# On the Inviscid Limit of the 3D Navier-Stokes Equations with Generalized
Navier-slip Boundary Conditions111This research is supported in part by NSFC
10971174, and Zheng Ge Ru Foundation, and Hong Kong RGC Earmarked Research
Grants CUHK-4041/11P, CUHK-4042/08P, a Focus Area Grant from The Chinese
University of Hong Kong, and a grant from Croucher Foundation.

Yuelong Xiao222Institute for Computational and Applied mathematics, Xiangtan
University.$\ {}^{,{\ddagger}}$ and Zhouping Xin333The Institute of
Mathematical Sciences, The Chinese University of Hong Kong.

###### Abstract

In this paper, we investigate the vanishing viscosity limit problem for the
3-dimensional (3D) incompressible Navier-Stokes equations in a general bounded
smooth domain of $R^{3}$ with the generalized Navier-slip boundary conditions
(1.11). Some uniform estimates on rates of convergence in
$C([0,T],L^{2}(\Omega))$ and $C([0,T],H^{1}(\Omega))$ of the solutions to the
corresponding solutions of the idea Euler equations with the standard slip
boundary condition are obtained.

##  1 Introduction

Let $\Omega\subset R^{3}$ be a bounded smooth domain. We consider the Navier-
Stokes equations

$\displaystyle\partial_{t}u^{\varepsilon}-\varepsilon\Delta
u^{\varepsilon}+u^{\varepsilon}\cdot\nabla u^{\varepsilon}+\nabla
p^{\varepsilon}=0,\textrm{ in }\Omega$ (1.1) $\displaystyle\nabla\cdot
u^{\varepsilon}=0,\textrm{ in }\Omega$ (1.2)

and the Euler equations

$\displaystyle\partial_{t}u+u\cdot\nabla u+\nabla p=0,\textrm{ in }\Omega$
(1.4) $\displaystyle\nabla\cdot u=0,\textrm{ in }\Omega$

We are interested in the vanishing viscosity limit problem: do the solutions
of the Navier-Stokes equations (1.1),(1.2) converge to that of the Euler
equations (1.4) with the same initial data as $\varepsilon$ vanishes?.

The vanishing viscosity limit problem for the Navier-Stokes equations is a
classical issue and has been well studied when the domain has no boundaries,
various convergence results have been obtained, see for instances[13, 14, 16,
19, 20, 27, 34].

However, in the presence of a physical boundary, the situations is much more
complicated, and the problem becomes challenging due to the boundary layers.
This is so even in the case that the corresponding solution to the initial
boundary value problem of the Euler equations remains smooth.

Indeed, in presence of a boundary $\partial\Omega$, one of the most important
physical boundary conditions for the Euler equations (1.4) is the following
slip boundary condition, i.e.,

$u\cdot n=0\ {\rm on}\ \partial\Omega$ (1.5)

and the initial boundary value problem of the Euler equations (1.4) with the
slip boundary condition (1.5) has a smooth solution at least locally in time.
Corresponding to the slip boundary condition (1.5) for the Euler equations,
there are different choices of boundary conditions for the Navier-Stokes
equations, one is the mostly-used no-slip boundary condition, i.e.,

$u^{\varepsilon}=0,\ {\rm on}\ \Omega$ (1.6)

This Dirichlet type boundary condition was proposed by Stokes ([33]) assuming
that fluid particles are adherent to the boundary due to the positive
viscosity. Although the well-posedness of the smooth solution to the initial
boundary value problem of the Navier-Stokes equations with the no-slip
boundary condition can be established quiet easily (at least local in time),
the asymptotic convergence of the solution to the corresponding solution of
the Euler equations (1.4) with the boundary condition (1.5) as the viscosity
coefficient $\varepsilon$ tends to zero is one of major open problems except
special cases (see [15, 29, 31, 32]) due to the possible appearance of the
boundary layers. In general, only some sufficient conditions are obtained for
the convergence in $L^{2}(\Omega)$, see [21, 23, 38]. However, there are some
new results in 2D case announced recently in [46].

Another class of boundary conditions for the Navier-Stokes equations is the
Navier-slip boundary conditions, i.e,

$u^{\varepsilon}\cdot n=0,\ [2(S(u^{\varepsilon})n)+\gamma
u^{\varepsilon}]_{\tau}=0,\ {\rm on}\ \partial\Omega$ (1.7)

where $2S(u^{\varepsilon})=(\nabla u^{\varepsilon}+(\nabla
u^{\varepsilon})^{T})$ is the viscous stress tensor, and $\gamma$ is a smooth
given function. Such conditions were proposed by Navier in [30], which allow
the fluid to slip at the boundary, and have important applications for
problems with rough boundaries, prorated boundaries, and interfacial boundary
problems, see [6, 9, 35, 37]. The well-posedness of the Naver-Stokes equations
(1.1), (1.2) with the Navier-slip boundary conditions and related results have
been established, see [9, 43] and the references therein.

In a recent paper [25], the Navier-slip boundary condition (1.7) is written to
the following generalized one

$u^{\varepsilon}\cdot n=0,\
[2(S(u^{\varepsilon})n)+Au^{\varepsilon}]_{\tau}=0,\ {\rm on}\ \partial\Omega$
(1.8)

with $A$ is a smooth symmetric tensor. The well-posedness of the Naver-Stokes
equations (1.1), (1.2) with the generalized Navier-slip boundary condition is
indeed similar to the classical one.

On the other hand, the following vorticity-slip boundary condition

$u^{\varepsilon}\cdot n=0,\ n\times(\omega^{\varepsilon})=\beta
u^{\varepsilon}\ {\rm on}\ \partial\Omega$ (1.9)

with $\beta$ is a smooth function is introduced in [43], where
$\omega=\nabla\times u$ is the vorticity of the fluid.

It is noticed that

$(2(S(v)n)-(\nabla\times v)\times n)_{\tau}=GD(v)_{\tau}\ {\rm on}\
\partial\Omega$ (1.10)

where $GD(v)=-2S(n)v$ (see[43]). Hence the generalized slip condition (1.8) is
equivalent to the following generalized vorticity-slip condition

$u^{\varepsilon}\cdot n=0,\
n\times(\omega^{\varepsilon})=[Bu^{\varepsilon}]_{\tau}\ {\rm on}\
\partial\Omega$ (1.11)

with $B$ a given smooth symmetric tensor on the boundary. The equivalence can
also be seen for instance in [1, 2, 3, 7, 39, 41].

