarizing, in a compact form, theStokes flow problem (1.1) can be recast as the evolution problem dfdt = Φ( f ( t )) , t ≥ , f (0) = f , (5.17)where Φ : H s ( R ) → H s − ( R ) is defined, cf. (5.9), by Φ( f ) := 1 µ + − µ − h β ( f ) | ( − f ′ , ⊤ i . (5.18)Observe that, due to (5.16), Φ ∈ C ∞ ( H s ( R ) , H s − ( R )) , (5.19)and that Φ maps bounded sets in H s ( R ) to bounded sets in H s − ( R ) . Linearization, localization, and proof of the main result
We are going to prove that the nonlinear and nonlocal problem (5.17) is parabolic in H s ( R ) in the sense that the Fréchet derivative ∂ Φ( f ) , generates an analytic semigroup in L ( H s − ( R )) for each f ∈ H s ( R ) . This property then enables us to use the abstract existence resultsfrom [14] in the proof of our main result Theorem 1.1. Theorem 6.1.
For any f ∈ H s ( R ) , the Fréchet derivative ∂ Φ( f ) , considered as an un-bounded operator in H s − ( R ) with dense domain H s ( R ) , generates an analytic semigroup in L ( H s − ( R )) . The proof of Theorem 6.1 requires some preparation. To start, fix f ∈ H s ( R ) , s ′ ∈ (3 / , s ) ,and let β := β ( f ) := ( β , β ) ⊤ . We have β ∈ H s − ( R ) . Differentiating the relations (5.18) and (5.15), we get ∂ Φ( f )[ f ] = 1 µ + − µ − h ∂β ( f )[ f ] | ( − f ′ , ⊤ i − β f ′ µ + − µ − (6.1)and (1 + 2 a µ D ( f ))[ ∂β ( f )[ f ]] = 2 a µ ∂G ( f )[ f ] − a µ ∂ D ( f )[ f ][ β ] . (6.2)For the computation of ∂ D ( f )[ f ][ β ] and ∂G ( f )[ f ] we use the relation ∂B n, ( f )[ f ][ h ] = nB n, ( f , f )[ f, f , . . . f , h ] − B n +2 , ( f , f , f )[ f, f , . . . , f , h ] , n ∈ N , see [17, Lemma C.4]. Additionally we use Lemma 4.1 (iii) to rewrite this expression as ∂B n, ( f )[ f ][ h ] = h (cid:0) nB n − , ( f )[ f ′ ] − B n +1 , ( f )[ f ′ ] (cid:1) + R ,n [ f, h ]= h (cid:0) nB n − , ( f )[ f ′ ] + ( n − B n +1 , ( f )[ f ′ ] (cid:1) + R ,n [ f, h ] , where nB n − , ( f ) := 0 for n = 0 and k R ,n [ f, h ] k H s − ≤ C k h k H s − k f k H s ′ , WO-PHASE STOKES FLOW 19 with a constant C independent of f ∈ H s ( R ) and h ∈ H s − ( R ) . Using these relations, weinfer from (3.5) that ( ∂ D ( f )[ f ][ β ]) i = 1 π (cid:8) B i + k − , [ f ′ β k ] + β k (cid:0) ( i + k − f ′ B i + k − , + ( i + k − f ′ B i + k − , − ( i + k − B i + k − , − ( i + k − B i + k, (cid:1) [ f ′ ] (cid:9) + R ,i [ f ] (6.3)for i = 1 , , where we used the shorthand notation B n,m := B n,m ( f ) and k R ,i [ f ] k H s − ≤ C k f k H s ′ , f ∈ H s ( R ) . (6.4)Taking the derivative of (5.6), the same arguments yield πσ − ∂G i ( f )[ f ] = T i, ( f )[ f ] + T i, ( f )[ f ] + R ,i [ f ] , i = 1 , , (6.5)where T , ( f )[ f ] :=( B , − B , )[( a + φ + f ′ a ) f ′ ] + B , [(3( φ + f ′ a ) − a ) f ′ ]+ B , [( φ + f ′ a + a ) f ′ ] ,T , ( f )[ f ] := φ (3 f ′ B , − B , − f ′ B , + 2 B , − f ′ B , )[ f ′ ]+ φ ( − B , − f ′ B , + 6 B , + 2 f ′ B , − B , )[ f ′ ] ,T , ( f )[ f ] := − B , [( φ + f ′ a + a ) f ′ ] + ( B , − B , )[( a + φ + f ′ a ) f ′ ]+ B , [( φ + f ′ a − a ) f ′ ] ,T , ( f )[ f ] := φ ( B , + 6 f ′ B , − B , − f ′ B , + B , )[ f ′ ]+ φ ( f ′ B , − B , − f ′ B , + 6 B , + f ′ B , )[ f ′ ] , (6.6)cf. [17, Eq. (3.7)-(3.9)]. Here we used the shortened notation a i := a i ( f ) and φ i := φ i ( f ) for i = 1 , and k R ,i [ f ] k H s − ≤ C k f k H s ′ , f ∈ H s ( R ) . (6.7)In order to prove Theorem 6.1 we consider the path Ψ : [0 , −→ L ( H s ( R ) , H s − ( R )) defined by Ψ( τ )[ f ] := 1 µ + − µ − hB ( τ )[ f ] | ( − τ f ′ , ⊤ i − τ β f ′ µ + − µ − (6.8)for τ ∈ [0 , and f ∈ H s ( R ) , where B ( τ )[ f ] is defined by (1 + 2 τ a µ D ( f ))[ B ( τ )[ f ]] = 2 a µ ( ∂G ( τ f )[ f ] − τ ∂ D ( f )[ f ][ β ]) . (6.9)Theorem 4.2, (6.3)–(6.7), and Lemma 4.1 (ii) ensure that B : [0 , −→ L (cid:0) H s ( R ) , H s − ( R ) (cid:1) is well-defined, and kB ( τ )[ f ] k H s − ≤ C k f k H s , τ ∈ [0 , , f ∈ H s ( R ) , (6.10) with C independent of f and τ . We also note that both paths B and Ψ are continuousand Ψ(1) = ∂ Φ( f ) . Besides, since B (0) = 2 a µ ∂G (0) = (cid:16) , − a µ σ H ◦ ddξ (cid:17) ⊤ , where H = π − B , is the Hilbert transform, we observe that Ψ(0) is the Fourier multiplier
Ψ(0) = − σ µ + + µ − ) H ◦ ddξ = − σ µ + + µ − ) (cid:16) − d dξ (cid:17) / . (6.11)We next locally approximate the operator Ψ( τ ) , τ ∈ [0 , , by certain Fourier multipli-ers A j,τ , cf. Theorem 6.2 below. For this purpose, given ε ∈ (0 , , we choose N = N ( ε ) ∈ N and a so-called finite ε -localization family, that is a set { ( π εj , ξ εj ) | − N + 1 ≤ j ≤ N } such that • π εj ∈ C ∞ ( R , [0 , , − N + 1 ≤ j ≤ N , and N X j = − N +1 ( π εj ) = 1; • supp π εj is an interval of length ε for all | j | ≤ N − and supp π εN ⊂ {| ξ | ≥ /ε } ; • π εj · π εl = 0 if [ | j − l | ≥ , max {| j | , | l |} ≤ N − or [ | l | ≤ N − , j = N ]; • k ( π εj ) ( k ) k ∞ ≤ Cε − k for all k ∈ N , − N + 1 ≤ j ≤ N ; • ξ εj ∈ supp π εj , | j | ≤ N − . The real number ξ εN plays no role in the analysis below. To each ε -localization family we asso-ciate a norm on H r ( R ) , r ≥ , which is equivalent to the standard norm. Indeed, given r ≥ and ε ∈ (0 , , there exists a constant c = c ( ε, r ) ∈ (0 , such that c k f k H r ≤ N X j = − N +1 k π εj f k H r ≤ c − k f k H r , f ∈ H r ( R ) . (6.12)To introduce the aforementioned Fourier multipliers A j,τ , we first define the coefficientfunctions α τ , β τ : R −→ R , τ ∈ [0 , , by the relations α τ := σ µ + + µ − ) (cid:0) a ( τ f ) + τ f ′ a ( τ f ) (cid:1) , β τ := − τ β µ + − µ − . (6.13)We now set A j,τ := A εj,τ := − α τ ( ξ εj ) (cid:16) − d dξ (cid:17) / + β τ ( ξ εj ) ddξ , | j | ≤ N − , A N,τ := A εN,τ := − σ µ + + µ − ) (cid:16) − d dξ (cid:17) / . (6.14)We obviously have A j,τ ∈ L ( H s ( R ) , H s − ( R )) , − N + 1 ≤ j ≤ N , τ ∈ [0 , .The following estimate of the localization error is the main step in the proof of Theorem 6.1. WO-PHASE STOKES FLOW 21
Theorem 6.2.
