The spin-coating process : analysis of the free boundary value problem

Universität Konstanz

The spin-coating process

Analysis of the free boundary value problems

Robert Denk Matthias Geissert

Matthias Hieber Jürgen Saal Okihiro Sawada

Konstanzer Schriften in Mathematik

(vormals: Konstanzer Schriften in Mathematik und Informatik)

Nr. 262, Februar 2010 ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-104438

URL: http://kops.ub.uni-konstanz.de/volltexte/2010/10443/

PROBLEM

ROBERT DENK, MATTHIAS GEISSERT, MATTHIAS HIEBER, J ¨URGEN SAAL, OKIHIRO SAWADA

Abstract. In this paper, an accurate model for the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coatong process is being described as a one-phase free boundary value problem for Newtonian fluids in the rotational setting. The method presented is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation will be so-called maximal regularity approach. In order to pursue this one needs to determine the precise regularity classes for the associated inhomogenous linearized equations. This is being achieved by applying the Newton polygon method to the boundary sumbol.

1. Introduction

The spin-coating process may be roughly speaking described as follows: it is a method of placing a small drop of coating material, in liquid form, on the center of a disc, which is then spun rapidly about its axis. The drop is then driven by two competing forces: centrifugal forces cause the liquid to be thrown radially outwards, whereas surface tension forces will work against this spreading. For large centrifugal forces, the coating material film thins.

Of particular interest is the situation where the coating material is a polymer dissolved in a solvent.

As the film thins, the solvent evaporates and the solution viscosity increases, reducing the radial flow.

Eventually, the viscosity becomes so large that relative motion virtually ceases and the process is completed by evaporating the residual solvent.

Spin-coating has many applications. The process is used, for example, in manufacturing micro- electronic devices or magnetic storage discs. In all cases a uniform layer is required and essential.

It has to be stressed that complete mathematical models describing all the above effects do not seem to exist.

In order to develop an accurate model and to investigate it rigorously from an analytical point of view, we describe the spin-coating process as a one-phase free boundary value problem for a Newtonian fluid subject to surface tension and rotational effects.

More precisely, let Γ0 ⊂ Ω(0) be a surface which bounds a region Ω(0) filled with a viscous, in- compressible fluid. Denoting by Γ(t) the position of the boundary at timet, Γ(t) is then the interface seperating the fluid occupying the region Ω(t) and its complement. In the following, the normal on Γ(t) is denoted byν(t,·) and V(t,·) andκ(t,·) denote the normal velocity and mean curvature of Γ(t), re- spectively. Assume that the free surface may be described as the graph of a height functionh. Thus, the region Ω(t) describing the fluid is occupying may be represented as Ω(t) :={(x, y);x∈R², y∈(0, h)}, where h=h(t, x) is the height function. The boundary of Ω(t) splits into the free surface on the top part Γ⁺(t) ={(x, h(t, x)) :x∈R²) and the bottom part Γ⁻(t) ={(x,0) :x∈R²}, the interface of the fluid with a solid phase. In the situation of spin-coating it is natural to consider the case where Γ⁺(0) is close to a plane, i.e. Γ(0) is a graph overR² given by a functionh0. Then the motion of the fluid is

Key words and phrases. spin-coating, Navier-Stokes, free boundary, surface tension, Navier slip, Newton polygon.

1

governed foru= (v, w)^T withv= (u1, u2)^T by the following set of equations:

(1.1)

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ρ(∂tu+ (u· ∇)u) =ν∆u− ∇q−ρ[2ω×u+ω×(ω×(x, y))] in Ω(t)

div u = 0 in Ω(t)

−T ν =σHν on Γ⁺(t)

V =u·ν on Γ⁺(t)

v =ch^α∂3v, on Γ⁻(t)

w = 0 on Γ⁻(t)

u(0) =u0 in Ω(0)

h(0) =h0+δ inR².

The first equation represents the equation for momentum subject to Coriolis and centrifugal forces given by 2ω×uand ω×(ω×(x, y)), respectively. Here ρ, ν and ω denote the density and viscosity of the fluid andω the speed of rotation. The second equation is the condition that the fluid is incompressible.

The third equation says that there is a jump of the stress tensor T =µ(∇u+ (∇u)^T)−qI

in normal direction at the interface Γ which is determined by its mean curvatureκand by the surface tensionσ. Further,V denotes the velocity of the free surface Γ in normal direction. The fifth and sixth equations above describe wetting phenomena at the bottom part Γ⁻(t) of Ω(t). Note that the classical Dirichlet condition holds only for the third componentw of the fluid velocityu. In the case, where a contact line exists and the liquid on a solid substrate spreads and displaces the surrounding fluid, say gas, it is well known that the classical homogeneous Dirichlet condition foruleads to a nonintegrable singularity at the contact line, see [HS71] and [DD74]. This singularities can be relieved by allowing relative motion, i.e. slip, between the liquid and the solid near the contact line. This means that the condition of no penetration is retained and tangential relative motion is allowed. The Navier slip conditionon Γ⁻ demands that the velocity at the interface to be proportional to its normal derivative:

v=k(h)∂3v.

The functionk(·) describes the slip parameter and depends on the heighth. In the fifth equation above we assume thatkis of the formk(h) =ch^α, wherec andαare positive constants.

