# An Eccentric Eclipsing Binary: CG Aur

E. Sipahi<EMAIL_ADDRESS>H. A. Dal  Ege University, Science
Faculty, Department of Astronomy and Space Sciences, 35100 Bornova, İzmir,
Turkey

###### Abstract

In this study, we present CG Aur’s photometric observations obtained in the
observing seasons 2011 and 2012, the first available multi-colour light
curves. Their shape indicates that the system is an Algol binary. The light
curve analyses reveal that CG Aur is a detached binary system with an
effective temperature difference between the components, approximately 1000 K.
The first estimate of the absolute dimensions of the components indicated that
the system locates on the main sequence in the HR diagram. The primary
component is slightly evolved from the ZAMS.

###### keywords:

techniques: photometric — (stars:) binaries: eclipsing — stars: early-type —
stars: individual: (CG Aur)

††journal: New Astronomy

##  1 Introduction

CG Aur ($V=11^{m}.43$) is classified as an eclipsing binary in the SIMBAD
database. The system was first discovered by Hoffleit (1935), while Kanda
(1939) gave the first light elements. The photographic light curve of the
system was obtained by Kurochkin (1951). The system was observed by Zakirov
(1995) and the colour indexes of $U-B$ and $B-V$ were given as 0m.36 and 0m.66
in this study, respectively. Presenting the $O-C$ analyses of the system, Wolf
et al. (2011) indicated that CG Aur is an interesting triple and eccentric
eclipsing system showing a slow apsidal motion with an important relativistic
contribution as well as the rapid LITE caused by a third body orbiting with a
very short period of 1.9 years. According to them, CG Aur probably belongs to
an important group of other early-type and triple eclipsing systems with a
very short third-body orbital period.

The apsidal motion in the eccentric eclipsing binaries has been used for
decades to test the models of stellar structure and evolution. CG Aur analyzed
here has some properties, which make the system an important ”astrophysical
laboratories” for studying the stellar structure and evolution. The lack of
the multi-colour light curves makes CG Aur an interesting system for including
it to our photometric programme. We followed the observations of the system
and obtained the light curves in 2011 and 2012, and we discussed the light and
colour variations. CG Aur is important due to not only being a member of the
group of the triple systems, but also having an eccentric orbit.

##  2 Observations

The observations were acquired with a thermoelectrically cooled ALTA U+42
2048$\times$2048-pixel CCD camera attached to a 40-cm Schmidt-Cassegrain MEADE
telescope at Ege University Observatory. Using exposure times of 100 s in B
filter and 40 s in V, R and I filters, the BVRI-band observations were
recorded over five nights in 2011 and four nights in 2012. Calibration images
(bias frames and twilight sky flats) were taken intermittently during
observations to correct for pixel to pixel variations on the chip. CCD
observations were reduced as follows: Bias and dark frames were subtracted
from the science frames and then corrected for the flat-fielding. These
reduced CCD images were used to obtain the differential magnitudes of the
program stars. We used GSC 1857 833 and GSC 1857 736 as a comparison and check
stars shown in Figure 1, respectively. There was no variation observed in the
brightnesses of the comparison star.

During the observations, we obtained one primary and one secondary times of
minimum light. These minima times and their errors were determined using the
method of Kwee & van Woerden (1956) and are presented in Table 1. In order to
calculate the phases of the photometric data of CG Aur, the following linear
ephemeris was used:

$JD~{}(Hel.)~{}=~{}24~{}55983.2658(6)~{}+~{}1^{d}.8048588(3)~{}\times~{}E.$
(1)

The V-light and the colour curves obtained in this study are shown in Figure
2. The shape of the light curves indicates that CG Aur is an Algol type binary
and reveals that the primary minimum, which lasts $\sim$6 hours, is deeper
than the secondary one. The mean depths of the eclipses in B, V, R, and I
filters are 0m.434, 0m.390, 0m.382, and 0m.360 in the primary minimum and
0m.168, 0m.202, 0m.218, and 0m.235 in the secondary minimum, respectively. The
$B-V$, $V-R$ and $V-I$ colour curves of the system are also displayed in
Figure 2. The system is slightly redder at the primary and bluer at the
secondary minimum which is consistent with the spectral types of the
components.

##  3 Light Curve Analysis

Photometric analysis of CG Aur was carried out using the PHOEBE V.0.31a
software (Prša & Zwitter, 2005). The software uses the version 2003 of the
Wilson-Devinney Code (Wilson & Devinney, 1971; Wilson, 1990). The BVRI light
curves were analysed simultaneously assuming the ”detached” configuration. In
the process of the computation, we initially adopted the following fixed
parameters: the mean temperature of the primary component ($T_{1}$), the
linear limb-darkening coefficients of $x_{1}$ and $x_{2}$ for various bands
(van Hamme, 1993), the gravity-darkening exponents of $g_{1}$, $g_{2}$ (Lucy,
1967) and the bolometric albedo coefficients of $A_{1}$, $A_{2}$ (Rucinski,
1969). The adjustable parameters commonly employed are the orbital inclination
($i$), the mean temperature of the secondary component ($T_{2}$), the
potentials of the components ($\Omega_{1}$ and $\Omega_{2}$) and the
monochromatic luminosity of the primary component ($L_{1}$). The third light
($L_{3}$) was used also as free parameter to check for the third light
contribution. We used the $U-B$ and $B-V$ values given by Zakirov (1995) and
determined the dereddened colours of the system as $(U-B)_{0}=0^{m}.06$,
$(B-V)_{0}=0^{m}.24$. Then, we took JHK magnitudes of the system
($J=10^{m}.261$, $H=10^{m}.036$, $K=9^{m}.955$) from the 2MASS Catalogue
(Cutri et al., 2003). Using these magnitudes, we derived dereddened colours as
a $(J-H)_{0}=0^{m}.225$ and $(H-K)_{0}=0^{m}.081$ for the system. Using the
calibrations given by Tokunaga (2000), we derived the temperature of the
primary component as 7650 K and 7475 K depending on the UBV and JHK dereddened
colours, respectively. Both of them indicate the same spectral range which is
A5-F0. We adopted the mean temperature of the primary component as 7650 K for
the light curve analyse. To find a photometric mass ratio, the solutions are
obtained for a series of fixed values of the mass ratio from $q=0.4$ to 1.0 in
increments of 0.1. The sum of the squared residuals ($\Sigma res^{2}$) for the
corresponding mass ratios are plotted in Figure 3, where the lowest value of
($\Sigma res^{2}$) was found at about $q=0.7$. The photometric elements for
the mass-ratio of 0.7 are listed in Table 2, and the corresponding light
curves are plotted in Figure 4 as continuous lines. The Roche Lobe geometry of
the system is displayed for the phase of 0.25 in Figure 5.

Although there is no available radial velocity curve of the system, we tried
to estimate the absolute parameters of the components. Considering its
spectral type, using the calibration of Tokunaga (2000), we estimate the mass
of the primary component as approximately 1.78 $M_{\odot}$, and the mass of
the secondary component is computed from the estimated mass ratio of the
system. Using Kepler’s third law, we calculate the semi-major axis ($a$), and
also the mean absolute radii of the components. We can calculate the distance
of the system by using primary and secondary component separately, via their
photometric and absolute properties. We adopted bolometric corrections from
Tokunaga (2000), while calculating the distance. Photometric properties of the
components lead to an average distance of 460 pc. All the estimated absolute
parameters are listed in Table 3. In Figure 6, we plot the components in the
$log~{}(M/M_{\odot})$-$log~{}(R/R_{\odot})$ and
$log~{}(T_{eff})$-$log~{}(L/L_{\odot})$ planes. The continuous and dotted
lines represent the ZAMS and TAMS theoretical model developed by Girardi et
al. (2000). All the tracks are taken from Girardi et al. (2000) for the stars
with $Z=0.02$. In the figure, the open circles represent the secondary
component, while the filled circles represent the primary component. Both of
the components locate in the main-sequence band, while the more massive
component is more evolved.

##  4 Summary

We have obtained the multi-colour CCD photometry for the interesting eclipsing
binary CG Aur. Based on these observations, we presented the first BVRI light
curves of the system and analysed them to find the parameters obtained from
the orbital solution, considering also the Roche configuration of the system.
The results allow us to draw the following conclusions.

$\bullet$ The physical and geometrical parameters of the components have been
derived. Orbital parameters indicate that CG Aur is a detached binary system
with a little temperature difference of approximately 1000 K.

$\bullet$ Our photometric model describes CG Aur as an Algol type eclipsing
binary in which the more massive and hotter primary component is the larger
one.

$\bullet$ In order to discuss the present evolutionary status of the
components of CG Aur, both of them were plotted on the
$log~{}(M/M_{\odot})$-$log~{}(R/R_{\odot})$ and
$log~{}(T_{eff})$-$log~{}(L/L_{\odot})$ planes. The components of the system
located inside the Main-Sequence band. The primary component is slightly
evolved from the Zero-Age Main-Sequence (ZAMS), while the secondary component
is still very close to ZAMS. CG Aur is a detached eclipsing binary and its
components are Main-Sequence dwarfs, the evolutionary states of the components
of the system could be estimated from single-star evolutionary models. We
compared the physical parameters of the system with those inferred from the
evolutionary tracks for single stars of masses in the range of 1.8 and 1.3
$M_{\odot}$, taken from Girardi et al. (2000) for solar composition. There is
a good agreement between the estimated masses and evolutionary masses for the
components of the system.

$\bullet$ The detailed $O-C$ study of the system were presented by Wolf et al.
(2011). In this study, we obtained a small amount of the third light
contribution to the total light from the light curve solution of the system
for the first time. According to our solution, the luminosity fraction of the
third body was found maximum in I filter, while the least contribution was
determined in B filter, which indicates its cool nature. The spectral type of
the third body should be late type.

$\bullet$ As said by Wolf et al. (2011), CG Aur is an interesting triple and
eccentric eclipsing system showing the slow apsidal motion with the important
relativistic contribution as well as the rapid LITE caused by a third body
orbiting with the very short period of 1.9 years. CG Aur is shown to be an
Algol-type system of special interest because of its nature. The system
belongs to the important group of other early-type and triple eclipsing
systems with a very short third-body orbital period (e.g. IM Aur, IU Aur, FZ
CMa, AO Mon) as said by Wolf et al. (2011), and also it has an eccentric
orbit. CG Aur is important due to not only being a member of the group of the
triple systems, but also having an eccentric orbit, while short period
binaries with periods less than a week generally have circular orbits. It is
recommended that more attention should be paid to this system.

$\bullet$ In future work, spectroscopic observations should be made to obtain
radial velocity curves, which will allow a better discussion of the absolute
dimensions of the components and the evolutionary status of CG Aur. New
timings of this eclipsing binary are also necessary to improve the LITE
parameters of the system.

## Acknowledgment

The author acknowledge generous allotments of observing time at the Ege
University Observatory. We also thank the referee for useful comments that
have contributed to the improvement of the paper.

## References

  * Cutri et al. (2003) Cutri, R.M., Skrutskie, M.F., van Dyk, S. et al., 2003, The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC Infrared Science Archive (http://irsa.ipac.caltech.edu/applications/Gator/).
  * Girardi et al. (2000) Girardi, L., Bressan, A., Bertelli, G., Chiosi, C., 2000, A&AS 141, 371
  * Hoffleit (1935) Hoffleit, D., 1935, Harvard Obs. Bull., 901
  * Kanda (1939) Kanda, S., 1939, Tokyo Astr. Obs. Bull. 393, 786
  * Kurochkin (1951) Kurochkin, N.E., 1951, Variable Stars 8, 351
  * Kwee & van Woerden (1956) Kwee, K.K., van Woerden, H., 1956, BAN 12, 327
  * Lucy (1967) Lucy, L.B., 1967, Z. Astrophys, 65, 89
  * Prša & Zwitter (2005) Prša, A., Zwitter, T., 2005, ApJ, 628, 426
  * Rucinski (1969) Rucinski, S.M., 1969, AcA, 19, 245
  * Tokunaga (2000) Tokunaga, A.T., 2000, ”Allen’s Astrophysical Quantities”, Fouth Edition, ed. A.N.Cox (Springer), p.143
  * van Hamme (1993) van Hamme, W., 1993, AJ, 106, 2096
  * Wilson (1990) Wilson, R.E., 1990, ApJ, 356, 613
  * Wilson & Devinney (1971) Wilson, R.E., Devinney, E.J., 1971, ApJ, 166, 605
  * Wolf et al. (2011) Wolf, M., Lehký, M., Šmelcer, L., Kučáková, H., Kocián, R., 2011, New Astronomy, 16, 402
  * Zakirov (1995) Zakirov, M.M., Astronomy Letters, 1995, Volume 21, Issue 5, 675

Figure 1: The program star CG Aur and comparison and check stars on the sky
plane, which is in size of $23^{\prime}.27\times 23^{\prime}.27$. Figure 2: CG
Aur’s V-light and colour curves. Figure 3: The variation of the sum of
weighted squared residuals versus mass ratio. Figure 4: CG Aur’s light curves
observed in BVRI bands and the synthetic curves derived from the light curve
solutions in each band. Figure 5: CG Aur’s Roche geometry, while the system is
at the phase of 0.25. Figure 6: The places of the components of CG Aur in the
$log~{}(M/M_{\odot})$-$log~{}(R/R_{\odot})$ plane (upper panel) and
$log~{}(T_{eff})$-$log~{}(L/L_{\odot})$ plane (bottom panel). In the figures,
the filled circles represent the primary, while open circles represent the
secondary component. The continuous and dotted lines represent the ZAMS and
TAMS theoretical model developed by Girardi et al. (2000). All the tracks
taken from Girardi et al. (2000) are derived for the stars with $Z=0.02$.