Let µ > be given and fix s ′ ∈ (3 / , s ) . Then there exist ε ∈ (0 , and aconstant K = K ( ε ) such that k π εj Ψ( τ )[ f ] − A j,τ [ π εj f ] k H s − ≤ µ k π εj f k H s + K k f k H s ′ (6.15) for all − N + 1 ≤ j ≤ N , τ ∈ [0 , , and f ∈ H s ( R ) . Before proving Theorem 6.2 we first present some auxiliary lemmas which are used in theproof. We start with an estimate for the commutator [ B n,m ( f ) , ϕ ] (we will apply this estimatein the particular case ϕ = π εj , − N + 1 ≤ j ≤ N ). Lemma 6.3.
Let n, m ∈ N , s ∈ (3 / , , f ∈ H s ( R ) , and ϕ ∈ C ( R ) with uniformly continu-ous derivative ϕ ′ be given. Then, there exist a constant K that depends only on n, m, k ϕ ′ k ∞ , and k f k H s such that k ϕB n,m ( f, . . . , f )[ f, . . . , f, h ] − B n,m ( f, . . . , f )[ f, . . . , f, ϕh ] k H ≤ K k h k (6.16) for all h ∈ L ( R ) .Proof. This result is a particular case of [1, Lemma 12]. (cid:3)
The results in Lemma 6.4-Lemma 6.8 below describe how to “freeze the coefficients” of themultilinear operators B n,m . For these operators, this technique has been first developed in [16]in the study of the Muskat problem.
Lemma 6.4.
Let n, m ∈ N , / < s ′ < s < , and ν ∈ (0 , ∞ ) be given. Let further f ∈ H s ( R ) and ω ∈ { } ∪ H s − ( R ) . For any sufficiently small ε ∈ (0 , , there is a constant K dependingonly on ε, n, m, k f k H s , and k ω k H s − (if ω = 1 ) such that (cid:13)(cid:13)(cid:13) π εj ωB n,m ( f )[ h ] − ω ( ξ εj )( f ′ ( ξ εj )) n [1 + ( f ′ ( ξ εj )) ] m B , [ π εj h ] (cid:13)(cid:13)(cid:13) H s − ≤ ν k π εj h k H s − + K k h k H s ′− for all | j | ≤ N − and h ∈ H s − ( R ) .Proof. See [1, Lemma 13]. (cid:3)
We now provide a similar result as in Lemma 6.4, the difference to the latter being that thelinear argument of B n,m is now multiplied by a function a that also needs to be frozen at ξ εj . Lemma 6.5.
Let n, m ∈ N , / < s ′ < s < , and ν ∈ (0 , ∞ ) be given. Let fur-ther f ∈ H s ( R ) , a ∈ H s − ( R ) , and ω ∈ { } ∪ H s − ( R ) . For any sufficiently small ε ∈ (0 , ,there is a constant K only depending on ε, n, m, k f k H s , k a k H s − , and k ω k H s − (if ω = 1 )such that (cid:13)(cid:13)(cid:13) π εj ωB n,m ( f )[ ah ] − a ( ξ εj ) ω ( ξ εj )( f ′ ( ξ εj )) n [1 + ( f ′ ( ξ εj )) ] m B , [ π εj h ] (cid:13)(cid:13)(cid:13) H s − ≤ ν k π εj h k H s − + K k h k H s ′− for all | j | ≤ N − and h ∈ H s − ( R ) .Proof. See [17, Lemma D.5]. (cid:3)
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 ∈ N , / < s ′ < s < , and ν ∈ (0 , ∞ ) be given. Let further f ∈ H s ( R ) and ω ∈ H s − ( R ) . For any sufficiently small ε ∈ (0 , , there is a constant K depending onlyon ε, n, m, k f k H s , and k ω k H s − such that k π εN ωB n,m ( f )[ h ] k H s − ≤ ν k π εN h k H s − + K k h k H s ′− for h ∈ H s − ( R ) .Proof. See [1, Lemma 14]. (cid:3)
Lemma 6.7 is the counterpart of Lemma 6.6 in the case when ω = 1 . Lemma 6.7.
Let n, m ∈ N , / < s ′ < s < , and ν ∈ (0 , ∞ ) be given. Let fur-ther f ∈ H s ( R ) . For any sufficiently small ε ∈ (0 , , there is a constant K depending onlyon ε, n, m, and k f k H s such that k π εN B ,m ( f )[ h ] − B , [ π εN h ] k H s − ≤ ν k π εN h k H s − + K k h k H s ′− and k π εN B n,m ( f )[ h ] k H s − ≤ ν k π εN h k H s − + K k h k H s ′− , n ≥ , for all h ∈ H s − ( R ) .Proof. See [1, Lemma 15]. (cid:3)
Finally, Lemma 6.8 below is the analogue of Lemma 6.5 corresponding to the case j = N . Lemma 6.8.
