#  $H^{2}$-regularity for stationary and non-stationary Bingham
problems with perfect slip boundary condition

Takeshi Fukao  Faculty of Advanced Science and Technology, Ryukoku University,
1-5 Yokotani, Seta Oe-cho, Otsu-shi, 520-2194 Shiga, Japan
<EMAIL_ADDRESS>and  Takahito Kashiwabara  Graduate School of
Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, 153-8914
Tokyo, Japan<EMAIL_ADDRESS>
###### Abstract.

$H^{2}$-spatial regularity of stationary and non-stationary problems for
Bingham fluids formulated with the pseudo-stress tensor is discussed. The
problem is mathematically described by an elliptic or parabolic variational
inequality of the second kind, to which weak solvability in the Sobolev space
$H^{1}$ is well known. However, higher regularity up to the boundary in a
bounded smooth domain seems to remain open. This paper indeed shows such
$H^{2}$-regularity if the problems are supplemented with the so-called perfect
slip boundary condition. For the stationary Bingham–Stokes problem, the key of
the proof lies in a priori estimates for a regularized problem avoiding
investigation of higher pressure regularity, which seems difficult to get in
the presence of a singular diffusion term. The $H^{2}$-regularity for the
stationary case is then directly applied to establish strong solvability of
the non-stationary Bingham–Navier–Stokes problem, based on discretization in
time and on the truncation of the nonlinear convection term.

###### Key words and phrases:

Bingham fluid; variational inequality; regularity up to boundary; slip
boundary condition

###### 2010 Mathematics Subject Classification:

Primary: 76D03, 76A05.

TF acknowledges the support from the JSPS KAKENHI Grant-in-Aid for Scientific
Research(C), Japan, Grant Number 21K03309. TK acknowledges the support from
the JSPS KAKENHI Grant-in-Aid for Early-Career Scientists, Japan, Grant Number
20K14357.

##  1\. Introduction

Bingham fluid, one of non-Newtonian fluids, is a model describing the motion
of viscoplastic materials, which behave like viscous fluid if shear stress
magnitude exceeds a threshold, called the yield stress, and like rigid bodies
otherwise. Examples of viscoplastic materials are sauces, pastes, crude oils,
drilling muds, slurries, lava, etc., understanding of which is important in
several fields of industry or physics.

Let $\Omega\subset\mathbb{R}^{d}\,(d=2,3)$ be a bounded domain with the smooth
boundary $\Gamma:=\partial\Omega$. We consider the following boundary value
problem:

(1.1)
$\displaystyle-\operatorname{div}(\bm{T}^{D}-p\bm{I})=\bm{f},\quad\operatorname{div}\bm{u}=0\quad\text{in}\quad\Omega,$
(1.2)
$\displaystyle\quad\begin{cases}\bm{T}^{D}=\nu\nabla\bm{u}+g\dfrac{\nabla\bm{u}}{|\nabla\bm{u}|}&\text{if}\quad\nabla\bm{u}\neq\bm{O},\\\
|\bm{T}^{D}|\leq g&\text{if}\quad\nabla\bm{u}=\bm{O},\end{cases}$ (1.3)
$\displaystyle\bm{u}\cdot\bm{n}=0,\quad(\bm{T}^{D}\bm{n})_{\tau}=\bm{0}\quad\text{on}\quad\Gamma,$

which we refer to as the _stationary Bingham–Stokes_ problem (note that it
reduces to the standard Stokes equations if $g=0$), and the following initial
boundary value problem:

(1.4) $\displaystyle\frac{\partial\bm{u}}{\partial
t}+(\bm{u}\cdot\nabla)\bm{u}-\operatorname{div}(\bm{T}^{D}-p\bm{I})=\bm{f},\quad\operatorname{div}\bm{u}=0\quad\text{in}\quad
Q:=(0,T)\times\Omega,$ (1.5)
$\displaystyle\quad\begin{cases}\bm{T}^{D}=\nu\nabla\bm{u}+g\dfrac{\nabla\bm{u}}{|\nabla\bm{u}|}&\text{if}\quad\nabla\bm{u}\neq\bm{O},\\\
|\bm{T}^{D}|\leq g&\text{if}\quad\nabla\bm{u}=\bm{O},\end{cases}$ (1.6)
$\displaystyle\bm{u}\cdot\bm{n}=0,\quad(\bm{T}^{D}\bm{n})_{\tau}=\bm{0}\quad\text{on}\quad\Sigma:=(0,T)\times\Gamma,$
(1.7) $\displaystyle\bm{u}(0)=\bm{u}_{0}\quad\text{in}\quad\Omega,$

which we refer to as the _non-stationary Bingham–Navier–Stokes_ problem. Here,
the unknown functions $\bm{T}^{D}$, $\bm{u}$, and $p$ mean the deviatoric part
of the stress tensor ($\nu>0$ is the viscosity constant), the velocity, and
the pressure of a material, respectively. Then $\bm{T}=-p\bm{I}+\bm{T}^{D}$
constitutes the total stress tensor with $\bm{I}$ the identity matrix. For
matrices $\bm{A},\bm{B}\in\mathbb{R}^{d\times d}$, we define the inner product
and magnitude by $\bm{A}:\bm{B}=\sum_{i,j=1}^{d}A_{ij}B_{ij}$ and
$|\bm{A}|=(\bm{A}:\bm{A})^{1/2}$, respectively. The external force $\bm{f}$
and the yield stress $g\geq 0$ are given functions.

Let us discuss the stationary problem for a while. Compared with the standard
Bingham fluid model (see e.g. [13, Section VI.1.3]), problem (1.1)–(1.2) is
simplified in the sense that we exploit the _pseudo stress tensor_ , i.e, we
use $\nabla\bm{u}=(\partial u_{i}/\partial x_{j})_{1\leq i,j\leq d}$ instead
of the rate-of-strain tensor
$\bm{D}(\bm{u})=(\nabla\bm{u}+(\nabla\bm{u})^{\top})/2$ in the constitutive
law. This facilitates technical arguments in our analysis (especially, the
coercivity relation is simpler than the one obtained from Korn’s inequality,
cf. (2.2) below), still retaining mathematical difficulties arising from the
Bingham nonlinearity.

The boundary condition (1.3) is the perfect slip boundary condition, where
$\bm{n}$ is the outer unit normal to $\Gamma$ and the subscript $\tau$ means
the tangential part of a vector (namely,
$\bm{v}_{\tau}=\bm{v}-(\bm{v}\cdot\bm{n})\bm{n}$). In view of (1.2), it
reduces to

(1.8)
$\bm{u}\cdot\bm{n}=0,\quad(\nabla\bm{u}\,\bm{n})_{\tau}=\bm{0}\quad\text{on}\quad\Gamma.$

By adopting the theory presented in the classical literature [13, Chapter VI],
we can easily find that (1.1)–(1.3) is formulated as a variational inequality
problem and that it admits a unique weak solution
$\bm{u}\in\bm{H}^{1}(\Omega)$. However, the higher regularity, e.g.,
$\bm{u}\in\bm{H}^{2}(\Omega)$ seems much less understood. For the scalar
Bingham problem without the divergence-free constraint (also known as Bingham
flows in cylinders), the $H^{2}$-regularity is established by [7, Theorem 15]
using a sophisticated estimate for a resolvent-like problem for the Laplace
operator. From another perspective, partial interior $C^{1}$-regularity, as
well as interior $H^{2}$-regularity (i.e.
$\bm{u}\in\bm{H}^{2}_{\mathrm{loc}}(\Omega)$), is obtained in [14, 15] for the
stationary Bingham flow with the convective term. See also [25] which mentions
some interior $H^{2}$-regularity for the non-stationary problem. More
recently, the $C^{1}(\Omega)$-regularity is established for the scalar case by
[35].