Table 1: The times of minimum light for CG Aur. HJD (24 00000 +) | Sigma | Type | Filter
---|---|---|---
55983.2661 | 0.0010 | I | B
55983.2655 | 0.0006 | I | V
55983.2659 | 0.0004 | I | R
55983.2656 | 0.0004 | I | I
56002.2782 | 0.0010 | II | B
56002.2668 | 0.0018 | II | V
56002.2732 | 0.0020 | II | R
56002.2682 | 0.0008 | II | I
Table 2: The parameters obtained from the light curve analysis.

Parameter | Value | Parameter | Value
---|---|---|---
$T_{0}$ | 24 50014.6248 |  $P$ (day) | 1.8048588
$q$ | 0.7 |  $i$ (∘) | 87.74$\pm$0.01
$e^{1}$ | 0.124 |  $w$ (∘) | 291.05$\pm$0.01
$T_{1}$ (K) | 7650 |  $T_{2}$ (K) | 6600$\pm$27
$\Omega_{1}$ | 5.617$\pm$0.001 | $\Omega_{2}$ | 6.850$\pm$0.002
L1/LT $(B)$ | 0.862$\pm$0.015 | L3/LT $(B)$ | 0.0007$\pm$0.0002
L1/LT $(V)$ | 0.836$\pm$0.014 | L3/LT $(V)$ | 0.0020$\pm$0.0002
L1/LT $(R)$ | 0.814$\pm$0.014 | L3/LT $(R)$ | 0.0031$\pm$0.0002
L1/LT $(I)$ | 0.792$\pm$0.012 | L3/LT $(I)$ | 0.0080$\pm$0.0002
$g_{1}$, $g_{2}$ | 0.32, 0.32 |  $A_{1}$, $A_{2}$ | 0.5, 0.5
Phase shift | 0.0146$\pm$0.0002 |  $x_{1,bol}$, $x_{2,bol}$ | 0.522, 0.481
$x_{1,B}$, $x_{2,B}$ | 0.596, 0.656 |  $x_{1,R}$, $x_{2,R}$ | 0.421, 0.434
$x_{1,V}$, $x_{2,V}$ | 0.518, 0.532 |  $x_{1,I}$, $x_{2,I}$ | 0.329, 0.351
$<r_{1}>$ | 0.209$\pm$0.001 | $<r_{2}>$ | 0.126$\pm$0.001

  * 1

Taken from Wolf et al. (2011)

Table 3: The estimated absolute parameters derived for CG Aur. Parameter | Primary |  | Secondary
---|---|---|---
Mass ($M_{\odot}$) | 1.78 |  | 1.25
Radius ($R_{\odot}$) | 2.09 |  | 1.26
Luminosity ($L_{\odot}$) | 13.49 |  | 2.72
$M_{bol}$ (mag) | 1.92 |  | 3.65
$log~{}(g)$ | 4.05 |  | 4.33
$d$ (pc) |  | 460 |


# On an inhomogeneous slip-inflow boundary value problems for a steady viscous
compressible channel flows

Wengang Yang  School of Mathematics and Statistics, Wuhan University, Wuhan,
Hubei Province, 430072, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

We prove the existence and uniqueness of strong solutions to the steady
isentropic compressible Navier-Stokes equations with inflow boundary
conditions for density and mixed boundary conditions for the velocity around a
shear flow. In particular, the Dirichlet boundary condition on inflow and
outflow part of the boundary while the full Navier boundary conditions on the
wall $\Gamma_{0}$ for the velocity filed are considered. For our result, there
are no restrictions on the amplitude of friction coefficients $\alpha$, and
the appropriately large hypothesis for the viscous coefficients $\mu$ is
enough. One of the substantial ingredients of our proof is an elegant
transformation induced by the flow field. With the help of this
transformation, we can overcome the difficulties caused by the hyperbolicity
of continuity equation, establish the a priori estimates for a linearized
system and apply the fixed point arguments.

###### Key words and phrases:

inhomogeneous boundary conditions, compressible Navier-Stokes system, strong
solution

##  1\. introduction and main results

In this paper, we consider the steady isentropic compressible Navier-Stokes
equations with inflow boundary condition in a two-dimensional tube
$\Omega=(0,1)\times(0,1)$ near a shear flow. It is well known that the
Navier–Stokes equations for a steady isentropic compressible viscous flow is a
mixed system of hyperbolic-elliptic type. The momentum equations are an
elliptic system in the velocity, while the continuity equation is hyperbolic
in the density. Therefore, it is necessary to prescribe the density on the
part of inflow boundary $(\mathbf{u}\cdot\mathbf{n}<0)$ where $\mathbf{n}$ is
the outward unit normal to the part of inflow boundary. The inflow boundary
value problem considered in this article reads as follows:

(1.1) $\displaystyle\operatorname{div}(\rho\mathbf{u})$ $\displaystyle=0$
$\displaystyle\text{ in }\Omega,$
$\displaystyle\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{u})-\mu\Delta\mathbf{u}-(\mu+\nu)\nabla\operatorname{div}{\bf
u}+\nabla P$ $\displaystyle=0$ $\displaystyle\text{ in }\Omega,$
$\displaystyle\mathbf{u}$ $\displaystyle=\mathbf{u}_{0}$ $\displaystyle\text{
on }\Gamma_{\text{in }}\cup\Gamma_{\text{out }},$ $\displaystyle\rho$
$\displaystyle=\rho_{\text{in }}$ $\displaystyle\text{ on }\Gamma_{\text{in
}},$ $\displaystyle\mathbf{u}\cdot\mathbf{n}$ $\displaystyle=0$
$\displaystyle\text{ on }\Gamma_{0},$ $\displaystyle 2\mu\mathbf{n}\cdot
D({\bf u})\cdot\tau+\alpha{\bf u}\cdot\tau$ $\displaystyle=b$
$\displaystyle\text{ on }\Gamma_{0},$

where ${\bf u}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is the unknown
velocity filed of the fluid and $\rho:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is
the unknown density. The $D(\bf{u})$ denotes the symmetric part of the
velocity gradient, more precisely,

$D({\bf u})=\frac{1}{2}(\nabla{\bf u}+(\nabla{\bf u})^{\mathrm{T}}).$

The constant viscosity coefficients $\mu$ and $\nu$ satisfy $\mu>0,\nu+\mu\geq
0$ and $\alpha\geq 0$ is a friction coefficient. The pressure P is usually
determined through the equation of states. In the case of isentropic flows,
the pressure P is given by $P(\rho)=\rho^{\gamma}$ with $\gamma>1$ being the
heat ratio. Next, ${\bf n}$ and $\tau$ are outer normal and tangent vectors to
$\partial\Omega$. The boundary $\partial\Omega$ is naturally divided into the
inflow part $\Gamma_{in}$, the outflow part $\Gamma_{out}$, and the
impermeable wall $\Gamma_{0}$, more precisely,

$\displaystyle\Gamma_{in}$ $\displaystyle=\\{x_{1}=0,0\leq x_{2}\leq 1\\},$
$\displaystyle\Gamma_{out}$ $\displaystyle=\\{x_{1}=1,0\leq x_{2}\leq 1\\},$
$\displaystyle\Gamma_{0}$ $\displaystyle=\\{0\leq x_{1}\leq
0,x_{2}=0\\}\cup\\{0\leq x_{1}\leq 0,x_{2}=1\\}.$

The flow of a viscous, compressible fluid were governed by the compressible
Navier-Stokes system, which has been extensively studied. The general
existence results for weak solution to stationary compressible Navier-Stokes
system are given by the work of Lions[15] under the assumption that $\gamma>1$
in two dimensions and $\gamma\geq\frac{5}{3}$ in three dimensions. Later, this
results was improved with the weaker assumptions on the specific heat ration
$\gamma>\frac{3}{2}$ in three dimensions by Feireisl[4] for nonsteady
equations. By adapting Feireisl’s nonsteady approach, the authors in [19] show
that $\gamma>\frac{3}{2}$ is also valid to the case of steady system. An
overview of these results is given in the monograph [20, 5]. Several refined
results for steady compressible flow were given in [7, 27, 25]. Let us remark
that all those results were obtained under various homogeneous boundary
condition.

The strong solutions of steady compressible Navier-Stokes equations with
homogeneous boundary condition has been studied in [2, 10, 18, 28, 29]. As to
the existence of strong solutions to the stationary Navier-Stokes system with
the inhomogeneous boundary conditions, the authors in [11] proved the
existence of strong solutions to the stationary problems with an inflow
boundary conditions for the density and Dirichlet boundary condition for the
velocity in a smooth two dimension domain $\Omega$ under the assumption that
the Reynolds number is small. A mass of researches showed that the regularity
of the strong solutions is restricted by the geometry of the boundary[14]. In
[12, 13, 21], the authors studied the existence and regularity of solutions to
an inflow boundary value problem under the assumption that the viscosity
coefficient $\mu$ is large enough on a polygon domain. In [22], T. Piasecki
proved the existence of strong solution around a constant equilibrium with an
inflow boundary condition for the density and the full Navier boundary
conditions for the velocity filed under the assumption that the friction
coefficient $\alpha$ is large enough. Later, the authors obtained similar
results in a cylinder domains [23]. It is worth mentioning that the validity
for compressible perturbation of a Poiseuille type flow under the same
boundary conditions as before is also obtained in [17]. In contrast to the
boundary conditions prescribed above, the authors established the existence of
strong solutions near the constant state ${\bf u}=0,\rho=1$ with Dirichlet
boundary condition on $\Gamma_{in}$ and $\Gamma_{out}$ while slip without
friction boundary conditions on the wall $\Gamma_{0}$ in [9]. Recently, the
authors in [6] studied the existence of weak solutions to the stationary
compressible Navier–Stokes system for arbitrarily large boundary data under
additional physical hypotheses called molecular hypothesis and positive
compressibility in 2D or 3D domains. One can refer to [3, 26] for more results
concerning the existence of strong solutions with inhomogeneous boundary
conditions.

The goal of this paper is to investigate the existence of strong solutions
near the shear flow to the steady isentropic compressible Navier-Stokes system
with inflow boundary conditions in a square. We impose the Dirichlet boundary
condition on inflow and outflow part of the boundary, and the full Navier
boundary conditions on the wall $\Gamma_{0}$ for the velocity. Due to the
hyperbolicity of the density in the continuity equation, it is natural to
prescribe the density on the inflow part of the boundary. Let us remark here
that we do not need any restrictions on the amplitude of friction coefficients
$\alpha$, and an assumption that $\mu>\frac{1}{2\pi^{2}}$ is enough. It is
worth noting that most of the results discussed above investigated the
structural stability around a constant equilibrium, while our background
solution is a shear flow.

Let us introduce the perturbed flow. Consider the shear flow
$(\rho_{0}=1,U_{0})$, with $U_{0}=(1+x_{2},0)=(\bar{U},0)$. It is obviously
that the shear flow $(\rho,U_{0})$ satisfy the following system:

$\displaystyle\operatorname{div}(\rho_{0}{U_{0}})$ $\displaystyle=0$
$\displaystyle\text{ in }\Omega,$
$\displaystyle\operatorname{div}(\rho_{0}{U_{0}}\otimes{U_{0}})-\mu\Delta{U_{0}}-(\mu+\nu)\nabla\operatorname{div}U_{0}+\nabla
P_{0}$ $\displaystyle=0$ $\displaystyle\text{ in }\Omega,$
$\displaystyle{U_{0}}$ $\displaystyle={U_{0}}$ $\displaystyle\text{ on
}\Gamma_{\text{in }}\cup\Gamma_{\text{out }},$ $\displaystyle\rho_{0}$
$\displaystyle=1$ $\displaystyle\text{ on }\Gamma_{\text{in }},$
$\displaystyle{U_{0}}\cdot\mathbf{n}$ $\displaystyle=0$ $\displaystyle\text{
on }\Gamma_{0},$ $\displaystyle 2\mu\mathbf{n}\cdot D({U_{0}})\cdot\tau+\alpha
U_{0}\cdot\tau$ $\displaystyle=\tilde{b}$ $\displaystyle\text{ on
}\Gamma_{0},$

where $\tilde{b}$ is determined when $U_{0}$ and $\alpha$ are given. Our aim
here is to study the existence and uniqueness of strong solution in a square
around the shear flow $(\rho_{0},U_{0})$. To formulate our main results, it’s
convenient to define the quantity $D_{0}$ as

(1.2) $D_{0}:=\|{\bf
u}_{0}-U_{0}\|_{{2-\frac{1}{p},p};\Gamma_{in}\cap\Gamma_{out}}+\|b-\tilde{b}\|_{1-\frac{1}{p},p;\Gamma_{0}}+\|\rho_{in}-1\|_{{1,p};\Gamma_{in}}$

which measure the distance of the initial data away from the shear flow
$U_{0}$.

######  Theorem 1.1.

Suppose that $2<p<\infty$, if $D_{0}$ given by (1.2) is small enough and the
viscosity coefficients $\mu>\frac{1}{2\pi^{2}}$, then there exists a unique
solution $({\bf u},\rho)\in W^{2,p}(\Omega)\times W^{1,p}(\Omega)$ to the
system $(\ref{S-NS})$ such that

(1.3) $\|{\bf u}-U_{0}\|_{{2,p};\Omega}+\|\rho-1\|_{{1,p};\Omega}\leq
E(D_{0}),$

where $E(D_{0})$ can be arbitrarily small provided that the quantity $D_{0}$
is small enough.

