Well-posedness and qualitative behaviour of solutions for a two-phase Navier–Stokes-Mullins–Sekerka system

Well-posedness and qualitative behaviour of solutions for a two-phase Navier–Stokes-Mullins–Sekerka system

HELMUTABELS

Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, Universit¨atsstraße 31, D-93053 Regensburg, Germany

E-mail:helmut.abels@mathematik.uni-regensburg.de

MATHIASWILKE

Institut f¨ur Mathematik, Martin-Luther-Universit¨at Halle-Wittenberg Theodor-Lieser-Str. 5, D-06120 Halle, Germany

E-mail:mathias.wilke@mathematik.uni-halle.de

[Received 30 December 2011 and in revised form 14 September 2012]

We consider a two-phase problem for two incompressible, viscous and immiscible fluids which are separated by a sharp interface. The problem arises as a sharp interface limit of a diffuse interface model. We present results on local existence of strong solutions and on the long-time behavior of solutions which start close to an equilibrium. To be precise, we show that as time tends to infinity, the velocity field converges to zero and the interface converges to a sphere at an exponential rate.

2010 Mathematics Subject Classification: Primary 35R35; Secondary 35Q30, 76D27, 76D45, 76T99.

Keywords: Two-phase flow, Navier–Stokes system, Free boundary problems, Mullins–Sekerka equation, convergence to equilibria.

1. Introduction

We study the flow of two incompressible, viscous and immiscible fluids inside a bounded domain

˝ Rⁿ,nD 2; 3. The fluids fill domains˝^C.t /and˝ .t /,t > 0, respectively, with a common interface .t /between both fluids. The flow is described in terms of the velocityvW.0;1/˝! Rⁿand the pressurepW.0;1/˝ !Rin both fluids in Eulerian coordinates. We assume the fluids to be of Newtonian type, i.e., the stress tensors of the fluids are of the formT .v; p/D2^˙Dv pI in˝^˙.t /with constant viscosities^˙ > 0and2Dv D rvC rv^T. Moreover, we consider the case with surface tension at the interface. In this model the densities of the fluids are assumed to be the same and for simplicity set to one. For the evolution of the phases we take diffusional effects into account and consider a contribution to the flux that is proportional to the negative gradient of the chemical potential. Precise assumptions are made below. This is motivated e.g. from studies of spinodal decomposition in certain polymer mixtures, cf. [28].

To formulate our model we introduce some notation first. Denote by .t / the unit normal of .t /that points outside˝^C.t /and byV andH the normal velocity and scalar mean curvature of .t /with respect to .t /. ByŒŒwe denote the jump of a quantity across the interface in direction of .t /, i.e.,

ŒŒf .x/D lim

h!0 f .xCh .t // f .x h .t //

for x2 .t /:

c European Mathematical Society 2013

Then our model is described by the following system

@tvCv rv divT .v; p/D0 in˝^˙.t /fort > 0; (1.1) divvD0 in˝^˙.t /fort > 0; (1.2) mD0 in˝^˙.t /fort > 0; (1.3) .t /ŒŒT .v; p/DH .t / on .t /fort > 0; (1.4) V .t /vj .t /D mŒŒ .t / r on .t /fort > 0; (1.5) j .t /DH on .t /fort > 0; (1.6) together with the initial and boundary conditions

vj@˝ D0 on@˝fort > 0; (1.7) ˝mrj@˝ D0 on@˝fort > 0; (1.8)

˝^C.0/D˝₀^C; (1.9)

vjtD0Dv0 in˝; (1.10)

wherev0; ˝₀^C are given initial data satisfying@˝₀^C\@˝ D ;and where; m > 0are a surface tension and a mobility constant, respectively. Here and in the following it is assumed thatvand do not jump across .t /, i.e.,

ŒŒvDŒŒD0 on .t /fort > 0:

Equations (1.1)–(1.2) describe the conservation of linear momentum and mass in both fluids and (1.4) is the balance of forces at the boundary. The equations forvare complemented by the non- slip condition (1.7) at the boundary of˝. The conditions (1.3), (1.8) describe together with (1.5) a continuity equation for the masses of the phases, and (1.6) relates the chemical potentialto the L2-gradient of the surface area, which is given by the mean curvature of the interface.

For m D 0 the velocity field vis independent of . In this case, (1.5) describes the usual kinematic condition that the interface is transported by the flow of the surrounding fluids and (1.1)–

(1.10) reduces to the classical model of a two-phase Navier–Stokes flow as for example studied by Denisova and Solonnikov [10] and K¨ohne et al. [23], where short time existence of strong solutions is shown. On the other hand, ifm > 0, the equations (1.3), (1.6), (1.8) withv D 0 define the Mullins–Sekerka flow of a family of interfaces. This evolution describes the gradient flow for the surface area functional with respect to theH ¹.˝/inner product. Therefore we will also call (1.1)–

(1.10) the Navier–Stokes/Mullins–Sekerka system.