Compared to the case of no-slip boundary condition, the asymptotic behavior of
solutions to the Navier-Stokes equations with the Navier-slip boundary
conditions as $\varepsilon\rightarrow 0$ is relatively easy to establish. Some
strong convergence results in 2D (see [12, 26]) and various weak convergence
results in 3D (see [17, 24, 25]) are also established, where it has been shown
that

$\|u^{\varepsilon}-u\|^{2}+\varepsilon\int_{0}^{t}\|u^{\varepsilon}-u\|_{1}^{2}dt\leq
c\varepsilon^{\frac{3}{2}}$ (1.12)

and

$\|u^{\varepsilon}-u\|_{L^{\infty}([0,T]\times\Omega)}\leq
c\varepsilon^{\frac{3}{8}(1-s)}$ (1.13)

for some $s>0$, see [25] for details.

For the homogeneous case, i.e, $B=0$ in (1.11), better convergence results are
available. The $H^{3}(\Omega)$ convergence and $H^{2}(\Omega)$ estimate on the
rate of convergence are obtained in [41] for flat domains. These results are
improved to $W^{k,p}(\Omega)$ in [2, 3]. However, such a strong convergence
can not be expected for general domains, except the case that the initial
vorticity vanishes on the boundary [42]. Note that the convergence in
$H^{2}(\Omega)$ or, $W^{k,p}(\Omega)$ (for $k\geq 2$), implies that limiting
solution of the Euler equation satisfies the boundary condition (1.11) with
$B=0$.

However, it is shown in [4, 5] that the solution to the Euler equations with
the slip boundary condition (1.5) can not satisfy the extra condition
$n\times\omega=0$ on the boundary in general, and satisfy $n\times\omega=0\
\rm{on}\ \partial\Omega$ only if $\omega\cdot n=0\ \rm{on}\ \partial\Omega$.
This implies that the only possible case for convergence of solutions to the
Navier-Stokes equations in $H^{2}(\Omega)$ is the one obtained in [42], where
the following estimate on the convergence rate

$\|u^{\varepsilon}-u\|^{2}+\varepsilon\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\int_{0}^{t}(\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\|u^{\varepsilon}-u\|_{2}^{2})dt\leq
c\varepsilon^{2}$ (1.14)

is obtained. Furthermore, a $W^{s,p}(\Omega)$ convergence result can also be
obtained in this case, see [10].

For general initial data and domains in the homogeneous case, i.e, $B=0$ in
(1.11), the best estimate on the rate of convergence established so far is in
$W^{1,p}(\Omega)$ in [40], which implies in the particular case $p=2$ that

$\|u^{\varepsilon}-u\|^{2}+\varepsilon\|u^{\varepsilon}-u\|_{1}^{2}\leq
c\varepsilon^{2-s}$ (1.15)

with $s=\frac{1}{2}$, where $u^{\varepsilon}$ and $u$ are solutions to (1.1),
(1.2), (1.9) and (1.4), (1.5) respectively.

However, all the strong convergence results depend essentially the homogeneous
property, i.e, $n\times\omega=0$ on $\partial\Omega$. Indeed, for general
$B$(not identically equal to zero on $\partial\Omega$), there is no any
estimate on the rate of convergence in $H^{k}(\Omega)$ (or $W^{k,p}(\Omega)$)
for $k\geq 1$, as far as the authors are aware. It should be noted that the
co-normal uniform estimates have been established in [28] for general domains
and general Navier-slip boundary conditions, which grantee the convergence of
solutions to the Navier-Stokes equations to that of the Euler equations in
$W^{1,\infty}(\Omega)$. Yet, some additional efforts are still needed to
obtain an estimate on the rate of convergence as in (1.14).

In a recent paper [44], we proposed the following slip boundary condition

$u^{\varepsilon}\cdot n=0,\ \omega^{\varepsilon}\cdot n=0,n\times(\Delta
u^{\varepsilon})=0\ {\rm on}\ \partial\Omega$ (1.16)

for the Navier-Stokes equations (1.1), (1.2) and obtained the following
estimate on the rate of convergence

$\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\|u^{\varepsilon}-u\|_{2}^{2}+\varepsilon\int_{0}^{t}(\|u^{\varepsilon}-u\|_{2}^{2}+\varepsilon\|u^{\varepsilon}-u\|_{3}^{2})dt\leq
c\varepsilon^{2-s}$ (1.17)

for any $s>0$.

In this paper, we investigate the vanishing viscosity problem for the Navier-
Stokes equations (1.1), (1.2) on a general bounded smooth domain
$\Omega\subset R^{3}$ with the generalized vorticity-slip boundary condition
(1.11). Our main result is the following estimate on the rate of convergence
for the solutions:

Theorem 1.1. Let $u(t)$ be the smooth solution of the boundary value problem
of Euler equation (1.4), (1.4), (1.5) on $[0,T]$ with $u(0)=u_{0}\in
H^{3}(\Omega)$ which satisfies the corresponding assumptions on the initial
data in Theorem 1 of [28] (see also Lemma 2.1 in next section), and
$u^{\varepsilon}(t)$ be the solution of the boundary value problem of the
Navier-Stokes equation (1.1), (1.2), (1.11) with the same initial data
$u_{0}$. Then, there is $T_{0}>0$ such that

$\|u^{\varepsilon}-u\|^{2}+\varepsilon\int_{0}^{T_{1}}\|u^{\varepsilon}-u\|_{1}^{2}dt\leq
c\varepsilon^{2-s},\ {\rm on}\ [0,T_{0}]$ (1.18)
$\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\int_{0}^{T_{1}}\|u^{\varepsilon}-u\|_{2}^{2}dt\leq
c\varepsilon^{1-s},\ {\rm on}\ [0,T_{0}]$ (1.19)

for any $s>0$ and $\varepsilon$ small enough. Consequently,

$\|u^{\varepsilon}-u\|_{1,p}^{p}\leq c_{p}\varepsilon^{1-s},\ {\rm on}\
[0,T_{0}]$ (1.20)

for $2\leq p<\infty$, any $s>0$ and $\varepsilon$ small enough, and

$\|u^{\varepsilon}-u\|_{L^{\infty}([0,T]\times\Omega)}\leq
c\varepsilon^{\frac{2}{5}(1-s^{\prime})}$ (1.21)

for any $s^{\prime}>0$.