On the top part, our problem differs from known one- or two-phase flow models through Coriolis andcentrifugal force. Wellposedness results in the non-rotating setting for one-phase flows with surface tension are due to Solonnikov [Sol87] [Sol99], [Sol04] and Shibata and Shimuzu [SS07]. In the setting of spin-coating it is natural to consider infinite layer-like domains. Note that the results cited do not cover this situation. An additional difficulty arising in infinite layers is the localization of the pressure termq. Our approach to circumvent this difficulty is a localization technique for thereduced Stokes system on two half spaces. Estimates for the solution of Laplace’s equation subject to various boundary conditions in Sobolev spaces of negative order will play a key role.

The case of an ocean of infinite extend bounded below by a solid surface and bounded above by a free surface was treated by Beale [Bea84], Tani [Tan96], and Tani and Tanaka [TT95]. The two-phase problem without rotational effects was investigated by Denisova in [Den91] and [Den94], by Tanaka in [Tan95] and by Pr¨uss and Simonett in [PS09].

Wellposedness results for the spin-coating system on the other hand seem not to be known and are the objectives of this paper. Keypoint of our approach are optimal regularity properties of the boundary symbol. They will be achieved by the so-called Newton polygon technique.

Our main result says that the above set of equations admits a unique, local, strong solution (u, p, h) in the space of maximal parabolic regularity provided the intitial data u0 and h0 belong to certain function spaces and are small. The precise regularity assertions will be given in Theorem 2.1 in the following section.

Some comments on notation and function spaces are in order. Lets∈R, 1< p <∞, Ω be a domain in R³ with smooth boundary ∂Ω = Γ and X be a Banach space. Then H_p^s(Ω, X) denotes the Bessel potential space of orders. The Slobodeckij spaceW_p^s(Ω, X) is defined asW_p^s(Ω, X) :=B_pp^s (Ω, X), where

B_pp^s (Ω, X) denotes the corresponding Besov space. Moreover, for T ∈(0,∞) setJ := (0, T). Then the space0W_p^s(J, X) is defined fors≥0 with s−1/p /∈N₀, as

0W_p^s(J, X) :=

({u∈W_p^s(J, X) :u(0) =. . .=u^(k)(0) = 0}, ifs∈(k+¹_p, k+ 1 +_p¹) fork∈N₀, W_p^s(J, X), if 0≤s < ¹_p.

The spaces0Hp^s(J, X) are defined analogously. The homogeneous versions of the above spaces will be denoted byHb_p^s(Ω, X) andcW_p^s(Ω, X). Moreover, we set⁰Hb_p¹(Ω,Γ) :={ϕ∈Hb_p¹(Ω) :γϕ= 0 on Γ}and

0Hb_p⁻¹(Ω,Γ) :=₀

Hb_p^1′(Ω,Γ)′

.

Here γ denotes the trace operator u7→u|^Γ. The trace operator γ depends of course on Ω,Γ and the smoothness of the underlying space. However, in order to simplify our notation, we always denote the trace operator byγ whenever no misunderstanding may occur.

For more information about the Navier-Stokes equations in fixed domains, we refer e.g. to [Gal94], [Ama00] and [FKS05] and in the rotational setting e.g. to [CT07], [GHH06] and [HS09].

2. Main Result

In this section, we prove that the free boundary value problem describing the spin-coating process, i.e.

on the free surface, the Navier-Stokes equations with surface tension in the rotational setting and the Navier-Stokes equations with Navier’s condition on the fixed bottom boundary are locally well posed.

Theorem 2.1. Let p >5. Then there existε >0 and T >0 such that for allu0∈Wp^2−2/p(Ω(0)) and allh0∈Wp^3−2/p(R²) withdivu0= 0 onΩ(0) and satisfying

k(u0, h0)k_Wp^2−2/p_(Ω(0))×Wp^3−2/p₍R²)< ε,

there exists a unique solution (u, q, h)of equation (1.1) within the regularity classes u ∈ H_p¹(J, Lp(Ω(t))³)∩Lp(J, H_p²(Ω(t))³),

q ∈ {p∈Lp(J,Hb_p¹(Ω(t))) :γq∈W_p^1/2−1/2p(J, Lp(Γ⁺(t)))∩Lp(J, W_p^1−1/p(Γ⁺(t)))}, h ∈ W_p^2−1/2p(J, Lp(Γ⁺(t)))∩H_p¹(J, W_p^2−1/p(Γ⁺(t)))∩Lp(J, W_p^3−1/p(Γ⁺(t))).

Note that due to our asumptionp >5 we have

h∈C(J, BU C²(Γ(t))) and∂th∈C(J, BU C¹(Γ(t))).

This implies that the normal of Ω(t), the normal velocity V of Γ(t) and its mean curvature are well defined and continuous. In particular, the equations on the free boundaries given in (1.1) can be understood pointwise. Foruwe have

u(t)∈BU C¹(Ω(t)) and ∇u(t)∈BU C(Ω(t)), t∈J.