On the other hand, it seems that there are no results of $H^{2}$-regularity
_up to the boundary_ when the divergence-free constraint is imposed, that is,
the pressure $p$ appears as an unknown variable. A major difficulty consists
in obtaining higher regularity than $L^{2}(\Omega)$ for $p$. In fact, as is
seen from the linear Stokes case, we usually need to establish higher
tangential regularity of $p$ before discussing the regularity of $\bm{u}$ in
the normal direction (see [11, pp. 24–25]). However, this strategy seems
infeasible for the Bingham case, which formally reads $\nabla
p=\bm{f}-\nu\Delta\bm{u}-\operatorname{div}(g\nabla\bm{u}/|\nabla\bm{u}|)$,
since the last term is too singular to be even a measurable function near the
region where $\nabla\bm{u}$ vanishes.

If the no-slip boundary condition is imposed, it is not trivial to settle this
issue. Thereby we suggest to replace it by the perfect slip one (1.3) or (1.8)
and indeed prove the regularity $\bm{u}\in\bm{H}^{2}(\Omega)$ up to the
boundary, which is the first objective of the present paper (see Corollary 3.1
below). In the proof, we introduce a regularized problem, transform it into a
half-space problem, and estimate tangential and normal second-order
derivatives. The key observation is that for (1.8) one can obtain the
regularity of velocity in the normal direction, knowing only lower regularity
for the pressure (in other words, higher regularity is shown only for $\bm{u}$
but not for $p$).

The latter part of this paper (Section 4) deals with the non-stationary
Bingham–Navier–Stokes problem. In this regard we notice that preceding studies
[6, 13, 16, 21, 24, 26, 28] considered only one spatial differentiability
whereas $H^{2}$-regularity in the two-dimensional whole space was obtained in
[23]. By directly applying Corollary 3.1, we will prove that there exists a
unique local-in-time solution $\bm{u}$ having the $H^{2}$-regularity—more
precisely, $\bm{u}\in W^{1,\infty}(0,T;\bm{L}^{2}(\Omega))\cap
L^{\infty}(0,T;\bm{H}^{2}(\Omega))$ in Theorem 4.1 and $\bm{u}\in
H^{1}(0,T;\bm{L}^{2}(\Omega))\cap L^{2}(0,T;\bm{H}^{2}(\Omega))$ in Theorem
4.2—depending on assumptions for the data $\bm{f},g$, and $\bm{u}_{0}$. The
proof is based on the discretization in time (also known as Rhothe’s method)
combined with the truncation technique for the convection term as in [3].

Briefly speaking, our well-posedness result tells us that, if the yield stress
$g$ is first-order differentiable with respect to space and time, then we can
construct an $H^{2}$-strong solution. As one of its possible applications and
future studies, we mention non-isothermal Bingham fluids involving heat
transfer, in which the yield stress and the viscosity parameter may depend on
temperature (cf. [20, 27]).

We use the standard Lebesgue spaces $L^{p}(\Omega)$ and Sobolev spaces
$W^{m,p}(\Omega)$ for $p\in[1,\infty]$ and $m\in\mathbb{N}$. Those spaces on
the boundary, in particular the Sobolev–Slobodeckij space $W^{s,p}(\Gamma)$
will also be utilized. Function spaces for vectors are indicated with bold
fonts, e.g., $\bm{L}^{p}(\Omega),\,\bm{W}^{m,p}(\Omega)$. Those for tensors
(matrices) are also indicated by the same symbols; nevertheless, in case
confusion may occur, we write $L^{p}(\Omega)^{d\times d}$ etc. Throughout this
paper, we let $C$ denote a generic positive constant that may differ at each
occurrence. When we need to specify its dependency on parameters, we indicate
$C=C(\Omega)$ etc.

##  2\. Weak formulations for the stationary problem

In this and next sections, let us consider the stationary Bingham–Stokes
problem (1.1)–(1.3). For simplicity in the presentation, we assume $\nu=1$.
Let us derive weak formulations and recall unique existence of a weak
solution, which is already known. For that purpose we introduce the following
Hilbert spaces:

$\displaystyle\bm{V}$
$\displaystyle=\\{\bm{v}\in\bm{H}^{1}(\Omega)\mid\bm{v}\cdot\bm{n}=0\;\text{
on }\Gamma\\},$ $\displaystyle\bm{V}_{\sigma}$
$\displaystyle=\\{\bm{v}\in\bm{V}\mid\operatorname{div}\bm{v}=0\\},\quad\|\bm{v}\|_{\bm{V}}=\|\bm{v}\|_{\bm{V}_{\sigma}}:=\|\bm{v}\|_{\bm{H}^{1}(\Omega)},$
$\displaystyle\bm{H}_{\sigma}$
$\displaystyle=\\{\bm{v}\in\bm{L}^{2}(\Omega)\mid\operatorname{div}\bm{v}=0,\quad\bm{v}\cdot\bm{n}=0\;\text{
on
}\Gamma\\},\quad\|\bm{v}\|_{\bm{H}_{\sigma}}:=\|\bm{v}\|_{\bm{L}^{2}(\Omega)},$
$\displaystyle L^{2}_{0}(\Omega)$ $\displaystyle=\Bigl{\\{}q\in
L^{2}(\Omega)\mid\int_{\Omega}q\,d\bm{x}=0\Bigr{\\}}.$

We regard
$\bm{V}_{\sigma}\subset\bm{H}_{\sigma}=\bm{H}_{\sigma}^{\prime}\subset\bm{V}_{\sigma}^{\prime}$
as a Gelfand triple, where the prime means the dual norm space. The
$L^{2}(\Omega)$-inner products for scalar-, vector-, and matrix-valued
functions are denoted by the same symbol $(\cdot,\cdot)$. The duality pairing
between $\bm{V}_{\sigma}^{\prime}$ and $\bm{V}_{\sigma}$ is written as
$\left<\cdot,\cdot\right>$.

With this setting, we show that a solution of the Bingham problem satisfies a
variational inequality.

######  Proposition 2.1.

(i) Let $(\bm{u},p)$ be a sufficiently smooth solution of (1.1)–(1.3). Then we
have

(2.1)
$(\nabla\bm{u},\nabla(\bm{v}-\bm{u}))+(g,|\nabla\bm{v}|-|\nabla\bm{u}|)\geq\left<\bm{f},\bm{v}-\bm{u}\right>\qquad\forall\bm{v}\in\bm{V}_{\sigma}.$

(ii) Let $\bm{f}\in\bm{V}_{\sigma}^{\prime}$ and $g\in L^{2}(\Omega),\,g\geq
0$. Then there exists a unique solution $\bm{u}\in\bm{V}_{\sigma}$ of (2.1).