The estimate (1.19) yields the desired estimates on the rate of convergence of
the solutions of the Navier-Stokes equations to that of the Euler equations in
$C([0,T],H^{1}(\Omega))$ norm for general domains with general Navier-slip
boundary conditions. It should be noted that the estimates (1.18) and (1.21)
improve (1.12) and (1.13), and (1.19), (1.20) are even better than (1.15)
stated in [40] for the special case $B=0$. Indeed, The estimate (1.19) is
optimal in the sense that $s$ can not be taken to be $0$ due to boundary
layers in general, see Remark 3.1. below for details.

These estimates are motivated by our recent work [44], where the first order
derivative estimates for the solutions are obtained for the Navier-Stokes
equations with a new vorticity boundary condition (1.16), and the work of
Masmoudi and Rousset in [28], where uniform estimates on
$\|u^{\varepsilon}\|_{W^{1.\infty}}$ of the solutions of the Navier-Stokes
equation with the general Navier-slip boundary condition are obtained.

Our approach is an elementary energy estimate for the difference of the
solutions between the Navier-Stokes equations and the Euler equations. Some
suitable integrating by part formulas in terms of the vorticity are
successfully used to get the optimal rate of convergence. In comparison to the
asymptotic analysis method associate to the boundary layers (see [18, 24, 25,
40]), we should not need any correctors near the boundary. Meanwhile, the
uniform regularity obtained in [28] plays an essential role in our analysis.

The rest of the paper is organized as follows: In the next section, we give a
preliminary on the well-posedness of the initial boundary value problem of the
Navier-Stokes equations (1.1),(1.2) and (1.11), and the local uniform
regularity of the solutions. Then, the estimates on the asymptotic convergence
(1.18), (1.19), (1.20) and (1.21) are established in section 3.

##  2 Preliminaries

Let $\Omega\subset R^{3}$ be a general bounded smooth domain. We begin by
considering the following Stokes problem:

$\displaystyle\alpha u-\Delta u+\nabla q=f\textrm{ in }\Omega$ (2.1)
$\displaystyle\nabla\cdot u=0\textrm{ in }\Omega$ (2.2) $\displaystyle u\cdot
n=0,n\times(\nabla\times u)=[Bu]_{\tau}\ {\rm on}\ \partial\Omega$ (2.3)

where $B$ is smooth symmetric tensor. Set

$H=\\{u\in L^{2}(\Omega);\nabla\cdot u=0,{\rm in}\ \Omega;\ u\cdot n=0;\ {\rm
on}\ \partial\Omega\\}$ $V=H^{1}(\Omega)\cap H$ $W=\\{u\in H^{2}(\Omega);\
n\times(\nabla\times u)=[Bu]_{\tau}\ {\rm on},\partial\Omega\\}$

It is well known that for any $v\in V$, one has

$\displaystyle\|v\|_{1}\leq c\|\nabla\times v\|$ (2.4)

Note that

$[Bu]_{\tau}\cdot\phi=Bu\cdot\phi$

for any $\phi$ satisfying $\phi\cdot n=0$ on the boundary. We associate the
Stokes problem (2.1)-(2.3) with the following bilinear form

$\displaystyle
a_{\alpha}(u,\phi)=\alpha(u,\phi)+\int_{\partial\Omega}Bu\cdot\phi+\int_{\Omega}(\nabla\times
u)\cdot(\nabla\times\phi)$ (2.5)

Note that

$\displaystyle|\int_{\partial\Omega}Bu\cdot u|\leq c\|u\|\|u\|_{1}$ (2.6)

for all $u\in V$ and (2.4). It follows that $a_{\alpha}(u,\phi)$ is a positive
definite symmetric bilinear form if $\alpha$ large enough, and is closed with
the domain $\mathcal{D}(a_{\alpha})=V$. Similar to the discussions in [43],
one can define the self-adjoint operator associated with $a_{\alpha}$ as

$A=\alpha I-P\Delta$

with the domain $\mathcal{D}(A)=V\cap W$, which implies

$\displaystyle\|v\|_{2}\leq c\|v\|+\|P\Delta v\|,\ \forall v\in\mathcal{D}(A)$
(2.7)

Now, we turn to the boundary value problem of the Navier-Stokes equations
(1.1), (1.2) on $\Omega\subset R^{3}$ with the generalized vorticity-slip
boundary condition (1.11).

By using the Galerkin method based on the orthogonal eigenvectors of $A$,
noting the energy equation

$\frac{1}{2}\frac{d}{dt}\|u^{\varepsilon}\|^{2}+\varepsilon\|\nabla\times
u^{\varepsilon}\|^{2}+\varepsilon\int_{\partial\Omega}Bu^{\varepsilon}\cdot
u^{\varepsilon}=0$ (2.8)

valid for approximate solutions, and (2.4), (2.6), one can obtain the global
existence of weak solutions to the the boundary value problem of Navier-Stokes
equations (1.1), (1.2), (1.11) (see the corresponding definition in, for
instance, [43]).

Noting the energy equation

$\frac{1}{2}\frac{d}{dt}(\|\nabla\times
u^{\varepsilon}\|^{2}+\int_{\partial\Omega}Bu^{\varepsilon}\cdot
u^{\varepsilon})+\varepsilon\|P\Delta
u^{\varepsilon}\|^{2}+(u^{\varepsilon}\cdot\nabla u^{\varepsilon},P\Delta
u^{\varepsilon})=0$ (2.9)

for approximate solutions, (2.4), and (2.6)-(2.8), one can show that if
$u^{\varepsilon}_{0}\in V$, then there is a maximum time interval
$[0,T^{\varepsilon})$ such that the weak solutions are the unique strong one
on $[0,T^{\varepsilon})$ (see the corresponding definition in, for instance,
[43]).

It follows from Lemma 3.10 in [43] that

$(2(S(u)n)-\omega\times n)_{\tau}=GD(u)_{\tau}$ (2.10)

with $GD(u)=-2S(n)u$.

