数理解析研究所講究録
第 1905 巻 2014 年 73-89

73

Mathematical analysis of spin-coat model:
maximal regularity theory and method of
Newton polygon
Okihiro Sawada*

Abstract

An accurate model for the spin-coating process is presented and investigated from view point of the mathematical analysis. The method
is based on the Hanzawa transform for the free boundary value problem to quasilinear evolution equations on a fixed layer domain. For
solving, the maximal regularity approach is used. This is achieved byapplying the Newton polygon method to the boundary symbol with
Weis’ Fourier multiplier theory.

1

Introduction

This note is a brief survey of the results related to [DGHSSII], mainly.
Problem We will analyze the spin-coating process, mathematically.
coating has many applications, for example, in manufacturing micro-electronic
devices or magnetic storage. Various models describing certain aspects havebeen developed in the engineering sciences as well as from physical or chemical point of views. The spin-coating process basically has the following threestages:$Spin-\backslash$1. Deposition of the coating fluid onto the substrate.
2. Acceleration of the substrate up to its final.

3. Spinning of the substrate at a constant speed.
*Applied Physics Course, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan$E$-mail address: okihiro@gifu-u.ac.jp

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In what follows, we zoom into the detail of stage 3, in particular, thefluid viscous forces dominate the thinning. We consider the Navier-Stokes
equations (one-phase flow) in an infinite layer-like domain. Also, we shall
impose the free boundary on the top, and Navier’s partially-slip boundaryon the bottom. The equations are as$(SCP)$$\{\begin{array}{l}\rho(U_{t}+(U, \nabla)U)-\mu\triangle U+\nabla P=-\rho[2\Lambda\cross U+\chi_{R}\Lambda\cross(\Lambda\cross x)] in \cup\{t\}\cross\Omega(t) ,t\in(0,T)\nabla\cdot U=0 in \cup\{t\}\cross\Omega(t) ,t\in(0,T)-\mathbb{T}\nu=\sigma\kappa\nu, V=U\cdot\nu on t\in(0,T)\cup\{t\}\cross\Gamma_{+}(t) ,U’=c(\delta+h)^{\alpha}\partial_{3}U’, U^{3}=0 on (0, T)\cross\Gamma_{-},U|_{t=0}=U_{0} (\nabla\cdot U_{0}=0) in \Omega(0) ,h|_{t=0}=h_{0} (|h_{0}|<\delta/2) in \mathbb{R}^{2}.\end{array}$Notations of diﬀerential are asfor$i=1$ , 2, 3,,,. For vector fields$(or, (a, b)$ sometimes) by the scalar product inand we denoteor . The vector field $U=(U’, U^{3})=(U^{1}(t, x), U^{2}(t, x), U^{3}(t, x))$ denotesby the unknown velocity of the liquid at time $t\in(0, T)$ for some $T>0$
and location $x\in\Omega(t)$ ; the scalar fields $P=P(t, x)$ and $h=h(t, x’)$ standfor the unknown pressure and the unknown amplitude from the average of, respectively. The region of the fluid at time is represented
the region$:=\{x=(x’, x_{3});x’\in \mathbb{R}^{2},$ $x_{3}\in(0,$ $\delta+h(t,$bywhich is an infinite
layer-like domain in. The boundary of region is splited into two parts,$:=\Gamma_{-}:=\{(x’, 0);x’\in \mathbb{R}^{2}\}$, and thethe bottom is denoted by$:=\{(x’, h(t, x x’\in \mathbb{R}^{2})$
top surface stands for} at each time . On
the top surface we have used the outer normal and its tangential vector
those are defined by explicitly (we will give them in below); the curvatureon the top surface will also be done as well. The stress tensor is denoted by$:=\mathbb{T}(U, P)$$:=\nabla U+(\nabla U)^{T}$ is the deformation, wheretensor and is theidentity matrix;
stands for the transposed matrix$M.$of
We put the following as positive constants: is the density coeﬃcient,is the viscosity, is the surface-tension, is the average of height; andare some positive constants due to Navier’s partially-slip boundary condition;
we usually takein experimentation, associated with the study of anonintegrable singularity at the contact line in [DD74, HS71]. Also, is the$U_{t}$$\nabla$$:=(\partial_{1}, \partial_{2}, \partial_{3})$$:=(\partial_{1}, \partial_{2})$$\triangle$$\partial_{t}$$:=\partial/\partial t,$$\partial_{i}$$:=\partial/\partial x_{i}$$:= \sum_{i=1}^{3}\partial_{i}^{2}$$a\cdot b$

$b$

$a$

$\nabla’$$:=\partial_{t}U,$$\mathbb{R}^{2}$$\mathbb{R}^{3}$$\delta$$t$$\Omega(t)$$\Omega(t)$$x$$\mathbb{R}^{3}$$:=\mathbb{R}^{2}$$\Gamma_{-}(t)$$t$$\Gamma_{+}(t)$$v$$\tau,$$h$$\kappa$$\mathbb{T}$$:=\mathbb{S}(U)+\mathbb{I}P$$3\cross 3$$\mathbb{I}$$\mathbb{S}(U)$$M^{T}$$\mu$$\rho$$\delta$$\sigma$$c$$\alpha\sim 5/3$