This paper will be organized as follows. In Section 2, we linearized the
original problem (1.1) around the shear flow $(\rho_{0},U_{0})$. Next, we
introduce an elegant transformation to overcome the difficulties caused by the
hyperbolicity of the continuity equation. In Section 3, we establish the basic
and higher order energy estimates to the linearized equations. In order to
overcome the difficulties caused by the nonlinearity for the Navier–Stokes
system, we derive the uniform $W^{2,p}(\Omega)\times
W^{1,p}(\Omega),(2<p<\infty)$ estimate for the solution of the linearized
system. An iteration scheme is also developed to study the existence and
uniqueness of strong solution. In Section 4, we construct approximated
solutions by a Galerkin method and obtain the existence and regularity of weak
solution to the linearized system. In the Section 5, we show that the
iteration scheme designed in the Section 3 is a contraction and prove our main
result Theorem 1.1 finally.

##  2\. Reformulation of problem

It is convenient to convert the inhomogeneous boundary conditions of the
original problems into homogeneous one. To this end, introduce a function
$\tilde{\bf u}\in W^{2,p}(\Omega)$ such that

(2.1) $\displaystyle\tilde{\mathbf{u}}=\mathbf{u}_{0}-U_{0}$ $\displaystyle
on\,\Gamma_{in}\cup\Gamma_{out},$
$\displaystyle\tilde{\mathbf{u}}\cdot\mathbf{n}=0$ $\displaystyle
on\,\Gamma_{0},$ $\displaystyle 2\mu\mathbf{n}\cdot D(\tilde{\bf
u})\cdot\tau+\alpha\tilde{\bf u}\cdot\tau=0$ $\displaystyle\text{ on
}\Gamma_{0}.$

Indeed, one can construct $\tilde{\bf u}$ as a solution of the following
Lam$\acute{e}$ system

$-\mu\Delta\tilde{\bf u}-(\mu+\nu)\nabla\operatorname{div}\tilde{\bf
u}=0\quad\quad\text{ in }\Omega,\\\ $

with boundary condition (2.1).Consequently,

(2.2) $\|\tilde{\bf u}\|_{W^{2,p}(\Omega)}\leq C|{\bf
u}_{0}-U_{0}|_{W^{2-\frac{1}{p},p}(\Gamma_{in}\cap{\Gamma_{out}})}.$

We denotes the differences

${\mathbf{v}}=\mathbf{u}-\tilde{\mathbf{u}}-U_{0},\,w=\rho-1.$

Then, one can yields $(\bar{\mathbf{u}},\bar{\rho})$ satisfy the following
system by straightforward computation,

(2.3) $\displaystyle\ \bar{U}\partial_{1}w+({\bf
v}+\tilde{\mathbf{u}})\cdot\nabla w+\operatorname{div}{\bf v}$
$\displaystyle=F({\bf v},w)$ $\displaystyle\text{ in }\Omega,$ $\displaystyle\
\bar{U}\partial_{1}{\bf v}+{\bf{v}}\cdot\nabla U_{0}+\gamma\nabla
w-\mu\Delta{\bf v}-(\mu+\nu)\nabla\operatorname{div}{\bf v}$
$\displaystyle=G({\bf v},w)$ $\displaystyle\text{ in }\Omega,$ $\displaystyle\
{\bf v}$ $\displaystyle=0$ $\displaystyle\text{ on }\Gamma_{\text{in
}}\cup\Gamma_{\text{out }},$ $\displaystyle\ w$ $\displaystyle=\rho_{\text{in
}}-1$ $\displaystyle\text{ on }\Gamma_{\text{in }},$ $\displaystyle\ {\bf
v}\cdot\mathbf{n}$ $\displaystyle=0$ $\displaystyle\text{ on }\Gamma_{0},$
$\displaystyle 2\mu\mathbf{n}\cdot D({\bf v})\cdot\tau+\alpha{\bf v}\cdot\tau$
$\displaystyle=B$ $\displaystyle\text{ on }\Gamma_{0},$

where

(2.4) $\displaystyle B=$ $\displaystyle b-\tilde{b},$ $\displaystyle F({\bf
v},w)=$
$\displaystyle-\operatorname{div}\tilde{\mathbf{u}}-w\operatorname{div}({\bf
v}+\tilde{\mathbf{u}}),$ $\displaystyle G({\bf v},w)=$
$\displaystyle-(w+1)\left({\bf
v}+\tilde{\mathbf{u}}+U_{0}\right)\cdot\nabla\tilde{\mathbf{u}}-(w+1)({\bf
v}+\tilde{\mathbf{u}})\cdot\nabla{\bf v}$ $\displaystyle-wU_{0}\cdot\nabla{\bf
v}-\tilde{\bf{u}}\cdot\nabla U_{0}-w({\bf v}+\tilde{\mathbf{u}})\cdot\nabla
U_{0}$
$\displaystyle+\mu\Delta\tilde{\mathbf{u}}+(\mu+\nu)\nabla\operatorname{div}\tilde{\mathbf{u}}+\gamma\left[(w+1)^{\gamma-1}-1\right]\nabla
w.$

A straightforward computation gives the estimates of $F$ and $G$ as follows.

######  Lemma 2.1.

Let $F({\bf v},w)$ and $G({\bf v},w)$ be defined by (2.4).Then we have

(2.5) $\displaystyle\|F({\bf v},w)\|_{W^{1,p}}+\|G({\bf v},w)\|_{L^{p}}$
$\displaystyle\leq C[(\|{\bf v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})^{3}+(\|{\bf
v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})^{2}]$ $\displaystyle+E(D_{0})(\|{\bf
v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})+\|\tilde{\bf u}\|_{W^{2,p}}.$

###### Proof.

The estimates is almost obviously by the embedding
$W^{1,p}(\Omega)\hookrightarrow L^{\infty}(\Omega)$, one can easily check it
term by term. In fact, the term $w\operatorname{div}{\bf v}$ in $F({\bf v},w)$
is estimated as

(2.6) $\displaystyle\|w\operatorname{div}{\bf v}\|_{1;p}$
$\displaystyle\leq\|w\|_{\infty}\|\nabla{\bf v}\|_{p}+\|\nabla
w\|_{p}\|\nabla{\bf v}\|_{\infty}+\|w\|_{\infty}\|\nabla^{2}{\bf v}\|_{p}$
$\displaystyle\leq C(\|{\bf v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})^{2}.$

The most subtle term in $G({\bf v},w)$ is controlled by

(2.7) $\displaystyle\|w{\bf v}\cdot\nabla{\bf v}\|_{p}$
$\displaystyle\leq\|w\|_{\infty}\|{\bf v}\|_{p}\|\nabla{\bf v}\|_{\infty}$
$\displaystyle\leq C\|{\bf v}\|_{2;p}^{2}\|w\|_{1;p}$ $\displaystyle\leq
C(\|{\bf v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})^{3},$

hence, the results follows. ∎

We want to prove the existence of solution by fixed point theorem. However, a
direct application of fixed point arguments may fail due to the term ${\bf
v}\cdot\nabla w$ on the left hand side of (2.3) which would cause the loss of
regularity on the right hand side of the system. Fortunately, we can introduce
a good enough transformation such that the problematic term disappear in the
new coordinates. Indeed, consider the following transformation
$\psi:(z_{1},z_{2})\mapsto(x_{1},x_{2})$ satisfy

(2.8) $\begin{cases}x_{1}=\psi^{1}(z_{1},z_{2})=z_{1},\\\
x_{2}=\psi^{2}(z_{1},z_{2})=z_{2}+\int_{0}^{z_{1}}\tilde{v}(t,\psi^{2}(t;z_{2}))dt,\\\
\end{cases}$

where $\tilde{v}=\frac{({\bf v+\tilde{\bf u}})^{(2)}}{\bar{U}+({\bf
v+\tilde{\bf u}})^{(1)}}$. Thanks to the smallness assumption, the
transformation $\psi:\tilde{\Omega}\rightarrow\Omega$ is a diffeomorphism.
Furthermore, if we denote

(2.9) ${\bf E}=\left(\begin{array}[]{ccc}0&0\\\
\tilde{v}(\psi(z))&\partial_{z_{2}}\int_{0}^{z_{1}}\tilde{v}(\psi(t,z_{2}))dt\\\
\end{array}\right),$

with $\nabla\psi={\bf Id}+{\bf E}$, we have the following lemma.

######  Lemma 2.2.

Suppose that $\psi$ is defined in (2.8), then $\psi$ is a diffeomorphism such
that $\psi(\tilde{\Omega})=\Omega$, moreover

(2.10) $\|{\bf E}\|_{W^{1,p}(\tilde{\Omega})}\leq E,$

here $E=E(\|{\bf v}\|_{2,p},\|\tilde{\bf u}\|_{2,p})$ can be arbitrary small
provided $\|{\bf v}\|_{2,p},\|\tilde{\bf u}\|_{2,p}$ small enough.

###### Proof.

Since $\bar{U}$ is strictly positive and $\|{\bf v}\|_{2,p},\|\tilde{\bf
u}\|_{2,p}$ is small, there exists a positive constant $c$, such that

$\bar{U}+({\bf v}+\tilde{\bf u})^{(1)}\geq c>0,$

it follows that $\tilde{v}\in W^{2,p}$ and

(2.11) $\|\tilde{v}\|_{2,p}\leq E(\|{\bf v}\|_{2,p},\|\tilde{\bf u}\|_{2,p}).$

At this stage,the first part is standard, one can refer to [23], and the
estimate (2.10) for $E_{21}$ is immediately. On the other hand,

(2.12) $\displaystyle|E_{22}|^{p}$
$\displaystyle\leq|\int_{0}^{z_{1}}\nabla\tilde{v}(\psi(t,z_{2}))\cdot\partial_{z_{2}}\psi(t,z_{2})dt|^{p}$
$\displaystyle\leq|z_{1}|^{p-1}\int_{0}^{z_{1}}|\nabla\tilde{v}(\psi(t,z_{2}))\cdot\partial_{z_{2}}\psi(t,z_{2})|^{p}dt$
$\displaystyle\leq
C\|\nabla\tilde{v}\|_{\infty}^{p}\int_{0}^{z_{1}}|1+E_{22}|^{p}dt,$

Integrating it over $\Omega$ gives

(2.13) $\|E_{22}\|_{p}\leq C\|\nabla\tilde{v}\|_{\infty}\|1+E_{22}\|_{p}.$

Differentiate $E_{22}$ with respect to $z_{1}$ we have

$\displaystyle\partial_{z_{1}}E_{22}=\nabla\tilde{v}(\psi(z_{1},z_{2}))\cdot\partial_{z_{2}}\psi(z_{1},z_{2}),$

which gives

(2.14) $\|\partial_{z_{1}}E_{22}\|_{p}\leq
C\|\nabla\tilde{v}\|_{\infty}\|1+E_{22}\|_{p}\leq E,$

similarly, differentiate $E_{22}$ with respect to $z_{2}$ we have

$\displaystyle\partial_{z_{2}}E_{22}$
$\displaystyle=\partial_{z_{2}}\int_{0}^{z_{1}}\partial_{x_{i}}\tilde{v}(\psi(t,z_{2}))\partial_{z_{2}}\psi^{i}(t,z_{2})dt$
$\displaystyle=\int_{0}^{z_{1}}\partial_{x_{i},x_{j}}\tilde{v}(\psi(t,z_{2}))\partial_{z_{2}}\psi^{j}(t,z_{2})\partial_{z_{2}}\psi^{i}(t,z_{2})dt$
$\displaystyle+\int_{0}^{z_{1}}\partial_{x_{i}}\tilde{v}(\psi(t,z_{2}))\partial^{2}_{z_{2}}\psi^{i}(t,z_{2})dt$
$\displaystyle:=I_{1}+I_{2},$

it follows that

$\displaystyle|I_{1}|^{p}$
$\displaystyle=|\int_{0}^{z_{1}}\partial_{x_{i},x_{j}}\tilde{v}(\psi(t,z_{2}))\partial_{z_{2}}\psi^{j}(t,z_{2})\partial_{z_{2}}\psi^{i}(t,z_{2})dt|^{p}$
$\displaystyle\leq\|\partial_{z_{2}}\psi\|^{2p}_{\infty}|z_{1}|^{p-1}\int_{0}^{z_{1}}|\nabla_{x}^{2}\tilde{v}|^{p}dt$
$\displaystyle\leq C\int_{0}^{z_{1}}|\nabla_{x}^{2}\tilde{v}|^{p}dt$

and

$\displaystyle|I_{2}|^{p}$
$\displaystyle=|\int_{0}^{z_{1}}\partial_{x_{i}}\tilde{v}(\psi(t,z_{2}))\partial^{2}_{z_{2}}\psi^{i}(t,z_{2})dt|^{p}$
$\displaystyle\leq
C\|\nabla\tilde{v}\|_{\infty}\int_{0}^{z_{1}}|\partial_{z_{2}}E_{22}|^{p}$

which implies

(2.15) $\|\partial_{z_{2}}E_{22}\|_{p}\leq
C(\|\tilde{v}\|_{2,p}+\|\nabla\tilde{v}\|_{\infty}\|\partial_{z_{2}}E_{22}\|_{p}),$

finally, combine all those inequality together the estimate (2.10) for
$E_{22}$ is immediate and the proof thus is completed. ∎

######  Remark 2.3.

Since the transformation $\psi$ is a diffeomorphism, there exists a inverse
transform $\phi$ such that $\psi\circ\phi={\bf Id}$, and

(2.16) $\nabla\phi=\nabla\psi^{-1}={\bf Id}+\tilde{\bf E}.$

Denote $J=|\nabla\psi|$, then $\nabla\psi^{-1}$ can be explicitly computed as

(2.17)
$\nabla\psi^{-1}=\frac{1}{J}\left(\begin{array}[]{ccc}1+\partial_{z_{2}}\int_{0}^{z_{1}}\tilde{v}(\psi(t,z_{2}))dt&0\\\
-\tilde{v}(\psi(z))&1\\\ \end{array}\right),$

hence, the estimates (2.10) also holds for $\tilde{\bf E}$. We also denote the
transform $\psi$ as $\psi_{{\bf v}+\tilde{\bf u}}$ to emphasize that
$\psi_{{\bf v}+\tilde{\bf u}}$ is induced by $\tilde{v}$ through (2.8).