Then, the boundary conditions (1.7), (1.8) and (1.9) can be written to (1.11),
and (1.8) is equivalent to (1.11). Hence, one has the following uniform
regularity result for the solutions to the Navier-Stokes equations:

Lemma 2.1.(Masmoudi and Rousset [28]) Let $m>6$ be an integer and $\Omega$ be
a $C^{m+2}$ domain. Consider $u_{0}\in E^{m}\cap H$ such that $\nabla u_{0}\in
W^{1,\infty}_{co}$. Then there is $T_{m}>0$ such that for all sufficient small
$\varepsilon$, there is a unique solution $u^{\varepsilon}\in
C([0,T_{m}],E^{m})$ to the Navier-Stokes problem (1.1), (1.2), (1.11) with
$u^{\varepsilon}(0)=u_{0}$. Moreover, there is a constant $C$ such that

$\|u^{\varepsilon}\|_{H_{co}^{m}(\Omega)}+\|\nabla
u^{\varepsilon}\|_{H_{co}^{m-1}(\Omega)}+\|\nabla
u^{\varepsilon}\|_{W_{co}^{1,\infty}(\Omega)}+\varepsilon\int_{0}^{t}\|\nabla^{2}u^{\varepsilon}\|_{H_{co}^{m-1}(\Omega)}^{2}dt\leq
C$ (2.11)

on $[0,T_{m}]$.

Where $H_{co}^{m}(\Omega)$ and $W_{co}^{1,\infty}(\Omega)$ are co-norm vector-
spaces, $\|\cdot\|_{H_{co}^{m}(\Omega)}$ and
$\|\cdot\|_{W_{co}^{1,\infty}(\Omega)}$ denote the corresponding norms, and

$E^{m}=\\{u\in H_{co}^{m}(\Omega)|\nabla u\in H_{co}^{m-1}(\Omega)\\}$

Since the notations are rather complicate to be expressed, we omit it here and
refer to Masmoudi and Rousset [28] for the details. This uniform regularity
implies in particular the following uniform bound

$\|u^{\varepsilon}\|_{W^{1,\infty}}\leq C,\ {\rm on}\ [0,T_{m}],$

which plays an essential role in our estimates.

##  3 Convergence of the solutions

We now turn to the purpose of this paper to establish the convergence with a
rate for the solutions $u^{\varepsilon}$ to $u$. We start with the basic
$L^{2}-$estimate.

Theorem 3.1. Let $u_{0}\in H^{3}(\Omega)$ satisfy the assumptions stated in
Theorem 1.1. and $u(t)$ be the solution to the Euler equations (1.4), (1.4),
(1.5) on $[0,T]$ with $u(0)=u_{0}$, and $u(t)=u^{\varepsilon}(t)$ be the
solution to the Navier-Stokes problem (1.1), (1.2), (1.11) with
$u^{\varepsilon}(0)=u_{0}$. Then

$\|u^{\varepsilon}-u\|^{2}+\varepsilon\int_{0}^{T_{0}}\|u^{\varepsilon}-u\|_{1}^{2}dt\leq
c\varepsilon^{2-s}\ {\rm on}\ [0,T_{0}]$ (3.1)

for any $s>0$ and $\varepsilon$ small enough, where $T_{0}=\min\\{T,T_{m}\\}$.
Consequently,

$\|u^{\varepsilon}-u\|_{L^{\infty}([0,T]\times\Omega)}\leq
c\varepsilon^{\frac{2}{5}(1-s^{\prime})}$ (3.2)

for any $s^{\prime}>0$.

Proof: Note that $u^{\varepsilon}-u$ satisfies

$\displaystyle\partial_{t}(u^{\varepsilon}-u)-\varepsilon\Delta(u^{\varepsilon}-u)+\Phi+\nabla(p^{\varepsilon}-p)=\varepsilon\Delta
u,\textrm{ in }\Omega$ (3.3) $\displaystyle\nabla\cdot
u^{\varepsilon}=0,\textrm{ in }\Omega$ (3.4)
$\displaystyle(u^{\varepsilon}-u)\cdot n=0,\
n\times(\omega^{\varepsilon}-\omega)=[B(u^{\varepsilon}-u)+Bu]_{\tau}-n\times\omega\
{\rm on}\ \partial\Omega$ (3.5)

where $\omega=\nabla\times u$, $\omega^{\varepsilon}=\nabla\times
u^{\varepsilon}$,

$\Phi=u\cdot\nabla(u^{\varepsilon}-u)+(u^{\varepsilon}-u)\cdot\nabla
u+(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$

Then, the following identity holds

$\frac{1}{2}\frac{d}{dt}\|u^{\varepsilon}-u\|^{2}+\varepsilon\|\nabla\times(u^{\varepsilon}-u)\|^{2}+\mathcal{B}_{0}+(\Phi,u^{\varepsilon}-u)=\varepsilon(\Delta
u,u^{\varepsilon}-u)$ (3.6)

where

$\mathcal{B}_{0}=\varepsilon\int_{\partial\Omega}n\times(\omega^{\varepsilon}-\omega)(u^{\varepsilon}-u)=\varepsilon\int_{\partial\Omega}(B(u^{\varepsilon}-u)+Bu-n\times\omega)(u^{\varepsilon}-u)$

Note that

$\int_{\partial\Omega}n\times(\omega^{\varepsilon}-\omega)(u^{\varepsilon}-u)=\int_{\partial\Omega}(B(u^{\varepsilon}-u)+Bu-n\times\omega)(u^{\varepsilon}-u)$
$\leq c\int_{\partial\Omega}(|(u^{\varepsilon}-u)|^{2}+|(u^{\varepsilon}-u)|)$
$\leq
c(\|u^{\varepsilon}-u\|\|\omega^{\varepsilon}-\omega\|+|u^{\varepsilon}-u|_{L(\partial\Omega)})$

It follows from the trace theorem that

$|u^{\varepsilon}-u|_{L(\partial\Omega)}\leq|u^{\varepsilon}-u|_{L^{q}(\partial\Omega)}\leq
c\|u^{\varepsilon}-u\|_{H^{s}(\Omega)}$ (3.7)

for any $q>1$ and then any $s>0$.