$V$$\alpha$75$:=(0,0,\Lambda_{0})$ , where
velocity vector field of the top surface,is theangular velocity of rotation,inis a smooth cut-oﬀ function as
andinfor some $R>0.$
We usethe exterior product in. So, $2\Lambda\cross U=2\Lambda_{0}(-U^{2}, U^{1},0)$stands for the Coriolis term,
is an unbounded function
as the term of centrifugal force. In order investigate the fluid dynamics
around the origin in terms of -theory, we use the cut-oﬀ. The initial$U_{0}=U_{0}(x)$$h_{0}=h_{0}(x’)$
data(and then the initial region) are given
andas some functions enjoying a certain regularity. We assume that is relatively
large (and initial amplitude of the top surface
is relatively small) so that$|h_{0}|<\delta/2$ , that is to say, the top surface does not touch to the bottom at leastin some short time-interval. We have imposed the compatibility condition:. Up to the situation, we will add further compatibility conditionson$\Lambda$$\Lambda_{0}\in \mathbb{R}$$\chi_{R}\equiv 1$$\chi_{R}$$\chi_{R}\equiv 0$$B_{R}$$B_{R+1}^{c}$$\mathbb{R}^{3}$$\cross as$$\Lambda\cross(\Lambda\cross x)=-\Lambda_{0}^{2}(x’, 0)$$L_{p}$$\chi_{R}$$\Omega(O)$$\delta$$h_{0}$$\nabla\cdot U_{0}=0$$U_{0}.$

Known results We now list-up several known results related to (SCP).
Mathematical analysis of the Navier-Stokes equations in a layer-like domain
with nonhomogeneous Dirichlet boundary condition was studied by AbeShibata [AS03], Abels-Wiegner [AW05] and Abels $[Abels05a, Abels05b]$ . The
proof is based on the semigroup theory due to the Helmholtz decompostion
by Simader-Sohr [SS92] and the resolvent estimate by Farwig-Sohr [FS94]; seealso Farwig [Farwig03]. In the case of more general situation, the reader may
find the recent development in [Abe04, AY10, Abels06, AbelslO, Kagei08,
Kagei12, Shibata13] and the references therein.
For more information about mathematical theory on the Navier-Stokes
equations in fixed domains in various setting, we refer to e.g., [Amann00,
FKS05, Galdi94, GHHS12]. Also, in the rotational setting in the whole space
the reader may find recent works by active researcher in e.g., [BMNOI, CT07,
GHH06, HS10, IT13].
A model problem for the free boundary in a bounded domain (onephase flow, meaning, the out-side is adapted) with a surface tension has
been observed by Solonnikov [Solonnikov87, Solonnikov99, Solonnikov04],
and Shibata-Shimuzu [SS07]. Note that the surface tension plays a role forgetting the smoothing eﬀect. Without surface tension, even the time-local
solvability is rather tough problem. Hataya obtained the precise analysis on
it in [Hataya09]; see also Hosono-Kawashima [HK06].
The two-phase problem is a hot issue. The case of an ocean with infinite depth and bounded above by a free surface was treated by Beale[Beale84]. After Beale’s work, its improvement in several direction have been
done by, for instance, Beale-Nishida [BN85], Allain [Allain85, Allain87], Tani
[Tani96] and Tani-Tanaka [TT95]. The two-phase problem of the Navier-

76
Stokes equations was investigated by Denisova [Denisova91, Denisova94],
Sylvester [Sylvester90, Sylvester96], Tanaka [Tanaka93, Tanaka95] and Pr\"ussSimonett [PS09, PS10]. So far, many researcher continue to study related
topics; see [AR09, LST12, Maekawa07, SSII] and the reference therein.
Main theorem We now state the main results in this note, that is, the timelocal existence and uniqueness of strong solutions to (SCP) in the suitablesetting of initial data and a rotation speed. The definition of function spaces
will be given in Section 3.
$R,$
Theorem 1.1 ([DGHSSII]). Let $p>5$ , and let
be positive constants. Then there exist $\epsilon>0$ and $T>0$ such that forwith $\nabla\cdot U_{0}=0$ on
all $U_{0}\in W_{p}^{2-2/p}(\Omega(O))$ and all
satisfying$\rho,$$\mu,$$\delta,$$\sigma,$$\Lambda_{0}$$c,$$\alpha,$$h_{0}\in W_{p}^{3-2/p}(\mathbb{R}^{2})$$\Omega(O)$$\Vert U_{0}\Vert_{W_{p}^{2-2/p}(\Omega(0))}+\Vert h_{0}\Vert_{W_{p}^{3-2/p}(\mathbb{R}^{2})}<\epsilon,$

there$exi_{\mathcal{S}}ts$

a unique solution $(U, P, h)$ to (SCP) within the regularity classes$U\in H_{p}^{1}(0, T;L_{p}(\Omega(t))^{3})\cap L_{p}(0, T;H_{p}^{2}(\Omega(t))^{3})$,$P\in\{P\in L_{p}(\hat{H}_{p}^{1}(\Omega(t))):P\in W_{p}^{1/2-1/2p}(L_{p}(\Gamma_{+}(t)))\cap L_{p}(W_{p}^{1-1/p}(\Gamma_{+}(t)))\},$$h\in W_{p}^{2-1/2p}(L_{p}(\mathbb{R}^{2}))\cap H_{p}^{1}(W_{p}^{2-1/p}(\mathbb{R}^{2}))\cap L_{p}(W_{p}^{3-1/p}(\mathbb{R}^{2}))$.