Some comments of our approach how to prove the above result are in order: first, by applying the Hanzawa transform to problem (1.1), we obtain a set of equations on a fixed layer-like domainD :=

R²×(0, δ) of fixed height δ with top and bottom boundaries Γ⁺ =R²× {δ} and Γ⁻ =R²× {0}. In section 3 we will verify that the set of equations forρ=ν= 1 on the fixed domainD are of the form

(2.1)

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∂tu−∆u+∇q =χRΛ×(Λx)−2ω×u+F(u, q, h) in J×D,

divu =Fd(u, h) in J×D,

−γT(u, q)νD−σ∆xhνD =G+(u, q, h) onJ×Γ⁺,

∂th−γw =H(u, h) onJ×Γ⁺,

γv−γcδ^α∂yv =G₋(u, h) onJ×Γ⁻,

γw = 0 onJ×Γ⁻,

u|^t=0 =u0 in D,

h|^t=0 =h0 in R².

for certain functionsF, F2, G+, G₋ and H. Splitting the normal stress into its tangential and normal component, the linearization of (2.1) leads to the following linear inhomogeneous problem

(2.2)

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∂tu−∆u+∇q =f1 in J×D, divu =fd in J×D, γ∂yv+γ∇^xw =gv onJ×Γ⁺, γ2∂yw−γq−σ∆xh =gw onJ×Γ⁺,

∂th−γw =fh onJ×Γ⁺, γv−γcδ^α∂yv =g₋ onJ×Γ⁻, γw = 0 onJ×Γ⁻, u|^t=0 =u0 in D, h|^t=0 =h0 onR²

Note that in the first line above the Coriolis term 2ω×u is neglected. It will be included lateron, however, in the definition ofF1.

Secondly, we show maximalLp-regularity for the linearized problem (2.2) in Section 4. To this end, we split the original problem into two model problems defined on half spaces, use the equivalence of the Stokes problem and the reduced Stokes problem explained in Appendix B as well as the Newton polygon technique explained in Appendix C. Finally, a fixed-point argument yields the existence of a unique solution (u, p, h) to equation (1.1) belonging to the regularity class described in Theorem 2.1.

A second comment about the regularity class seems also to be in order: assume that the linear problem (2.2) admits a solution (u, p, h) satisfying

u∈H_p¹(J, Lp(D)³)∩Lp(J, H_p²(D)³), q∈Lp(J,Hb_p¹(D)), then the right hand sidesf1 andfd need to satisfyf1∈Lp(J×Ω) and

fd∈H_p¹(J,0Hb_p⁻¹(D))∩Lp(J, H_p¹(D)),

since the operator div maps Lp into H_p⁻¹. By trace theory, u0 necessarily belongs to Wp^2−2/p(D).

Moreover, the trace ofubelongs to the class

Y0:=W_p^1/2−1/2p(J, Lp(Γ⁺))∩Lp(J, W_p^2−1/p(Γ⁺)), and that of∇uto

Y1:=W_p^1/2−1/2p(J, Lp(Γ⁺)³)∩Lp(J, W_p^1−1/p(Γ⁺)³).

Thusgv∈Y1and if in additionq∈Wp^1/2−1/2p(J, Lp(Γ⁺))∩Lp(J, Wp^1−1/p(Γ⁺)), then alsogw∈Y1. The equation forhis defined onY0; hencehshould naturally belong to

W_p^2−1/2p(J, Lp(Γ⁺))∩H_p¹(J, W_p^2−1/p(Γ⁺)).

The fourth equation above is defined in Y1 and contains the term ∆x. Thus hshould also belong to Lp(J, Wp^3−1/p(Γ⁺)) and the natural space forhis

W_p^2−1/2p(J, Lp(Γ⁺))∩H_p¹(J, W_p^2−1/p(Γ⁺))∩Lp(J, W_p^3−1/p(Γ⁺)).

This also impliesh0∈Wp^3−2/p(R²).

Having thus observed that the above regularity for h is necessary for u and q solving (2.2) and belonging to the above regularity classes, our main result shows that under these assumptions the nonlinear problem admits a unique solution in the above regularity classes.

3. Hanzawa transformation

In this section we transform the problem (1.1) to a problem on the fixed domainD =R²×(0, δ) for someδ > 0. The top and bottom boundary of D are given by Γ⁺ = R²× {δ} and Γ⁻ = R²× {0}, respectively. To this end, we define forJ := (0, T)

Θ :J×R²×(0, δ)→ [

t∈J

{t} ×Ω(t), Θ(t, x, y) := (t, x,yh(t, x)

δ )

as well asθ(t, x, y) := (x, yh(t, x)/δ). Thus Θ(t, x, y) = (t, θ(t, x, y)) for allt∈J,x∈R²andy∈(0, δ).

We then define the transformed variables by

(Θ^∗u)(t, x, y) :=u(Θ(t, x, y)), v(t, x, y) :=

(Θ^∗u)1(t, x, y) (Θ^∗u)2(t, x, y)

w(t, x, y) := (Θ^∗u3)(t, x, y) :=u3(Θ(t, x, y)), π(t, x, y) := (Θ^∗q)(t, x, y) :=q(Θ(t, x, y)).

Then, the Jacobian of Θ and its inverse are of the form

DΘ =



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1 0 0 0

0 1 0 0

0 0 1 0

y∂th/δ y∂1h/δ y∂2h/δ h/δ





 andDΘ⁻¹=



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1 0 0 0

0 1 0 0

0 0 1 0

−^yδh²∂th −h^yδ2∂1h −^yδh²∂2h δ/h





.

By means of this coordinate transformation we obtain the assertions:

Θ^∗∂tu=∂tv−y

h(∂yv)∂th, Θ^∗∂ju=∂jv−y

h(∂yu)∂jh, j = 1,2, Θ^∗∂_j²u=∂_j²v−2y

h(∂j∂yv)∂jh+ y²

h²(∂_y²v)(∂jh)²−y(∂yv) h∂_j²h−2(∂jh)² h²

!