The motivation to consider (1.1)–(1.10) withm > 0is twofold: First of all, the modified system gives a regularization of the classical modelmD0since the transport equation for the evolution of the interface is replaced by a third order parabolic evolution equation (cf. also the effect ofm > 0 in (1.13) below). Secondly, (1.1)–(1.10) appears as sharp interface limit of the following diffuse interface model, introduced by Hohenberg and Halperin [20] and rigorously derived by Gurtin et

al. [19]:

@tvCv rv div.2.c/Dv/C rpD "div.rc˝ rc/ in˝.0;1/; (1.11)

divvD0 in˝.0;1/; (1.12)

@tcCv rcDm in˝.0;1/; (1.13)

D" ¹f⁰.c/ "c in˝.0;1/; (1.14)

vj@˝ D0 on@˝.0;1/; (1.15)

@ncj_@˝ D@nj_@˝ D0 on@˝.0;1/; (1.16)

.v; c/jtD0D.v0; c0/ in˝: (1.17) Herecis the concentration of one of the fluids, where we note that a partial mixing of both fluids is assumed in the model, andf is a suitable “double-well potential”, e.g., f .c/ D c².1 c/². Moreover," > 0is a small parameter related to the interface thickness,is the so-called chemical potential andm > 0 is the mobility. We refer to [2,8] for some analytic results for this model and to [18,22] for results for a non-Newtonian variant of this model. For some results on the sharp interface limit of (1.11)–(1.17) we refer to A. and R¨oger [5, Appendix] and A., Garcke, and Gr¨un [4].

The purpose of this paper is to prove existence of strong solutions of (1.1)–(1.10) locally in time.

Moreover, we will prove stability of spheres, which are equilibria for the systems. (More precisely, we show dynamic stability of the solutionsv0,; p const., and˝^C.t /DBR.x/˝for all t > 0.) Existence of weak solutions for large times and general initial data was shown in [5].

In the following we will assume that˝ Rⁿ,nD2; 3, is a bounded domain withC⁴-boundary and that^˙; m; > 0are constants. One essential feature of (1.1)–(1.10) is the coupling of lower order between the velocity fieldvand the chemical potentialin equation (1.5). Indeed, we will obtain functions in the regularity classes2Lp.JIW_p².˝n .///and

v2H₂¹ JIL2.˝/ⁿ

\L2

JIH₂² ˝n ./n :

Taking the trace to .t /yieldsrj 2Lp.JIWp^{1 1=p}. .//ⁿ/and by complex interpolation and Sobolev embeddings we obtain

v2H₂¹ JIL2.˝/ⁿ

\L2

JIH₂² ˝n ./n ,!Lq

JIW_p¹ ˝n ./n

; whereq > pandp62.nC2/=n. This shows that the trace

vj 2Lq

JIW_p^{1 1=p} ./n

possesses more regularity with respect to time compared to rj . We make essential use of this fact by applying the following strategy for the proof of local-in-time well-posedness. After parameterizing the free interface .t /via the Hanzawa transform by a height functionh, the basic idea is to reduce (1.1)–(1.10) to a single equation forh. To this end we first assume that the interface, henceh, is given. Then we solve the (transformed) two-phase Navier–Stokes equations to obtain a solution operatorvD SNS.h/. Doing the same for the (transformed) two-phase Mullins–Sekerka equations, this yields a solution operator D SM S.h/. Finally, we consider the transformed evolution equation (1.5) for the height functionh and replacevand bySNS.h/andSM S.h/,

respectively, to obtain a single equation forh. This quasilinear parabolic equation in turn can be solved by parabolic theory. The only point one has to take care of is that the solution operatorSNS

is nonlocal in time and space. Therefore one has to deal with a parabolic equation with local leading part and lower order perturbations which are nonlocal (in time and space). Having solved the single equation forh one readily computes the velocity, the pressure and the chemical potential by the solution operators obtained before.

Let us comment on the choice of anL²-setting for the Navier–Stokes part, while the equations for the height functionhand the chemical potential are treated by anL^p-theory,p > 2. One advantage is that the optimal regularity result for the two-phase Navier–Stokes equations with a given interface (see TheoremA.1) is more or less easy to prove since it relies solely on resolvent estimates inL². Another benefit is the reduction of the regularity of the initial velocity and the compatibility conditions att D0. For instance, ifp D 2, then there is no compatibility condition for the initial valuev0coming from the jump of the stress tensor, that is equation (1.4).

The structure of the paper is as follows: First we introduce some basic notation and auxiliary results in Section2. Then we will prove that for a given sufficiently smooth interface .t /the Navier–Stokes part of the system, i.e., (1.1)–(1.2), (1.4), (1.7), (1.10) possesses for sufficiently small times a unique strong solutionvinL²-Sobolev spaces, which are second order in space and first order in time. This result is proved using a coordinate transformation to the initial domains˝₀^˙ which goes back to Hanzawa and applying the contraction mapping principle. A key tool in our analysis will be a maximalL²-regularity result for the linearized Stokes system, which is proved in the appendix. Afterwards in Section4we prove that the full system possesses a strong solution locally in time for sufficiently smooth initial data by reducing the whole system to a single equation for the height functionh(see above). Then in Section5we prove stability of the stationary solutions that are given byv0,; p const:and .t / @Br.x0/˝ and we show that.v.t /; .t //

converges to an equilibrium ast ! 1at an exponential rate.

2. Preliminaries

2.1 Notation and Function Spaces

IfX is a Banach space, r > 0,x 2 X, thenBX.x; r/ denotes the (open) ball in X aroundx with radiusr. We will often write simplyB.x; r/instead ofBX.x; r/ifX is well known from the context.