Let n, m ∈ N , / < s ′ < s < , and ν ∈ (0 , ∞ ) be given. Let fur-ther f ∈ H s ( R ) , a ∈ H s − ( R ) , and ω ∈ { } ∪ H s − ( R ) . For any sufficiently small ε ∈ (0 , ,there is a constant K depending on ε, n, m, k f k H s , k a k H s − , and k ω k H s − (if ω = 1 ) suchthat k π εN ωB n,m ( f )[ ah ] k H s − ≤ ν k π εN h k H s − + K k h k H s ′− for all h ∈ H s − ( R ) .Proof. See [17, Lemma D.6]. (cid:3)
We are now in a position to prove Theorem 6.2.
Proof of Theorem 6.2.
Fix µ > and let ε ∈ (0 , . Let further { ( π εj , ξ εj ) | − N + 1 ≤ j ≤ N } be a finite ε -localization family. We choose a second family { χ εj | − N + 1 ≤ j ≤ N } with thefollowing properties: • χ εj ∈ C ∞ ( R , [0 , and χ εj = 1 on supp π εj , − N + 1 ≤ j ≤ N ; • supp χ εj is an interval of length ε , | j | ≤ N − , and supp χ εN ⊂ {| x | ≥ /ε − ε } .In the arguments that follow we repeatedly use the estimate k gh k H s − ≤ C ( k g k ∞ k h k H s − + k h k ∞ k g k H s − ) (6.17)which holds for g, h ∈ H s − ( R ) and s ∈ (3 / , , with a constant C independent of g and h .Below we denote by C constants that do not depend on ε and by K constants that maydepend on ε . We need to approximate the linear operators (cid:2) f
7→ B ( τ )[ f ] − τ f ′ B ( τ )[ f ] (cid:3) and [ f β f ′ ] , see (6.8)-(6.9), where we set B ( τ ) =: ( B ( τ ) , B ( τ )) ⊤ . The proof is dividedin several steps. WO-PHASE STOKES FLOW 23
Step 1.
We consider the operator [ f β f ′ ] . Since χ εj π εj = π εj , (6.17) yields k π εj ( β f ′ ) − β ( ξ εj )( π εj f ) ′ k H s − ≤ C k χ εj ( β − β ( ξ εj )) k ∞ k ( π εj f ) ′ k H s − + K k f k H s ′ for | j | ≤ N − and k π εN ( β f ′ ) k H s − ≤ C k χ εN β k ∞ k ( π εN f ) ′ k H s − + K k f k H s ′ . From (5.16) we have β ∈ C s − / ( R ) and β ( ξ ) → for | ξ | → ∞ . Hence, if ε is sufficientlysmall, then k π εj ( β f ′ ) − β ( ξ εj )( π εj f ) ′ k H s − ≤ µ | µ + − µ − | k π εj f k H s + K k f k H s ′ , | j | ≤ N − , k π εN ( β f ′ ) k H s − ≤ µ | µ + − µ − | k π εN f k H s + K k f k H s ′ . (6.18)The approximation procedure for (cid:2) f
7→ B ( τ )[ f ] − τ f ′ B ( τ )[ f ] (cid:3) is more involved. Step 2.
We prove there exists a constant C B such that k π εj B ( τ )[ f ] k H s − ≤ C B k π εj f k H s + K k f k H s ′ (6.19)for all − N + 1 ≤ j ≤ N , τ ∈ [0 , , and f ∈ H s ( R ) . To start, we infer from (6.9) that (1 + 2 τ a µ D ( f ))[ π εj B ( τ )[ f ]] = 2 a µ π εj ∂G ( τ f )[ f ] − τ a µ π εj ∂ D ( f )[ f ][ β ]+ 2 τ a µ (cid:0) D ( f )[ π εj B ( τ )[ f ]] − π εj D ( f )[ B ( τ )[ f ]] (cid:1) . (6.20)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 − -estimate for the op-erators B m,n provided in Lemma 4.1 (ii). So we get k π εj ∂G ( τ f )[ f ] k H s − + k π εj ∂ D ( f )[ f ][ β ] k H s − ≤ C k π εj f k H s + K k f k H s ′ , (6.21)and similarly, using (3.5) and (6.10) with s replaced by s ′ , k D ( f )[ π εj B ( τ )[ f ]] − π εj D ( f )[ B ( τ )[ f ]] k H s − ≤ K kB ( τ )[ f ] k ≤ K k f k H s ′ . (6.22)The estimate (6.19) follows now from (6.20)–(6.22) and Theorem 4.2. Step 3.