Due to our assumption $p>5$ , by Sobolev embedding we have$h\in C(O, T;BUC^{2}(\mathbb{R}^{2}))$and$\partial_{t}h\in C(0, T; BUC^{} (\mathbb{R}^{2}))$.

is enough, that
See e.g., [Amann95]. This implies that the regularity ofis to say, the normal velocityofand its mean curvature are welldefined and continuous. In particular, the equations in the region and on thefree boundaries can be understood in the classical sense, point-wisely. For, we also note that$\Omega(t)$$V$$\Gamma_{+}(t)$$\kappa$$U$$U(t)\in BUC^{1}(\Omega(t))$

and$\nabla U(t)\in BUC(\Omega(t))$,$t\in(O, T)$.

The uniqueness of solutions obtained by Theorem 1.1 comes from thedual backward equations; this technique was developed by Lions-Masmoudi[LMOI]. We deal with the nonlinear terms regarded as perturbation from thesolutions to the linearized equations, as in Beale-Nishida [BN85].
We next state the existence and uniqueness results of another type; thestrong solution exists in an arbitrary time-interval if the initial data and therotation speed are suﬃciently small.

77
$R,$
Theorem 1.2 ([DGHSSII]). Let $p>5,$ $T>0$ , and let
be positive constants. Then there exists
such that for all
and allonwithsatisfying.
$\rho,$$\epsilon>0$$\alpha$$c,$$W_{p}^{2-2/p}(\Omega(O))$$h_{0}\in W_{p}^{3-2/p}(\mathbb{R}^{2})$$\mu,$$\delta,$$\sigma,$$U_{0}\in$$\nabla\cdot U_{0}=0$$\Omega(O)$$\Vert U_{0}\Vert_{W_{p}^{2-2/p}(\Omega(0))}+\Vert h_{0}\Vert_{W_{p}^{3-2/p}(\mathbb{R}^{2})}+|\Lambda_{0}|<\epsilon,$

there exists a unique solution $(U, P, h)$ to (SCP) in $(0, T)$ within the same
regularity classes in Theorem 1.1.

2

Hanzawa transformIn this section we shall derive the linearized equations of (SCP). Hanzawa[Hanzawa81] introduced the transformation for fixing the region to solve the
Stefan problem. So, the transform treated here is called Hanzawa transform’,
we use this terminology, throughout this paper.
We shall change the free boundary problem (SCP) into a problem
in the fixed domain. The top and bottom boundaries ofare given by, respectively. To this end,
and
we define$\Omega(t)$$D$$D:=\mathbb{R}^{2}\cross(0, \delta)$$\Gamma_{+}:=\mathbb{R}^{2}\cross\{\delta\}$$\Theta$:$\Gamma_{-}:=\mathbb{R}^{2}\cross\{0\}$$(0, T) \cross \mathbb{R}^{2}\cross(0, \delta)arrow\bigcup_{t\in(0,T)}\{t\}\cross\Omega(t))$

as well as$\theta(t, x’, y)$

all $t\in(O, T)$ ,$:=(x’, yh(t, x’)/\delta)$$x’\in \mathbb{R}^{2}$$\Theta(t, x’, y)$$:=(t, x’, \frac{yh(t,x’)}{\delta})$. Hence, $\Theta(t, x’, y)=(t, \theta(t, x’, y))$ for

and $y\in(0, \delta)$ . We define the transformed variables by$u’(t, x’, y):=[(\Theta^{*}U^{2})(t,x’,y)(\Theta^{*}U^{1})(t,x’,y)]:=U’(\Theta(t, x’, y$$u^{3}(t, x’, y):=(\Theta^{*}U^{3})(t, x’, y):=U^{3}(\Theta(t, x’, y$$q(t, x’, y):=(\Theta^{*}P)(t, x’, y):=P(\Theta(t, x’, y$At $t=0$ , we also modify the initial velocity as$u_{0}(x’, y) :=(\theta^{*}U_{0})(x’, y) :=U_{0}(\theta(0, x’, y$Then,$D\Theta$(the Jacobian of$\Theta$) is of the form

$D\Theta=(\begin{array}{llll}1 0 0 00 1 0 00 0 1 0y\partial_{t}h/\delta y\partial_{1}h/\delta y\partial_{2}h/\delta h/\delta\end{array})$