, j= 1,2, Θ^∗∂yu= δ

h∂yv, Θ^∗∂_y²u= δ²

h²∂_y²v, Θ^∗∆u= [∆x+δ²

h²∂_y²]v−2y

h∇^x∂yv∇^xh+y²

h²|∇^xh|²∂_y²v−y

h(∂yv)∆h+ 2 y

h²(∂yv)|∇^xh|², Θ^∗(u· ∇)u= ((Θ^∗u)·(∇^x, δ

h)∂y)(Θ^∗u)−y

h(∂y(Θ^∗u))(v· ∇^x)h.

Θ^∗∇q= (∇^x,δ

h∂y)π−y

h(∂yπ)(∇^x,0)^T.

The fourth equation of (1.1) is transformed via the outer normalν given by

ν:= 1

p1 +|∇^xh|²(−∂1h,−∂2h,1)^T into

∂th= Θ^∗Vν

p1 +|∇^xh|²= Θ^∗ν·wp

1 +|∇^xh|²=−∇^xh·v+w.

In order to compute the transformed stress tensor on Γ⁺, we note first that the outer normalν at the free surface and the outer normalνD= (0,0,1)^T at Γ⁺ are related through

ν = 1

p1 +|∇^xh|²(I+K)^TνD, withK:=



 0 0 0

0 0 0

−∂1h −∂2h 0



.

Employing this representation, we compute the transformed stress tensor on the upper boundary to be equal to

Θ^∗T(u, q) =∇(Θ^∗u)Dθ⁻¹+Dθ^−T∇(Θ^∗u)^T−Iq.

WritingDθ⁻¹ asDθ⁻¹=I_δ/h(I+K) withI_δ/h:=





1 0 0

0 1 0

0 0 δ/h



,we obtain

Θ^∗T(u, q)ν = 1 p1 +|∇^xh|²

∇(Θ^∗u)(Iδ/h(I+K)) + (Iδ/h(I+K))^T∇(Θ^∗u)^T−Iπ

(I+K)^TνD

= 1

p1 +|∇^xh|²

(∇^x,δ

h∂3)(Θ^∗u) + ((∇^x, δ

h∂3)(Θ^∗u))^T −Iπ

νD

+ 1

p1 +|∇^xh|²

− ∇^x(Θ^∗u)∇^xh+δ

h|∇^xh|²∂3(Θ^∗u)−((∇^x,δ

h∂3)v)^T∇^xh

−δ

h∂3w(∇^xh,0)^T+ δ

h(∇^xh, ∂3v)(∇^xh,0)^T +π(∇^xh,0)^T . The mean curvatureκis given by

κ=−∇^x· ∇^xh p1 +|∇^xh|²

!

=− ∆xh p1 +|∇^xh|²+

X2 j,k=1

∂jh∂kh

(1 +|∇^xh|²)³²∂j∂kh.

The transformed Navier-slip condition on the lower boundary reads as v= Θ^∗w^′ =ch^αΘ^∗∂yu=ch^αδ

h∂y(Θ^∗u) =cδh^α−1∂y(Θ^∗u).

Summarizing, the equation (1.1) reads in transformed coordinates as

(3.1)

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∂tu−∆u+∇π =χRω×(ω×(x, y)) +F1(u, π, h) inJ×D,

divu =Fd(u, h) inJ×D,

γT(u, π)νD−σ∆xhνD =G+(u, π, h) onJ×Γ⁺,

∂th−γw =H(u, h) onJ×Γ⁺,

γv−γcδ^α∂yv =G₋(u, h) onJ×Γ⁻,

γw = 0 onJ×Γ⁻,

u|^t=0 =u0 inD,

h|^t=0 =h0+δ inR²,

where the functions on the right hand side above are given by F1(u, p, h) := y

h(∂yu)∂th+ (δ²/h²−1)∂_y²u−2y

h∇^x∂yu∇^xh+y²

h²|∇^xh|²∂_y²u− y

h(∂yu)∆xh + 2 y

h²(∂yu)|∇^xh|²+y

h(∂yπ)(∇^xh,0)^T−(u·(∇^x,δ

h∂y))u+y

h(∂yu)v· ∇^xh + (1− δ

h)(0,0, ∂yπ)^T −2ω×u Fd(u, h) := (1− δ

h)∂yu+y

h∂yv· ∇^xh G⁺(u, p, h) := (1−δ/h)∂y(v,2w) +σ( 1

p1 +|∇^xh|² −1)∆xh·νD−σ X2 j,k=1

∂jh∂kh

(1 +|∇^xh|²)^3/2∂j∂kh·νD

− σ

p1 +|∇^xh|²(∆xh− X2 j,k=1

∂jh∂kh

1 +|∇^xh|²∂j∂kh)(∇^xh,0)^T −[−∇^xu∇^xh+δ

h|∇^xh|²∂yu

−((∇^x,δ

h∂y)v)^T∇^xh−δ

h∂yw(∇^xh,0)^T+ δ

h(∇^xh, ∂yv)(∇^xh,0)^T+π(∇^xh,0)^T] H(u, h) :=−∇^xh·v

G⁻(u, h) :=cδ(h^α−1−δ^α−1)∂yu.