Given τ ∈ [0 , and − N + 1 ≤ j ≤ N , let B j,τ ∈ L ( H s ( R ) , H s − ( R ) ) denote theFourier multipliers B j,τ := a µ σ π a ( τ f )( ξ εj ) B , ◦ ( d/dξ ) − a ( τ f )( ξ εj ) B , ◦ ( d/dξ ) ! , | j | < N, and B N,τ := a µ σ π − B , ◦ ( d/dξ ) ! . We next prove that given ν > , we have k π εj B ( τ )[ f ] − B j,τ [ π εj f ] k H s − ≤ ν k π εj f k H s + K k f k H s ′ (6.23)for all − N + 1 ≤ j ≤ N , τ ∈ [0 , , f ∈ H s ( R ) and all sufficiently small ε . To start, wemultiply (6.9) by π εj and get π εj B ( τ )[ f ] = 2 a µ π εj (cid:2) ∂G ( τ f )[ f ] − τ (cid:0) D ( f )[ B ( τ )[ f ]] + ∂ D ( f )[ f ][ β ] (cid:1)(cid:3) (6.24) We consider the terms on the right hand side of (6.24) one by one. To deal with the first termwe recall (6.5)-(6.7). Repeated use of Lemma 6.4 and Lemma 6.5 then shows that k a µ π εj ∂G ( τ f )[ f ] − B j,τ [ π εj f ] k H s − ≤ ν k π εj f k H s + K k f k H s ′ (6.25)for | j | ≤ N − , while Lemma 6.6, Lemma 6.7, and Lemma 6.8 yield k a µ π εN ∂G ( τ f )[ f ] − B N,τ [ π εN f ] k H s − ≤ ν k π εN f k H s + K k f k H s ′ (6.26)provided that ε is sufficiently small.We estimate the second term on the right of (6.24) and let | j | ≤ N − first. Combining (3.5),Lemma 6.4, Lemma 6.5, (6.10) with s replaced by s ′ , and (6.19) we obtain k π εj D ( f )[ B ( τ )[ f ]] k H s − ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π εj B , B , B , B , ! f ′ B ( τ )[ f ] f ′ B ( τ )[ f ] ! − f ′ ( ξ εj )(1 + f ′ ( ξ εj )) f ′ ( ξ εj ) f ′ ( ξ εj ) f ′ ( ξ εj ) ! B , [ π εj B ( τ )[ f ]] B , [ π εj B ( τ )[ f ]] ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s − + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π εj B , B , B , B , ! B ( τ )[ f ] B ( τ )[ f ] ! − f ′ ( ξ εj )(1 + f ′ ( ξ εj )) f ′ ( ξ εj ) f ′ ( ξ εj ) f ′ ( ξ εj ) ! B , [ π εj B ( τ )[ f ]] B , [ π εj B ( τ )[ f ]] ! (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s − ≤ ν | a µ | k π εj f k H s + K k f k H s ′ (6.27)provided that ε 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 k π εN D ( f )[ B ( τ )[ f ]] k H s − ≤ ν | a µ | k π εN f k H s + K k f k H s ′ (6.28)provided that ε is sufficiently small.It remains to consider the term π εj ∂ D ( f )[ f ][ β ] on the right of (6.24). To this end we arguesimilarly 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 | ≤ N − , respectively fromLemma 6.6 and Lemma 6.8 if j = N , that k π εj ∂ D ( f )[ f ][ β ] k H s − ≤ ν | a µ | k π εj f k H s + K k f k H s ′ (6.29)provided that ε is sufficiently small. The estimate (6.23) follows now from (6.24)-(6.29). WO-PHASE STOKES FLOW 25
Step 4.