The usual L^p-Sobolev spaces are denoted by W_p^k.˝/ for k 2 N₀; 1 6 p 6 1, and H^k.˝/ D W₂^k.˝/. MoreoverW_p;0^k .˝/andH₀^k.˝/denote the closure ofC₀¹.˝/inW_p^k.˝/, H^k.˝/, respectively. The vector-valued variants are denoted byW_p^k.˝IX /andH^k.˝IX /, where X is a Banach space. The usual Besov spaces are denoted byB_p;q^s .Rⁿ/,s 2 R,1 6 p; q 6 1, cf., e.g., [7,36]. If˝ Rⁿis a domain, B_p;q^s .˝/is defined by restriction of the elements of B_p;q^s .Rⁿ/to˝, equipped with the quotient norm. We refer to [7,36] for the standard results on interpolation of Besov spaces and Sobolev embeddings. We only note thatB_p;q^s .˝/andW_p^k.˝/

are retracts ofB_p;q^s .Rⁿ/andW_p^k.Rⁿ/, respectively, because of the extension operator constructed in Stein [35, Chapter VI, Section 3.2] for bounded Lipschitz domains. In particular, we have

W_p^k₀.˝/; W_p^kC1₁ .˝/

;pDB_p;p^kC.˝/ if 1

p D 1 p0

C p1

; k2N₀; (2.1)

for all 2 .0; 1/, cf. [36, Section 2.4.2 Theorem 1]. We also denoteW_p^kC.˝/D B_p;p^kC.˝/for k2N₀, 2.0; 1/,16p61. Furthermore, we define

L²_.0/.˝/Dn

f 2L².˝/W Z

˝

f .x/ dxD0o

; L².˝/D˚

f 2C₀¹.˝/ⁿWdivf D0 ^L

2.˝/ⁿ

:

In order to derive some suitable estimates we will use vector-valued Besov spaces B_q;1^s .IIX /, wheres2.0; 1/,16q61,I is an interval, andXis a Banach space. They are defined as

B_q;1^s .IIX /D˚

f 2L^q.IIX /W kfk_B^s_{q;1.IIX /}<1 ; kfk_Bq;1^s _{.IIX /}D kfkL^q.IIX /C sup

0<h61

khf .t /kL^q.IhIX /;

wherehf .t /D f .tCh/ f .t /andIhD ft 2I WtCh 2Ig. Moreover, we setC^s.IIX /D B_1;1^s .IIX /,s2.0; 1/. Now letX0; X1be two Banach spaces. Usingf .t / f .s/DRt

s d

dtf . / d it is easy to show that for16q0< q161

W_q¹₁.IIX1/\L^q0.IIX0/ ,!B_q;1 .IIX/; 1

q D 1 q0

C q1

; (2.2)

where 2.0; 1/andXD.X0; X1/Œ orXD.X0; X1/;r,16r61. Furthermore, B_q;1 .IIX / ,!C ^1q.IIX / for all0 < < 1; 16q61with 1

q > 0; (2.3) cf., e.g., [32]. Furthermore, fors 2 .0; 1/we define H^s.0; TIX / D B_2;2^s .0; TIX /, wheref 2 B_2;2^s .0; TIX /if and only iff 2L².0; TIX /and

kfk²_B^s

2;2.0;TIX / D kfk_L²2.0;TIX /C Z T

0

Z T

0

kf .t / f . /k_X²

jt j^2sC1 dt d <1:

In the following we will use that Z T

0

Z T

0

kf .t / f . /k_X²

jt j^2sC1 dt d 6 Z T

0

Z T

0

jt j^{2.s0 s/ 1}dt d kfk²

C^s0.Œ0;T IX /

6Cs⁰;sT^{2.s0 s/C1}kfk²

C^s0.Œ0;T IX /

for all0 < s < s⁰61, which implies

kfk_H^s.0;TIX /6Cs;s⁰T¹²kfk_Cs0

.Œ0;T IX / for allf 2C^s0.Œ0; T IX / (2.4) provided that0 < s < s⁰61,0 < T 61.

Furthermore, we note that the space of boundedk-times continuously differentiable functions fWU X ! Y with bounded derivatives are denoted byBC^k.UIY /, whereX; Y are Banach spaces and U is an open set. Moreover, f 2 C^k.UIY / if for every x 2 U there is some neighborhoodV ofxsuch thatfjV 2BC^k.VIY /.

We will frequently use the following multiplication result for Besov spaces:

kfgkB_p;max^s

.q1;q2/ 6Cr;s;p;qkfkB_p1;q1^r kgkB_p;q2^s (2.5) for allf 2B_p^r_1;q1.Rⁿ/; g2B_p;q^s ₂.Rⁿ/provided that16p6p161,16q1; q261,r > _pⁿ

1, and

rCn

1

p1 C ¹_p 1

C < s6r;

cf. [21, Theorem 6.6]. SinceW_p^s.Rⁿ/DB_p;p^s .Rⁿ/for everys2.0;1/nN, this implies that kfgk_Wp^s_.Rⁿ_/6Cs;pkfk_Wp^s_.Rⁿ_/kgk_Wp^s_.Rⁿ_/ for allf; g2W_p^s.Rⁿ/ (2.6) provided thats ⁿ_p > 0,16p61. Concerning composition operators, we note that

G.f /2B_p;q^s .Rⁿ/ for allG2C¹.R/withG.0/D0; f 2B_p;q^s .Rⁿ/ (2.7) provided that agains ⁿ_p > 0,1 6 p; q 6 1. This implies thatf ¹ 2 B_p;q^s .˝/for all f 2 B_p;q^s .˝/such thatjfj > c0 > 0if ˝ is a bounded Lipschitz domain. Moreover, the mapping f 7!G.f /is bounded onB_p;q^s .Rⁿ/under the previous conditions. We refer to Runst [29] for an overview, further results, and references. Furthermore, using the boundedness off 7!G.f /one can easily derive that

G./2C¹ B_p;q^s .Rⁿ/ for anyG2C¹.R/withG.0/D0. To this end one uses

G f .x/Ch.x/

DG f .x/

CG⁰ f .x/

C Z 1

0

G⁰⁰ f .x/Ct h.x/

dt h.x/² together with (2.6) and the fact that.G⁰⁰.f Ct h//t2Œ0;1is bounded inB_p;q^s .Rⁿ/.