78
We can easily compute its inverse. By means of this coordinate transform,
we assert for $j=1$ , 2$\Theta^{*}\partial_{t}U=\partial_{t}u-\frac{y}{h}(\partial_{y}u)\partial_{t}h,$$\Theta^{*}\partial_{j}U=\partial_{j}u-\frac{y}{h}(\partial_{y}u)\partial_{j}h,$$\Theta^{*}\partial_{j}^{2}U=\partial_{j}^{2}u-2\frac{y}{h}(\partial_{j}\partial_{y}u)\partial_{j}h+\frac{y^{2}}{h^{2}}(\partial_{y}^{2}u)(\partial_{j}h)^{2}-y(\partial_{y}u)\frac{h\partial_{j}^{2}h-2(\partial_{j}h)^{2}}{h^{2}},$$\Theta^{*}\partial_{y}U=\frac{\delta}{h}\partial_{y}u,$$\Theta^{*}\partial_{y}^{2}U=\frac{\delta^{2}}{h^{2}}\partial_{y}^{2}u,$$\Theta^{*}\Delta U=[\triangle^{J}+\frac{\delta^{2}}{h^{2}}\partial_{y}^{2}]u-2\frac{y}{h}(\nabla’h, \nabla’)\partial_{y}u+\frac{y^{2}}{h^{2}}|\nabla’h|^{2}\partial_{y}^{2}u$$- \frac{y}{h}(\partial_{y}u)\triangle’h+2\frac{y}{h^{2}}(\partial_{y}u)|\nabla’h|^{2},$$\Theta^{*}(U\cdot\nabla)U=((\Theta^{*}u)\cdot(\nabla’, \frac{\delta}{h})\partial_{y})(\Theta^{*}u)-\frac{y}{h}(\partial_{y}(\Theta^{*}u))(u’\cdot\nabla’)h,$$\Theta^{*}\nabla P=(\nabla’, \frac{\delta}{h}\partial_{y})q-\frac{y}{h}(\partial_{y}q)(\nabla’, 0)h.$Here
as well astransformed via the outer normal$\nabla=(\nabla’, \partial_{y})$$\triangle=\Delta’+\partial_{y}^{2}$

$\nu$

of

. The fourth equation of (SCP) is$\Gamma_{+}(t)$given by$\nu:=\frac{1}{\sqrt{1+|\nabla’h|^{2}}}(-\partial_{1}h, -\partial_{2}h, 1)$

into$\partial_{t}h=\Theta^{*}V_{\nu}\sqrt{1+|\nabla’h|^{2}}=\Theta^{*}.\nu\cdot u^{3}\sqrt{1+|\nabla’h|^{2}}=-(u’\cdot\nabla’)h+u^{3}$

In order to compute the transformed stress tensor on , we note first that theand the outer normal $v_{D}=(0,0,1)$
outer normal on the free surface
on $\Gamma+are$ related through$\Gamma_{+}$$\nu$$\nu=\frac{1}{\sqrt{1+|\nabla’ h|^{2}}}(\mathbb{I}+K)\nu_{D}$$\Gamma_{+}(t)$with$K:=(\begin{array}{lll}0 0 00 0 0-\partial_{1}h -\partial_{2}h 0\end{array})$

Employing this representation, we compute the transformed stress tensor on
the top boundary to be equal to$\Theta^{*}\mathbb{T}(U, P)=\nabla(\Theta^{*}u)D\theta^{-1}+D\theta^{-T}\nabla(\Theta^{*}u)-\mathbb{I}q.$

79

Writing$D\theta^{-1}$as$D\theta^{-1}=I_{\frac{\delta}{h}}(\mathbb{I}+K)$

with$:=(\begin{array}{lll}1 0 00 1 00 0 \delta/h\end{array})$$I_{\frac{\delta}{h}}$, we obtain$\Theta^{*}\mathbb{T}(U, P)v$$= \frac{1}{\sqrt{1+|\nabla’h|^{2}}}[\nabla(\Theta^{*}u)(I_{\frac{\delta}{h}}(\mathbb{I}+K))+(I_{\frac{\delta}{h}}(\mathbb{I}+K))^{T}\nabla(\Theta^{*}u)-\mathbb{I}q](\mathbb{I}+K)^{T}\nu_{D}$$= \frac{1}{\sqrt{1+|\nabla’h|^{2}}}[(\nabla’, \frac{\delta}{h}\partial_{y})(\Theta^{*}u)+((\nabla’, \frac{\delta}{h}\partial_{y})(\Theta^{*}u))-\mathbb{I}q]\nu_{D}$$+ \frac{1}{\sqrt{1+|\nabla’h|^{2}}}[-\nabla’(\Theta^{*}u)\nabla’h+\frac{\delta}{h}|\nabla’h|^{2}\partial_{y}(\Theta^{*}u)-((\nabla’, \frac{\delta}{h}\partial_{y})u’)(\nabla’h, 0)$-$\frac{\delta}{h}\partial_{y}u^{3}(\nabla’h, 0)+\frac{\delta}{h}(\nabla’h, \partial_{y}u’)(\nabla’h, 0)+q(\nabla’h, 0)].$

Analogously, the mean curvature$\kappa=-\nabla’$$\kappa$of$\Gamma_{+}(t)$is given by$( \frac{\nabla’h}{\sqrt{1+|\nabla’ h|^{2}}})=-\frac{\triangle’h}{\sqrt{1+|\nabla’ h|^{2}}}+\sum_{j,k=1}^{2}\frac{\partial_{j}h\partial_{k}h}{(1+|\nabla h|^{2})^{\frac{3}{2}}}\partial_{j}\partial_{k}h.$

The transformed Navier’s partially slip condition on the bottom reads as$u’= \Theta^{*}U’=c(\delta+h)^{\alpha}\Theta^{*}\partial_{y}U’=c(\delta+h)^{\alpha}\frac{\delta}{h}\partial_{y}(\Theta^{*}u)=c\delta(\delta+h)^{\alpha-1}\partial_{y}(\Theta^{*}u)$.