We are now in a position to localize the operators (cid:2) f
7→ B ( τ )[ f ] − τ f ′ B ( τ )[ f ] (cid:3) . Theestimate (6.23) shows that, choosing ε sufficiently small, we have (cid:13)(cid:13)(cid:13) π εj B ( τ )[ f ] + a µ σ π a ( τ f )( ξ εj ) B , [( π εj f ) ′ ] (cid:13)(cid:13)(cid:13) H s − ≤ µ | µ + − µ − | k π εj f k H s + K k f k H s ′ (6.30)for | j | ≤ N − and (cid:13)(cid:13)(cid:13) π εN ( B ( τ )[ f ] + a µ σ π B , [( π εN f ) ′ ] (cid:13)(cid:13)(cid:13) H s − ≤ µ | µ + − µ − | k π εN f k H s + K k f k H s ′ . (6.31)Moreover, for | j | ≤ N − , we write in view of χ εj π εj = π εj (cid:13)(cid:13)(cid:13) π εj f ′ B ( τ )[ f ] − a µ σ π f ′ ( ξ εj ) a ( τ f )( ξ εj ) B , [( π εj f ) ′ ] (cid:13)(cid:13)(cid:13) H s − ≤ k χ εj ( f ′ − f ′ ( ξ εj )) π εj B ( τ )[ f ] k H s − + C (cid:13)(cid:13)(cid:13) π εj B ( τ )[ f ] − a µ σ π a ( τ f )( ξ εj ) B , [( π εj f ) ′ ] (cid:13)(cid:13)(cid:13) H s − . The first term on the right hand side may be estimated by using (6.10) (with s replacedby s ′ ), (6.17), (6.19), and the fact that f ′ ∈ C s − / ( R ) . For the second term we rely on (6.23).Hence, if ε is sufficiently small then (cid:13)(cid:13)(cid:13) π εj f ′ B ( τ )[ f ] − a µ σ π f ′ ( ξ εj ) a ( τ f )( ξ εj ) B , [( π εj f ) ′ ] (cid:13)(cid:13)(cid:13) H s − ≤ µ | µ + − µ − | k π εj f k H s + K k f k H s ′ . (6.32)For j = N , it follows from (6.10) (with s replaced by s ′ ), (6.17), (6.19), and the fact that f ′ vanishes at infinity that k π εN f ′ B ( τ )[ f ] k H s − ≤ µ | µ + − µ − | k π εN f k H s + K k f k H s ′ . (6.33)The desired claim (6.15) follows now from (6.8), (6.18), (6.30), and (6.32) if | j | ≤ N − ,respectively from (6.8), (6.18), (6.31), and (6.33) if j = N. (cid:3) We now investigate the Fourier multipliers A j,τ found in Theorem 6.2. We recall thedefinitions (5.12), (6.13), and (6.14) and observe that as f ′ , β , a i ( τ f ) ∈ H s − ( R ) , i = 1 , and τ ∈ [0 , , there is a constant η ∈ (0 , such that η ≤ α τ ≤ η and | β τ | ≤ η , τ ∈ [0 , . Based on this, it can be shown as in [16, Proposition 4.3], that there is a constant κ ≥ suchthat for all ε ∈ (0 , , − N + 1 ≤ j ≤ N , and τ ∈ [0 , we have • λ − A j,τ ∈ L ( H s ( R ) , H s − ( R )) is an isomorphism for all Re λ ≥ , (6.34) • κ k ( λ − A j,τ )[ f ] k H s − ≥ | λ | · k f k H s − + k f k H s , f ∈ H s ( R ) , Re λ ≥ . (6.35)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 ′ ∈ (3 / , s ) and let κ ≥ be the constant in (6.35). Theorem 6.2with µ := 1 / κ implies that there are ε ∈ (0 , , a constant K = K ( ε ) > and boundedoperators A j,τ ∈ L ( H s ( R ) , H s − ( R )) , − N + 1 ≤ j ≤ N and τ ∈ [0 , , satisfying κ k π εj Ψ( τ )[ f ] − A j,τ [ π εj f ] k H s − ≤ k π εj f k H s + 2 κ K k f k H s ′ , f ∈ H s ( R ) . Moreover, (6.35) yields κ k ( λ − A j,τ )[ π εj f ] k H s − ≥ | λ | · k π εj f k H s − + 2 k π εj f k H s for all − N + 1 ≤ j ≤ N , τ ∈ [0 , , Re λ ≥ , and f ∈ H s ( R ) . The latter estimates combinedlead us to κ k π εj ( λ − Ψ( τ ))[ f ] k H s − ≥ κ k ( λ − A j,τ )[ π εj f ] k H s − − κ k π εj Ψ( τ )[ f ] − A j,τ [ π εj f ] k H s − ≥ | λ | · k π εj f k H s − + k π εj f k H s − κ K k f k H s ′ . Summing over j , we deduce from (6.12), Young’s inequality, and the interpolation prop-erty (4.4) that there exist constants κ ≥ and ω > such that κ k ( λ − Ψ( τ ))[ f ] k H s − ≥ | λ | · k f k H s − + k f k H s (6.36)for all τ ∈ [0 , , Re λ ≥ ω , and f ∈ H s ( R ) .From (6.11) we also deduce that ω − Ψ(0) ∈ L ( H s ( R ) , H s − ( R )) is an isomorphism. Thistogether with method of continuity [3, Proposition I.1.1.1] and (6.36) implies that also ω − Ψ(1) = ω − ∂ Φ( f ) ∈ L ( H s ( R ) , H s − ( R )) (6.37)is an isomorphism. The estimate (6.36) (with τ = 1 ) and (6.37) finally imply that ∂ Φ( f ) generates an analytic semigroup in L ( H s − ( R )) , cf. [3, Chapter I], and the proof is complete. (cid:3) We are now in a position to prove the main result, for which we can exploit abstract theoryfor fully nonlinear parabolic problems from [14].