4. Maximal regularity for the linearized problem

It is the aim of this section to prove maximal regularity estimates for the linearized problem (2.2). To this end, we introduce the function spaceFassociated with the right hand side of (2.2) as

F(J, D) :=F₁(J, D)×F_d(J, D,Γ+)×G₊(J,Γ⁺)×H(J,Γ⁺)×G₋(J,Γ⁻)×I₁(D)×I₂(Γ⁺), where

F₁(J, D) := Lp(J, Lp(D)³),

F_d(J, D,Γ⁺) := H_p¹(J,0Hb_p⁻¹(D,Γ⁺))∩H_p^1/2(J, Lp(D))∩Lp(J, H_p¹(D)), G₊(J,Γ⁺) := W_p^1/2−1/2p(J, Lp(Γ⁺)³)∩Lp(J, W_p^1−1/p(Γ⁺)³),

H(J,Γ⁺) := W_p^1−1/2p(J, Lp(Γ⁺))∩Lp(J, W_p^2−1/p(Γ⁺)), G₋(J,Γ⁻) := Wp^1/2−1/2p(J, Lp((Γ⁻)²)∩Lp(J, Wp^1−1/p(Γ⁻)²),

I₁(D) := W_p^2−2/p(D)³, I₂(Γ⁺) := W_p^3−2/p(Γ⁺), as well as the solution space

E(J, D) :=E₁(J, D)×E₂(J, D,Γ⁺)×E₃(J,Γ⁺) with

E₁(J, D) := H_p¹(J, Lp(D)³)∩Lp(J, H_p²(D)³),

E₂(J, D,Γ⁺) := {π∈Lp(J,Hb_p¹(D)) :π∈W_p^1/2−1/2p(J, Lp(Γ⁺))∩Lp(J, W_p^1−1/p(Γ⁺))}, E₃(J,Γ⁺) := W_p^2−1/2p(J, Lp(Γ⁺))∩H_p¹(J, W_p^2−1/p(Γ⁺))∩Lp(J, W_p^3−1/p(Γ⁺)).

Theorem 4.1. Let T >0,J := (0, T),p∈(1,∞) withp6= 3/2,3. Then there exists a unique solution (u, π, h)∈E(J, D) of the problem

(4.1)

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∂tu−∆u+∇π = f1 inJ×D, divu = fd inJ×D, γ∂yv+γ∇^xw = gv onJ×Γ⁺, γ∂yw−γp−σ∆xh = gw onJ×Γ⁺,

∂th−γw = fh onJ×Γ⁺, γv−cδ^αγ∂yv = g₋ onJ×Γ⁻, γw = 0 onJ×Γ⁻, u(0) = u0 inD, h(0) = h0 onΓ⁺

if and only if(f1, fd, g= (gv, gw), fh, g₋, u0, h0)belong toF(J, D)and satisfy the compatibility conditions gv(0) =γ∂yv0+γ∇^xw0 onΓ⁺ and g₋(0) =γv0−cδ^αγ∂yv0 on Γ⁻

in the casep >3 and

γw0= 0 on Γ⁻ and fd(0) = divu0

in the casep >3/2.

Note that we omittedδin the initial value because (u, p, h+δ) solves the linearized equations with the initial valueh0+δif and only if (u, p, h) solves these equations with the initial valueh0.

In order to prove Theorem 4.1, we consider first the problem

(4.2)

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∂tu−∆u+∇π = f1 in J×D, divu = fd in J×D, γ∂yv+γ∇^xw = 0 onJ×Γ⁺, 2γ∂yw−γπ−γσ∆xh = 0 onJ×Γ⁺,

∂th−γw = 0 onJ×Γ⁺, γv−cδ^αγ∂yv = 0 onJ×Γ⁻, γw = 0 onJ×Γ⁻, u|^t=0 = 0 in D, h|^t=0 = 0 on Γ⁺ Then the following result holds.

Lemma 4.2. Let p ∈ (1,∞), p 6= 3/2,3 and let f1 ∈ F₁(J, D) and fd ∈ F_d(J, D,Γ⁺) satisfying fd|^t=0= 0 if p >3/2. Then there exist Jτ := [0, τ] for some τ >0 and a solution(u, π, h)∈E(Jτ, D) of (4.2) inJτ satisfying

k(u, π, h)k^E(Jτ,D)≤C kf1k^F1(Jτ,D)+kfdk^Fd(Jτ,D,Γ⁺)

.

In order to prove Lemma 4.2 it suffices, thanks to Proposition B.1, to consider the reduced Stokes problem defined by

(4.3)

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∂tu−∆u+∇T2(u, h) = f1 inJ ×D, γ∂yv+γ∇^xw = 0 onJ×Γ⁺,

γdivu = gr inJ ×Γ⁺,

∂th−γw = 0 onJ×Γ⁺, γv−cδ^αγ∂yv = 0 onJ×Γ⁻, γw = 0 onJ×Γ⁻, u|^t=0 = 0 inD, h|^t=0 = 0 onR², wheref1∈F₁(J, D),gr∈G⁺(J,Γ⁺) satisfyinggr(0) = 0 ifp >3. Here

G⁺(J,Γ⁺) :=W_p^1/2−1/2p(J, Lp(Γ⁺))∩Lp(J, W_p^1−1/p(Γ⁺)), andT2(u, h) is defined byT2(u, h) :=p2, wherep2∈H_p¹(D) is the unique solution of

(4.4)

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∆p= 0 inD,

γ∂yp=γν(∆u− ∇divu) on Γ⁻, γp= 2γ∂yw−σ∆xh on Γ⁺,

which is guaranteed by Proposition A.5 provided (u, h)∈E₁×E₃. Hereγνu:=γν·u.