Finally, by standard methods these results directly carry over toW_p^s.˙ /; B_p;q^s .˙ / if˙ is an n-dimensional smooth compact manifold. Then G.0/ D 0 is no longer required since constant functions are inB_p;q^s .˙ /.

2.2 Coordinate Transformation and Linearized Curvature Operator

In the following let˙ ˝ be a smooth, oriented, compact and.n 1/-dimensional (reference) manifold with normal vector field˙. Moreover, for a given measurable “height function”hW˙ ! Rlet

hW˙ !RⁿWx 7!xCh.x/˙.x/:

Thenhis injective provided thatkhkL¹ 6afor some sufficiently smalla > 0, whereadepends on the maximal curvature of˙. Moreover, we chooseaso small that3a <dist.˙; @˝/. Then the so-calledHanzawa transformationis defined as

h.x; t /DxC d˙.x/=4a

h t; ˘.x/

˙ ˘.x/

; (2.8)

whered˙ is the signed distance function with respect to˙,˘.x/is the orthogonal projection onto

˙,2C¹.R/such that.s/D1forjsj< ¹₃ and.s/D0forjsj> ²₃ as well asj⁰.s/j64for

alls2R, andkhkL¹ < a. It is well-known thath.:; t /W˝ !˝ is aC¹-diffeomorphism. Hence

hWDh.˙ /Dh.˙ /is an oriented, compactC^k-manifold ifh2C^k.˙ /withkhkL¹.˙ /< a.

For the following let UD˚

h2W⁴

4 p

p .˙ /W khkL¹ < a ; (2.9) E_1;T DL^p 0; TIW⁴

1 p p .˙ /

\W_p¹ 0; TIW¹

1 p p .˙ /

; where3 < p6 ^2.nC2/

n ,0 < T <1. Furthermore, let

K.h/WDHhıh; (2.10)

whereHhW h !Rdenotes the mean curvature of h D h.˙ /, i.e., it is the sum of all principal curvatures.

LEMMA2.1 Let3 < p 6 ^2.nC2/

n andUW⁴

4 p

p .˙ /be as above. Then there are functions P 2C¹

U;L W⁴

1 p

p .˙ /; W²

1 p

p .˙ /

; Q2C¹ U; W²

1 p p .˙ / such that

K./DP ./CQ./ for all2U\W⁴

1 p p .˙ /:

Moreover, if˙ DSRWD@BR.0/, then

DK.0/DDWDDSR WD 1 n 1

n 1 R² CSR

: (2.11)

Proof. The proof follows essentially from the proof of [12, Lemma 3.1] and [12, Remark 3.2 a.].

To this end letf.Ul; 'l/ W 1 6 l 6 Lg be a localization system for˙, i.e.,˙ D SL

lD1Ul and 'lW. a; a/^{n 1} !Ul is a smooth local parametrization ofUl for alll D 1; : : : ; L. Moreover, let sD.s1; : : : ; sn 1/be the local coordinates ofUlwith respect to this parametrization and

l.s/WD 'l.s/

; Xl.s; r/WDX 'l.s/; r

; .s; r/2. a; a/ⁿ

be the local representations of; X, whereXW˙ . a; a/ ! Rⁿ withX.s; r/ D sCr˙.s/

and 2 U W⁴

4 p

p .˙ /. Then it follows from [12, Equations (3.4), (3.5), Remark 3.2 a.] that K./DP ./CQ./, whereP ./; Q./have the local representations

Pl./D 1 n 1

0

@

n 1X

j;kD1

pj k./@sj@skC

n 1X

iD1

pi./@si

1

A; Ql./D 1 n 1q./;

where

pj k./D 1 l³

l²w^{j k}./C

n 1X

l;mD1

w^{j l}./w^km./@sl@sm

;

pi./D 1 l³

0

@l²

n 1X

j;kD1

w^{j k} _{j k}ⁱ C

n 1X

j;lD1

w^{j l}w^ki _{j k}ⁿ@slC

n 1X

k;mD1

2w^km _nkⁱ @sm

n 1X

j;k;l;mD1

w^{j l}w^km _{j k}ⁱ @sl@sm 1 A;

q./D 1 l

n 1X

j;kD1

w^{j k}./ _{j k}ⁿ./; l D vu ut1C

n 1X

j;kD1

w^{j k}./@sj@sk;

i j k./D

n 1X

mD1

w^{i m}./@sj@skX@smXj.s;.s//; i¤n;

n

j k./D@sj@skX@snXj.s;.s//; wj k./.s/D@sjX @skXj.s;.s//; and.w^{j k}./.s//_j;kD1^{n 1} is the inverse of.w_{j k}./.s//_j;kD1^{n 1} .

Since˙ is smooth,X and@sjX @sjX are smooth. Thereforewj k./ 2 W⁴

4 p

p .˙ /because of (2.7). Since det..w_{j k}/_j;kD1^{n 1} />c0 > 0by construction, we obtainw^{j k}./ 2W⁴

4 p

p .˙ /for all j; kD1; : : : ; n 1because of (2.7).

Moreover,@sj2W³

4 p

p .˙ /and therefore

n 1X

j;kD1

w^{j k}./@sj@sk2W³

4 p p .˙ /

due to (2.6). Using (2.7) again, we obtainl 2 W³

4 p

p .˙ /. Proceeding this way, we finally obtain thatpj k./; pi./; q./2W³

4 p

p .˙ /for all2U. Now (2.5) implies that kauk

W² p1 p .˙ /

6Cpkak

W³ p4 p .˙ /

kuk

W² p1 p .˙ /

for alla2W³

4 p

p .˙ /; u2W²

1 p

p .˙ /. Hence P 2C¹

U;L W⁴

1 p

p .˙ /; W²

1 p

p .˙ /

; Q2C¹

U;L W²

1 p

p .˙ /

since the operators are compositions ofC¹-mappings. Moreover, (2.11) follows directly from the observations in the proof of [12, Lemma 3.1].