Summarizing, the transformed equations (TE) are$\{\begin{array}{ll}u_{t}-\triangle u+\nabla q=\chi_{R}\Lambda\cross(\Lambda\cross(x’, y))+F_{1}(u, q, h) in (0, T)\cross D,\nabla\cdot u=F_{d}(u, h) in (0, T)\cross D,\mathbb{T}(u, q)\nu_{D}-\sigma\triangle’h\nu_{D}=G_{+}(u, q, h) on (0, T)\cross r_{+},\partial_{t}h-u^{3}=H(u, h) on (0, T)\cross\Gamma+,u’-c\delta^{\alpha}\partial_{y}u’=G_{-}(u, h) on (0, T)\cross\Gamma_{-},u^{3}=0 on (0, T)\cross\Gamma_{-},u|_{t=0}=u_{0} in D,h|_{t=0}=h_{0} in \mathbb{R}^{2}.\end{array}$

For the sake of simplicity, we take $\rho=\mu=1$ . Here, the new functions on
the right hand side above are given by$F_{1}:=-(u, \nabla)u+\frac{y\partial_{t}h}{\delta+h}\partial_{y}u+(\frac{\delta^{2}}{(\delta+h)^{2}}-1)\partial_{y}^{2}u-\frac{2y}{\delta+h}(\nabla’h,\nabla’)\partial_{y}u$$+ \frac{y^{2}}{(\delta+h)^{2}}|\nabla’h|^{2}\partial_{y}^{2}u-\frac{y\triangle’h}{\delta+h}\partial_{y}u+\frac{2y}{(\delta+h)^{2}}|\nabla’h|^{2}\partial_{y}u+\frac{y\partial_{y}q}{\delta+h}(\nabla’h, 0)$

80$+ \frac{hu^{3}}{\delta+h}\partial_{y}u+\frac{y(u’,\nabla’)h}{\delta+h}\partial_{y}u+\frac{h}{\delta+h}(0,0, \partial_{y}q)$,$F_{d}:= \frac{h}{\delta+h}\partial_{y}u^{3}+\frac{y}{\delta+h}(\partial_{y}u’, \nabla’)h,$$G_{+}:=(1- \frac{\delta}{h})\partial_{y}(u’, 2u^{3})+\sigma(\frac{1}{\sqrt{1+|\nabla’h|^{2}}}-1)\triangle’h\cdot\nu_{D}$$- \sum_{j,k=1}^{2}\frac{\sigma\partial_{j}h\partial_{k}h\partial_{j}\partial_{k}h}{(1+|\nabla’ h|^{2})^{\frac{3}{2}}}\nu_{D}-\frac{\sigma}{\sqrt{1+|\nabla’h|^{2}}}(\triangle’h-\sum_{j,k=1}^{2}\frac{\partial_{j}h\partial_{k}h\partial_{j}\partial_{k}h}{1+|\nabla h|^{2}})(\nabla’h, 0)$$-[-( \nabla’h, \nabla’)u+\frac{\delta}{h}|\nabla’h|^{2}\partial_{y}u-((\frac{\delta}{h}\partial_{y})u’, \nabla’)\nabla’h-\frac{\delta}{h}(\partial_{y}u^{3})(\nabla’h, 0)$$+ \frac{\delta}{h}(\partial_{y}u’, \nabla’)h(\nabla’h, 0)+q(\nabla’h, 0)],$$H:=-(u’, \nabla’)h,$$G_{-}:=c\delta(h^{\alpha-1}-\delta^{\alpha-1})\partial_{y}u.$

Obviously, the toughest part in our analysis is to deal with the fourth equation (where $H$ appears in the right hand side), because that is a quasilinear
equation of hyperbolic type. In what follows, we rather discuss the transformed equations (TE) than (SCP).

3

Maximal regularity estimates$1<p<\infty,$
We first give the definition of function spaces. Letwith regular boundary
fora domain in
be an open interval,is the Lebesgue, and let $X$ be a Banach space. As usual,
the Bessel potential
space with integral exponent in . We denote byspace in of diﬀerential order and integral exponent as well as $H_{p}^{s}(J;X)$
is the space of $X$ -valued Bessel potential functions in . The Slobodeckij
denotes thespace $W_{p}^{s}(J;X)$ is defined as $W_{p}^{s}(J;X)$ $:=B_{p,p}^{s}(J;X)$ , where
corresponding Besov space. Moreover, for $T\in(0, \infty)$ , we sometimes omit$:=L_{p}(0, T;H_{p}^{s}(\Omega))$ ,the notation of the time-interval $J=(O, T)$ as
if no confusion occurs likely.
at $t=0$ defined
Let $oW_{p}^{s}(0, T;X)$ be the zero-trace subspace ofas
for $s\geq 0$ with
$J\subset \mathbb{R}$$s\in \mathbb{R},$$\mathbb{R}^{n}$$\Omega$$\Gamma$$:=\partial\Omega$$n\in \mathbb{N}$$L_{p}(\Omega)$$\Omega$$p$$\Omega$$s$$H_{p}^{S}(\Omega)$

$p$

$J$

$B_{p,p}^{s}$$L_{p}(H_{p}^{s}(\Omega))$$W_{p}^{s}(X)$$s-1/p\not\in \mathbb{N}_{0}:=\mathbb{N}U\{O\}$$0^{W_{p}^{s}(0,T;X):=}\{\begin{array}{l}\{u\in W_{p}^{s}(0, T;X) : u(O)=\ldots=u^{(k)}(0)=0\},if s\in(k+\frac{1}{p}, k+1+\frac{1}{p}) for k\in \mathbb{N}_{0},W_{p}^{s}(0, T;X) , if 0\leq s<\frac{1}{p}.\end{array}$