In order to construct a solution for equation (4.3), we employ a localization procedure to transfer the reduced Stokes problem to two problems in an half-space. More precisely, consider inD⁻:=R³₊the set of equations

(4.5)

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∂tu−∆u+∇T₂⁻(u) = f₋ inJ×D⁻, γv−cδ^αγ∂yv = 0 onJ×Γ⁻, γw = 0 onJ×Γ⁻, u(0) = 0 inD⁻,

whereT₂⁻(u) is defined byT₂⁻(u) :=p⁻₂, wherep⁻₂ is the unique solution of the equation ∆p = 0 in D⁻,

γ∂yp =γ∂ν(∆u− ∇divu) on Γ⁻. (4.6)

Note that by Proposition A.1 and Propostion A.4,T₂⁻ is well-defined due to the assumption onu.

The following lemma is a consequence of Theorem 5.1 and Proposition B.2(2).

Lemma 4.3. Let p∈(1,∞),p6= 3/2,3. Then there existsC >0 such that for f₋ ∈F₁(J, D⁻) there exists a unique solutionu₋∈E₁(J, D⁻)of equation (4.5)satisfying

ku₋k^E1(J,D⁻)≤Ckf₋k^F1(J,D⁻).

Similary as above, we consider the reduced system also inD⁺:=R²×(−∞, δ). It reads as

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∂tu−∆u+∇T₂⁺(u, h) = f+ inJ×D⁺, γ∂yv+γ∇^xw = 0 on J×Γ⁺,

γdivu = gr on J×Γ⁺,

∂th−γw = 0 inJ×Γ⁺, u(0) = 0 inD⁺, h(0) = 0 in Γ⁺, (4.7)

whereT₂⁺(u, h) is defined asT₂⁺(u, h) :=p⁺₂, wherep⁺₂ is the unique solution of ∆p = 0 in D⁺,

γ∂yp = 2γ∂yw−σ∆xh on Γ⁺. (4.8)

By Proposition A.1 and Propostion A.4,T₂⁺ is well-defined. Finally, Theorem 5.2 and Proposition B.1 imply the following result.

Lemma 4.4. Let p ∈ (1,∞), p 6= 3/2,3. Then there exists a constant C > 0 such that for f+ ∈ F₁(J, D⁺) and gr ∈ G⁺(J,Γ⁺) satisfying g(0) = 0 in the case p > 3, there exists a unique solution (u+, h)∈E₁(J, D⁺)×E₃(J,Γ⁺)of equation (4.7)satisfying

ku+k^E1(J,D⁺)+khk^E3(J,Γ⁺)≤C kf+k^F1(J,D⁺)+kgrk^G+(J,Γ+) .

Proof of Lemma 4.2. Letχ₋ ∈C^∞(R) be a cut-off function satisfying 0≤χ₋≤1 such that χ₋(y) =

1 for y≤δ/3, 0 for y≥2δ/3,

holds. Moreover, setχ+ := 1−χ₋. It follows from Lemma 4.4 and Lemma 4.3 that for f ∈F₁(J, D) andg∈G⁺(J,Γ⁺) there exist unique solutions (u+, h) of equation (4.7) andu₋ of (4.5), respectively.

Insertingu=χ+u++χ₋u₋ in (4.3), we obtain

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∂tu−∆u+∇T2(u, h) = f+S(f, g) inJ×D, γ∂yv+γ∇^xw = 0 onJ×Γ⁺,

∂th−γw = 0 in J×Γ⁺, γdivu = g onJ×Γ⁺, γv−cδ^αγ∂yv = 0 onJ×Γ⁻,

γw = 0 onJ×Γ⁻,

u(0) = 0 in D,

h(0) = 0 in R²,

whereS(f, g) is given by

S(f, g) :=∇T2(u, h)−χ+∇T₂⁺(u+, h)−χ₋∇T₂⁻(u₋)−(∆χ+)u+−(∆χ₋)u₋

−2(∇χ₋)(∇u₋)−2(∇χ+)(∇u+)

=

∇ T2(u, h)−χ+T₂⁺(u+, h)−χ₋T₂⁻(u₋) +

−(∆χ+)u+−(∆χ₋)u₋−2(∇χ₋)(∇u₋)−2(∇χ+)(∇u+) +

(∇χ+)T₂⁺(u+, h) + (∇χ₋)T₂⁻(u₋)

=:S1(f, g) +S2(f, g) +S3(f, g).

By H¨older’s inequality and Sobolev’s embedding theorem, for T0 > 0 there exist β > 0 and C > 0, independent ofT < T0, such that

kS2(f, g)k^F1(0,T,D)≤C

ku₋kLp(0,T;H_p¹(D))+ku+kLp(0,T;H_p¹(D))

≤CT^β ku+k^E1(0,T,D)+ku₋k^E1(0,T,D)

. Hence, by Propositon 4.4 and 4.3, we obtain

kS2(f, g)k^F1(0,T,D)≤CT^β kfk^F1(0,T,D)+kgkG⁺(0,T,Γ⁺))

.