By interpolation (see [36]), we have

$\|u^{\varepsilon}-u\|_{H^{s}(\Omega)}\leq
c\|u^{\varepsilon}-u\|^{(1-s)}\|u^{\varepsilon}-u\|_{1}^{s}\leq
c\|u^{\varepsilon}-u\|^{(1-s)}\|\omega^{\varepsilon}-\omega\|^{s}$ (3.8)

Note that

$\varepsilon\|u^{\varepsilon}-u\|^{(1-s)}\|\omega^{\varepsilon}-\omega\|^{s}\leq\delta\varepsilon\|\omega^{\varepsilon}-\omega\|^{2}+c_{\delta}(\|u^{\varepsilon}-u\|^{2}+\varepsilon^{2-s})$
(3.9)

for any $s\in(0,1)$ and

$\varepsilon\|u^{\varepsilon}-u\|\|\omega^{\varepsilon}-\omega\|\leq\varepsilon^{2}\|\omega^{\varepsilon}-\omega\|^{2}+\|u^{\varepsilon}-u\|^{2}$

Then, it holds that

$\mathcal{B}_{0}\leq
2\delta\varepsilon\|\omega^{\varepsilon}-\omega\|^{2}+c_{\delta}\|u^{\varepsilon}-u\|^{2}+\varepsilon^{2-s}$

for any $s\in(0,1)$ and $\varepsilon$ small enough.

Note also that

$(\Phi,u^{\varepsilon}-u)=((u^{\varepsilon}-u)\cdot\nabla
u,u^{\varepsilon}-u)\leq c\|u^{\varepsilon}-u\|^{2},$

and

$\varepsilon(\Delta
u,u^{\varepsilon}-u)\leq\|u^{\varepsilon}-u\|^{2}+c\varepsilon^{2}$

It follows that

$\frac{d}{dt}\|u^{\varepsilon}-u\|^{2}+\varepsilon\|\nabla\times(u^{\varepsilon}-u)\|^{2}\leq
c(\|u^{\varepsilon}-u\|^{2}+\varepsilon^{2-s})$ (3.10)

for any $s\in(0,1)$, and then for any $s>0$ and $\varepsilon$ small enough.
Note that $u^{\varepsilon}(0)-u(0)=0$. Then, (3.1) follows from (3.10) and the
Gronwall lemma.

Consequently, by using the Gagliardo-Nirenberg interpolation inequality, one
has

$\|u^{\varepsilon}-u\|_{L^{\infty}(\Omega)}\leq
c\|u^{\varepsilon}-u\|^{\frac{2}{5}}\|u^{\varepsilon}-u\|_{W^{1,\infty}(\Omega)}^{\frac{3}{5}}\leq
c\varepsilon^{\frac{2}{5}(1-s^{\prime})}$ (3.11)

for any $s^{\prime}>0$. The theorem is proved.

Next, we prove the major estimate in this paper.

Theorem 3.2. Let $u_{0}\in H^{3}(\Omega)$ satisfy the assumptions stated in
Theorem 1.1., and $u(t)$ be the solution to the Euler equations (1.4), (1.4),
(1.5) on $[0,T]$ with $u(0)=u_{0}$, and $u(t)=u^{\varepsilon}(t)$ be the
solution to the Navier-Stokes problem (1.1), (1.2), (1.11) with
$u^{\varepsilon}(0)=u_{0}$. Then

$\|u^{\varepsilon}-u\|_{1}^{2}+\varepsilon\int_{0}^{T}\|u^{\varepsilon}-u\|_{2}^{2}dt\leq
c\varepsilon^{1-s}\ {\rm on}\ [0,T_{0}]$ (3.12)

for any $s>0$ and $\varepsilon$ small enough, where $T_{0}=\min\\{T,T_{m}\\}$.

Proof: Let $s$ be the same as in Theorem 3.1.. Note the smoothness of the
solutions and

$\partial_{t}(u^{\varepsilon}-u)\cdot n=0\ \rm{on}\ \partial\Omega$

It follows from (3.3),(3.4) and (3.5) that

$\frac{1}{2}\frac{d}{dt}\|(\omega^{\varepsilon}-\omega)\|^{2}+\varepsilon\|P\Delta(u^{\varepsilon}-u)\|^{2}-(\Phi,P\Delta(u^{\varepsilon}-u))$
$=\int_{\partial\Omega}\partial_{t}(u^{\varepsilon}-u)\cdot(n\times(\omega^{\varepsilon}-\omega))-\varepsilon(\Delta
u,P\Delta(u^{\varepsilon}-u))$

where $P$ is the Lerray projection,

$\Phi=u\cdot\nabla(u^{\varepsilon}-u)+(u^{\varepsilon}-u)\cdot\nabla
u+(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$

and that

$\int_{\partial\Omega}\partial_{t}(u^{\varepsilon}-u)\cdot(n\times(\omega^{\varepsilon}-\omega))=\int_{\partial\Omega}\partial_{t}(u^{\varepsilon}-u)\cdot(B(u^{\varepsilon}-u)+Bu-n\times\omega)$
$=\frac{1}{2}\frac{d}{dt}(\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)+2\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega))-\mathcal{B}_{1}$

where

$\mathcal{B}_{1}=\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot\partial_{t}(Bu-n\times\omega)$

It follows from Theorem 3.1. and (3.9)

$|\mathcal{B}_{1}|\leq
c\int_{\partial\Omega}|(u^{\varepsilon}-u)|\leq\delta\|(\omega^{\varepsilon}-\omega)\|^{2}+c\varepsilon^{1-s}$

Note also that

$\varepsilon|(\Delta
u,P\Delta(u^{\varepsilon}-u))|\leq\frac{\varepsilon}{2}\|P\Delta(u^{\varepsilon}-u)\|^{2}+c\varepsilon$

and that

$-(\Phi,P\Delta(u^{\varepsilon}-u))=(P\Phi,-\Delta(u^{\varepsilon}-u))$
$=(\nabla\times\Phi,(\omega^{\varepsilon}-\omega))+\int_{\partial\Omega}n\times(\omega^{\varepsilon}-\omega)\cdot
P\Phi$
$=(\nabla\times\Phi,(\omega^{\varepsilon}-\omega))+\int_{\partial\Omega}(B(u^{\varepsilon}-u)+Bu-n\times\omega)\cdot
P\Phi$

Then, we get

$\frac{1}{2}\frac{d}{dt}E+\frac{\varepsilon}{2}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
N+BN+BNL+c(\|(\omega^{\varepsilon}-\omega)\|^{2}+\varepsilon^{1-s})$ (3.13)

for $\varepsilon$ small enough, where

$E=\|(\omega^{\varepsilon}-\omega)\|^{2}-(\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)+2\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega))$
$N=|(\nabla\times\Phi,(\omega^{\varepsilon}-\omega))|$
$BN=|\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot P\Phi|$
$BNL=|\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi|$

The term $N$ can be estimated easily by using the Sobolev inequalities and the
known uniform bounds for $\|u^{\varepsilon}\|_{\infty}$ and $\|\nabla
u^{\varepsilon}\|_{\infty}$. The term $BN$ can be estimated after integrating
by part properly. While, the estimate for the leading order term $BNL$ is
rather complicated due to the possible appearance of boundary layers. We now
carry out these estimates.