###### Proof.

(i) It follows from integration by parts that
$(\bm{T}^{D}(\bm{u}),\nabla\bm{v})=(\bm{f},\bm{v})$ for all
$\bm{v}\in\bm{V}_{\sigma}$, and hence

$\displaystyle(\nabla\bm{u},\nabla(\bm{v}-\bm{u}))+(g,|\nabla\bm{v}|-|\nabla\bm{u}|)-(\bm{f},\bm{v}-\bm{u})$
$\displaystyle=(g,|\nabla\bm{v}|-|\nabla\bm{u}|)-(\bm{T}^{D}(\bm{u})-\nabla\bm{u},\nabla(\bm{v}-\bm{u})).$

The right-hand side $\geq 0$ because
$(\bm{T}^{D}(\bm{u})-\nabla\bm{u}):\nabla\bm{v}\leq g|\nabla\bm{v}|$ and
$(\bm{T}^{D}(\bm{u})-\nabla\bm{u}):\nabla\bm{u}=g|\nabla\bm{u}|$ in $\Omega$
by (1.2).

(ii) Observe that the functional $j$ defined by $j(\bm{v})=(g,|\nabla\bm{v}|)$
is continuous from $\bm{V}$ to $\mathbb{R}$ and convex, and that the following
coercivity holds:

(2.2) $\|\nabla\bm{v}\|_{\bm{L}^{2}(\Omega)}^{2}\geq
C\|\bm{v}\|_{\bm{H}^{1}(\Omega)}^{2}\qquad\forall\bm{v}\in\bm{V}.$

Then a well-known existence theory of elliptic variational inequalities (see
[18, Theorem I.4.1]) implies the desired result. ∎

######  Remark 2.1.

There holds a Poincaré inequality $\|\bm{v}\|_{\bm{L}^{2}(\Omega)}\leq
C\|\nabla\bm{v}\|_{\bm{L}^{2}(\Omega)}$ for $\bm{v}\in\bm{H}^{1}(\Omega)$ such
that $\bm{v}\cdot\bm{n}=0$ on $\Gamma$, which may be proved in a similar way
to [5, Lemma 2.3]. In fact, a constant vector field $\bm{v}$ satisfying
$\bm{v}\cdot\bm{n}=0$ on $\Gamma$ must be zero since the Gauss map $\bm{n}$ is
surjective from a compact manifold $\Gamma$ onto the $(d-1)$-dimensional
sphere.

The converse statement of Proposition 2.1 is not trivial to prove. We achieve
it by constructing a multiplier based on a regularized variational inequality
introduced for $\epsilon>0$ as follows:

(2.3)
$(\nabla\bm{u}_{\epsilon},\nabla(\bm{v}-\bm{u}_{\epsilon}))+(g,\sqrt{|\nabla\bm{v}|^{2}+\epsilon^{2}}-\sqrt{|\nabla\bm{u}_{\epsilon}|^{2}+\epsilon^{2}})\geq\left<\bm{f},\bm{v}-\bm{u}_{\epsilon}\right>\qquad\forall\bm{v}\in\bm{V}_{\sigma}.$

######  Proposition 2.2.

(i) Let $\bm{f}\in\bm{V}_{\sigma}^{\prime}$ and $g\in L^{2}(\Omega),\,g\geq
0$. Then there exists a unique solution $\bm{u}_{\epsilon}\in\bm{V}_{\sigma}$
of (2.3). Moreover, it satisfies the following variational equation:

(2.4)
$(\nabla\bm{u}_{\epsilon},\nabla\bm{v})+\Big{(}g\frac{\nabla\bm{u}_{\epsilon}}{\sqrt{|\nabla\bm{u}_{\epsilon}|^{2}+\epsilon^{2}}},\nabla\bm{v}\Big{)}=\left<\bm{f},\bm{v}\right>\qquad\forall\bm{v}\in\bm{V}_{\sigma}.$

(ii) We have $\|\bm{u}_{\epsilon}\|_{\bm{H}^{1}(\Omega)}\leq
C\|\bm{f}\|_{\bm{V}_{\sigma}^{\prime}}$. Moreover, when $\epsilon\to 0$,
$\bm{u}_{\epsilon}$ converges to the solution $\bm{u}$ of (2.1) strongly in
$\bm{V}_{\sigma}$.

###### Proof.

(i) Unique existence of $\bm{u}_{\epsilon}$ is immediately obtained as in
Proposition 2.1(ii). Let $j_{\epsilon}:\bm{V}\to\mathbb{R}$ be defined by
$j_{\epsilon}(\bm{v})=(g,\sqrt{|\nabla\bm{v}|^{2}+\epsilon^{2}})$. It is
Gâteaux differentiable and we get

$j_{\epsilon}^{\prime}(\bm{v})[\bm{w}]=\lim_{s\to
0}\frac{j_{\epsilon}(\bm{v}+s\bm{w})-j_{\epsilon}(\bm{v})}{s}=\int_{\Omega}g\frac{\nabla\bm{v}}{\sqrt{|\nabla\bm{v}|^{2}+\epsilon^{2}}}:\nabla\bm{w}\,d\bm{x}\quad(\bm{v},\bm{w}\in\bm{V}).$

Now take $\bm{u}_{\epsilon}+s\bm{v}$ with $s\gtrless 0$ as a text function in
(2.3) to find

$(\nabla\bm{u}_{\epsilon},\nabla\bm{v})+\frac{j_{\epsilon}(\bm{u}_{\epsilon}+s\bm{v})-j_{\epsilon}(\bm{u}_{\epsilon})}{s}\gtreqless\left<\bm{f},\bm{v}\right>.$

Making $s\to 0$ deduces the desired equality.

(ii) The $\bm{H}^{1}(\Omega)$-estimate for $\bm{u}_{\epsilon}$ immediately
results from taking $\bm{v}=\bm{u}_{\epsilon}$ in (2.4) together with the
coercivity (2.2). Next, taking $\bm{v}=\bm{u}_{\epsilon}$ in (2.1) and
$\bm{v}=\bm{u}$ in (2.3) respectively and adding the resulting inequalities,
we obtain

$\|\nabla(\bm{u}-\bm{u}_{\epsilon})\|_{\bm{L}^{2}(\Omega)}^{2}\leq
j_{\epsilon}(\bm{u})-j(\bm{u})+j(\bm{u}_{\epsilon})-j_{\epsilon}(\bm{u}_{\epsilon})\leq\epsilon\|g\|_{L^{1}(\Omega)},$

where we have used
$0\leq\sqrt{|\bm{A}|^{2}+\epsilon^{2}}-|\bm{A}|=\epsilon^{2}/(\sqrt{|\bm{A}|^{2}+\epsilon^{2}}+|\bm{A}|)\leq\epsilon$.
This shows $\bm{u}_{\epsilon}\to\bm{u}$ strongly in $\bm{V}_{\sigma}$ as
$\epsilon\to 0$. ∎

For the regularized problem, it is easy to recover the associated pressure.