Observe that the most important property of the transform $\psi$ is

(2.18)
$\partial_{z_{1}}=\frac{\partial{x_{1}}}{\partial{z_{1}}}\partial_{x_{1}}+\frac{\partial{x_{2}}}{\partial{z_{1}}}\partial_{x_{2}}=\partial_{x_{1}}+\tilde{v}\partial_{x_{2}},$

where $\tilde{v}=\frac{({\bf v+\tilde{\bf u}})^{(2)}}{\bar{U}+({\bf
v+\tilde{\bf u}})^{(1)}}$. Hence we can change variable and rewrite the system
(2.3) in coordinates z, which leads to

(2.19) $\displaystyle((\bar{U}+({\bf v+\tilde{\bf u}})^{(1)})\circ\psi_{{\bf
v}+\tilde{\bf u}})\partial_{z_{1}}w+\operatorname{div}_{z}{\bf
v}=\tilde{F}({\bf v},w)\quad\text{ in }\tilde{\Omega},$
$\displaystyle(\bar{U}\circ\psi_{{\bf v}+\tilde{\bf u}})\partial_{z_{1}}{\bf
v}+{\bf{v}}\cdot\nabla_{z}U_{0}+\gamma\nabla_{z}w-\mu\Delta_{z}{\bf
v}-(\mu+\nu)\nabla\operatorname{div}_{z}{\bf v}=\tilde{G}({\bf
v},w)\quad\text{ in }\tilde{\Omega},$ $\displaystyle{\bf v}=0\quad\text{ on
}\tilde{\Gamma}_{\text{in }}\cup\tilde{\Gamma}_{\text{out }},$ $\displaystyle
w=\rho_{\text{in }}-1\quad\text{ on }\tilde{\Gamma}_{\text{in }},$
$\displaystyle{\bf v}\cdot\mathbf{n}=0\quad\text{ on }\tilde{\Gamma}_{0},$
$\displaystyle 2\mu\mathbf{n}\cdot D_{z}({\bf v})\cdot\tau+\alpha{\bf
v}\cdot\tau=\tilde{B}\quad\text{ on }\tilde{\Gamma}_{0}.$

Here $\tilde{B}=B-2\mu{\bf n}\cdot R({\bf v},D)\cdot\tau$, and the function
$R(\cdot,\cdot)$ with the first variable denotes a function and the second is
a differential operator representing the differences of the differential
operator acting on the function in x-coordinates and z-coordinates. For
instance, $R({\bf v},D)=D_{x}{\bf v}-D_{z}{\bf v}$, and we also have

(2.20) $\displaystyle\tilde{F}({\bf v},w)=$ $\displaystyle F({\bf v},w)-R({\bf
v},\operatorname{div})$ $\displaystyle\tilde{G}({\bf v},w)=$ $\displaystyle
G({\bf v},w)-\bar{U}R({\bf v},\partial_{1})-{\bf v}\cdot R(U_{0},\nabla)$
$\displaystyle-$ $\displaystyle\gamma R(w,\nabla)+\mu R({\bf
v},\Delta)+(\mu+\nu)R({\bf v},\nabla\operatorname{div}).$

######  Lemma 2.4.

Let $\tilde{F}$ and $\tilde{G}$ are given by (2.20).Then we have

(2.21) $\displaystyle\|\tilde{F}({\bf v},w)\|_{W^{1,p}}+\|\tilde{G}({\bf
v},w)\|_{L^{p}}$ $\displaystyle\leq C[(\|{\bf
v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})^{3}+(\|{\bf
v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})^{2}]$ $\displaystyle+E(D_{0})(\|{\bf
v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})+\|\tilde{\bf u}\|_{W^{2,p}}$

###### Proof.

A direct computation shows that under the change of variable
$\psi:(z_{1},z_{2})\mapsto(x_{1},x_{2})$, there holds

(2.22) $\displaystyle\partial_{z_{i}}v^{l}$
$\displaystyle=\partial_{x_{i}}v^{l}+E_{ik}\partial_{x_{k}}v^{l},$
$\displaystyle\partial^{2}_{z_{i}z_{j}}v^{l}$
$\displaystyle=\partial^{2}_{x_{i}x_{j}}v^{l}+E_{ik}\partial^{2}_{x_{j}x_{k}}v^{l}+\partial_{x_{j}}E_{ik}\partial_{x_{k}}v^{l}$
$\displaystyle+E_{jk}\partial^{2}_{x_{i}x_{k}}v^{l}+E_{jm}E_{ik}\partial^{2}_{x_{k}x_{m}}v^{l}+E_{jm}\partial_{x_{m}}E_{ik}\partial_{x_{k}}v^{l}$

where $1\leq i,j,l\leq 2$ are integers, and the repeated subscribe always sum
over it’s index set. The above estimates combine with (2.5) and Lemma 2.1 the
results is immediate. ∎

##  3\.  a priori estimates

In this section, we construct a sequence that will converge to a solution of
the nonlinear system (2.19).The solution sequence is defined as follows:

(3.1) $\displaystyle((\bar{U}+({{\bf v}^{n}+\tilde{\bf
u}})^{(1)})\circ\psi_{{\bf v}^{n}+\tilde{\bf
u}})\partial_{z_{1}}w^{n+1}+\operatorname{div}_{z}{\bf v}^{n+1}=\tilde{F}({\bf
v}^{n},w^{n})\quad\text{ in }\tilde{\Omega},$
$\displaystyle(\bar{U}\circ\psi_{{\bf v}^{n}+\tilde{\bf
u}})\partial_{z_{1}}{\bf
v}^{n+1}+{\bf{v}}^{n+1}\cdot\nabla_{z}U_{0}+\gamma\nabla_{z}w^{n+1}$
$\displaystyle-\mu\Delta_{z}{\bf
v}^{n+1}-(\mu+\nu)\nabla\operatorname{div}_{z}{\bf v}^{n+1}=\tilde{G}({\bf
v}^{n},w^{n})\quad\text{ in }\tilde{\Omega},$ $\displaystyle{\bf
v}^{n+1}=0\quad\text{ on }\tilde{\Gamma}_{\text{in
}}\cup\tilde{\Gamma}_{\text{out }},$ $\displaystyle
w^{n+1}=\rho_{in}-1\quad\text{ on }\tilde{\Gamma}_{in}$ $\displaystyle{\bf
v}^{n+1}\cdot\mathbf{n}=0\quad\text{ on }\tilde{\Gamma}_{0},$ $\displaystyle
2\mu\mathbf{n}\cdot D_{z}({\bf v}^{n+1})\cdot\tau+\alpha{\bf
v}^{n+1}\cdot\tau=\tilde{B}({\bf v}^{n})\quad\text{ on }\tilde{\Gamma}_{0},$

where $\tilde{B}({\bf v}^{n})=B-2\mu{\bf n}\cdot R({\bf v}^{n},D)\cdot\tau$.
To show the existence of the solutions to (3.1), we firstly deal with the
following linear system:

(3.2) $\displaystyle((\bar{U}+\bar{\bf v}^{(1)})\circ\psi_{\bar{\bf
v}})\partial_{z_{1}}w+\operatorname{div}{\bf v}=f\quad\text{ in
}\tilde{\Omega},$ $\displaystyle(\bar{U}\circ\psi_{\bar{\bf
v}})\partial_{z_{1}}{\bf v}+{\bf{v}}\cdot\nabla U_{0}+\gamma\nabla
w-\mu\Delta{\bf v}-(\mu+\nu)\nabla\operatorname{div}{\bf v}=g\quad\text{ in
}\tilde{\Omega},$ $\displaystyle{\bf v}=0\quad\text{ on
}\tilde{\Gamma}_{\text{in }}\cup\tilde{\Gamma}_{\text{out }},$ $\displaystyle
w=w_{in}\quad\text{ on }\tilde{\Gamma}_{\text{in }},$ $\displaystyle{\bf
v}\cdot\mathbf{n}=0\quad\text{ on }\tilde{\Gamma}_{\text{0 }},$ $\displaystyle
2\mu\mathbf{n}\cdot D_{z}({\bf v})\cdot\tau+\alpha{\bf
v}\cdot\tau=\tilde{B}\quad\text{ on }\tilde{\Gamma}_{0},$

where $f,g\in L^{2}(\Omega)$, $w_{in}\in L^{2}(\Gamma_{in}),\tilde{B}\in
L^{2}(\Gamma_{0})$ are given functions. Without the risk of confusing, we
readily remove the superscript of the domains.

###  3.1. Solution of mass equation

One of the most important advantage of changing variable is that we can define
a operator $S:v\mapsto w$ as the solution of equation

(3.3) $\begin{cases}((\bar{U}+\bar{\bf v}^{(1)})\circ\psi_{\bar{\bf
v}})\partial_{z_{1}}w=v\quad\quad\text{in}\,\Omega,\\\
w=w_{in}(z_{2})\quad\quad on\,\Gamma_{in}.\end{cases}$

Indeed,for a continuous function $v$, setting

(3.4) $S(v)(z):=w_{in}(z_{2})+\int_{0}^{z_{1}}\frac{v}{(\bar{U}+\bar{\bf
v}^{(1)})\circ\psi_{\bar{\bf v}}}(t,z_{2})dt,$

then it is easy to verfy that $S(v)$ satisfies (3.3). Moreover,we have the
following estimates.

######  Lemma 3.1.

Let $S$ be defined in (3.4), $w_{in}\in L^{2}(\Gamma_{in})$ is given then

(3.5) $\|S(v)\|_{L^{\infty}(L^{2})(\Omega)}\leq
C(|w_{in}|_{L^{2}(\Gamma_{in})}+\|v\|_{L^{2}(\Omega)})$

###### Proof.

Given $z_{1}$, let $\Omega_{z_{1}}$ denote the $z_{1}$-cut of $\Omega$, then

(3.6) $\displaystyle|S(v)|^{2}_{L^{2}(\Omega_{z_{1}})}$
$\displaystyle=\int_{0}^{1}[w_{in}(z_{2})+\int_{0}^{z_{1}}\frac{v}{(\bar{U}+\bar{\bf
v}^{(1)})\circ\psi_{\bar{\bf v}}}(t,z_{2})dt]^{2}dz_{2}$ $\displaystyle\leq
C[|w_{in}|^{2}_{L^{2}(\Gamma_{in})}+\|v\|_{L^{2}(\Omega)}],$

where we have used

$\bar{U}+({\bf v}+\tilde{\bf u})^{(1)}\geq c>0,$

for some positive constant $c>0$, and the estimates (3.5) follows. ∎

######  Remark 3.2.

By density arguments one can easily extend $S$ to $L^{2}(\Omega)$ which also
preserves the same estimates as (3.5).

###  3.2. $H^{1}$ estimates for linear system

######  Lemma 3.3.

Let $\bar{\bf v}\in W^{2,p}(\Omega)$ and $\|\bar{\bf v}\|_{2,p}$ be small
enough, the viscous coefficients $\mu>\frac{1}{2\pi^{2}}$. Suppose that $({\bf
v},w)$ be a solution to the system (3.2) with given $(f,g,\tilde{B},w_{in})\in
L^{2}(\Omega)\times V^{*}\times L^{2}(\Gamma_{0})\times L^{2}({\Gamma_{in}})$,
then

(3.7) $\|{\bf v}\|_{H^{1}}+\|w\|_{L^{\infty}(L^{2})}\leq
C(\|f\|_{L^{2}}+\|g\|_{V^{*}}+\|\tilde{B}\|_{L^{2}(\Gamma_{0})}+\|w_{in}\|_{L^{2}({\Gamma_{in}})})$

where

$V:=\\{{\bf v}\in W^{1,2}(\Omega):{\bf
v}=0\,\text{on}\,\Gamma_{in}\cup\Gamma_{out},\,{\bf v}\cdot
n|_{\Gamma_{0}}=0\\}$

and $V^{*}$ is the dual space of V.