Estimates on $N$:

The term $N$ is estimated as follows:

Note that

$\nabla\times\Phi=\nabla\times(\omega\times(u^{\varepsilon}-u)+(\omega^{\varepsilon}-\omega)\times
u+(\omega^{\varepsilon}-\omega)\times(u^{\varepsilon}-u)$
$=[\omega,u^{\varepsilon}-u]+[\omega^{\varepsilon}-\omega,u]+[\omega^{\varepsilon}-\omega,\omega^{\varepsilon}-\omega]$

where

$[\varphi,\psi]=\psi\cdot\nabla\varphi-\varphi\cdot\nabla\psi$

And

$|((u^{\varepsilon}-u)\cdot\nabla\omega,\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$
$|(\omega\cdot\nabla(u^{\varepsilon}-u),\omega^{\varepsilon}-\omega)|\leq
c\|u^{\varepsilon}-u\|_{1}\|\omega^{\varepsilon}-\omega\|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

It follows that

$|([\omega,u^{\varepsilon}-u],\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

On the other hand,

$(u\cdot\nabla(\omega^{\varepsilon}-\omega),\omega^{\varepsilon}-\omega)=0$
$((\omega^{\varepsilon}-\omega)\nabla u,\omega^{\varepsilon}-\omega)\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

It follows that

$|([\omega^{\varepsilon}-\omega,u],\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

Note that

$((u^{\varepsilon}-u)\cdot\nabla(\omega^{\varepsilon}-\omega),\omega^{\varepsilon}-\omega)=0$
$((\omega^{\varepsilon}-\omega)\cdot\nabla(u^{\varepsilon}-u),\omega^{\varepsilon}-\omega)$
$\leq\|\nabla(u^{\varepsilon}-u)\|_{\infty}\|\omega^{\varepsilon}-\omega\|^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

(here and below we will use the uniform regularity that $\|\nabla
u^{\varepsilon}\|_{\infty}\leq c$). It follows that

$|([\omega^{\varepsilon}-\omega,u^{\varepsilon}-u],\omega^{\varepsilon}-\omega)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

Hence

$N\leq c\|\omega^{\varepsilon}-\omega\|^{2}$ (3.14)

Estimates on $BN$:

Next, we estimate the term

$BN=|\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot P\Phi|$

Note that

$P\Phi=\Phi+\nabla p_{\Phi}$

involves a scalar function $p_{\Phi}$, which is difficult to estimate on the
boundary. We will transform it to an estimate on $\Omega$ by an integrating by
part formula.

To this end, we first extend $n,B$ to the interior of $\Omega$ as follows:

$n(x)=\varphi(r(x))\nabla(r(x))$ $B(x)=\varphi(r(x))B(\Pi x)$

where

$r(x)=\min_{y\in\partial\Omega}d(x,y)$ $\Pi x=y_{x}\in\partial\Omega$

such that

$r(x)=d(x,y_{x})$

which is uniquely defined on $\Omega_{\sigma}=\\{x\in\Omega,\ r(x)\leq
2\sigma\\}$ for some $\sigma>0$, and $\varphi(s)$ is smooth and compactly
supported in $[0,2\sigma)$ such that

$\varphi(0)=1,\ {\rm on}\ [0,\sigma]$

Then, we can deduce the estimate of the boundary term $BN$ to an interior
estimate on $\Omega$ by the Stokes formula

$\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot
P\Phi=\int_{\partial\Omega}(n\times B(u^{\varepsilon}-u)\cdot(n\times P\Phi)$
$=(n\times B(u^{\varepsilon}-u),\nabla\times\Phi)-(\nabla\times(n\times
B(u^{\varepsilon}-u)),P\Phi)$

since $P\Phi\cdot n=0$ on the boundary, and $\nabla\times
P\Phi=\nabla\times\Phi$.

Note that

$|(n\times B(u^{\varepsilon}-u),(u^{\varepsilon}-u)\cdot\nabla\omega)|\leq
c\|u^{\varepsilon}-u\|^{2}\leq c\|\omega^{\varepsilon}-\omega\|^{2}$
$|(n\times B(u^{\varepsilon}-u),\omega\cdot\nabla(u^{\varepsilon}-u))|\leq
c\|u^{\varepsilon}-u\|_{1}^{2}\leq c\|\omega^{\varepsilon}-\omega\|^{2}$

It follows that

$|(n\times B(u^{\varepsilon}-u),[\omega,u^{\varepsilon}-u])|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

Note that $u\cdot n=0$ on $\partial\Omega$, $\nabla\cdot u=0$. Then

$|(n\times B(u^{\varepsilon}-u),u\cdot\nabla(\omega^{\varepsilon}-\omega))|$
$=|(\omega^{\varepsilon}-\omega,u\cdot\nabla(n\times
B(u^{\varepsilon}-u)))|\leq c\|u^{\varepsilon}-u\|_{1}^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

and

$|(n\times B(u^{\varepsilon}-u),(\omega^{\varepsilon}-\omega)\cdot\nabla
u)|\leq c\|\omega^{\varepsilon}-\omega\|^{2}$

It follows that

$|(n\times B(u^{\varepsilon}-u),[\omega^{\varepsilon}-\omega,u])|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

Similarly

$|(n\times
B(u^{\varepsilon}-u),(u^{\varepsilon}-u)\cdot\nabla(\omega^{\varepsilon}-\omega))|=|(\omega^{\varepsilon}-\omega,(u^{\varepsilon}-u)\cdot\nabla(n\times
B(u^{\varepsilon}-u)))|$ $\leq c\|u^{\varepsilon}-u\|_{1}^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

and

$|(n\times
B(u^{\varepsilon}-u),(\omega^{\varepsilon}-\omega)\cdot\nabla(u^{\varepsilon}-u))|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

It follows that

$|(n\times
B(u^{\varepsilon}-u),[\omega^{\varepsilon}-\omega,u^{\varepsilon}-u])|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

Hence

$|(n\times B(u^{\varepsilon}-u),\nabla\times\Phi)|\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$ (3.15)

Note also that

$|(\nabla\times(n\times
B(u^{\varepsilon}-u)),P\Phi)|\leq\|\nabla\times(n\times
B(u^{\varepsilon}-u))\|\|\Phi\|$ $\leq c\|(u^{\varepsilon}-u))\|_{1}^{2}\leq
c\|\omega^{\varepsilon}-\omega\|^{2}$

Hence, we have

$BN\leq c\|\omega^{\varepsilon}-\omega\|^{2}$ (3.16)

Estimate on $\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi$:

Finally, we estimate the leading order term on the boundary

$BNL=|\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi|$

It should be noted that the estimate is trivial if

$[Bu]_{\tau}-n\times\omega=0$

which is that the Euler solution satisfies the same boundary condition (1.11)
just as the Navier-Stokes solution does so there is no strong boundary layer.
This applies to the cases treated in [41, 42]. Here,
$[Bu]_{\tau}-n\times\omega$ may be not equal zero, then boundary layers may
occur. Additional efforts are needed to overcome the new difficulties.