COROLLARY2.2 LetKbe as in (2.10). Then K2C¹

E_1;T \UIH¹⁴ 0; TIL2.˙ /

\L2 0; TIH¹².˙ / : Moreover, for every" > 0; 0 < T0<1there is someC > 0such that

kKkBC¹.E1;T\U_"IH¹⁴.0;TIL2.˙ //\L2.0;TIH¹².˙ ///6C for all0 < T 6T0, whereU_"D fa2UW kakL¹.˙ /6a "g.

Proof. We use that

K.h/D X

j˛j62

a˛.x; h;rsh/@^˛_sh for allh2C².˙ /, wherea˛W˙RR^{n 1}!Ris smooth. Since

E_1;T ,!B

2 3

p;1 0; TIW²

1 p p .˙ /

\B

1 3

p;1 0; TIW³

1 p p .˙ / due to (2.2) and

B

2

p;13 0; TIW²

1 p p .˙ /

,!C¹³ Œ0; T IC⁰.˙ / due to (2.3) andp > 3, we conclude that

a˛.x; h;rsh/2C¹³ Œ0; T IC⁰.˙ /

for allh2E_1;T \U and for allj˛j62. Moreover, the mapping

U\E_1;T 3h7!a˛.x; h;rsh/2C¹³ Œ0; T IC⁰.˙ / isC¹sincea˛are smooth. Furthermore, we conclude that

ka˛.x; h;rsh/@^˛_svk

H¹⁴.0;TIL2.˙ //6C"ka˛.x; h;rsh/@^˛_svk

B 1

p;13 .0;TILp.˙ //

6C"ka˛.x; h;rsh/k

C¹³.Œ0;T IC⁰.˙ //kvk

B 1

p;13 .0;TIW¹ p1 p .˙ //

6C"ka˛.x; h;rsh/k

C¹³.Œ0;T IC⁰.˙ //kvk_E1;T

for allj˛j 6 2,v 2 E_1;T,h 2 E_1;T \U_"," > 0. Since multiplication is smooth (if bounded), it follows that

K2BC¹

E_1;T \U_"IH¹⁴ 0; TIL2.˙ / for any" > 0. Finally, we use thata˛.x; h;rsh/2BUC.Œ0; T IC¹.˙ //and

E_1;T \U_"3h7!a˛.x; h;rsh/2BUC Œ0; T IC¹.˙ / is inC¹with bounded derivative. Hence

a˛.x; h;rsh/r_s^˛h2Lp 0; TIW¹

1 p p .˙ /

,!L2 0; TIH¹².˙ /

for everyh 2 U_" \E_1;T," > 0and the mappingh 7! K.h/is in BC¹ with respect to the corresponding spaces. Altogether we have proved the corollary.

3. Two-Phase Navier–Stokes System for Given Interface

In this section we assume that the family of interfacesf .t /gt >0 is known and we will solve the system (1.1), (1.2), (1.4), (1.7), (1.10) together with the jump conditionŒŒvD0.

For the following let˙ ˝be a smooth compact.n 1/-dimensional reference manifold as in the previous section. Moreover, we assume that there is a domain˝e^C₀ ˝such that˙D@e˝^C₀. Moreover, we assume that

.t /D˚

xCh.t; x/˙.x/Wx2˙ DW h.t /

for someh2U\E_1;T, where

E_1;T WDW_p¹.JIX0/\Lp.J; X1/;

J DŒ0; T , and

X0DW¹

1 p

p .˙ /; X1DW⁴

1 p p .˙ /;

forp >max.^nC3₂ ; 3/D3,nD2; 3, and˙.x/is the exterior normal on@e˝^C₀ D˙. HereUis as in (2.9).

For givenh2E_1;T lethQDEh2eE_1;T, where

EWE_1;T !eE_1;T WDW_p¹ JIW_p¹.˙a/

\Lp JIW_p⁴.˙a/

is a continuous extension operator and˙a D fx 2 ˝ W dist.x; ˙ / < ag. Then by Lion’s trace method of real interpolation, we have

eE_1;T ,!BUC.Œ0; T IXe/; eX DW⁴

3 p

p .˙a/ ,!C².˙a/; (3.1) sincep > ^nC3₂ . Moreover, if we equipE_1;T andeE_1;T with the norms

kuk_E1;T D kuk

Wp¹.JIW¹ p1

p .˙ //\Lp.J;W⁴ 1p p .˙ //

C ku.0/kX; kuke^E1;T D kuk_Wp¹_.JIWp¹_{.˙a//\Lp.J;Wp}⁴_.˙a//C ku.0/ke^X;

then the operator norm of the embedding (3.1) is bounded inT > 0. Additionally, we have e

E_1;T ,!C^{1 p1} Œ0; T IW_p¹.˙a/ : Interpolation with (3.1) implies

eE_1;T ,!C Œ0; T IB^2C

n p p;1 .˙a/

,!C Œ0; T IC².˙a/

for some > 0sincep > ^nC3₂ . Here again all operator norms of the embeddings are bounded in T > 0. We will need the following technical lemma:

LEMMA 3.1 For every" > 0the extension operatorE above can be chosen such that for every 0 < T <1

sup

06t6T

h t; ˘./

Eh.t;/_C1.˙a/6"khk_E1;T:

Proof. First of all, sinceEh.t; x/Dh.t; x/for allx2˙,t 2Œ0; T , sup

06t6T

h t; ˘./

Eh.t;/_C1.˙_a0/6a⁰ sup

06t6T

kEh.t;/k_C²_.˙a0/ 6C a⁰khk_E1;T

for any0 < a⁰6a, whereCis independent of0 < T <1. Hence, if, for given" > 0,a⁰is chosen sufficiently small, we have

sup

06t6T

h t; ˘./

Eh.t;/_C1.˙_a0/6"khk_E1;T: (3.2)

If we now defineE⁰WE_1;T !eE_1;T by .E⁰h/.t; x/D.Eh/

t; ˘.x/C a⁰

ad˙.x/˙ ˘.x/

for allx2˙a; t 2Œ0; T ;

thenE⁰WE_1;T !eE_1;T is an extension operator, which satisfies the statement of the lemma.

For technical reasons, we modify the Hanzawa transformationhto eh.x; t /DxC.d˙.x/=a/h.t; x/Q ˙ ˘.x/

; wherehQDEh2Ee1;T is the extension ofhto˝ as above. Then

eh.:; t / h.:; t /

C¹.˝/6Ch.:; t /Q h ˘.:/; t_C1.˙a/

for all06t 6T, whereC is independent ofhand0 < T <1. If we now choose" > 0in (3.2) sufficiently small,eh.:; t /W˝ !˝ is again aC¹-diffeomorphism for every0 6t 6T. This can be shown by applying the contraction mapping principle to

xD_h¹ h.x/ eh.x/Cy

for giveny 2˝, which is equivalent toeh.x/Dy. Moreover,eh.˙; t /Dh.˙; t /D .t /for all06t 6T.

Now let

Fh;tDeh.:; t /ıeh.:; 0/ ¹:

ThenF_h;tW˝!˝withF_h;t.˝₀^˙/D˝^˙.t /andF_h;t. 0/D .t /, where 0D .0/D@˝^C.0/.

Moreover,FhD.Fh;t/t2Œ0;T 2BUC.Œ0; T IW⁴

3 p

p .˝//\W_p¹.0; TIW_p¹.˝//and

kFh1 Fh2k_C.Œ0;T IC².˝//6Ckh1 h2k_E1;T; (3.3) kFh1 Fh2k_Wp¹_.0;TIWp¹_.˝//6Ckh1 h2k_E1;T; (3.4) for allkhjk_E1;T 6R,j D1; 2, whereC is independent ofhj and0 < T <1. SinceFh;0DId˝

for allh2E_1;T, (3.3) implies

kFh1 Fh2kBUC.Œ0;T IC².˝//6C Tkh1 h2k_E1;T: (3.5)

Now we consider

@tvCv rv ^˙vC r QpD0 in˝^˙.t /; t2.0; T /;

divvD0 in˝^˙.t /; t2.0; T /;

ŒŒvD0 on .t /; t2.0; T /;

ŒŒ .t /T .v;p/Q DH .t / .t / on .t /; t2.0; T /;

vj_@˝ D0 on@˝; t 2.0; T /;

vjtD0Dv0 on˝^˙.t /; t 2.0; T /:

Defining

u.x; t /Dv Ft;h.x/; t

; q.x; t /D Qp Ft;h.x/; t

; the latter system can be transformed to

@tu ^˙uC rqDa^˙.hIDx/.u; q/C@tF_h r_hu u r_hu inQ_T^˙; (3.6) divuDTr

I A.h/

ru

DWg.h/u inQ_T^˙; (3.7)

ŒŒuD0 on 0;T; (3.8)

ŒŒ 0 T .u; q/Dt .hIDx/.u; q/CHeh on 0;T; (3.9)

uj@˝ D0 on@˝T; (3.10)

ujtD0Dv0 on˝₀^˙; (3.11)

whereQ^˙_T D.0; T /˝₀^˙,˝₀ D˝n. 0[˝₀^C/, 0;T D.0; T / 0,@˝T D.0; T /@˝. Here a^˙.hIDx/.u; q/D^˙divh.rhu/ ^˙divruC.r rh/q;

r_hDA.h/r; divhuDTr.r_hu/; A.h/DDF_t;h^T; _hD A.h/ ₀ jA.h/ ₀j; t .h; Dx/.u; q/DŒŒ. ₀ h/.2^˙Du qI /C2hsym.ru rhu/;

HQh.x/DH .t /.Fh;t.x// .t /.Fh;t.x// for allx2 0: In the following letYT DY_T¹Y_T², where

Y_T¹Dn

u2BUC Œ0; T IH¹.˝/ⁿ

\H¹ 0; TIL2;.˝/

Wuj_˝˙

0 2L2 0; TIH².˝₀^˙/ⁿo

; Y_T²Dn

q2L2 0; TIL2;.0/.˝/

W rqj_˝˙

0 2L2 .0; T /˝₀^˙no : The main result of this section is:

THEOREM 3.2 LetR > 0,h0 2 U. Then there is someT0 D T0.R/ > 0 such that for every 0 < T 6T0andh2 E_1;T \UwithhjtD0 D h0andv0 2 H₀¹.˝/ⁿ\L2;.˝/,n D 2; 3, with maxfkhk_E1;T;kv0k_H¹

0.˝/g6Rthere is a unique solution.u; p/DWF_T.h; v0/2YTof (3.6)–(3.11).