###### Proof.

For any $\phi\in V$ we have

$\displaystyle\int_{\Omega}(-\mu\Delta{\bf
v}-(\mu+\nu)\nabla\operatorname{div}{\bf v})\cdot\phi dz$
$\displaystyle=\int_{\Omega}(2\mu D({\bf
v}):\nabla\phi+\nu\operatorname{div}{\bf v}\operatorname{div}\phi)dz$
$\displaystyle-\int_{\partial\Omega}2\mu{\bf n}\cdot D({\bf v})\cdot\phi ds.$

Then multiplying (3.2) by ${\bf v}$ and integrating over $\Omega$ we have

(3.8) $\displaystyle-$
$\displaystyle\frac{1}{2}\int_{\Omega}\partial_{z_{1}}(\bar{U}\circ\psi_{\bar{\bf
v}})|{\bf v}|^{2}dz+\int_{\Omega}({\bf v}\cdot\nabla U_{0})\cdot{\bf
v}dz-\gamma\int_{\Omega}w\operatorname{div}{\bf v}dz$ $\displaystyle+$
$\displaystyle 2\mu\int_{\Omega}|D({\bf
v})|^{2}dz+\nu\int_{\Omega}|\operatorname{div}{\bf
v}|^{2}dz+\int_{\Gamma_{0}}\alpha|{\bf
v}|^{2}ds=\int_{\Gamma_{0}}\tilde{B}{\bf v}\cdot\tau+\int_{\Omega}g\cdot{\bf
v}.$

For the first term, we have

(3.9) $|\partial_{z_{1}}(\bar{U}\circ\psi_{\bar{\bf
v}})|=|\partial_{x_{1}}\bar{U}+E_{1k}\partial_{x_{k}}\bar{U}|=|E_{12}|\leq
E(\|{\bar{\bf v}}\|_{W^{2,p}}).$

As to the second term in the left hand side of (3.8), by applying the
Poincar$\acute{e}$’s inequality we get

$|\int_{\Omega}({\bf v}\cdot\nabla U_{0})\cdot{\bf
v}dz|=|\int_{\Omega}v^{(1)}v^{(2)}dz\leq\frac{1}{2}\|{\bf
v}\|^{2}_{L^{2}}\leq\frac{1}{2}\|\nabla{\bf v}\|^{2}_{L^{2}}.$

The well known Korn inequality in [17, 8] shows that

(3.10) $\mu\pi^{2}\|\nabla{\bf v}\|^{2}_{L^{2}}\leq 2\mu\int_{\Omega}|D({\bf
v})|^{2}dz+\nu\int_{\Omega}|\operatorname{div}{\bf v}|^{2}dz,$

here the appearance of constant $\pi$ concerning the best constant in Korn
inequality. Involving in the continuity equation in (3.2) we have

(3.11) $\displaystyle-\int_{\Omega}w\operatorname{div}{\bf v}$ $\displaystyle
dz=\int_{\Omega}w((\bar{U}+\bar{\bf v}^{(1)})\circ\psi_{\bar{\bf
v}})\partial_{z_{1}}wdz-\int_{\Omega}wfdz$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{\Omega}\partial_{z_{1}}((\bar{U}+\bar{\bf
v}^{(1)})\circ\psi_{\bar{\bf
v}})w^{2}dz-\frac{1}{2}\int_{\Gamma_{in}}((\bar{U}+\bar{\bf
v}^{(1)})\circ\psi_{\bar{\bf v}})w_{in}^{2}ds$
$\displaystyle+\frac{1}{2}\int_{\Gamma_{out}}((\bar{U}+\bar{\bf
v}^{(1)})\circ\psi_{\bar{\bf v}})w^{2}ds-\int_{\Omega}wfdz$

Due to the smallness assumption of ${\bar{\bf v}}$, the integral over
$\Gamma_{out}$ will be nonnegative, and we also have

(3.12) $|\partial_{z_{1}}(\bar{\bf v}^{(1)}\circ\psi_{\bar{\bf
v}})|=|\partial_{x_{1}}\bar{\bf v}^{(1)}+E_{1k}\partial_{x_{k}}\bar{\bf
v}^{(1)}|\leq E(\|{\bar{\bf v}}\|_{W^{2,p}}).$

Combing all those estimates together we arrive at

(3.13) $\displaystyle\ [\mu\pi^{2}$
$\displaystyle-(\frac{1}{2}+\frac{E(\|\bar{\bf
v}\|_{W^{2,p}})}{2})]|\nabla{\bf v}\|^{2}_{L^{2}}$
$\displaystyle\leq\|g\|_{V^{*}}\|{\bf
v}\|_{L^{2}}+|\tilde{B}|_{L^{2}({\Gamma_{0}})}\|{\bf
v}\|_{L^{2}({\Gamma_{0}})}+E\|w\|^{2}_{L^{2}}$
$\displaystyle+C|w_{in}|^{2}_{L^{2}(\Gamma_{in})}+\|w\|_{L^{2}}\|f\|_{L^{2}}$

Observe that the left hand side of (3.13) will be positive provided $\|{\bf
v}\|_{W^{2,p}}$ is small enough and the viscosity efficiencies
$\mu>\frac{1}{2\pi^{2}}$. Finally by substituting $v=f-\operatorname{div}{\bf
v}$ in (3.5) yields

(3.14) $\|w\|_{L^{\infty}(L^{2})}\leq
C(|w_{in}|_{L^{2}(\Gamma_{in})}+\|\nabla{\bf
v}\|_{L^{2}(\Omega)}++\|f\|_{L^{2}(\Omega)}).$

Combing (3.13) and (3.14) by using trace theorem and Poincar$\acute{e}$’s
inequality we get the estimates (3.7). ∎

Next, we want to show higher regularity up to boundary by the method of
difference quotient.

######  Lemma 3.4.

Suppose that $({\bf v},w)\in H^{1}\times L^{\infty}(L^{2})$ be a solution to
the the system (3.2) with $(f,g,\tilde{B},w_{in})\in H^{1}(\Omega)\times
L^{2}(\Omega)\times H^{\frac{1}{2},2}(\Gamma_{0})\times H^{1}({\Gamma_{in}})$,
then we have

(3.15) $\|w\|_{H^{1}(\Omega)}+\|{\bf v}\|_{H^{2}(\Omega)}\leq
C(\|f\|_{H^{1}}+\|g\|_{L^{2}}+|\tilde{B}|_{H^{\frac{1}{2},2}(\Gamma_{0})}+|w_{in}|_{H^{1}({\Gamma_{in}})}).$

###### Proof.

Let us firstly focus on the interior estimates. By using the standard $L^{2}$
theory of elliptic system, we only need to find a bound of
$\|\partial_{z_{2}}w\|_{L^{2}}$. To this end, define the difference quotient
operator along $z_{2}$ direction $\Delta_{2}^{-h}$ as

$\Delta_{2}^{-h}\phi:=-\frac{\phi(z_{1},z_{2}-h)-\phi(z_{1},z_{2})}{h}.$

Effecting $\Delta_{2}^{h}$ on the mass equation in (3.2), we have

(3.16) $(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1}}(\Delta_{2}^{h}w)+\partial_{z_{1}}w^{h}\Delta_{2}^{h}(\bar{U}+\bar{\bf
v}^{(1)})+\operatorname{div}(\Delta_{2}^{h}{\bf v})=\Delta_{2}^{h}f,$

where $w^{h}(z_{1},z_{2})=w(z_{1},z_{2}+h)$. Hence, in order to find a bound
of $\|\partial_{z_{2}}w\|_{L^{2}}$, it is sufficient to find a bound of
$\|\partial_{z_{2}}{\bf v}\|^{2}_{H^{1}}$. Since $({\bf v},w)\in H^{1}\times
L^{\infty}(L^{2})$ be a solution of the system (3.2), which means

(3.17) $\displaystyle\int_{\Omega}\bar{U}\partial_{z_{1}}{\bf v}$
$\displaystyle\cdot\phi dz+\int_{\Omega}({\bf v}\cdot\nabla U_{0})\cdot\phi
dz-\gamma\int_{\Omega}w\operatorname{div}\phi dz+\mu\int_{\Omega}\nabla{\bf
v}:\nabla\phi dz$ $\displaystyle+(\mu+\nu)\int_{\Omega}\operatorname{div}{\bf
v}\operatorname{div}\phi dz=\int_{\Omega}g\cdot\phi dz$

holds for any $\phi\in C_{c}^{\infty}(\Omega)$. There is no risk of confusing,
we may remove the superposition of $\psi$. Replace $\phi$ in (3.17) by
$\Delta_{2}^{-h}\phi$, then we have

(3.18) $\displaystyle-$
$\displaystyle\gamma\int_{\Omega}w\operatorname{div}(\Delta_{2}^{-h}\phi)dz+\mu\int_{\Omega}\nabla{\bf
v}:\nabla(\Delta_{2}^{-h}\phi)dz+(\mu+\nu)\int_{\Omega}\operatorname{div}{\bf
v}\operatorname{div}(\Delta_{2}^{-h}\phi)dz$ $\displaystyle=$
$\displaystyle\gamma\int_{\Omega}(\Delta_{2}^{h}w)\operatorname{div}\phi
dz-\mu\int_{\Omega}\nabla(\Delta_{2}^{h}{\bf v}):\nabla\phi
dz-(\mu+\nu)\int_{\Omega}\operatorname{div}(\Delta_{2}^{h}{\bf
v})\operatorname{div}\phi dz$ $\displaystyle=$
$\displaystyle-\int_{\Omega}\bar{U}\partial_{z_{1}}{\bf
v}\cdot(\Delta_{2}^{-h}\phi)dz-\int_{\Omega}({\bf v}\cdot\nabla
U_{0})\cdot(\Delta_{2}^{-h}\phi)dz+\int_{\Omega}g\cdot\Delta_{2}^{-h}\phi.$

For any interval $I\subset\subset(0,1)$, selecting a cut-off function
$\eta(t)\in C_{c}^{\infty}(0,1)$, such that $0\leq\eta\leq 1$ and $\eta\equiv
1$ in $I$. Substituting $\phi=\eta^{2}(z_{2})\Delta_{2}^{h}{\bf v}$ in (3.18)
and combing with (3.16), a direct computation shows that

(3.19)
$\displaystyle\gamma\int_{\Omega}(\Delta_{2}^{h}w)\operatorname{div}\phi dz$
$\displaystyle=\gamma\int_{\Omega}(\Delta_{2}^{h}w)[2\eta(\Delta_{2}^{h}{\bf
v})\cdot\nabla\eta+\eta^{2}\operatorname{div}\Delta_{2}^{h}{\bf v}]dz$
$\displaystyle=\gamma\int_{\Omega}(\Delta_{2}^{h}w)[2\eta(\Delta_{2}^{h}{\bf
v})\cdot\nabla\eta-\eta^{2}(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1}}(\Delta_{2}^{h}w)]dz$
$\displaystyle+\gamma\int_{\Omega}(\Delta_{2}^{h}w)[\eta^{2}\Delta_{2}^{h}f-\eta^{2}\partial_{z_{1}}w^{h}\Delta_{2}^{h}(\bar{U}+\bar{\bf
v}^{(1)})]dz$
$\displaystyle=\gamma\int_{\Omega}(\Delta_{2}^{h}w)[2\eta(\Delta_{2}^{h}{\bf
v})\cdot\nabla\eta+\eta^{2}\Delta_{2}^{h}f-\eta^{2}\partial_{z_{1}}w^{h}\Delta_{2}^{h}(\bar{U}+\bar{\bf
v}^{(1)})]dz$
$\displaystyle+\frac{\gamma}{2}\int_{\Omega}\partial_{z_{1}}(\bar{U}+\bar{\bf
v}^{(1)})|\eta\Delta_{2}^{h}w|^{2}-\frac{\gamma}{2}\int_{\Gamma_{out}}(\bar{U}+\bar{\bf
v}^{(1)})|\eta\Delta_{2}^{h}w|^{2}ds,$
$\displaystyle+\frac{\gamma}{2}\int_{\Gamma_{in}}(\bar{U}+\bar{\bf
v}^{(1)})|\eta\Delta_{2}^{h}w|^{2}ds$

and

(3.20) $\displaystyle-\mu$
$\displaystyle\int_{\Omega}\nabla(\Delta_{2}^{h}{\bf v}):\nabla\phi
dz-(\mu+\nu)\int_{\Omega}\operatorname{div}(\Delta_{2}^{h}{\bf
v})\operatorname{div}\phi dz$ $\displaystyle=$
$\displaystyle-\mu\int_{\Omega}\eta^{2}|\nabla(\Delta_{2}^{h}{\bf
v})|^{2}dz-(\mu+\nu)\int_{\Omega}\eta^{2}|\operatorname{div}(\Delta_{2}^{h}{\bf
v})|^{2}dz$ $\displaystyle+$ $\displaystyle
2\mu\int_{\Omega}\eta\nabla(\Delta_{2}^{h}{\bf
v}):(\nabla\eta\Delta_{2}^{h}{\bf
v})dz+2(\mu+\nu)\int_{\Omega}\eta\operatorname{div}(\Delta_{2}^{h}{\bf
v})\Delta_{2}^{h}{\bf v}\cdot\nabla\eta dz,$

Combing (3.18)-(3.20), we obtain

(3.21) $\displaystyle\int_{\Omega}\eta^{2}|\nabla(\Delta_{2}^{h}{\bf v})|^{2}$
$\displaystyle\leq C(\delta\|\eta\nabla(\Delta_{2}^{h}){\bf
v}\|^{2}_{L^{2}}+\|{\bf
v}\|^{2}_{H^{1}}+\|\partial_{z_{2}}f\|^{2}_{L^{2}}+\|g\|^{2}_{L^{2}}+|\partial_{z_{2}}w_{in}|^{2}_{L^{2}(\Gamma_{in})})$
$\displaystyle\leq
C(\|f\|^{2}_{H^{1}}+\|g\|^{2}_{L^{2}}+\|\tilde{B}\|^{2}_{L^{2}(\Gamma_{0})}+\|w_{in}\|^{2}_{H^{1}({\Gamma_{in}})}),$

here $\delta$ is a arbitrary small constant. Letting $h\rightarrow 0$ in
$(\ref{def-quo-5})$, we get

$\|\eta\partial_{z_{2}}{\bf v}\|^{2}_{{H^{1}}(\Omega)}\leq
C(\|f\|^{2}_{{H^{1}}(\Omega)}+\|g\|^{2}_{{L^{2}}(\Omega)}+\|\tilde{B}\|^{2}_{L^{2}(\Gamma_{0})}+\|w_{in}\|^{2}_{H^{1}({\Gamma_{in}})}),$

denote $\Omega^{\prime}=(0,1)\times I$, which implies that

(3.22) $\|\partial_{z_{2}}{\bf v}\|^{2}_{{H^{1}}(\Omega^{\prime})}\leq
C(\|f\|^{2}_{{H^{1}}(\Omega)}+\|g\|^{2}_{{L^{2}}(\Omega)}+\|\tilde{B}\|^{2}_{L^{2}(\Gamma_{0})}+\|w_{in}\|^{2}_{H^{1}({\Gamma_{in}})}).$