81
The spaces $0^{H_{p}^{s}(0,T;X)}$ are defined, analogously. The homogeneous versionof the above spaces will be denoted by. We
andfurthermore set
and$\hat{H}_{p}^{s}(0, T;X)$$0\hat{H}_{p}^{1}(0, T;\Gamma)$$:=\{\varphi\in\hat{H}_{p}^{1}(\Omega) :$\hat{W}_{p}^{s}(0, T;X)$\gamma\varphi=0 on \Gamma\}$$0^{\hat{H}_{p}^{-1}(0,T;\Gamma)} :=(^{0}\hat{H}_{p}^{1},(0, T;\Gamma))’$Here,denotes the trace operator$p’=p/(p-1)$ is the H\"older
conjugate exponent of , and $X’$ means the topological dual space of $X.$
It is the aim of this section to prove maximal regularity estimates for the
linearized problem of (TE).
To this end, we introduce the function space associated with the righthand side of (TE) as$u\mapsto u|_{\Gamma},$$\gamma$$p$$\mathbb{F}$$\mathbb{F}:=\mathbb{F}_{1}\cross \mathbb{F}_{d}\cross \mathbb{G}_{+}\cross \mathbb{H}\cross \mathbb{G}_{-}\cross \mathbb{I}_{u}\cross \mathbb{I}_{h}$with$\mathbb{F}_{1} :=\mathbb{F}_{1}(0, T;D) :=L_{p}(0, T;L_{p}(D)^{3})$,$\mathbb{F}_{d} :=\mathbb{F}_{d}(0, T;D) :=H_{p}^{1}(_{0}\hat{H}_{p}^{-1}(D))\cap H_{p}^{1/2}(L_{p}(D))\cap L_{p}(H_{p}^{1}(D))$$\mathbb{G}_{+}:=\mathbb{G}_{+}(0, T;\Gamma^{+}):=W_{p}^{1/2-1/2p}(L_{p}(\Gamma^{+})^{3})\cap L_{p}(W_{p}^{1-1/p}(\Gamma^{+})^{3})$$\mathbb{H}:=\mathbb{H}(0, T;\mathbb{R}^{2}) :=W_{p}^{1-1/2p}(L_{p}(\mathbb{R}^{2}))\cap L_{p}(W_{p}^{2-1/p}(\mathbb{R}^{2}))$

,

,

,

$\mathbb{G}_{-}:=\mathbb{G}_{-}(0, T;\Gamma_{-}):=W_{p}^{1/2-1/2p}(L_{p}(\Gamma_{-})^{2})\cap L_{p}(W_{p}^{1-1/p}(\Gamma_{-})^{2})$,$\mathbb{I}_{u}:=\mathbb{I}_{u}(D):=W_{p}^{2-2/p}(D)^{3},$$\mathbb{I}_{h}:=\mathbb{I}_{h}(\mathbb{R}^{2}):=W_{p}^{3-2/p}(\mathbb{R}^{2})$.

We also put the similar solution classes:$\mathbb{E}:=\mathbb{E}_{u}\cross \mathbb{E}_{q}\cross \mathbb{E}_{h}$with$\mathbb{E}_{u}$$:=H_{p}^{1}(0, T;L_{p}(D)^{3})\cap L_{p}(0, T;H_{p}^{2}(D)^{3})$,$\mathbb{E}_{q}:=\{q\in L_{p}(0, T;\hat{H}_{p}^{1}(D)):q\in W_{p}^{1/2-1/2p}(L_{p}(\Gamma^{+}))\cap L_{p}(W_{p}^{1-1/p}(\Gamma^{+}))\},$$\mathbb{E}_{h}:=W_{p}^{2_{\backslash }-1/2p}(L_{p}(\mathbb{R}^{2}))\cap H_{p}^{1}(W_{p}^{2-1/p}(\mathbb{R}^{2}))\cap L_{p}(W_{p}^{3-1/p}(\mathbb{R}^{2}))$.

We are now position to state the maximal regularity estimates for thesolutions to the linearized (TE) around the trivial solution when$\Lambda_{0}=0.$

Proposition 3.1. Let
there exists a unique olution ($\Lambda_{0}=0,$$\mathcal{S}$$T>0,$ $p\in(1, \infty)$ with $p\neq 3/2$ , 3.

$u, q, h)\in \mathbb{E}$Thento (TE) if and only if the terms in82
the right hand side and initial data
the following compatibility conditions$(F_{1}, F_{d}, G_{+}, H, G_{-}, u_{0}, h_{0})\in \mathbb{F}$$G_{+}(0)=\partial_{y}u_{0}’+\nabla’u_{0}^{3}$$u_{0}^{3}=0$on$\Gamma_{-},$

on$\Gamma_{+},$$F_{d}(0)=\nabla\cdot u_{0}$$G_{-}(O)=u_{0}’-c\delta^{\alpha}\partial_{y}u_{0}’$on$\Gamma_{-}$and satisfy

if $p>3,$

if $p>3/2.$

We can derive the maximal regularity estimates in above classes as follows:
Lemma 3.2. Let $p\in(1, \infty)$ with $p\neq 3/2$ , 3, $\Lambda=F_{1}=F_{d}=G_{+}=G_{-}=$$u_{0}=h_{0}=0$ , and let. Then there exist $T>0,$ $C>0$ and a solution$(u, q, h)\in \mathbb{E}$of (TE) in $(0, T)$ satisfying$H\in \mathbb{H}$$\Vert(u, q, h)\Vert_{\mathbb{E}}\leq C\Vert H\Vert_{\mathbb{H}}.$