In order to estimateS1(f, g), note thatϕdefined by ϕ:=T2(u, h)−χ+T₂⁺(u+, h)−χ₋T₂⁻(u₋) solves the equation

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∆ϕ = −div∇(χ+T2(u+, h) +χ₋T₂⁻(u₋)) in D,

γ∂yϕ = 0 on Γ⁻,

γϕ = 0 on Γ⁺.

It follows from Proposition A.5 that

kϕk^Lp(0,T;H_p¹(D))≤Ck −div∇(χ+T2(u+, h) +χ₋T₂⁻(u₋))kLp(0,T;0Hp⁻¹(D,Γ⁺)). Forϕ∈⁰H_p¹(D,Γ⁺) we estimate the above right hand side further as

|hdiv∇(χ+T2(u+, h) +χ₋T2(u₋)), ϕi|

=|h(∆χ+)T2(u+, h) + 2(∇χ+)(∇T2(u+, h)), ϕi+h(∆χ₋)T2(u₋) + 2(∇χ₋)(∇T2(u₋), ϕi|

≤C

kT₂⁺(u+, h)k^Lp(0,T;H¹_p(R²×(δ/3,2δ/3)))+kT₂⁻(u₋)k^Lp(0,T;H_p¹(R²×(δ/3,2δ/3)))

kϕk^Lp(D).

Consider the solutions of (4.6) and (4.8). Their Fourier transform are given by ˆ

p⁻(ξ, y) = (i ξ

|ξ|∂yvˆ₋(ξ,0) +kξkwˆ₋(ξ,0))e^−|ξ|y, ˆ

p⁺(ξ, y) =

2∂ywˆ+(ξ, δ) +σ|ξ|²ˆh(ξ)

e^{−|ξ|(δ−y)} 1 kξk.

By Mikhlin’s multiplier theorem there exists a constantC >0, independent ofu+,u₋ andhsuch that kT₂⁻(u₋)kLp(0,T;H_p¹(R²×(1/3,2/3)))≤C kγ∂yu₋kLp(0,T;Lp(R²))+kγu₋kLp(0,T;Lp(R²))

, kT₂⁺(u+, h)k^Lp(0,T;H_p¹(R²×(1/3,2/3)))≤C kγ∂yu+kLp(0,T;Lp(∂D⁺))+k∆xhk^Lp(0,T;Lp(R²))

. Hence,

kwk^Lp(0,T;H_p¹(D))≤C kγ∂yu+k^Lp(0,T;Lp(∂D⁺))+kγu₋k^Lp(0,T;Lp(R²))

+kγ∂yu₋k^Lp(0,T;Lp(R²))+k∆xhk^Lp(0,T;Lp(R²))

. By similar arguments as above we obtain

kS1(f, g)k^F1(0,T;D) ≤ CT^β kfk^F1(0,T,D)+kgkG⁺(0,T,Γ⁺)

, kS3(f, g)k^F1(0,T,D) ≤ CT^β kfk^F1(0,T,D)+kgkG⁺(0,T,Γ⁺). . Summarizing, it follows that

kS(f, g)k^F1(0,T,D)≤ 1

2kfk^F1(0,T,D)+kgk^G+(0,T,Γ+))

providedT is small enough. The assertion thus follows by a Neumann series argument.

Proof of Theorem 4.1. Suppose that the dataf1, fd, gv, gw, fh, g₋, u0, h0satisfies the assumptions given in Theorem 4.1. In a first step we show that the existence of a triple (˜u,π,˜ ˜h)∈E(J, D) satisfying the last seven equations of (4.1). Keeping the notation introduced in the proof of Lemma 4.2, we define further cut-off functionsχ1, χ2∈C^∞(R) satisfying

χ1(y) =

(1 for y≤²3δ,

0 for y≥⁵6δ, and χ2(y) =

(1 for y≥ ¹3δ, 0 for y≤ ¹6δ.

By [DPZ08, Theorem 2.1] (see also [DPZ08, Example 3.7]), there exists (w+,˜h)∈E₁×E₃satisfying

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∂tw+−∆w+ = 0 in J×D⁺,

∂t˜h−w+ = fh onJ×Γ⁺, 2γ∂yw+−σ∆x˜h = e^∆x·(γ∂yw0−σ∆xh˜0) onJ×Γ⁺,

w+(0) = χ1w0 in D⁺,

˜h(0) = h0 on Γ⁺.

Heree^∆x· refers to the time variablet. We further set π+(t, x, y) = e^{−√−∆x(δ−y)}

gw−e^∆x·(γ∂yw0−σ∆xh˜0) (t, x).

Then (w+, π+,˜h) is a solution of the equation

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∂t˜h−w+ = fh on J×Γ⁺, 2γ∂yw+−γπ+−σ∆xh˜ = gw on J×Γ⁺,

w+(0) = χ1w0 in D⁺,

˜h(0) = h0 on Γ⁺. Furthermore, denote byv+ the solution of the equation

∂tv+−∆v+ = 0 in J×D⁺, γ∂yv+ = gv−γ∇^xw+ on J×Γ⁺,

v+(0) = χ1v0 in D⁺.