On the other hands, (3.16) implies that

(3.23) $\|\partial_{z_{2}}w\|_{L^{2}(\Omega^{\prime})}\leq
C(\|\partial_{z_{2}}{\bf
v}\|_{{H^{1}}(\Omega^{\prime})}+\|f\|_{{H^{1}}(\Omega)}+|\partial_{z_{2}}w(0,z_{2})|_{L^{2}(\Gamma_{in})}),$

which combing with the standard $H^{2}$ estimates of elliptic system gives the
interior estimate of (3.15) in $\Omega^{\prime}$.
As to the boundary estimates near
$\Gamma_{0}^{-}:=\\{(z_{1},z_{2})|0<z_{1}<1,z_{2}=0\\}$, we extend the domain
$\Omega$ to $\Omega^{*}=(0,1)\times(-1,1)$, and denote the even extension of
$w,v_{1}$, the odd extension of $v_{2}$ with respect to $z_{1}=0$ as
$w^{*},v_{1}^{*},v_{2}^{*}$ respectively. More precisely,

$\displaystyle v_{1}^{*}(z_{1},z_{2})$
$\displaystyle=\begin{cases}v_{1}(z_{1},z_{2})\quad\quad\,0<z_{2}<1,\\\
\hat{v}_{1}(z_{1},z_{2})=v_{1}(z_{1},-z_{2})\quad-1<z_{2}<0,\end{cases}$
$\displaystyle v_{2}^{*}(z_{1},z_{2})$
$\displaystyle=\begin{cases}v_{2}(z_{1},z_{2})\quad\quad\,\,0<z_{2}<1,\\\
\hat{v}_{2}(z_{1},z_{2})=-v_{2}(z_{1},-z_{2})\quad-1<z_{2}<0.\end{cases}$

Due to the boundary condition in (3.2), we have $v_{2}=0$ on $\Gamma_{0}^{-}$
and the Navier boundary conditions preserve this symmetry, that is to say

$2\mu\mathbf{n}\cdot D({\bf v})\cdot\tau+\alpha{\bf
v}\cdot\tau=2\mu\hat{\mathbf{n}}\cdot D({\hat{\bf
v}})\cdot\tau+\alpha{\hat{\bf v}}\cdot\tau\quad\text{on}\,\,\Gamma_{0}^{-},$

where $\hat{\mathbf{n}}$ is the outer normal vector of the domain
$(0,1)\times(-1,0)$ on $\Gamma_{0}^{-}$, which implies that

$\partial_{z_{2}}v_{1}^{*}=-\frac{\tilde{B}}{\mu}+\frac{\alpha}{\mu}v_{1}+\partial_{1}v_{2}\quad\text{on}\,\,\Gamma_{0}^{-}.$

Hence the extended function $(v_{1}^{*},v_{2}^{*})$ still belongs to
$H^{1}(\Omega^{*})$, and it is easy to check that $({\bf v}^{*},w^{*})$
satisfy the system (3.2) in the senses of $\mathfrak{D}^{\prime}(\Omega^{*})$
with corresponding extension of source terms. It follows that the estimates
near the boundary $\Gamma_{0}^{-}$ is converted to an interior one, a slight
modification of the above proof derives the desired estimates. An identical
argument can be applied on
$\Gamma_{0}^{+}:=\\{(z_{1},z_{2})|0<z_{1}<1,z_{2}=1\\}$ and we get the global
regularity as claimed. ∎

###  3.3. $W^{2,p}$ estimates of linear system

The crucial point to establish the $W^{2,p}$ estimates is to find a bound of
$\|\partial_{z_{2}}w\|_{L^{p}}$, a key observation shows that it can be
bounded by $\|\operatorname{curl}\partial_{z_{2}}{\bf v}\|_{L^{p}}$. To this
end, we introduce the following lemma concerning the so called Bogovskii
operator which is proved by Bogovskii in [1].

######  Lemma 3.5.

Suppose that $\Omega\subset\mathbb{R}^{n}$ is starlike with respect to some
ball contained in it, and that $1<p<\infty$ and $n\geq 2$, Then, the exists a
constant $C>0$, depending only on $n,p$ and $\Omega$, such that for any $h\in
L^{p}(\Omega)$ with $\int_{\Omega}h(x)dx=0$, there is a vector field
$\omega\in W_{0}^{1,p}$ satisfying

$\displaystyle\begin{cases}\operatorname{div}\omega=h\quad&\text{in}\quad\Omega,\\\
\omega=0\quad&\text{on}\quad\partial\Omega.\end{cases}$

and

$\|\omega\|_{W^{1,p}(\Omega)}\leq C\|h\|_{L^{p}(\Omega)},$

where the constant $C$ depends only on $p$ and $\Omega$.

By constructing a proper test function defined by Bogovskii operator, we can
establish the following results.

######  Lemma 3.6.

Assume that $({\bf v},w)$ be a solution of (3.2) with
$(f,g,\tilde{B},w_{in})\in W^{1,p}(\Omega)\times L^{p}(\Omega)\times
H^{\frac{1}{2},2}(\Gamma_{0})\times W^{1,p}(\Gamma_{in})$, then for any
$p\in(2,\infty)$ we have

(3.24) $\displaystyle\|$
$\displaystyle\partial_{z_{2}}w\|_{L^{p}}+|\partial_{z_{2}}w|_{L^{p}(\Gamma_{out})}$
$\displaystyle\leq C(\|\operatorname{curl}\partial_{z_{2}}{\bf
v}\|_{L^{p}}+\|f\|_{W^{1,p}}+\|g\|_{L^{p}}+|\tilde{B}|^{2}_{H^{\frac{1}{2},2}(\Gamma_{0})}+|w_{in}|_{W^{1,p}({\Gamma_{in}})})$

###### Proof.

Setting $\phi=\mathfrak{B}(\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}-M)$,
then by the properties of Bogovskii operator we have

(3.25) $\|\phi\|_{W^{1,\frac{p}{p-1}}(\Omega)}\leq
C\|\partial_{z_{2}}w\|^{p-1}_{L^{p}(\Omega)},$

where $M=\int_{\Omega}\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}dz$. Next,
differentiate the momentum equation in (3.2) with respect to $z_{2}$ and
multiply the resulting equations by $\phi$ we have

(3.26) $\displaystyle-$
$\displaystyle\gamma\int_{\Omega}|\partial_{z_{2}}w|^{p}dz+\gamma
M\int_{\Omega}|\partial_{z_{2}}w|dz+(2\mu+\nu)\int_{\Omega}\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}\operatorname{div}\partial_{z_{2}}{\bf
v}dz$ $\displaystyle+$
$\displaystyle\mu\int_{\Omega}\partial_{z_{2}}(\operatorname{curl}{\bf
v}):\nabla\phi
dz+(\mu+\nu)M\int_{\Omega}\operatorname{div}\partial_{z_{2}}{\bf
v}dz+\int_{\Omega}(\partial_{z_{2}}\bar{U})\partial_{z_{1}}{\bf v}\cdot\phi
dz$ $\displaystyle+$
$\displaystyle\int_{\Omega}\bar{U}(\partial_{z_{1},z_{2}}{\bf v})\cdot\phi
dz+\int_{\Omega}\partial_{z_{2}}{\bf v}\nabla U_{0}\phi dz+\int_{\Omega}{\bf
v}\cdot\nabla(\partial_{z_{2}}U_{0})\phi
dz=-\int_{\Omega}g\partial_{z_{2}}\phi dz$

Consequently,

(3.27) $\displaystyle\gamma\int_{\Omega}$
$\displaystyle|\partial_{z_{2}}w|^{p}dz-(2\mu+\nu)\int_{\Omega}\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}\operatorname{div}\partial_{z_{2}}{\bf
v}dz$ $\displaystyle\leq(\|\operatorname{curl}\partial_{z_{2}}{\bf
v}\|_{L^{p}}+\|g\|_{L^{p}}+\|{\bf
v}\|_{W^{1,p}})\|\partial_{z_{2}}w\|^{p-1}_{L^{p}}$
$\displaystyle+(\|\partial_{z_{2}}w\|_{L^{1}}+\|\nabla^{2}{\bf
v}\|_{L^{1}})\|\partial_{z_{2}}w\|^{p-1}_{L^{p-1}}.$

To control the second term of the l.h.s of (3.27), we differentiate the mass
equation in (3.2) to obtain

(3.28) $\partial_{z_{2}}(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1}}w+(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1},z_{2}}w+\operatorname{div}\partial_{z_{2}}{\bf
v}=\partial_{z_{2}}f$

and hence

(3.29) $\displaystyle-$
$\displaystyle\int_{\Omega}\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}\operatorname{div}\partial_{z_{2}}{\bf
v}dz$
$\displaystyle=\int_{\Omega}\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}[\partial_{z_{2}}(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1}}w+(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1},z_{2}}w-\partial_{z_{2}}f]dz$
$\displaystyle=\int_{\Omega}\partial_{z_{2}}(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}\partial_{z_{1}}wdz-\int_{\Omega}\partial_{z_{2}}w|\partial_{z_{2}}w|^{p-2}\partial_{z_{2}}f$
$\displaystyle-\frac{1}{p}\int_{\Omega}\partial_{z_{1}}(\bar{U}+\bar{\bf
v}^{(1)})|\partial_{z_{2}}w|^{p}+\int_{\Gamma_{out}}(\bar{U}+\bar{\bf
v}^{(1)})|\partial_{z_{2}}w|^{p}ds-\int_{\Gamma_{in}}(\bar{U}+\bar{\bf
v}^{(1)})|\partial_{z_{2}}w|^{p}ds.$

Recall that $\bar{U}(z_{1},z_{2})=\bar{U}\circ\psi_{\bar{\bf v}}$ and the
definition of $\psi_{\bar{\bf v}}$, due to the smallness of
$\|\partial_{z_{1}}\bar{\bf v}^{(1)}\|_{\infty}$, we have

$|\partial_{z_{1}}(\bar{U}+\bar{\bf v}^{(1)})|\leq E,$

which combine with $(\ref{w2p-2})$ and $(\ref{w2p-4})$, after using Young’s
inequality and (3.15) we arrive at

(3.30)
$\displaystyle\int_{\Omega}|\partial_{z_{2}}w|^{p}dz+\int_{\Gamma_{out}}|\partial_{z_{2}}w|^{p}ds$
$\displaystyle\leq(\|\operatorname{curl}\partial_{z_{2}}{\bf
v}\|^{p}_{L^{p}}+\delta\|\partial_{z_{1}}w\|^{p}_{L^{p}}+\|g\|^{p}_{L^{p}}+\|f\|^{p}_{W^{1,p}}+|w_{in}|^{p}_{W^{1,p}}(\Gamma_{in})),$

here $\delta$ is a small constant depending on $\|{\bar{\bf v}}\|_{2,p}$,
which yields $(\ref{w2p})$. ∎

The $W^{2,p}$ estimates is done if $\|\operatorname{curl}\partial_{z_{2}}{\bf
v}\|_{L^{p}}$ is well controlled. Roughly speaking, by taking
$\operatorname{curl}$ on the both side of momentum equations in $(\ref{Lin-
sys})$, One can find $\operatorname{curl}{\bf v}$ satisfies a Laplace systems,
and we can expect there is a good estimate in the interior. But unfortunately,
it’s not clear about the behavior of $\operatorname{curl}{\bf v}$ near
$\Gamma_{in}$ and $\Gamma_{out}$. Nevertheless, the authors have proved by a
delicate construction in [9] the following beautiful results,

(3.31) $\|\operatorname{curl}{\bf v}\|_{W^{1,p}(\Omega)}\leq C(\epsilon\|{\bf
v}\|_{W^{2,p}}+\|f\|_{W^{1,p}}+\|g\|_{L^{p}}+\|w_{in}\|_{W^{1,p}({\Gamma_{in}})}),$

where $\epsilon$ is a sufficiently small constant. Since (3.2) has exactly the
same main terms as the linear system in [9], one checks easily that the
estimate (3.31) still holds true for the system (3.2) by adapting the
constructions developed in [9]. It’s immediately to get the $W^{2,p}$
estimates by using $(\ref{w2p-key})$.

######  Lemma 3.7.

For any $p\in(2,\infty)$, assume that $({\bf v},w)$ be a solution of (3.2)
with $(f,g,\tilde{B},w_{in})\in W^{1,p}(\Omega)\times L^{p}(\Omega)\times
W^{1-\frac{1}{p},p}(\Gamma_{0})\times W^{1,p}(\Gamma_{in})$, then we have

(3.32) $\displaystyle\|w\|_{W^{1,p}(\Omega)}$
$\displaystyle+\|\partial_{z_{1}}w\|_{W^{1,p}(\Omega)}+\|{\bf
v}\|_{W^{2,p}(\Omega)}$ $\displaystyle\leq
C(\|f\|_{W^{1,p}}+\|g\|_{L^{p}}+\|\tilde{B}\|_{W^{1-\frac{1}{p},p}(\Gamma_{0})}+\|w_{in}\|_{W^{1,p}({\Gamma_{in}})})$

###### Proof.

The same extension introduced in the proof of Lemma 3.4 shows that the
boundary estimates can be converted into the interior one, hence we only
demonstrate the interior estimates for the convenience. By standard $W^{2,p}$
estimates of elliptic system, we have

(3.33) $\|{\bf v}\|_{W^{2,p}(\Omega)}\leq C(\|{\bf
v}\|_{W^{1,p}}+\|g\|_{L^{p}}+\|\tilde{B}\|_{W^{1-\frac{1}{p},p}(\Gamma_{0})}+\|\nabla
w\|_{L^{p}})$

By virtue of mass equation in (3.2) we have

(3.34) $\|{\bf v}\|_{W^{2,p}(\Omega)}\leq
C(\|f\|_{W^{1,p}}+\|g\|_{L^{p}}+\|\tilde{B}\|_{W^{1-\frac{1}{p},p}(\Gamma_{0})}+\|{\bf
v}\|_{W^{1,p}}+\|\partial_{z_{2}}w\|_{L^{p}})$

which combing with the estimates (3.24) and (3.31) after using interpolation
inequalities gives the results. ∎

##  4\. Solution of linear system

In this section, the existence of weak solution to the system (3.2) is given
by Galerkin method. Having the weak solution at hand, we can show easily that
this solution is also strong if the data has the appropriate regularity.