Proof of Theorem 1.1. Well-posedness:
Given α ∈ (0 , , T > , and a Banach space X weset C αα ((0 , T ] , X ) := n f : (0 , T ] −→ X (cid:12)(cid:12)(cid:12) k f k C αα := sup t k f ( t ) k + sup s = t k t α f ( t ) − s α f ( s ) k| t − s | α < ∞ o . The property (5.19) together with Theorem 6.1 shows that the assumptions of [14, Theo-rem 8.1.1] are satisfied for the evolution problem (5.17). According to this theorem, (5.17)has, for each f ∈ H s ( R ) , a local solution f ( · ; f ) such that f ∈ C([0 , T ] , H s ( R )) ∩ C ([0 , T ] , H s − ( R )) ∩ C αα ((0 , T ] , H s ( R )) , where T = T ( f ) > and α ∈ (0 , is fixed (but arbitrary). This solution is unique withinthe set [ α ∈ (0 , C αα ((0 , T ] , H s ( R )) ∩ C([0 , T ] , H s ( R )) ∩ C ([0 , T ] , H s − ( R )) . We improve the uniqueness property by showing that the solution is unique within
C([0 , T ] , H s ( R )) ∩ C ([0 , T ] , H s − ( R )) . WO-PHASE STOKES FLOW 27
Indeed, let f now be any solution to (5.17) in that space, let s ′ ∈ (3 / , s ) be fixed andset α := s − s ′ ∈ (0 , . Using (4.4), we find a constant C > such that k f ( t ) − f ( t ) k H s ′ ≤ C | t − t | α , t , t ∈ [0 , T ] , (6.38)which shows in particular that f ∈ C αα ((0 , T ] , H s ′ ( R )) . The uniqueness statement of [14,Theorem 8.1.1] applied in the context of (5.17) with Φ ∈ C ∞ ( H s ′ ( R ) , H s ′ − ( R )) establishesthe uniqueness claim. This unique solution can be extended up to a maximal existencetime T + ( f ) , see [14, Section 8.2]. Finally, [14, Proposition 8.2.3] shows that the solution mapdefines a semiflow on H s ( R ) which, according to [14, Corollary 8.3.8], is smooth in the openset { ( t, f ) | < t < T + ( f ) } . This proves (i). Parabolic smoothing:
The uniqueness result established in (i) enables us to use a parametertrick applied also to other problems, cf., e.g., [4, 9, 16, 18], in order to establish (iia) and (iib).The proof details are similar to those in [15, Theorem 1.2 (v)] or [1, Theorem 2 (ii)] andtherefore we omit them.
Global existence:
We prove the statement by contradiction. Assume there exists a maximalsolution f ∈ C([0 , T + ) , H s ( R )) ∩ C ([0 , T + ) , H s − ( R )) to (5.17) with T + < ∞ and such that sup [0 ,T + ) k f ( t ) k H s < ∞ . (6.39)Recalling that Φ maps bounded sets in H s ( R ) to bounded sets in H s − ( R ) , we get sup t ∈ [0 ,T + ) (cid:13)(cid:13)(cid:13) dfdt ( t ) (cid:13)(cid:13)(cid:13) H s − = sup t ∈ [0 ,T + ) k Φ( f ( t )) k H s − < ∞ . (6.40)Let s ′ ∈ (3 / , s ) be fixed. Arguing as above, see (6.38), from the bounds (6.39) and (6.40) weget that f : [0 , T + ) −→ H s ′ ( R ) is uniformly continuous. Applying [14, Theorem 8.1.1] to (5.17)with Φ ∈ C ∞ ( H s ′ ( R ) , H s ′ − ( R )) , we may extend the solution f to a time interval [0 , T ′ + ) with T + < T ′ + and such that f ∈ C([0 , T ′ + ) , H s ′ ( R )) ∩ C ([0 , T ′ + ) , H s ′ − ( R )) . Since by (iib) (with s replaced by s ′ ) we have f ∈ C ((0 , T ′ + ) , H s ( R )) , this contradicts themaximality property of f and the proof is complete. (cid:3) Appendix A. The hydrodynamic double-layer potential near Γ Given f ∈ H ( R ) and β ∈ H ( R ) , we let ( w, q ) be given by (2.6) and (2.7). We recall thedefinitions (3.3) of D ( f ) and (3.2) of the operators B n,m . Lemma A.1.