Hence, by construction the triple (u+= (v+, w+), π+,h) satisfies the equation˜

∂t˜h−w+ = fh on J×Γ⁺, γ∂yv++γ∇^xw+ = gv on J×Γ⁺, 2γ∂yw+−γπ⁺−σ∆x˜h = gw on J×Γ⁺,

u+(0) = χ1u0 in D⁺,

˜h(0) = ˜h0 on Γ⁺. Furthermore, by [DHP07, Theorem 2.1] there exists a functionu₋ satisfying

∂tu₋−∆u₋ = 0 in J×D⁻, γv₋−cδ^αγ∂yv₋ = g₋ on J×Γ⁻, γw₋ = 0 on J×Γ⁻, u₋(0) = χ2u0 in D⁻,

providedγw0= 0 on Γ⁻ ifp >3/2 andγv0−cδ^αγ∂yv0=g₋(0) on Γ⁻ ifp >3.

Setting ˜u:=χ+u++χ₋u₋ and ˜p:=χ+p+ and chossing ˜has above, we see that the triple (˜u,p,˜ ˜h)∈ E(J, D⁺) satisfies the last seven equations of (4.1). Moreover, givenf1, fdas in Theorem 4.1, it follows from Lemma 4.2 that there exists a solution of (4.2) in some small time intervall Jτ = (0, τ). By repeating these arguments, we obtain a solution of (4.1) on an arbitrary time intervallJ.

Finally, the uniqueness of the solution to (4.1) follows by the following duality argument. To this end, assume that (u, h, p) satisfies

(4.9)

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∂tu−∆u+∇π = 0 inJ×D, divu = 0 inJ×D, γ∂yv+γ∇^xw = 0 onJ×Γ⁺, 2γ∂yw−γπ−σ∆xh = 0 onJ×Γ⁺,

∂th−γw = 0 onJ×Γ⁺, γv−cδ^αγ∂yv = 0 onJ×Γ⁻, γw = 0 onJ×Γ⁻, u(0) = 0 inD, h(0) = 0 on Γ⁺ and for ˜f ∈F₁^′ let (˜u,˜h,p) be the solution of the˜ dual backward problem

(4.10)

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−∂tu˜−∆˜u+∇π˜ = f˜ in J×D, div ˜u = 0 in J×D, γ∂y˜v+γ∇^xw˜ = 0 onJ×Γ⁺, 2γ∂yw˜−γ˜π−σ∆x˜h = 0 onJ×Γ⁺,

−∂t˜h−γw˜ = 0 onJ×Γ⁺, γ˜v−cδ^αγ∂yv˜ = 0 onJ×Γ⁻, γw˜ = 0 onJ×Γ⁻,

˜

u(T) = 0 in D,

˜h(T) = 0 on Γ⁺ Integration by parts yields

0 =h∂tu−∆u+∇π,u˜i^J×D=hu,−∂tu˜−∆˜ui^J×D+hu(s),u(s)˜ i^D|^T0 − hγ∂yu, γu˜iJ×Γ⁺

− hγu, γ˜ui^J×Γ⁻+hγu, γ∂yu˜i^J×Γ⁺+hγu, γ∂yu˜i^J×Γ⁻+hγπ, γw˜i^J×Γ⁺+hγπ, γw˜i^J×Γ⁻

= :I1+. . .+I8.

Sinceu(0) = ˜u(T) = 0, we haveI2= 0. The two equations on Γ⁻ implyI4+I6+I8 = 0. Moreover, the equations describing the tangential stresses on Γ⁺, integration by parts, and the divergence free conditions yield

I5+I3=hγv, γ∂yv˜i^J×Γ⁺− hγ∂yv, γv˜i^J×Γ⁺+hγw, γ∂yw˜i^J×Γ⁺− hγ∂yw, γw˜i^J×Γ⁺

=hγv,−γ∇^xw˜i^J×Γ⁺+hγ∇^xw, γv˜i^J×Γ⁺+hγw, γ∂yw˜i^J×Γ⁺− hγ∂yw, γw˜i^J×Γ⁺

=hγ∇^xv, γw˜iJ×Γ⁺− hγw, γ∇^x˜viJ×Γ⁺+hγw, γ∂yw˜iJ×Γ⁺− hγ∂yw, γw˜iJ×Γ⁺

=− hγ∂yw, γw˜i^J×Γ⁺+hγw, γ∂yw˜i^J×Γ⁺+hγw, γ∂yw˜i^J×Γ⁺− hγ∂yw, γw˜i^J×Γ⁺

=hγw,2γ∂yw˜i^J×Γ⁺− h2γ∂yw, γw˜i^J×Γ⁺.

The equations for the normal stress on Γ⁺, the equations for the normal velocity of Γ⁺ as well as integration by parts yield

I5+I3=hγw, γ˜π+σ∆^′˜hiJ×Γ⁺− hγπ+σ∆^′h, γw˜iJ×Γ⁺

=hγw, γ˜πiJ×Γ⁺− hγπ, γw˜iJ×Γ⁺+h∂th, σ∆x˜hiJ×Γ⁺+hσ∆xh, ∂th˜iJ×Γ⁺

=hγw, γ˜πiJ×Γ⁺− hγπ, γw˜iJ×Γ⁺− h∂t∇^xh, σ∇^x˜hiJ×Γ⁺+hσ∇^x∂th,∇˜hiJ×Γ⁺

=hγw, γ˜πiJ×Γ⁺− hγπ, γw˜iJ×Γ⁺. Summarizing, we obtain

0 =hu,−∂tu˜−∆˜ui^J×D+hγw, γπ˜iJ×Γ⁺ =hu,−∂tu˜−∆˜u+∇π˜i^J×D=hu,f˜i^J×D, f˜∈F₁^′, which implies thatu≡0. Now, it is not difficult to show thath= 0 and π= 0.