###  4.1. Weak solution

######  Definition 4.1.

We call $(w,{\bf v})$ is a weak solution of (3.2) if for any $\phi\in V$ there
holds

(4.1) $\displaystyle\int_{\Omega}[\bar{U}\partial_{z_{1}}$ $\displaystyle
w+{{\bf v}\cdot\nabla U_{0}}]\phi
dz-\gamma\int_{\Omega}w\operatorname{div}\phi dz-\int_{\Gamma_{0}}\alpha({\bf
v}\cdot\tau)(\phi\cdot\tau)ds$ $\displaystyle+\int_{\Omega}(2\mu D({\bf
v}):\nabla\phi+\nu\operatorname{div}{\bf
v}\operatorname{div}\phi)dz=\int_{\Omega}g\phi
dz+\int_{\Gamma_{0}}\tilde{B}(\phi\cdot\tau)ds.$

and for any $\eta\in\tilde{C}^{\infty}(\Omega)$, there holds

(4.2) $\displaystyle-\int_{\Omega}\eta\partial_{z_{1}}(\bar{U}+\bar{\bf
v}^{(1)})wdz$
$\displaystyle-\int_{\Omega}\partial_{z_{1}}\eta(\bar{U}+\bar{\bf
v}^{(1)})wdz$ $\displaystyle=\int_{\Omega}[f-\operatorname{div}{\bf v}]\eta
dz+\int_{\Gamma_{in}}(\bar{U}+\bar{\bf v}^{(1)})w_{in}\eta ds$

where $\eta\in\tilde{C}^{\infty}(\Omega)$ means that $\eta\in
C^{\infty}(\Omega)$ and $\eta|_{\Gamma_{out}}=0$.

we want to apply the Galerkin method to prove the existence of weak solution.
To this end, introduce an orthonormal basis of $\\{\omega_{k}\\}\subset V$ and
finite dimensional subspace
$V^{N}=\\{\Sigma_{i=1}^{N}c_{i}\omega_{i}:c_{i}\in\mathbb{R}\\}$. We look for
the approximate velocity field of the form ${\bf
v}^{N}=\Sigma_{i=1}^{N}c_{i}\omega_{i}$. Note that the solution of the
continuity equation has been given by the operator $S$ defined in (3.4).Now we
proceed with the Galerkin scheme.Taking $g=g^{N},{\bf v}={\bf
v}^{N}=\Sigma_{i=1}^{N}c_{i}\omega_{i},\phi=\omega_{k},k=1,2...N$ and
$w=w^{N}=S(f^{N}-\operatorname{div}{\bf v}^{N})$, where $f^{N},g^{N}$ are
orthogonal projections of $f,g$ on $V^{N}$. We arrive at a system of N
equations

(4.3) $B^{N}({\bf v}^{N},\omega_{k})=0,\quad k=1,2...N,$

where $B^{N}:V^{N}\times V^{N}\rightarrow\mathbb{R}$ is defined as

$\displaystyle B^{N}({\bf v}^{N},\phi)$ $\displaystyle=\int_{\Omega}\\{{\bf
v}^{N}\bar{U}\partial_{z_{1}}\phi+({\bf
v}^{N}\cdot\nabla\bar{U}_{0})\cdot\phi+2\mu D({\bf
v}^{N}):\nabla\phi+\nu\operatorname{div}{\bf
v}^{N}\operatorname{div}\phi\\}dz$ $\displaystyle-\int_{\Omega}g^{N}\cdot\phi
dz-\gamma\int_{\Omega}S(f^{N}-\operatorname{div}{\bf
v}^{N})\operatorname{div}\phi dz$ $\displaystyle+\int_{\Gamma_{0}}[\alpha({\bf
v}^{N}\cdot\tau)-\tilde{B}](\phi\cdot\tau)d\sigma.$

Now, if ${\bf v}^{N}$ satisfies (4.3) for $k=1,2...N$, then the pair $({\bf
v}^{N},w^{N})$ satisfies (3.2) for $(\phi,\eta)\in V\times\tilde{C}^{\infty}$
with $w=w^{N}$. we call such a pair an approximation solution. In order to
show the existence of approximation solution, we apply the following results
for finite dimensional Hilbert space[20].

######  Lemma 4.2.

Let X be a finite dimensional Hilbert space and let $P:X\rightarrow X$ be a
continuous operator satisfying

$\exists M>0:\,(p(\xi),\xi)>0\,\,for\,\,\|\xi\|=M,$

then there is at leats one $\xi^{\star}$ such that $\|\xi^{\star}\|\leq M$ and
$P(\xi^{\star})=0$.

The existence of approximation solution is guaranteed by the following lemma.

######  Lemma 4.3.

Let $f,g\in L^{2}(\Omega)$, $w_{in}\in L^{2}(\Gamma_{in}),\tilde{B}\in
L^{2}(\Gamma_{0})$. If $\|\bar{\bf v}\|_{W^{2,p}}$ is small enough and the
viscous coefficients $\mu>\frac{1}{2\pi^{2}}$, then there exists ${\bf
v}^{N}\in V^{N}$ fulfills (4.3). Furthermore, there exists a positive constant
M indecent of N such that

(4.4) $\|{\bf v}^{N}\|_{H^{1}}\leq M.$

###### Proof.

Define $P^{N}:V^{N}\rightarrow V^{N}$ as

(4.5)
$P^{N}(\xi^{N})=\Sigma_{k=1}^{N}B^{N}(\xi^{N},\omega_{k})\omega_{k}\,\,\text{for}\,\,\xi^{N}\in
V^{N}.$

According to Lemma 4.3, we need to show that $(p(\xi^{N}),\xi^{N})>0$ on some
sphere in $V^{N}$. Since $B^{N}(\cdot,\cdot)$ is linear with respect to the
second variable, we have

(4.6) $\displaystyle(p^{N}(\xi^{N}),\xi^{N})=B^{N}(\xi^{N},\xi^{N})$
$\displaystyle=2\mu\int_{\Omega}|D(\xi^{N})|^{2}dz+\nu\int_{\Omega}|\operatorname{div}\xi^{N}|^{2}dz$
$\displaystyle+\int_{\Omega}(\xi^{N}\cdot\nabla
U_{0})\cdot\xi^{N}dz-\int_{\Omega}g^{N}\cdot\xi^{N}$
$\displaystyle-\frac{1}{2}\int_{\Omega}\partial_{z_{1}}(\bar{U}\circ\psi_{\bar{\xi}^{N}})|\xi^{N}|^{2}dz-\int_{\Gamma_{0}}\tilde{B}\xi^{N}\cdot\tau$
$\displaystyle+\int_{\Gamma_{0}}\alpha|\xi^{N}|^{2}ds-\gamma\int_{\Omega}S(f^{N}-\operatorname{div}\xi^{N})\operatorname{div}\xi^{N}dz.$

Due to Korn inequality similar as in the proof of Lemma 3.3, we only need to
find a bound on the lats term in (4.6). Denote
$\eta^{N}=S(f^{N}-\operatorname{div}\xi^{N})$, then

$\int_{\Omega}\eta^{N}\operatorname{div}\xi^{N}dz=\int_{\Omega}f^{N}\eta^{N}dz-\int_{\Omega}(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1}}\eta^{N}\eta^{N}dz,$

the first term is controlled by

(4.7)
$\int_{\Omega}f^{N}\eta^{N}dz\leq\|f\|_{L^{2}}\|\eta^{N}\|_{L^{2}}\leq\|f\|_{L^{2}}(\|f\|_{L^{2}}+\|\xi^{N}\|_{H^{1}}+|w_{in}|_{L^{2}})$

where we have used (3.5). With the second integral we have

$\displaystyle-\int_{\Omega}(\bar{U}+\bar{\bf
v}^{(1)})\partial_{z_{1}}\eta^{N}\eta^{N}dz$
$\displaystyle=\frac{1}{2}\int_{\Gamma_{in}}(\bar{U}+\bar{\bf
v}^{(1)})(\eta^{N})^{2}d\sigma-\frac{1}{2}\int_{\Gamma_{out}}(\bar{U}+\bar{\bf
v}^{(1)})(\eta^{N})^{2}d\sigma$
$\displaystyle+\frac{1}{2}\int_{\Omega}\partial_{z_{1}}(\bar{U}+\bar{\bf
v}^{(1)})(\eta^{N})^{2}dz$ $\displaystyle\leq
C|w_{in}|^{2}_{L^{2}}+E\|\eta^{N}\|^{2}_{L^{2}}.$

Combine all those estimates together, we get

$(p^{N}(\xi^{N})\geq C(\|\xi^{N}\|^{2}_{H^{1}}-D\|\xi^{N}\|_{H^{1}}-D^{2}),$

where
$D=\|f\|_{L^{2}}+\|g\|_{V^{*}}+\|\tilde{B}\|_{L^{2}(\Gamma_{0})}+\|w_{in}\|_{L^{2}({\Gamma_{in}})}$.
Hence there exists a constant $M$ such that $p^{N}(\xi^{N})>0$ for some
$\|\xi^{N}\|=M$. Applying Lemma 4.3 we conclude that there is a $\xi^{\star}$
such that $P^{N}(\xi^{\star})=0$ and $\|\xi^{\star}\|\leq M.$ Finally, since
$\\{\omega_{k}\\}$ is the basis of $V^{N}$, $P^{N}(\xi^{\star})=0$ implies
$B^{N}(\xi^{\star},\omega_{k})=0$, that is to say $\xi^{\star}$ is an
approximation solution. ∎

Since the system is linear, the uniform estimate (4.4) immediately gives the
existence of weak solution. The results is in the following.

######  Lemma 4.4.

Let $f,g\in L^{2}(\Omega)$, $w_{in}\in L^{2}(\Gamma_{in}),\tilde{B}\in
L^{2}(\Gamma_{0})$. If $\|\bar{\bf v}\|_{W^{2,p}}$ is small enough and the
viscous coefficients $\mu>\frac{1}{2\pi^{2}}$, then the system (3.2) has a
unique weak solution $({\bf v},w)\in H^{1}\times L^{\infty}(L^{2})$, which
satisfies the estimate (3.7).

###### Proof.

The above lemma shows that the approximation solution ${\bf v}^{N}$ satisfy

$\|{\bf v}^{N}\|_{H^{1}}\leq M$

which combined with (3.5) gives

$\|{\bf v}^{N}\|_{H^{1}}+\|w\|_{L^{\infty}(L^{2})}\leq M.$

Hence there exists a pair $({\bf v},w)\in H^{1}\times L^{\infty}(L^{2})$ such
that

${\bf v}^{N}\rightharpoonup{\bf v}\quad\text{in}\,H^{1},$

and

$w^{N}\rightharpoonup w\quad\text{in}\,L^{\infty}(L^{2}).$

Since the system (3.2) is linear, passing to the limit in (4.1) for ${{\bf
v}^{N},w^{N}}$, it follows that ${\bf v}$ satisfy (4.1) with $w$. Similarly,
taking the limit in $(\ref{def-weak-w})$ it is easy to verify
$w=S(f-\operatorname{div}{\bf v})$, the boundary condition on $w$ is
guaranteed to hold by the definition of the operator $S$. It’s obvious that
$({\bf v},w)$ satisfies the estimate (3.7). The proof is thus complete. ∎

###  4.2. Strong solution

In this section, we will show that the weak solution given above is also
strong by using symmetric extension methods.

######  Lemma 4.5.

Let $f\in W^{1,p}(\Omega),g\in L^{p}(\Omega),w_{in}\in
W^{1,p}(\Gamma_{in})\tilde{B}\in W^{1-\frac{1}{p},p}(\Gamma_{0})$. If
$\|\bar{\bf v}\|_{W^{2,p}}$ is small enough and the viscous coefficients
$\mu>\frac{1}{2\pi^{2}}$ , then the system (3.2) has a unique strong solution
$({\bf v},w)\in W^{2,p}\times W^{1,p}$, which satisfies the estimate (3.32).

###### Proof.

Since (3.2) is a linear system, the a priori estimates (3.32) will deduce the
regularity of the weak solution in the interior. In order deal with the
singularity of the boundary at the junctions of $\Gamma_{0}$ with
$\Gamma_{in}$ and $\Gamma_{out}$, we still apply the symmetric extension
methods introduced in the proof of lemma (3.4). Hence we can extend the weak
solution on the negative values of $z_{2}$, and using the estimates (3.32)
show that the extended solution has the same regularity. ∎

##  5\. proof of main results

According to the analysis in Section 4. We can define the solution operator of
linear systems as

(5.1) $T:W^{2,p}(\Omega)\times W^{1,p}(\Omega)\rightarrow
W^{2,p}(\Omega)\times W^{1,p}(\Omega)$

such that $({\bf v}^{n+1},w^{n+1})=T({\bf v}^{n},w^{n})$ is the solution of
the linear system (3.1). In the following we want to show that T is a
contraction mapping in some proper subspace, hence $T$ has a unique solution
of (2.19).