######  Proposition 2.3.

If $\bm{f}\in\bm{L}^{2}(\Omega)$, then for the solution
$\bm{u}_{\epsilon}\in\bm{V}_{\sigma}$ constructed in Proposition 2.2, there
exists a unique $p_{\epsilon}\in L^{2}_{0}(\Omega)$ such that

(2.5)
$(\nabla\bm{u}_{\epsilon},\nabla\bm{v})+\Big{(}g\frac{\nabla\bm{u}_{\epsilon}}{\sqrt{|\nabla\bm{u}_{\epsilon}|^{2}+\epsilon^{2}}},\nabla\bm{v}\Big{)}-(p_{\epsilon},\operatorname{div}\bm{v})=(\bm{f},\bm{v})\qquad\forall\bm{v}\in\bm{V}.$

Moreover, $p_{\epsilon}$ satisfies

(2.6) $\|p_{\epsilon}\|_{L^{2}(\Omega)}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega)}+\|g\|_{L^{2}(\Omega)}).$

###### Proof.

Unique existence of $p_{\epsilon}$ follows from an abstract theory for
variational equations in a mixed form (see [17, Lemma I.4.1(ii)]), if we
notice that the inf-sup condition

$C\|q\|_{L^{2}(\Omega)}\leq\sup_{\bm{v}\in\bm{H}^{1}_{0}(\Omega)}\frac{(q,\operatorname{div}\bm{v})}{\|\bm{v}\|_{\bm{H}^{1}(\Omega)}}\quad\forall
q\in L^{2}_{0}(\Omega)$

holds (see e.g. [17, p. 81]). Since $\bm{H}^{1}_{0}(\Omega)\subset\bm{V}$, we
also obtain

$\displaystyle C\|p_{\epsilon}\|_{L^{2}(\Omega)}$
$\displaystyle\leq\sup_{\bm{v}\in\bm{V}}\frac{(p_{\epsilon},\operatorname{div}\bm{v})}{\|\bm{v}\|_{\bm{H}^{1}(\Omega)}}=\sup_{\bm{v}\in\bm{V}}\frac{(\nabla\bm{u}_{\epsilon},\nabla\bm{v})+\Big{(}g\frac{\nabla\bm{u}_{\epsilon}}{\sqrt{|\nabla\bm{u}_{\epsilon}|^{2}+\epsilon^{2}}},\nabla\bm{v}\Big{)}-(\bm{f},\bm{v})}{\|\bm{v}\|_{\bm{H}^{1}(\Omega)}}$
$\displaystyle\leq\|\nabla\bm{u}_{\epsilon}\|_{\bm{L}^{2}(\Omega)}+\|g\|_{L^{2}(\Omega)}+\|\bm{f}\|_{\bm{L}^{2}(\Omega)},$

which combined with Proposition 2.2(ii) proves the $L^{2}(\Omega)$-estimate
for the pressure. ∎

With the help of the regularized problem, we are now able to establish a
counterpart to the “converse” statement of Proposition 2.1 (cf. [18, Theorem
II.6.3]).

######  Proposition 2.4.

Let $\bm{f}\in\bm{L}^{2}(\Omega)$, $g\in L^{2}(\Omega),\,g\geq 0$, and
$\bm{u}\in\bm{V}_{\sigma}$ be the solution of (2.1). Then there exist
$\bm{\lambda}\in\bm{L}^{\infty}(\Omega)=L^{\infty}(\Omega)^{d\times d}$ and
$p\in L^{2}_{0}(\Omega)$ such that

(2.7)
$\displaystyle(\nabla\bm{u}+g\bm{\lambda},\nabla\bm{v})-(p,\operatorname{div}\bm{v})=(\bm{f},\bm{v})\qquad\forall\bm{v}\in\bm{V},$
$\displaystyle\operatorname{Tr}\bm{\lambda}=0,\quad|\bm{\lambda}|\leq 1,\quad
g\bm{\lambda}:\nabla\bm{u}=g|\nabla\bm{u}|\quad\text{a.e.\ in}\quad\Omega.$

In particular, if we set $\bm{T}^{D}:=\nabla\bm{u}+g\bm{\lambda}$ then
$(\bm{T}^{D},\bm{u},p)$ solves (1.1)–(1.3).

###### Proof.

In (2.5) we set
$\bm{\lambda}_{\epsilon}:=\nabla\bm{u}_{\epsilon}/\sqrt{|\nabla\bm{u}_{\epsilon}|^{2}+\epsilon^{2}}\in\bm{L}^{\infty}(\Omega)$
to get

(2.8)
$\displaystyle(\nabla\bm{u}_{\epsilon}+g\bm{\lambda}_{\epsilon},\nabla\bm{v})-(p_{\epsilon},\operatorname{div}\bm{v})=(\bm{f},\bm{v})\qquad\forall\bm{v}\in\bm{V},$
$\displaystyle\operatorname{Tr}\bm{\lambda}_{\epsilon}=0,\quad|\bm{\lambda}_{\epsilon}|\leq
1\quad\text{a.e.\ in}\quad\Omega.$

In view of this and (2.6), we can choose a sub-sequence of
$\\{\bm{\lambda}_{\epsilon}\\},\\{p_{\epsilon}\\}$ (denoted by the same
symbols) and some $\bm{\lambda}\in\bm{L}^{\infty}(\Omega),p\in
L^{2}_{0}(\Omega)$ such that $\operatorname{Tr}\bm{\lambda}=0$,
$|\bm{\lambda}|\leq 1$ a.e. in $\Omega$ and

$\displaystyle\bm{\lambda}_{\epsilon}\rightharpoonup\bm{\lambda}\quad\text{weakly-$*$
in}\quad\bm{L}^{\infty}(\Omega),\qquad p_{\epsilon}\rightharpoonup
p\quad\text{weakly in}\quad L^{2}(\Omega).$

Then, letting $\epsilon\to 0$ in (2.8) we obtain (2.7). Next, taking
$\bm{v}=\bm{0},2\bm{u}$ in (2.1) we have
$\|\nabla\bm{u}\|_{\bm{L}^{2}(\Omega)}^{2}+(g,|\nabla\bm{u}|)=(\bm{f},\bm{u})$,
which combined with (2.7) for $\bm{v}=\bm{u}$ leads to
$(g,|\nabla\bm{u}|)=(g\bm{\lambda},\nabla\bm{u})$. Since $|\bm{\lambda}|\leq
1$, this holds if and only if $g\bm{\lambda}:\nabla\bm{u}=g|\nabla\bm{u}|$
a.e. in $\Omega$. Consequently, we have
$g\bm{\lambda}=g\nabla\bm{u}/|\nabla\bm{u}|$ where $\nabla\bm{u}\neq\bm{O}$.
Now we find from (2.7) that (1.1) holds in the distributional sense (or in
$\bm{H}^{-1}(\Omega)$) and that the second condition in (1.3) holds in the
sense of $\bm{H}^{-1/2}(\Gamma)$. ∎

######  Remark 2.2.

Uniqueness of $\bm{\lambda}$ (thus that of $p$ neither) may not be true; see
[19, p. 354].

##  3\. $H^{2}$-regularity for the stationary problem

We continue to focus on the stationary problems and to assume $\nu=1$ as in
the previous section. The following $\bm{H}^{2}$-regularity theorem on the
regularized problem is the main result of this section.

######  Theorem 3.1.

Let $\Omega$ be of $C^{3,1}$-class, $\bm{f}\in\bm{L}^{2}(\Omega)$, $g\in
H^{1}(\Omega)$ be non-negative, and $\epsilon\in(0,1]$. Then the solution of
$\bm{u}_{\epsilon}$ of (2.3) satisfies
$\bm{u}_{\epsilon}\in\bm{H}^{2}(\Omega)$,
$(\nabla\bm{u}_{\epsilon}\bm{n})_{\tau}=\bm{0}$ a.e. on $\Gamma$, and

$\|\bm{u}_{\epsilon}\|_{\bm{H}^{2}(\Omega)}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega)}+\|g\|_{H^{1}(\Omega)}),$

where the constant $C=C(d,\Omega)$ is independent of $\epsilon$.