Note that one can apply the lemma of superposition for the solutions
to the linearized equations of (TE). For making this note short, we skip its
detail of other cases.
In order to derive the maximal regularity estimates we consider the corresponding resolvent equations. By Laplace transform (roughly speaking,: $x’\mapsto\xi’\in$:and Fourier transform (roughly speaking,, we deduce the Stokes resolvent problem$\mathcal{L}^{-1}$$\mathcal{F}^{-1}$$\partial_{t}\mapsto\lambda\in \mathbb{C})$$\mathbb{R}^{2})$(RP)$\{\begin{array}{ll}\lambda\hat{u}+|\xi’|^{2}\hat{u}-\partial_{y}^{2}\hat{u}+(i\xi’, \partial_{y})\hat{q}=\hat{F}_{1} in D,i\xi’\cdot\hat{u}’+\partial_{y}\hat{u}^{3}=\hat{F}_{d} in D,\lambda\hat{h}+\hat{u}^{3}=\hat{H} on r_{+},\partial_{y}\hat{u}’+i\xi’\hat{u}^{3}=\hat{G}_{+}’ on r_{+},2\partial_{y}\hat{u}^{3}-\hat{q}+\sigma|\xi’|^{2}\hat{h}=\hat{G}_{+}^{3} on \Gamma_{+},\hat{u}’-c\delta^{\alpha}\partial_{y}\hat{u}’=\hat{G}_{-} on \Gamma_{-},\hat{u}^{3}=0 on \Gamma_{-}.\end{array}$Here, we take $\Lambda=0$ and $i:=\sqrt{-1}.$
To solve (RP), we put ‘Ansatz’ for$\hat{h}=\hat{h}(\xi’)$$(\hat{u},\hat{q},\hat{h})$for$\lambda\in \mathbb{C}$as follows:,$\hat{q}(\xi’, y)=\hat{\psi}_{-}(\xi’)e^{-|\xi’|y}+\hat{\psi}_{+}(\xi’)e^{|\xi’|y},$$\hat{u}’(\xi’, y)=\hat{\phi}_{-}’(\xi’)e^{-\omega y}-\int_{0}^{\delta}k_{-}(\xi’, y, s)i\xi’\hat{q}(\xi’, s)ds$$+ \hat{\phi}_{+}’(\xi’)e^{\omega y}-\int_{0}^{\delta}\ell_{-}(\xi’, y, s)i\xi’\hat{q}(\xi’, s)ds,$$\hat{u}^{3}(\xi’, y)=\hat{\phi}_{-}^{3}(\xi’)e^{-\omega y}-\int_{0}^{\delta}k_{+}(\xi’, y, s)\partial_{y}\hat{q}(\xi’, s)ds$

83$+ \hat{\phi}_{+}^{3}(\xi’)e^{\omega y}-\int_{0}^{\delta}\ell_{+}(\xi’, y, s)\partial_{y}\hat{q}(\xi’, s)ds.$

Here, for$\lambda\in \mathbb{C}$and$\xi’\in \mathbb{R}^{2}$

we denoted$k_{\pm}( \xi’, y, s):=\frac{1}{2\omega}(e^{-\omega|y-s|}\pm e^{-\omega(y+s)})$,$:=\sqrt{\lambda+|\xi’|^{2}}$$\omega$, and for $y,$ $s\in(0, \delta)$$\ell_{\pm}(\xi’, y, s):=\frac{1}{2\omega}(e^{\omega|y-s|}\pm e^{\omega(y+s)})$.

Our task is to find 9 functions
by equations
and boundary conditions.
In what follows, we take $\delta=c=\alpha=1$ for simplicity. Computing thematrix, we have e.g.,
inverse of$(\hat{h}, \psi \hat{\psi}_{+},\hat{\phi}_{-}^{1},\hat{\phi}_{+}^{1},\hat{\phi}_{-}^{2},\hat{\phi}_{+}^{2},\hat{\phi}_{-}^{3}, \hat{\phi}_{+}^{3})$$9\cross 9$$\hat{h}=\frac{\omega^{3}+\lambda|\xi’|+3\omega|\xi’|^{2}}{\lambda(\omega^{3}+\lambda|\xi’|+3\omega|\xi|^{2})+\sigma|\xi’|^{3}(\omega+|\xi’|)}\hat{H}=:\frac{m_{1}}{m_{2}}\hat{H}.$

Note that
contains theparabolic scale
at once. So,
and the hyperbolic scale
it is diﬃcult to apply the standard methods, at least directly, for guarantee
non-zero of the denominators and for deriving the resolvent estimate.
$m_{2}(\lambda, |\xi’|)=\lambda(\omega^{3}+\lambda|\xi’|+3\omega|\xi’|^{2})+\sigma|\xi’|^{3}(\omega+|\xi’|)$$(\lambda\sim|\xi’|^{2})$4$(\lambda\sim|\xi’|)$Method of Newton polygon