######  Lemma 5.1.

There is a positive constant $R>0$ such that the operator $T$ mapping $B_{R}$
into itself provided $D_{0}$ is small enough, where

$B_{R}=\\{({\bf v},w)\in W^{2,p}(\Omega)\times W^{1,p}:\|{\bf
v}\|_{W^{2,p}(\Omega)}+\|w\|_{W^{1,p}(\Omega)}\leq R\\}.$

###### Proof.

By (2.21) and (3.32) we have

(5.2) $\displaystyle\|T$ $\displaystyle({\bf v}^{n},w^{n})\|_{W^{2,p}\times
W^{1,p}}=\|({\bf v}^{n+1},w^{n+1})\|_{{W^{2,p}\times W^{1,p}}}$
$\displaystyle\leq C[(\|{\bf v}^{n}\|_{2,p}+\|w^{n}\|_{1,p})^{3}+(\|{\bf
v}^{n}\|_{2,p}+\|w^{n}\|_{1,p})^{2}]$ $\displaystyle+E(\|{\bf
v}\|_{W^{2,p}}+\|w\|_{W^{1,p}})+\|\tilde{\bf u}\|_{W^{2,p}}$
$\displaystyle\leq C(R^{3}+R^{2})+E(D_{0})$

for $R<1$, and hence for $R\leq 2E(D_{0})$ with $E(D_{0})\leq\frac{1}{16C}$,
we have

$\|({\bf v}^{n+1},w^{n+1})\|_{{W^{2,p}\times W^{1,p}}}\leq 2E(D_{0}),$

the proof thus is completed. ∎

######  Remark 5.2.

The same proof also shows that $\|\partial_{z_{1}}w^{n+1}\|_{1;p}\leq
E(D_{0})$, thanks to the second term in the r.h.s of (3.32),

In the next lemma we will show that $T$ is contraction.

######  Lemma 5.3.

There is a constant $0<E_{0}<1$ independent of n,such that

(5.3) $\|{\bf v}^{n+1}-{\bf
v}^{m+1}\|_{{2,p}:\Omega}+\|{w}^{n+1}-{w}^{m+1}\|_{{1,p}:\Omega}\leq
E_{0}[\|{\bf v}^{n}-{\bf
v}^{m}\|_{{2,p}:\Omega}+\|{w}^{n}-{w}^{m}\|_{{1,p}:\Omega}]$

holds for all $({\bf v}^{n},w^{n})\in B_{R}$.

###### Proof.

Making difference to derive the system of $({\bf v}^{n+1}-{\bf
v}^{m+1},{w}^{n+1}-{w}^{m+1})$, we have

(5.4) $\displaystyle((\bar{U}+({{\bf v}^{n}+\tilde{\bf
u}})^{(1)})\circ\psi_{{\bf v}^{n}+\tilde{\bf
u}})\partial_{z_{1}}(w^{n+1}-w^{m+1})+\operatorname{div}_{z}({\bf
v}^{n+1}-{\bf v}^{m+1})$ $\displaystyle=\tilde{F}({\bf
v}^{n},w^{n})-\tilde{F}({\bf v}^{m},w^{m})$
$\displaystyle-\partial_{z_{1}}w^{m+1}[((\bar{U}+({{\bf v}^{n}+\tilde{\bf
u}})^{(1)})\circ\psi_{{\bf v}^{n}+\tilde{\bf u}})-((\bar{U}+({{\bf
v}^{m}+\tilde{\bf u}})^{(1)})\circ\psi_{{\bf v}^{m}+\tilde{\bf u}})],$
$\displaystyle(\bar{U}\circ\psi_{{\bf v}^{n}+\tilde{\bf
u}})\partial_{z_{1}}({\bf v}^{n+1}-{\bf v}^{m+1})+({\bf v}^{n+1}-{\bf
v}^{m+1})\cdot\nabla_{z}U_{0}+\gamma\nabla_{z}(w^{n+1}-w^{m+1})$
$\displaystyle-\mu\Delta_{z}({\bf v}^{n+1}-{\bf
v}^{m+1})-(\mu+\nu)\nabla\operatorname{div}_{z}({\bf v}^{n+1}-{\bf
v}^{m+1})=\tilde{G}({\bf v}^{n},w^{n})-\tilde{G}({\bf v}^{m},w^{m})$
$\displaystyle-{\bf v}^{m+1}(\bar{U}\circ\psi_{{\bf v}^{n}+\tilde{\bf
u}}-\bar{U}\circ\psi_{{\bf v}^{m}+\tilde{\bf u}}),$ $\displaystyle{\bf
v}^{n+1}-{\bf v}^{m+1}=0,$ $\displaystyle w^{n+1}-w^{m+1}=0,$
$\displaystyle({\bf v}^{n+1}-{\bf v}^{m+1})\cdot\mathbf{n}=0,$ $\displaystyle
2\mu\mathbf{n}\cdot D_{z}({\bf v}^{n+1}-{\bf v}^{m+1})\cdot\tau+\alpha({\bf
v}^{n+1}-{\bf v}^{m+1})\cdot\tau=2\mu{\bf n}\cdot[R({v}^{m},D)-R({\bf
v}^{n},D)]\cdot\tau.$

Applying $W^{2,p}$ estimates to the $(\ref{diff-sys})$,we obtain

(5.5) $\displaystyle\|{\bf v}^{n+1}-{\bf
v}^{m+1}\|_{{2,p};\Omega}+\|w^{n+1}-w^{m+1}\|_{{1,p};\Omega}$
$\displaystyle\leq C\\{\|\partial_{z_{1}}w^{m+1}[((\bar{U}+({{\bf
v}^{n}+\tilde{\bf u}})^{(1)})\circ\psi_{{\bf v}^{n}+\tilde{\bf
u}})-((\bar{U}+({{\bf v}^{m}+\tilde{\bf u}})^{(1)})\circ\psi_{{\bf
v}^{m}+\tilde{\bf u}})]\|_{1,p}$ $\displaystyle+\|\tilde{F}({\bf
v}^{n},w^{n})-\tilde{F}({\bf v}^{m},w^{m})\|_{1,p}+\|{\bf
v}^{m+1}(\bar{U}\circ\psi_{{\bf v}^{n}+\tilde{\bf u}}-\bar{U}\circ\psi_{{\bf
v}^{m}+\tilde{\bf u}})\|_{p}$ $\displaystyle+\|\tilde{G}({\bf
v}^{n},w^{n})-\tilde{G}({\bf v}^{m},w^{m})\|_{p}+\|2\mu{\bf n}\cdot[R({\bf
v}^{m},D)-R({\bf v}^{n},D)]\cdot\tau\|_{1-\frac{1}{p},p;\Gamma_{0}}\\}$
$\displaystyle:=I_{1}+I_{2}+I_{3}+I_{4}+I_{5}$

Estimate of $I_{1}$. By the definition of $\psi,$ it continuously dependent on
the parameters ${\bf v}$, which implies that

(5.6) $|\psi_{{\bf v}^{n}+\tilde{\bf u}}-\psi_{{\bf v}^{m}+\tilde{\bf u}}|\leq
C|{\bf v}^{n}-{\bf v}^{m}|$

for some positive constant $C$. Hence,

(5.7) $\displaystyle|({{\bf v}^{n}+\tilde{\bf u}})^{(1)}\circ\psi_{{\bf
v}^{n}+\tilde{\bf u}}-({{\bf v}^{m}+\tilde{\bf u}})^{(1)}\circ\psi_{{\bf
v}^{m}+\tilde{\bf u}}|$ $\displaystyle\leq C[|\nabla({\bf v}^{n}+\tilde{\bf
u})^{(1)}||{\bf v}^{n}-{\bf v}^{m}|+|{\bf v}^{n}-{\bf v}^{m}|]$

Consequently,

(5.8) $\displaystyle I_{1}$
$\displaystyle\leq\|\partial_{z_{1}}w^{m+1}\|_{1,p}\|{\bf v}^{n}-{\bf
v}^{m}\|_{2,p}$ $\displaystyle\leq E(D_{0})\|{\bf v}^{n}-{\bf v}^{m}\|_{2,p}$

It’s easy to show that the same estimates holds for $I_{3}$.
Estimates of $I_{2}$. Note that (2.22) implies that

(5.9) $\displaystyle|R(\cdot,\partial)|\leq E\|\cdot\|_{1,p},$
$\displaystyle|R(\cdot,\partial^{2})|\leq E\|\cdot\|_{2,p}.$

Hence, to show the bounds of $I_{2}$, it sufficient to estimates $\|F({\bf
v}^{n},w^{n})-F({\bf v}^{m},w^{m})\|_{1,p}$.

(5.10) $\displaystyle F({\bf v}^{n},w^{n})-F({\bf v}^{m},w^{m})$
$\displaystyle=w^{n}\operatorname{div}({\bf
v}^{n}+{\bf\tilde{u}})-w^{m}\operatorname{div}({\bf v}^{m}+{\bf\tilde{u}})$
$\displaystyle=w^{n}\operatorname{div}{\bf v}^{m}-w^{m}\operatorname{div}{\bf
v}^{m}+(w^{n}-w^{m})\operatorname{div}{\bf\tilde{u}},$

and

(5.11) $\displaystyle w^{n}\operatorname{div}{\bf
v}^{n}-w^{m}\operatorname{div}{\bf v}^{m}$
$\displaystyle=w^{n}[\operatorname{div}_{z}{\bf v}^{n}+R({\bf
v}^{n},\operatorname{div})]-w^{m}[\operatorname{div}_{z}{\bf v}^{m}+R({\bf
v}^{m},\operatorname{div})]$
$\displaystyle=(w^{n}-w^{m})\operatorname{div}{\bf
v}^{n}+w^{m}\operatorname{div}({\bf v}^{n}-{\bf v}^{m})$
$\displaystyle=(w^{n}-w^{m})R({\bf v}^{n},\operatorname{div})+w^{m}R({\bf
v}^{n}-{\bf v}^{m},\operatorname{div}),$

then it is immediate that

(5.12) $\displaystyle\|F({\bf v}^{n},w^{n})-F({\bf v}^{m},w^{m})\|_{1,p}\leq
C(D_{0})(\|w^{n}-w^{m}\|_{1,p}+\|{\bf v}^{n}-{\bf v}^{m}\|_{2,p}).$

Estimates of $I_{4}$. Due to the same reason, we just need to estimates
$\|G({\bf v}^{n},w^{n})-G({\bf v}^{m},w^{m})\|_{p}$. For the convenience,
denote $\delta P^{\prime}(w)=P^{\prime}(w+1)-P^{\prime}(w)$, then

(5.13) $\displaystyle\delta P^{\prime}(w^{n})\nabla_{x}w^{n}-\delta
P^{\prime}(w^{m})\nabla_{x}w^{m}$ $\displaystyle=\delta
P^{\prime}(w^{n})\nabla_{x}(w^{n}-w^{m})-\nabla_{x}w^{m}(\delta
P^{\prime}(w^{m})-\delta P^{\prime}(w^{n}))$ $\displaystyle=\delta
P^{\prime}(w^{n})[\nabla_{z}(w^{n}-w^{m})+R(w^{n}-w^{m},\nabla)]$
$\displaystyle+[\nabla_{z}w^{m}+R(w^{m},\nabla)](\delta
P^{\prime}(w^{m})-\delta P^{\prime}(w^{n}))$

which gives

(5.14) $\|\delta P^{\prime}(w^{n})\nabla_{x}w^{n}-\delta
P^{\prime}(w^{m})\nabla_{x}w^{m}\|_{p}\leq
E(\|w^{n}\|_{1,p},\|w^{m}\|_{1,p})\|w^{n}-w^{m}\|_{1,p}.$

The estimates of reminder terms are similar, so we show one of them as a
example.

(5.15) $\displaystyle w^{n}{\bf v}^{n}\cdot\nabla_{x}{\bf v}^{n}-w^{m}{\bf
v}^{m}\cdot\nabla_{x}{\bf v}^{m}$ $\displaystyle=w^{n}{\bf
v}^{n}\cdot[\nabla_{z}{\bf v}^{n}+R({\bf v}^{n},\nabla)]-w^{m}{\bf
v}^{m}\cdot[\nabla_{z}{\bf v}^{m}+R({\bf v}^{m},\nabla)]$
$\displaystyle=(w^{n}-w^{m}){\bf v}^{n}\cdot\nabla_{z}{\bf v}^{n}+w^{m}({\bf
v}^{n}-{\bf v}^{m})\cdot\nabla_{z}{\bf v}^{n}+w^{m}{\bf
v}^{m}\cdot\nabla_{z}({\bf v}^{n}-{\bf v}^{m}),$
$\displaystyle+(w^{n}-w^{m}){\bf v}^{n}\cdot R({\bf v}^{n},\nabla)+w^{m}({\bf
v}^{n}-{\bf v}^{m})\cdot R({\bf v}^{n},\nabla)+w^{m}{\bf v}^{m}R({\bf
v}^{n}-{\bf v}^{m},\nabla),$

then, we have

(5.16) $\displaystyle\|G({\bf v}^{n},w^{n})-G({\bf v}^{m},w^{m})\|_{p}\leq
C(D_{0})(\|w^{n}-w^{m}\|_{1,p}+\|{\bf v}^{n}-{\bf v}^{m}\|_{2,p}).$

Estimates of $I_{5}$.

(5.17) $\displaystyle\|$ $\displaystyle 2\mu{\bf n}\cdot[R({\bf
v}^{m},D)-R({\bf v}^{n},D)]\cdot\tau\|_{1-\frac{1}{p},p;\Gamma_{0}}$
$\displaystyle\leq\|R({\bf v}^{m}-{\bf v}^{n},D)\|_{1,p}$ $\displaystyle\leq
C(E)\|{\bf v}^{n}-{\bf v}^{m}\|_{2,p}.$

Due to the smallness assumption of $C(\cdot)$, one can choose a small constant
$0<E_{0}<1$ independent of $n$ fulfills (5.3). ∎

Proof of Theorem 1.we have prove that the operator $T$ is contraction mapping
on $B_{R}\in W^{2,p}\times W^{1,p}$ for some small $R=R(D_{0})$. Hence the
Banach fixed point theorem gives existence of a unique fixed point in the ball
$B_{R}$. By the definition of $T$, the fixed point is exactly the solution of
the system (2.19). Finally, we can change variable to the original coordinate,
and in $x$ variable our solution satisfies the system (2.3). The proof is thus
complete.

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