######  Remark 3.1.

As for the pressure, although it is possible to show $p_{\epsilon}\in
H^{1}(\Omega)$, we were not able to get an $H^{1}(\Omega)$-estimate uniform in
$\epsilon$ (see Subsection 3.2.2).

This theorem allows us to derive an $\bm{H}^{2}$-regularity result for the
original variational inequality without regularization.

######  Corollary 3.1.

Under the same assumptions as in Theorem 3.1, the solution $\bm{u}$ of (2.1)
satisfies $\bm{u}\in\bm{H}^{2}(\Omega)$,
$(\nabla\bm{u}\,\bm{n})_{\tau}=\bm{0}$ a.e. on $\Gamma$, and
$\|\bm{u}\|_{\bm{H}^{2}(\Omega)}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega)}+\|g\|_{H^{1}(\Omega)})$.

###### Proof.

There exists a sub-sequence of $\\{\bm{u}_{\epsilon}\\}$ (denoted by the same
symbol) and some $\bm{u}^{*}\in\bm{H}^{2}(\Omega)$ such that
$\bm{u}_{\epsilon}\rightharpoonup\bm{u}^{*}$ weakly in $\bm{H}^{2}(\Omega)$ as
$\epsilon\to 0$. By the compactness, $\bm{u}_{\epsilon}$ converges to
$\bm{u}^{*}$ strongly in $\bm{H}^{1}(\Omega)$ and
$(\nabla\bm{u}^{*}\bm{n})_{\tau}=\bm{0}$ a.e. on $\Gamma$. Now Proposition
2.2(ii) and the uniqueness of the limit imply $\bm{u}=\bm{u}^{*}$ (this also
implies the $\bm{H}^{2}(\Omega)$-weak convergence of the whole sequence
$\\{\bm{u}_{\epsilon}\\}$). ∎

Theorem 3.1 is reduced to the following key proposition, which asserts local
$H^{2}$-regularity up to the boundary and inside the domain:

######  Proposition 3.1.

Under the same assumptions as in Theorem 3.1, let
$(\bm{u}_{\epsilon},p_{\epsilon})\in\bm{V}_{\sigma}\times L^{2}_{0}(\Omega)$
be as in Proposition 2.3.

(i) Each $\bm{x}_{0}\in\Gamma$ admits open neighborhoods $W_{\bm{x}_{0}}$ and
$U_{\bm{x}_{0}}$ such that $\overline{W_{\bm{x}_{0}}}\subset U_{\bm{x}_{0}}$
and $\nabla^{2}\bm{u}_{\epsilon}\in\bm{L}^{2}(\Omega\cap W_{\bm{x}_{0}})$.
Furthermore, there exists a constant
$C=C(d,\Omega,\bm{x}_{0},W_{\bm{x}_{0}},U_{\bm{x}_{0}})$, independent of
$\epsilon$, such that

$\|\nabla^{2}\bm{u}_{\epsilon}\|_{\bm{L}^{2}(\Omega\cap W_{\bm{x}_{0}})}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega\cap U_{\bm{x}_{0}})}+\|g\|_{H^{1}(\Omega\cap
U_{\bm{x}_{0}})}+\|\nabla\bm{u}_{\epsilon}\|_{\bm{L}^{2}(\Omega\cap
U_{\bm{x}_{0}})}+\|p_{\epsilon}\|_{L^{2}(\Omega\cap U_{\bm{x}_{0}})}).$

(ii) Each $\bm{x}_{0}\in\Omega$ admits open neighborhoods $W_{\bm{x}_{0}}$ and
$U_{\bm{x}_{0}}$ such that $\overline{W_{\bm{x}_{0}}}\subset U_{\bm{x}_{0}}$,
$\overline{U_{\bm{x}_{0}}}\subset\Omega$, and
$\nabla^{2}\bm{u}_{\epsilon}\in\bm{L}^{2}(\Omega\cap W_{\bm{x}_{0}})$.
Furthermore, the same estimate as in (i) holds.

###### Proof of Theorem 3.1.

The above proposition combined with a standard covering argument (note that
$\overline{\Omega}$ is compact) and with the estimate
$\|\bm{u}_{\epsilon}\|_{\bm{H}^{1}(\Omega)}+\|p_{\epsilon}\|_{L^{2}(\Omega)}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega)}+\|g\|_{L^{2}(\Omega)})$ (recall Proposition
2.2(ii) and (2.6)) proves the theorem. ∎

The rest of this section will be devoted to the proof of Proposition 3.1,
especially statement (i). Statement (ii), i.e., interior regularity may be
obtained in a similar (actually easier) manner to (i); see Remark 3.3 below.

###  3.1. Coordinate transformation to flatten the boundary

The fist step to show Proposition 3.1(i) is, as usual in a regularity theory
for elliptic problems, to localize the PDE using a cut-off function and to
transform it into a half-space problem.

####  3.1.1. Local coordinates

By assumption, $\Gamma=\partial\Omega$ is of $C^{3,1}$-class. Let us employ a
system of local coordinates that preserves the normal direction (see [36,
Theorem I.2.12] or [32, p. 361]). Namely, each $\bm{x}_{0}\in\Gamma$ admits an
open neighborhood $U_{\bm{x}_{0}}\subset\mathbb{R}^{d}$ such that the
following hold:

  * •

There exists a $C^{2,1}$-diffeomorphism $\bm{\varphi}:U_{\bm{x}_{0}}\to K$
satisfying $\bm{\varphi}(\bm{x}_{0})=\bm{0}$, where $K=(-1,1)^{d}$ is a cube.

  * •

For $K_{+}=\\{\bm{y}\in K\,:\,y_{d}>0\\}$ and $K_{0}=\\{\bm{y}\in
K\,:\,y_{d}=0\\}$, one has

$\bm{\varphi}(\Omega\cap U_{\bm{x}_{0}})=K_{+},\quad\bm{\varphi}(\Gamma\cap
U_{\bm{x}_{0}})=K_{0}.$

  * •

With the Jacobi matrix $\bm{\Phi}=\nabla_{\bm{x}}\bm{\varphi}$, (i.e.,
$\Phi_{ij}=\partial\varphi_{i}/\partial x_{j}$ for $1\leq i,j\leq d$ in the
component sense), one has

$\Phi_{di}=\Phi_{id}=0\quad(i\neq
d)\quad\text{and}\quad\Phi_{dd}=-1\quad\text{on}\quad\Gamma\cap
U_{\bm{x}_{0}}.$

  * •

The unit outward normal vector $\bm{n}$ to $\Gamma$ is mapped to that to
$K_{0}$ by $\bm{\varphi}$, that is,

(3.1) $\bm{n}\circ\bm{\varphi}^{-1}=(0,\dots,0,-1)^{\top}\quad\text{on}\quad
K_{0}.$

Writing $\bm{\psi}:=\bm{\varphi}^{-1}$, we attach the variables $\bm{x}$ and
$\bm{y}$ to the original and transformed coordinates, respectively, that is,

$\bm{y}=\bm{\varphi}(\bm{x})\Longleftrightarrow\bm{x}=\bm{\psi}(\bm{y})\qquad(\bm{x}\in
U_{\bm{x}_{0}},\;\bm{y}\in\bm{\varphi}(U_{\bm{x}_{0}})).$