Similar to the above, we have

$|\int_{\partial\Omega}(Bu-n\times\omega)\cdot
P\Phi|=|\int_{\partial\Omega}n\times(Bu-n\times\omega)\cdot n\times P\Phi|$
$=|(n\times(Bu-n\times\omega),\nabla\times\Phi)-(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)|$

The term $|(n\times(Bu-n\times\omega),\nabla\times\Phi)|$ is relatively easy
to be estimated since $\nabla\times\Phi$ does not involve the pressure
function.

Rewrite $\Phi$ as

$\Phi=\Phi_{1}+\Phi_{2}+\Phi_{3}$

where

$\Phi_{1}=(u^{\varepsilon}-u)\cdot\nabla u$
$\Phi_{2}=u\cdot\nabla(u^{\varepsilon}-u)$
$\Phi_{3}=(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$

Note that

$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{1})|=|(n\times(Bu-n\times\omega),\nabla\times((u^{\varepsilon}-u)\cdot\nabla
u))|$ $=|\int_{\partial\Omega}n\times((u^{\varepsilon}-u)\cdot\nabla
u)\cdot(n\times(Bu-n\times\omega))+(\nabla\times(n\times(Bu-n\times\omega)),(u^{\varepsilon}-u)\cdot\nabla
u)|$ $\leq
c(\int_{\partial\Omega}|u^{\varepsilon}-u|+\|u^{\varepsilon}-u\|)\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$

and

$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{3})|=|(n\times(Bu-n\times\omega),\nabla\times((u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)))|$
$=|\int_{\partial\Omega}n\times((u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u))\cdot(n\times(Bu-n\times\omega))+(\nabla\times(n\times(Bu-n\times\omega)),(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u))|$
$\leq
c(\int_{\partial\Omega}|u^{\varepsilon}-u|+\|u^{\varepsilon}-u\|_{1}^{2})\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$

since $u^{\varepsilon}-u$ is smooth, so that

$|\nabla(u^{\varepsilon}-u)|\leq c\|\nabla(u^{\varepsilon}-u)\|_{\infty}\leq
c,\ {\rm on}\ \partial\Omega$

To estimate $(n\times(Bu-n\times\omega),\nabla\times\Phi_{2})$, we note

$\nabla\times(\psi\cdot\nabla\phi)=\psi\cdot\nabla(\nabla\times\phi)+\nabla\psi^{\perp}\cdot\nabla\phi$

where $\nabla\psi^{\perp}$ is expressed in component by

$(\nabla\psi^{\perp}\cdot\nabla\phi)_{j}=(-1)^{j+1}\partial_{j+1}\psi\cdot\nabla\phi_{j+1}+(-1)^{j+2}\partial_{j+2}\psi\cdot\nabla\phi_{j+2}$
(3.17)

with the index modulated by 3. Hence, we have

$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{2})|=|(n\times(Bu-n\times\omega),\nabla\times(u\cdot\nabla(u^{\varepsilon}-u)))|$
$=|(n\times(Bu-n\times\omega),u\cdot\nabla(\omega^{\varepsilon}-\omega))+(n\times(Bu-n\times\omega),\nabla
u^{\perp}\cdot\nabla(u^{\varepsilon}-u))|$

Note that

$|(n\times(Bu-n\times\omega),u\cdot\nabla(\omega^{\varepsilon}-\omega))|=|(\omega^{\varepsilon}-\omega,u\cdot\nabla(n\times(Bu-n\times\omega)))|$
$=(\nabla\times(u^{\varepsilon}-u),u\cdot\nabla(n\times(Bu-n\times\omega)))$
$=|\int_{\partial\Omega}n\times(u^{\varepsilon}-u)\cdot(u\cdot\nabla(n\times(Bu-n\times\omega)))+(u^{\varepsilon}-u,\nabla\times(u\cdot\nabla(n\times(Bu-n\times\omega))))|$
$\leq c(\int_{\partial\Omega}|u^{\varepsilon}-u|+\|u^{\varepsilon}-u\|)\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$

Note also that

$(\partial_{j}u\cdot\nabla(u^{\varepsilon}-u)_{j},(n\times(Bu-n\times\omega))_{k})=$
$(\partial_{j}u,\nabla((u^{\varepsilon}-u)_{j}(n\times(Bu-n\times\omega))_{k})-(\partial_{j}u\cdot\nabla(n\times(Bu-n\times\omega))_{k},(u^{\varepsilon}-u)_{j}),$

and that

$|(\partial_{j}u,\nabla((u^{\varepsilon}-u)_{j}(n\times(Bu-n\times\omega))_{k})|=|\int_{\partial\Omega}(u^{\varepsilon}-u)_{j}(n\times(Bu-n\times\omega))_{k}\partial_{j}u\cdot
n|$ $\leq c\|u\|_{1,\infty}^{2}\int_{\partial\Omega}|u^{\varepsilon}-u|\leq
c\int_{\partial\Omega}|u^{\varepsilon}-u|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$

since $\nabla\cdot\partial_{j}u=0$, and that

$|(\partial_{j}u\cdot\nabla(n\times(Bu-n\times\omega))_{k},(u^{\varepsilon}-u)_{j})|\leq
c\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$

it follows that

$|(n\times(Bu-n\times\omega),\nabla\times\Phi_{2})|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$

Then, we get

$|(n\times(Bu-n\times\omega),\nabla\times\Phi)|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$ (3.18)

It remains to estimate $|(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)|$.
There is also a difficulty arising from $P\Phi$. To do it, it is noticed that