Moreover, for every" > 0

F_T 2BC¹.A";RB_H¹

0.0; R/IYT/;

where

A";RDn

h2B_E1;T.0; R/Wh.0/Dh0; sup

06t6T

kh.t /kL¹.˙ /6a "o :

We can formulate (3.6)–(3.11) as an abstract fixed-point equation

LwDG.wIh; v0/ inZT (3.12)

forw2YT, where

L.u; q/D 0 BB

@

@tu ^˙uC rq divu ŒŒ ₀ T^˙.u; q/

ujtD0

1 CC A;

G.u; qIh; v0/D 0 BB

@

a^˙.hIDx/uC@tFh rhu u rhu g.h/u _j˝j¹ R

˝g.h/u dx t .hIDx/.u; q/CHeh

v0

1 CC A

for allwD.u; q/2YT, whereZT DZ_T¹ Z_T² Z_T³ Z_T⁴, Z_T¹ DL2 .0; T /˝0n

; Z_T⁴ DH₀¹.˝/ⁿ\L2;.˝/;

Z_T² DL2 0; TIH_.0/¹ .˝0/

\H¹ 0; TIH_.0/¹.˝0/

; Z_T³ DL2 0; TIH

1 2 2 . 0/ⁿ

\H¹⁴ 0; TIL2. 0/ⁿ

;

andZ_T⁴ DH₀¹.˝/ⁿ\L2;.˝/. HereH_.0/¹ .˝0/DH¹.˝0/\L2;.0/.˝/is normed bykr kL2, H_.0/¹.˝/D.H_.0/¹ .˝//⁰.

First of all, let us note that (3.12) implies (3.6)–(3.11) except that (3.7) is replaced by divuDg.h/u 1

j˝j Z

˝

g.h/ dx:

But the latter equation implies (3.7), which can be seen as follows: LetK.t /D _j˝j¹ R

˝g.h.x; t // dx andv.x; t /Du.F_h¹.x; t /; t /. Thenv.t /2H₀¹.˝/for allt 2.0; T /and therefore

0D Z

˝

divv.x; t / dxD Z

˝

Tr

A h.x; t /

ru.x; t /

detDFh.x; t / dx DK.t /

Z

˝

detDF_h.x; t / dx

for allt 2Œ0; T . Since the last integral is positive, we obtainK.t /D0for allt 2.0; T /.

LEMMA3.3 LetR > 0," > 0, and letYT; ZT,h0be as above. Moreover, let A";RDn

h2E_1;T W sup

06t6T

kh.t /kL¹.˙ /6a "; h.0/Dh0;khk_E1;T 6Ro :

Then there is someT0 > 0such that for every0 < T 6 T0 the mapping G defined above is well-defined and

G 2C¹.BYT.0; R⁰/A";RB_H¹

0\L2;.0; R⁰/IZT/:

Moreover, there are someC; ˛ > 0such that

kG.w1Ih; v0/ G.w2Ih; v0/kZT 6C T^˛kw1 w2kYT

for everyw1; w22BYT.0; R⁰/,0 < T 6T0,h2A";R, andv02B_H¹

0\L2;.0; R/.

Proof. First of all, because of (3.3), for any" > 0there are someC; T0> 0such that kDxF_h idkBUC.Œ0;T IC¹.˝//6"khk_E1;T

for all0 < T 6 T0,khk_E1;T 6 R. HenceFt;hW˝ ! ˝ is aC²-diffeomorphism andDxFh is invertible with uniformly bounded inverse for theseh; T. Since matrix inversion is smooth on the set of invertible matrices,

A.h/DDF_h^T 2C Œ0; T IC¹.˝/

for some > 0ifkhk_E1;T 6R. Moreover, interpolation of (3.3) and (3.4) yields DFh2C^{12 2p1 C2} Œ0; T IC⁰.˝/

due to W_p¹.0; TIX / ,! C^{1 p1}.Œ0; T IX /, where the operator norm of the latter embedding is bounded in0 < T <1ifW_p¹.0; TIX /is normed by

kfk_Wp¹_{.0;TIX /}WD k.f; f⁰/kLp.0;TIX /C kf .0/kX: Here we have also used thatkfk_C1.˝/6Ckfk

1 2

C⁰.˝/kfk

1 2

C².˝/andW_p¹.˝/ ,!C⁰.˝/. Hence A.h/DDF_h^T 2C^{12 2p1 C2}.Œ0; T IC⁰ ˝/

: Furthermore,

A2BC¹.B_E1;T.0; R/IX / with (3.13)

X DC.Œ0; T IC¹ ˝/

\W_p¹ 0; TILp.˝/

\C^{12 2p1 C2}.Œ0; T IC⁰ ˝/

; (3.14)

again since matrix inversion is smooth.

Using the above observations, one easily obtains k.rh r/fk_L

2.0;TIH^k.˝^˙₀//6C Tkhk_E1;Tkfk_L

2.0;TIH^kC1.˝₀^˙//

for allf 2L2.0; TIH^k.˝₀^˙//,kD0; 1,khk_E1;T 6R. From this estimate, one derives ka^˙.hIDx/.u; q/k_L

2..0;T /˝₀^˙/6C Tkhk_E1;Tk.u; q/kYT; kg.h/uk_L

2.0;TIH¹.˝₀^˙//6C Tkhk_E1;Tkuk_L

2.0;TIH².˝₀^˙//; kvDF_hruk_L

2..0;T /˝₀^˙/6C T¹²kvk_L

1.0;TIWp¹.˝^˙₀//kuk_L

1.0;TIWp¹.˝₀^˙//

6C T¹²kvk_Y¹

Tkuk_Y¹

T; k@tFh ruk_L

2..0;T /˝₀^˙/D k.@tFh @tF0/ ruk_L

2..0;T /˝₀^˙/

6C T^{12 p1}khk_E1;Tkuk_L1.0;TIH¹_.˝//

6C T^{12 p1}khk_E1;Tkuk_Y¹

T;

where we have used (3.4) for the last estimate. Moreover, kt .h; Dx/uk_Z2

T

6C T^˛khk_E1;Tk.u; q/kYT (3.15) for some˛ > 0can be proved in the same way as in [1, Proof of Lemma 4.3]. In order to estimate g.h/u2H¹.0; TIH_.0/¹.˝//, we use that

g.h/u; '