The method of Newton polygon was developed by Euler, and is the way
to seek the direction for the biggest gradient of a complex-value function.
More precisely, taking
for $k>0$ , we observe which is the dominant
terms. We refer to the description of Newton polygon approach in [GV92]
and [DMV98].
In here, we give a brief idea of Newton polygon. We settle$\lambda\sim|\xi’|^{k}$$z:=|\xi’|\in \mathbb{R}_{+},$

So, 5 terms of$\omega$$:=\sqrt{\lambda+z^{2}},$$\lambda\sim z^{k}$for$m_{2}(\lambda, z)=\lambda(\omega^{3}+\lambda z+3\omega z^{2})+\sigma z^{3}(\omega+z)$$k>0.$

are of ordering as$\bullet 0<k<1\Rightarrow m_{2}\sim z^{k+3}+z^{2k+1}+z^{k+3} +z^{4} +z^{4},$$\bullet k=1 \Rightarrow m_{2}\sim z^{4} +z^{3} +z^{4} +z^{4} +z^{4},$$\bullet 1<k<2\Rightarrow m_{2}\sim z^{k+3}+z^{2k+1}+z^{k+3} +z^{4} +z^{4},$$\bullet k=2 \Rightarrow m_{2}\sim z^{5} +z^{5} +z^{5} +z^{4} +z^{4},$$\bullet k>2 \Rightarrow m_{2}\sim z^{5k/2}+z^{2k+1}+z^{3k/2+2}+z^{k/2+3}+z^{4}.$Note that the highest order terms dominate. In fact, putting $(r, s)$ the exponents of the terms, every point $(r, s)$ in above is in the convex hull$\lambda^{r}z^{s}$84$:=\{(0,0), (5/2, 0), (1, 3), (0,4)\}$ in clockwise direction;of 4 points
this convex hull $N(P)$ is so-called Newton polygon. In addition, all coeﬃcient of five terms is a positive real number. So, we can see that there exist
andfor allsuch thatandis the sector region in the complex. Here,plane.
Moreover, the method of Newton polygon leads us to derive the maximal
regularity estimates due to the Fourier multiplier theory by Kalton-Weis[KWOI]. Weis’ Fourier multiplier theory is a strong tool for mathematicalanalysis on fluid dynamics. Idneed, there are a lot of papers as applicationof it, for example, [DHP03, DHP07, DSS08, PS12, SSII].
be Fourier multiplier of symbol
Let$N(P)$$P$$m_{2}(\lambda, |\xi’|)\neq 0$$\varphi\in(0, \pi)$$\lambda_{0}>0$$\xi’\in \mathbb{R}^{2}$$\Sigma_{\varphi}$$M_{2}$$\lambda\in\lambda_{0}+\Sigma_{\varphi}$$:=\{\lambda\in \mathbb{C};|\arg\lambda|<\varphi\}$$:=\mathcal{L}^{-1}\mathcal{F}^{-1}m_{2}\mathcal{F}$

Lemma 4.1. If $1<p<\infty,$$m_{2}.$$r,$$s\geq 0$, then there exists$\rho_{0}\geq 0$such that$M_{2}\in Isom(\mathcal{K}_{p,\rho}^{r+5/2}(\overline{\mathcal{K}}_{p}^{s})\cap \mathcal{K}_{p_{)}\rho}^{r+1}(\mathcal{K}_{p}^{s+3})-\cap \mathcal{K}_{p,\rho}^{r}(\overline{\mathcal{K}}_{p}^{s+4}), \mathcal{K}_{p,\rho}^{r}(\overline{\mathcal{K}}_{p}^{s}))$for all$\rho\geq\rho_{0}$$0^{H_{p,\rho}^{r}(\mathbb{R}_{+};\overline{\mathcal{K}})}$, where$\mathcal{K}_{p,\rho}^{r}(\overline{\mathcal{K}}_{p}^{S})$$:=0^{\mathcal{K}_{p,\rho}^{r}}(\mathbb{R}_{+};\overline{\mathcal{K}}_{p}^{s}(\mathbb{R}^{2}))$$:=\{u;e^{-\rho t}\partial_{t}^{k}u\in L_{p}(\mathbb{R}_{+};\overline{\mathcal{K}})$,$\forall_{k}\in \mathbb{N}_{0},$;$\mathcal{K},$$\overline{\mathcal{K}}\in\{H,$$0\leq k\leq r,$$B_{p}$$u(O)=0\}.$The proof of this lemma is shown by [DGHSSII], we omit the detail. We
would emphasize that $(r, s)$ the diﬀerential orders in the image of isomorphism is added to the the corner of $N(P)$ in the domain. By this lemma, we,thereforethat for$prove\backslash$$H\in B_{p,p}^{1-1/2p}(L_{p}(\mathbb{R}^{2}))\cap L_{p}(B_{p,p}^{2-1/p}(\mathbb{R}^{2}))$$\mathcal{L}^{-1}\mathcal{F}^{-1}\frac{m_{1}}{m_{2}}\hat{H}=h\in B_{p,p}^{2-1/2p}(L_{p}(\mathbb{R}^{2}))\cap H_{p}^{1}(B_{p,p}^{2-1/p}(\mathbb{R}^{2}))\cap L_{p}(B_{p,p}^{3-1/p}(\mathbb{R}^{2}))$in $(0, T)$ for $T<\infty$ . This calculation completes the proof of Lemma 3.2.

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