$P\Phi=u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u+P\tilde{\Phi}$

where

$\tilde{\Phi}=2(u^{\varepsilon}-u)\cdot\nabla
u+(u^{\varepsilon}-u)\cdot\nabla(u^{\varepsilon}-u)$

with

$u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u=\nabla\times((u^{\varepsilon}-u)\times u)\in H$

since that $(u^{\varepsilon}-u)\cdot n=0$ and $u\cdot n=0$ on $\partial\Omega$
will imply

$(u^{\varepsilon}-u)\times u=\lambda n$

and

$H=\\{v=\nabla\times\phi;\ \phi\in H^{1}(\Omega),n\times\phi=0\\}$

Hence, we have

$P(u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u)=u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla u$

so that

$(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)=$
$(\nabla\times(n\times(Bu-n\times\omega)),u\cdot\nabla(u^{\varepsilon}-u)-(u^{\varepsilon}-u)\cdot\nabla
u)+(\nabla\times(n\times(Bu-n\times\omega)),P\tilde{\Phi})$

and

$\|P\tilde{\Phi}\|\leq c\|\tilde{\Phi}\|\leq
c(\|u\|_{1,\infty}+\|u^{\varepsilon}-u\|_{1,\infty})\|u^{\varepsilon}-u\|\leq
c\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$

These properties help us to complete the rest of estimates.

First, we have

$|(\nabla\times(n\times(Bu-n\times\omega)),P\tilde{\Phi})|\leq
c\|P\tilde{\Phi}\|\leq c\|\tilde{\Phi}\|\leq c\|u^{\varepsilon}-u\|\leq
c\varepsilon^{1-s}$

Next, note that

$|(\nabla\times(\times(Bu-n\times\omega)),u\cdot\nabla(u^{\varepsilon}-u))=$
$|(u^{\varepsilon}-u,u\cdot\nabla(\nabla\times(\times(Bu-n\times\omega))))|\leq
c\|u\|_{3}^{2}\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$

since $\nabla u=0$ in $\Omega$ and $u\cdot n=0$ on $\partial\Omega$.

Note also that

$|(\nabla\times(n\times(Bu-n\times\omega)),(u^{\varepsilon}-u)\nabla u)|\leq
c\|u\|_{2}^{2}\|u^{\varepsilon}-u\|\leq c\varepsilon^{1-s}$

Hence, we have

$|(\nabla\times(n\times(Bu-n\times\omega)),P\Phi)|\leq c\varepsilon^{1-s}$
(3.19)

Then, we conclude that

$BNL=|\int_{\partial\Omega}(Bu-n\times\omega)\cdot P\Phi|\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$ (3.20)

In conclusion, it follows from (3.13),(3.14),(3.16) and (3.20) that

$\frac{1}{2}\frac{d}{dt}E+\frac{\varepsilon}{2}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
c(\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon^{1-s})$ (3.21)

Recall that

$E=\|(\omega^{\varepsilon}-\omega)\|^{2}-(\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)+2\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega))$

and

$|\int_{\partial\Omega}B(u^{\varepsilon}-u)\cdot(u^{\varepsilon}-u)|\leq
c\int_{\partial\Omega}|(u^{\varepsilon}-u)|^{2}\leq
c\|u^{\varepsilon}-u\|\|\omega^{\varepsilon}-\omega\|\leq\|\omega^{\varepsilon}-\omega\|^{2}+c\|u^{\varepsilon}-u\|^{2}$

and

$|\int_{\partial\Omega}(u^{\varepsilon}-u)\cdot(Bu-n\times\omega)|\leq
c\int_{\partial\Omega}|(u^{\varepsilon}-u)|\leq\|\omega^{\varepsilon}-\omega\|^{2}+c\|u^{\varepsilon}-u\|^{2}$

It follows that

$\|\omega^{\varepsilon}-\omega\|^{2}+\varepsilon
c\int_{0}^{t}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
c\int_{0}^{t}\|\omega^{\varepsilon}-\omega\|^{2}+c\varepsilon^{1-s}$ (3.22)

By using the Gronwall Lemma, one gets that

$\|\omega^{\varepsilon}-\omega\|^{2}\leq c\varepsilon^{1-s}$ (3.23)

on $[0,T_{0}]$ for any given $s>0$ and $\varepsilon$ for small enough. Then

$\|u^{\varepsilon}-u\|_{1}^{2}\leq c\varepsilon^{1-s}$ (3.24)

and

$\varepsilon\int_{0}^{t}\|P\Delta(u^{\varepsilon}-u)\|^{2}\leq
c\varepsilon^{1-s}$ (3.25)

Combining this with (2.7) and Theorem 3.1 implies that

$\varepsilon\int_{0}^{t}\|(u^{\varepsilon}-u)\|_{2}^{2}\leq
c\varepsilon^{1-s}$ (3.26)

Note that

$\|\nabla(u^{\varepsilon}-u)\|_{p}^{p}\leq
c\|\nabla(u^{\varepsilon}-u)\|_{\infty}^{p-2}\|\nabla(u^{\varepsilon}-u)\|^{2}$

It follows that

$\|\nabla(u^{\varepsilon}-u)\|_{p}^{p}\leq c\varepsilon^{1-s}$

The theorem is proved.

Remark 3.1. The estimate (1.19) is optimal in the sense that $s$ can not be
taken to be $0$. Since if $s=0$, then (1.19) will imply that
$u^{\varepsilon}-u$ is uniform bounded in $L^{2}(0,T;H^{2}(\Omega))$. Then,
there is a subsequence such that $u^{\varepsilon_{n}}-u$ weakly convergence in
$H^{2}(\Omega)$ for a.e. $t\in[0,T]$. Note that $u^{\varepsilon_{n}}-u$
strongly convergence to $0$ in $H^{1}(\Omega)$ for all $t\in[0,T]$. It follows
that

$u^{\varepsilon_{n}}\rightarrow u\ weakly\ in\ H^{2}(\Omega),\ a.e.\ t$

Note that $W$ is closed in $H^{2}(\Omega)$, then is weakly closed. Hence,
$u\in W$ for a.e. $t\in[0,T]$, i.e.,

$[Bu]_{\tau}-n\times\omega=0,\ \ a.e.\ t$

Since $u$ is smooth, then it holds for all $t\in[0,T]$. This is impossible in
general, see [4, 5].

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