˝ D

u;div .I A.h/^T/'

˝ for all'2H_.0/¹ .˝/:

Therefore we obtain for all'2H_.0/¹ .˝/withkr'kL2.˝/D1 d

dt g.h/u; '

˝D

@tu;div .I A.h/^T/'

˝ .ru; .@tA.h/^T/'/˝

hF1.t /; 'i C hF2.t /; 'i;

where ˇˇ ˇˇ

@tu.t /;div

I A h.t /T '

˝

ˇˇ

ˇˇ6Ck@tu.t /kL2.˝/kI A.h/^Tk_L1.0;TIC¹_.˝//

6C Tk@tu.t /kL2.˝/khk_C.Œ0;TIC¹.˝//

and ˇˇ ˇˇ

ru.t /;

@t

I A h.t /^T '

˝

ˇˇ

ˇˇ6Ckuk

L1 0;TIH¹.˝/kk@tE h.t / k_W¹

p.˝/

6Ckuk_Y¹

Tk@tA h.t / k_Wp¹_.˝/

for allt 2.0; T /. Hence kF1k_L2.0;TIH ¹

.0//6C Tk@tukL2.˝.0;T //khk_C.Œ0;T IC¹.˝//

kF2k_L

2.0;TIH_.0/¹/6Ckuk_Y¹

Tk@tA.h.t //k_L

2.0;TIWp¹/6C T^{12 p1}kuk_Y¹

Tkhk_E1;T: and therefore

k@tg.h/uk_L2.0;TIH ¹

.0//6C.R/T^{min.;12 p1/}khk_E1;Tkuk_Y¹

T

for allh2 E_1;T withkhk_E1;T 6 R. Here we have used thatA2BC¹.A";RIX /, whereX is as in (3.14).

Finally, it remains to estimate the termHQh. To this end we use that HQhD

K.h/ı_h0¹ h; whereh0 WD Qh.; 0/j˙W˙ ! 0bijectively. Here

K2BC¹

A";RIH¹⁴ 0; TIL2.˙ /

\L2 0; TIH¹².˙ /

because of Corollary 2.2. Since h0 2 C².˙ /ⁿ is independent oft andh, the same is true for K./ı_h¹

0 with˙replaced by 0. Because of (3.13), we have forH .h/Q WD QH_hfor allh2A";R

HQ 2BC¹

B_E1;T.0; R/IH¹⁴ 0; TIL2. 0/

\L2 0; TIH¹². 0/ :

Altogether, since all terms inGare linear or bilinear in.u; q/andA.h/, these considerations imply thatG2BC¹.BYT.0; R/A";RB_H¹_.˝/ⁿ.0; R/IZT/and

kG.w1Ih; v0/ G.w2Ih; v0/kZT 6C T^˛0kw1 w2kYT (3.16) for allwj D .uj; qj/2 YT withkwjkYT 6 R,h 2 A";R,v0 2 BZ4.0; R/and0 < T 6 T0for some˛⁰> 0.

Proof of Theorem3.2:Let" > 0. Using Lemma3.3and choosingT0> 0sufficiently small, L ¹G.:Ih; v0/WBYT.0; R⁰/!BYT.0; R⁰/

becomes a contraction and is invertible ifh2A";Randkv0k_H¹_.˝/6R, where R⁰D2supnL ¹G.0Ih; v0/

YT Wh2A";R;kv0k_H¹_.˝/6Ro :

Hence for every.h; v0/2A";RBZ4.0; R/there is a uniquew DWF_T.h; v0/2BYT.0; R⁰/such that

wDL ¹G.wIh; v0/:

Moreover, (3.16) implies

L ¹DwG.wIh; v0/_L.Z

T/6C T^˛06 1 2 for allwj D.uj; qj/2YT withkwjkYT 6R,.h; v0/2A";RB_H¹

0\L2;.0; R/, and0 < T 6T0

ifT0is sufficiently small. Hence we can apply the implicit function theorem to F .wIh; v0/Dw L ¹G.wIh; v0/D0

and conclude that

F_T 2BC¹

A";RBZ4.0; R/IBYT.0; R⁰/ sinceDwF .wIh; v0/is invertible for allw2BYT.0; R⁰/,h2A";R; v02B_Z⁴

T.0; R/.

Finally we obtain that the mappingh7!.hu/ı.hjtD0/j˙ satisfies the conditions to apply the general result of [6]:

COROLLARY3.4 LetR; " > 0,T0 D T0.R/ > 0,A";R, andF_T be as in Theorem3.2. For every h2A";R,v02H₀¹.˝/ⁿ\L2;.˝/withkv0k_H¹

0.˝/6Rlet G_T.hIv0/WD.hu/ı.hjtD0/j˙;

where.u; p/ D F_T.h; v0/,0 < T 6 T0. Then there is someq > psuch thatG_T 2 C¹.A";R B_H¹

0\L2;.0; R/ILq.0; TIX0//. Moreover, if h1jŒ0;T⁰ D h2jŒ0;T⁰ for some0 < T⁰ 6 T, then G_T.h1Iv0/jŒ0;T⁰ D G_T.h2Iv0/jŒ0;T⁰, i.e., the mappingh 7! G_T.hIv0/is a Volterra map in the sense of [6].