We have w ± ∈ C (Ω ± , R ) , q ± ∈ C(Ω ± ) , w ± | Γ ◦ Ξ ∈ H ( R ) , and w ± = (cid:16) − D ( f )[ β ] ± β (cid:17) ◦ Ξ − on Γ , [ T ( w, q )]( ν ◦ Ξ − ) = 0 on Γ . ) (A.1) Proof.
For j = 0 , . . . , , let Z j ∈ C ( R \ { } ) be given by Z j ( y ) := y − j y j | y | , y ∈ R \ { } . Given φ ∈ H ( R ) , we define the function Z j [ φ ] : R \ Γ −→ R , j = 0 , . . . , , by Z j [ φ ]( x ) := Z R Z j ( r ) φ ds, x ∈ R \ Γ , r := x − ( s, f ( s )) . Recalling (2.8) , we have w = 1 π (cid:18) − (cid:18) Z Z Z Z (cid:19) [ f ′ β ] + (cid:18) Z Z Z Z (cid:19) [ β ] (cid:19) . It is shown in [17, Lemma A.1] that Z j [ φ ] ± ∈ C(Ω ± ) , with Z [ φ ] ± Z [ φ ] ± Z [ φ ] ± Z [ φ ] ± ◦ Ξ = B , ( f )[ φ ] B , ( f )[ φ ] B , ( f )[ φ ] B , ( f )[ φ ] ∓ π ω f ′ + 3 f ′ f ′ − f ′ − f ′ − f ′ − φ. (A.2)Consequently, w ± ∈ C(Ω ± , R ) , and the jump relations (A.2) imply (A.1) . Moreover, re-calling Corollary 4.4, we get w ± | Γ ◦ Ξ ∈ H ( R ) . Further, q ± ∈ C(Ω ± ) follows from [15,Lemma 2.1].Exchanging integration with respect to s and differentiation with respect to x by dominatedconvergence we find from (1.1b), (2.6), and (2.7) that ∂ l w j ( x ) = Z R ∂ l W i,kj ( r ) ν i β k ω ds, (A.3) ( T ( w, q )) jl ( x ) = Z R ( − δ jl Q i,k + ∂ l W i,kj + ∂ j W i,kj )( r ) ν i β k ω ds (A.4)for x ∈ R \ Γ and l, j = 1 , .For E ⊂ R open, Z ∈ C ( E ) , i = 1 , , we define rot Z := (rot Z , rot Z ) ∈ C( E, R ) by rot i Z := (cid:26) − ∂ Z if i = 1 , ∂ Z if i = 2 . With this notation, we find from integration by parts Z R (rot i Z j )( r ) ν i φω ds = Z R ( f ′ ∂ Z j ( r ) + ∂ Z j ( r )) φ ds = − Z R ∂ s ( Z j ( r )) φ ds = Z j [ φ ′ ] . Together with (A.3), (A.4), and the identities ∂ W i, = − ∂ W i, = − ∂ W i, = 1 π rot i Z ,∂ W i, = − ∂ W i, = ∂ W i, = 1 π rot i Z ,∂ W i, = − π rot i Z ,∂ W i, = 1 π rot i Z WO-PHASE STOKES FLOW 29 and Q i, = 1 π rot i ( −Z − Z ) , Q i, = 1 π rot i ( Z + Z ) , this yields w ± ∈ C (Ω ± , R ) and (A.1) . (cid:3) Acknowledgements:
The research leading to this paper was carried out while the secondauthor enjoyed the hospitality of DFG Research Training Group 2339 “Interfaces, ComplexStructures, and Singular Limits in Continuum Mechanics - Analysis and Numerics” at theFaculty of Mathematics of Regensburg University.
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Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Deutschland.
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