For a function $f$ defined in $U_{\bm{x}_{0}}$, its coordinate transformation
$\tilde{f}$ in $\bm{\varphi}(U_{\bm{x}_{0}})$ is given as
$\tilde{f}(\bm{y})=f(\bm{x})=f(\bm{\psi}(\bm{y}))$. Then (3.1) is rephrased as
$\tilde{\bm{n}}(\bm{y})=(0,\dots,0,-1)^{\top}$ for $y_{d}=0$. In addition, we
introduce the Jacobi matrix $\bm{\Psi}:=\nabla_{\bm{y}}\bm{\psi}$ and the
Jacobian $J:=\operatorname{det}\bm{\Psi}$. By the chain rule,
$\nabla_{\bm{x}}=\tilde{\bm{\Phi}}^{\top}\nabla_{\bm{y}}$. We will frequently
make use of the following estimates:

$\displaystyle\frac{1}{C}|\bm{\Psi}(\bm{y})|\leq|\bm{\Phi}(\bm{x})|\leq
C|\bm{\Psi}(\bm{y})|,\quad\frac{1}{C}\leq|J|\leq
C\quad(\bm{y}=\bm{\varphi}(\bm{x}),\,\bm{x}\in U_{\bm{x}_{0}}),$

where the constants $C>0$ depend only on $\Omega$, $\bm{x}_{0}$, and
$U_{\bm{x}_{0}}$. To describe tangential components and derivatives we use

$\bm{y}^{\prime}:=(y_{1},\dots,y_{d-1})^{\top},\quad\nabla^{\prime}:=(\partial_{1},\dots,\partial_{d-1})^{\top}.$

Since $\bm{\Psi}=\tilde{\bm{\Phi}}^{-1}$, we have

$|\nabla_{\bm{y}}\tilde{f}|=|\bm{\Psi}^{\top}\,\tilde{\bm{\Phi}}^{\top}\nabla_{\bm{y}}\tilde{f}|\leq
C|\tilde{\bm{\Phi}}^{\top}\nabla_{\bm{y}}\tilde{f}|.$

Hereafter we do not show the subscripts $\bm{x}$ or $\bm{y}$ in the gradient
operator explicitly when no confusion occurs.

As already noticed in Introduction, the gradient operator applied to a vector-
valued function $\bm{v}$ gives a matrix $\nabla\bm{v}$ defined by

$(\nabla\bm{v})_{ij}=\frac{\partial v_{i}}{\partial
x_{j}}\quad(i,j=1,\dots,d),\quad\text{that
is,}\quad\nabla\bm{v}=(\nabla\otimes\bm{v})^{\top}.$

In particular, because $|\bm{A}|=|\bm{A}^{\top}|$, one has

$|\nabla_{\bm{x}}\bm{v}|=|\nabla_{\bm{x}}\otimes\bm{v}|=|\tilde{\bm{\Phi}}^{\top}\nabla_{\bm{y}}\otimes\tilde{\bm{v}}|=|(\nabla_{\bm{y}}\tilde{\bm{v}})\tilde{\bm{\Phi}}|.$

We prefer, in the coordinate $\bm{y}$, the notation
$|\tilde{\bm{\Phi}}^{\top}\nabla_{\bm{y}}\otimes\tilde{\bm{v}}|$ to
$|(\nabla_{\bm{y}}\tilde{\bm{v}})\tilde{\bm{\Phi}}|$ for ease of viewing.

####  3.1.2. Reduction to a half-space problem

For simplicity in the notation, in the subsequent sections, we agree to omit
the subscript $\epsilon$ from the solution $(\bm{u}_{\epsilon},p_{\epsilon})$
of (2.5). Thus $(\bm{u},p)\in\bm{V}\times L^{2}_{0}(\Omega)$ satisfies
$\partial_{i}u_{i}=0$ and

$\int_{\Omega}\partial_{j}u_{i}\,\partial_{j}v_{i}\,d\bm{x}+\int_{\Omega}g\frac{\partial_{j}u_{i}\,\partial_{j}v_{i}}{\sqrt{|\nabla\bm{u}|^{2}+\epsilon^{2}}}\,d\bm{x}-\int_{\Omega}p\,\partial_{i}v_{i}\,d\bm{x}=\int_{\Omega}f_{i}v_{i}\,d\bm{x}\qquad\forall\bm{v}\in\bm{V},$

where we exploit Einstein’s summation convention, omitting the symbol
$\sum_{i,j=1}^{d}$.

For fixed $\bm{x}_{0}\in\Gamma$, we consider test functions
$\bm{v}\in\bm{V}_{\sigma}$ with $\operatorname{supp}\bm{v}\subset
U_{\bm{x}_{0}}$ and transform the above weak formulation using
$\bm{y}=\bm{\varphi}(\bm{x})$ together with $\partial/\partial
x_{j}=\tilde{\Phi}_{kj}\partial/\partial y_{k}$ (the summation convention is
applied to $k$ as well) to obtain

$\displaystyle\int_{K_{+}}\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}\,|J|\,d\bm{y}+\int_{K_{+}}\tilde{g}\,\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\,|J|\,d\bm{y}-\int_{K_{+}}\tilde{p}\,\tilde{\Phi}_{ki}\partial_{k}\tilde{v}_{i}\,|J|\,d\bm{y}=\int_{K_{+}}\tilde{f}_{i}\tilde{v}_{i}\,|J|\,d\bm{y}$
$\displaystyle\hskip
284.52756pt\forall\tilde{\bm{v}}\in\bm{H}^{1}(K_{+}),\;\tilde{v}_{d}=0\text{
on }K_{0},\;\operatorname{supp}\tilde{\bm{v}}\subset K,$
$\displaystyle\tilde{\Phi}_{ki}\partial_{k}\tilde{u}_{i}=0\quad\text{in
}K_{+}.$

Regarding $\tilde{\bm{v}}|J|$ as a new test function, we see that

(3.2)
$\displaystyle\int_{K_{+}}\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}\,d\bm{y}+\int_{K_{+}}\tilde{g}\,\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\,d\bm{y}-\int_{K_{+}}\tilde{p}\,\tilde{\Phi}_{ki}\partial_{k}\tilde{v}_{i}\,d\bm{y}=\int_{K_{+}}\tilde{F}_{i}\tilde{v}_{i}\,d\bm{y}$
$\displaystyle\hskip
227.62204pt\forall\tilde{\bm{v}}\in\bm{H}^{1}(K_{+}),\;\tilde{v}_{d}=0\text{
on
}K_{0},\;\operatorname{supp}\tilde{\bm{v}}\subset\bm{\varphi}(U_{\bm{x}_{0}}).$

Here,

$\tilde{F}_{i}:=\tilde{f}_{i}-\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\frac{\partial_{l}|J|}{|J|}-\tilde{g}\,\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\frac{\partial_{l}|J|}{|J|}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}+\tilde{p}\,\tilde{\Phi}_{ki}\frac{\partial_{k}|J|}{|J|}\quad(i=1,\dots,d)$

satisfies, because
$\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}/(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2})\leq
1$,

(3.3) $\|\tilde{\bm{F}}\|_{\bm{L}^{2}(K_{+})}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{L^{2}(K_{+})}+\|\tilde{\bm{u}}\|_{\bm{H}^{1}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}).$

###  3.2. Regularity in the tangential direction

The next step to show Proposition 3.1(i) is to establish regularity w.r.t.
$\bm{y}^{\prime}$ for the solution $(\tilde{\bm{u}},\tilde{p})$ of (3.2). If
we set
$\tilde{W}:=B(\bm{0};1/2)=\\{\bm{y}\in\mathbb{R}^{d}\mid|\bm{y}|<1/2\\}$, then
$W_{\bm{x}_{0}}:=\bm{\psi}(\tilde{W})$ is an open neighborhood of $\bm{x}_{0}$
and $\overline{W_{\bm{x}_{0}}}\subset U_{\bm{x}_{0}}$. Hence there exists a
cut-off function $\tilde{\theta}\in C_{0}^{\infty}(K)$ such that
$\tilde{\theta}\equiv 1$ in $\overline{\tilde{W}}$ and
$0\leq\tilde{\theta}\leq 1$ in $\mathbb{R}^{d}$. Tangential regularity of
$\tilde{\bm{u}}$ and that of $\tilde{p}$ are addressed below.

####  3.2.1. Estimates for $\nabla\nabla^{\prime}\tilde{\bm{u}}$ uniform in
$\epsilon$

######  Lemma 3.1.

We have $\nabla\nabla^{\prime}\tilde{\bm{u}}\in\bm{L}^{2}(K_{+}\cap\tilde{W})$
and

$\|\nabla\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+}\cap\tilde{W})}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega\cap U_{\bm{x}_{0}})}+\|g\|_{H^{1}(\Omega\cap
U_{\bm{x}_{0}})}+\|\nabla\bm{u}\|_{\bm{L}^{2}(\Omega\cap
U_{\bm{x}_{0}})}+\|p\|_{L^{2}(\Omega\cap U_{\bm{x}_{0}})}).$

The proof relies on the method of difference quotient, notation of which is
introduced as follows. The standard basis of $\mathbb{R}^{d}$ is denoted by
$\bm{e}_{1},\dots,\bm{e}_{d}$. For $\alpha=1,\dots,d-1$ and
$0<|h|<(1/2)\operatorname{dist}(\operatorname{supp}\tilde{\theta},\partial
K)$, the shift and difference quotient operators in the direction
$\bm{e}_{\alpha}$ are defined by

$(s_{h}^{\alpha}f)(\bm{y})=f(\bm{y}+h\bm{e}_{\alpha}),\qquad
D_{h}^{\alpha}f=\frac{s_{h}^{\alpha}f-f}{h}.$

Since it is clear that one direction $\alpha$ is fixed, we omit the
superscript $\alpha$ below. The following formulas are well known (see e.g.
[9, Chapter 9]):

$\displaystyle D_{h}(fg)$ $\displaystyle=f(D_{h}g)+(D_{h}f)(s_{h}g),$
$\displaystyle\int_{K_{+}}f\,(D_{-h}g)\,d\bm{y}$
$\displaystyle=\int_{K_{+}}(D_{h}f)\,g\,d\bm{y}\quad\text{if }f,g\in
L^{2}(K_{+})\text{ and
}\operatorname{supp}f\cup\operatorname{supp}g\subset\operatorname{supp}\tilde{\theta}$
$\displaystyle\|D_{h}f\|_{L^{p}(K_{+})}$ $\displaystyle\leq\|\nabla
f\|_{L^{p}(K_{+})}\quad(1\leq p\leq\infty)\quad\text{if }f\in
W^{1,p}(K_{+}),\;\operatorname{supp}f\subset\operatorname{supp}\tilde{\theta}.$

###### Proof of Lemma 3.1.

Take $\tilde{\bm{v}}=D_{-h}(\tilde{\theta}^{2}D_{h}\tilde{\bm{u}})$ in (3.2)
to get

$\displaystyle\int_{K_{+}}D_{h}(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj})\partial_{l}(\tilde{\theta}^{2}D_{h}\tilde{u}_{i})\,d\bm{y}+\int_{K_{+}}D_{h}\bigg{(}\tilde{g}\,\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{)}\partial_{l}(\tilde{\theta}^{2}D_{h}\tilde{u}_{i})\,d\bm{y}$
$\displaystyle\hskip
170.71652pt-\int_{K_{+}}\tilde{p}\,\tilde{\Phi}_{ki}\,\partial_{k}D_{-h}(\tilde{\theta}^{2}D_{h}\tilde{u}_{i})\,d\bm{y}=\;\int_{K_{+}}\tilde{F}_{i}D_{-h}(\tilde{\theta}^{2}D_{h}\tilde{u}_{i})\,d\bm{y},$

which is rephrased as

$J_{1}+J_{2}+J_{3}=J_{4}.$

We divide each $J_{a}$ into “principal” and “lower-order” parts in terms of
the number of differentiation or difference quotient applied to
$\tilde{\bm{u}}$. For example, for $a=1$ we have

$\displaystyle J_{1}$
$\displaystyle=\int_{K_{+}}\tilde{\theta}(\tilde{\Phi}_{kj}\partial_{k}D_{h}\tilde{u}_{i})\cdot\tilde{\theta}(\tilde{\Phi}_{lj}\partial_{l}D_{h}\tilde{u}_{i})\,d\bm{y}$
$\displaystyle\qquad+\int_{K_{+}}\bigg{[}D_{h}(\tilde{\Phi}_{kj}\tilde{\Phi}_{lj})s_{h}(\partial_{k}u_{i})\cdot\Big{(}\partial_{l}(\tilde{\theta}^{2})D_{h}\tilde{u}_{i}+\tilde{\theta}^{2}\partial_{l}D_{h}\tilde{u}_{i}\Big{)}+(\tilde{\Phi}_{kj}\partial_{k}D_{h}\tilde{u}_{i})\tilde{\Phi}_{lj}\cdot
2\tilde{\theta}(\partial_{l}\tilde{\theta})D_{h}\tilde{u}_{i}\bigg{]}\,d\bm{y}=:J_{11}+M_{1},$

and it follows that

(3.4) $\displaystyle J_{1}$
$\displaystyle=\int_{K_{+}}\tilde{\theta}^{2}\big{|}\tilde{\bm{\Phi}}^{\top}\nabla\otimes
D_{h}\tilde{\bm{u}}\big{|}^{2}\,d\bm{y}+M_{1}\geq C\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}+M_{1},$ $\displaystyle\hskip
85.35826pt|M_{1}|\leq
C\|\nabla\tilde{\bm{u}}\|_{L^{2}(K_{+})}(\|\nabla\tilde{\bm{u}}\|_{L^{2}(K_{+})}+\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}).$

For simplicity in the presentation, hereafter we omit the detailed expressions
of the lower-order terms $M_{a}$, only displaying their bounds based on
inequalities. Then, for $a=2,3,4$ we obtain

$\displaystyle J_{2}$
$\displaystyle=\int_{K_{+}}\tilde{\theta}^{2}\tilde{g}\,\tilde{\Phi}_{kj}D_{h}\bigg{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}\,\tilde{\Phi}_{lj}\partial_{l}D_{h}\tilde{u}_{i}\,d\bm{y}+M_{2}=:J_{21}+M_{2},$
$\displaystyle\hskip 85.35826pt|M_{2}|\leq
C\|\tilde{g}\|_{H^{1}(K_{+})}(\|\nabla\tilde{\bm{u}}\|_{L^{2}(K_{+})}+\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}),$ $\displaystyle J_{3}$
$\displaystyle=-\int_{K_{+}}\tilde{\theta}^{2}\tilde{p}\,D_{-h}D_{h}(\underbrace{\tilde{\Phi}_{ki}\partial_{k}\tilde{u}_{i}}_{=0})\,d\bm{y}+M_{3}=M_{3},$
$\displaystyle\hskip 85.35826pt|M_{3}|\leq
C\|\tilde{p}\|_{L^{2}(K_{+})}(\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}),$ $\displaystyle J_{4}$
$\displaystyle=\int_{K_{+}}\tilde{\theta}^{2}F_{i}(D_{-h}D_{h}\tilde{u}_{i})\,d\bm{y}+M_{4}\leq\|\bm{F}\|_{\bm{L}^{2}(K_{+})}\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+M_{4},$ $\displaystyle\hskip
85.35826pt|M_{4}|\leq\|\bm{F}\|_{\bm{L}^{2}(K_{+})}\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}.$

For $J_{21}$, a direct computation using
$1/\sqrt{a}-1/\sqrt{b}=(b-a)/(\sqrt{a}\sqrt{b}(\sqrt{a}+\sqrt{b}))$ shows

$\displaystyle\tilde{\Phi}_{kj}D_{h}\bigg{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}=\frac{\tilde{\Phi}_{kj}\partial_{k}D_{h}\tilde{u}_{i}}{s_{h}(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}$
$\displaystyle\hskip
28.45274pt-\frac{D_{h}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}})}{s_{h}(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}:\frac{s_{h}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}})+\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}}{s_{h}(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}+(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\,\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}.$

We multiply this by
$\tilde{\theta}^{2}\tilde{g}\cdot\tilde{\Phi}_{lj}\partial_{l}D_{h}\tilde{u}_{i}$
and add the resulting equations for $i,j=1,\dots,d$. Then, as a result of

$\frac{D_{h}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}})}{s_{h}(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}=\frac{(\tilde{\bm{\Phi}}^{\top}\nabla)\otimes
D_{h}\tilde{\bm{u}}}{s_{h}(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}+(D_{h}\tilde{\bm{\Phi}})\otimes\underbrace{s_{h}\bigg{[}\frac{\nabla\tilde{\bm{u}}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}}_{|\cdot|\leq
C}$

and the Cauchy–Schwarz inequality
$|\bm{P}|^{2}|\bm{Q}||\bm{R}|\geq(\bm{P}:\bm{Q})(\bm{R}:\bm{P})$, we deduce
that

$\displaystyle J_{21}$
$\displaystyle=\int_{K+}\tilde{\theta}^{2}\tilde{g}\frac{|\bm{B}|^{2}\cdot(s_{h}+\operatorname{id})(|\bm{A}|^{2}+\epsilon^{2})^{1/2}\cdot(|\bm{A}|^{2}+\epsilon^{2})^{1/2}-\big{[}\bm{B}:(s_{h}+\operatorname{id})\bm{A}\big{]}\big{[}\bm{A}:\bm{B}\big{]}}{s_{h}(|\bm{A}|^{2}+\epsilon^{2})^{1/2}\cdot(s_{h}+\operatorname{id})(|\bm{A}|^{2}+\epsilon^{2})^{1/2}\cdot(|\bm{A}|^{2}+\epsilon^{2})^{1/2}}\,d\bm{y}+M_{21}\geq
0+M_{21},$ $\displaystyle\hskip 85.35826pt|M_{21}|\leq
C\|\tilde{g}\|_{L^{2}(K_{+})}\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})},$

where $\operatorname{id}$ means an identity operator and we set
$\bm{A}:=\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}$ and
$\bm{B}:=\tilde{\bm{\Phi}}^{\top}\nabla\otimes D_{h}\tilde{\bm{u}}$.

Collecting the above estimates, applying the arithmetic-geometric inequality,
and absorbing $\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}$ to the right-hand side of
(3.4), we arrive at

(3.5) $\|\tilde{\theta}\,\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}^{2}+\|\tilde{g}\|_{H^{1}(K_{+})}^{2}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}+\|\tilde{p}\|_{L^{2}(K_{+})}^{2}),$

which implies

$\|\nabla D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+}\cap\tilde{W})}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}).$

Recalling that $D_{h}$ means $D_{h}^{\alpha}$ where $\alpha=1,\dots,d-1$ is
arbitrarily fixed, we take the limit $h\to 0$ to conclude
$\nabla\nabla^{\prime}\tilde{\bm{u}}\in\bm{L}^{2}(K_{+}\cap\tilde{W})$ and

(3.6)
$\|\nabla\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+}\cap\tilde{W})}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}).$

This completes the proof of Lemma 3.1. ∎

######  Remark 3.2.

The tangential estimate (3.6) combined with a trace inequality gives us

$\|\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{0}\cap\tilde{W})}\leq
C\|\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{H}^{1}(K_{+}\cap\tilde{W})}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}),$

which will be utilized in Subsection 3.3.4.

####  3.2.2. Estimates for $\nabla^{\prime}\tilde{p}$ (not uniform in
$\epsilon$)

Let us state an inf-sup condition adapted to the $\bm{y}$-coordinate.

######  Lemma 3.2.

There exist positive constants $C_{1}$ and $C_{2}$ depending only on $\Omega$
and $U_{\bm{x}_{0}}$ such that

$C_{1}\|\tilde{q}\|_{L^{2}(K_{+})}\leq\sup_{\tilde{\bm{v}}\in\bm{H}^{1}_{0}(K_{+})}\frac{\int_{K_{+}}\tilde{q}\,\tilde{\Phi}_{ki}\partial_{k}\tilde{v}_{i}\,d\bm{y}}{\|\tilde{\bm{v}}\|_{\bm{H}^{1}(K_{+})}}+C_{2}\bigg{|}\int_{K_{+}}\tilde{q}\,d\bm{y}\bigg{|}\qquad\forall\tilde{q}\in
L^{2}(K_{+}).$

###### Proof.

Given $\tilde{q}\in L^{2}(K_{+})$, we start from an inf-sup condition
associated with the Lipschitz domain $\Omega\cap U_{\bm{x}_{0}}$ (see [17,
Lemma I.4.1(ii)]) in the $\bm{x}$-coordinate:

$C\bigg{\|}q-\frac{1}{|\Omega\cap U_{\bm{x}_{0}}|}\int_{\Omega\cap
U_{\bm{x}_{0}}}q\,d\bm{x}\bigg{\|}_{L^{2}(\Omega)}\leq\sup_{\bm{v}\in\bm{H}^{1}_{0}(\Omega\cap
U_{\bm{x}_{0}})}\frac{\int_{\Omega\cap
U_{\bm{x}_{0}}}q\operatorname{div}\bm{v}\,d\bm{x}}{\|\bm{v}\|_{\bm{H}^{1}(\Omega\cap
U_{\bm{x}_{0}})}},$

where $|\cdot|$ means the $d$-dimensional measure. Obviously,

$C\|q\|_{L^{2}(\Omega)}\leq\sup_{\bm{v}\in\bm{H}^{1}_{0}(\Omega\cap
U_{\bm{x}_{0}})}\frac{\int_{\Omega\cap
U_{\bm{x}_{0}}}q\operatorname{div}\bm{v}\,d\bm{x}}{\|\bm{v}\|_{\bm{H}^{1}(\Omega\cap
U_{\bm{x}_{0}})}}+C\bigg{|}\int_{\Omega\cap
U_{\bm{x}_{0}}}q\,d\bm{x}\bigg{|}.$

Transforming this into the $\bm{y}$-coordinate, we obtain

$C\bigg{(}\int_{K_{+}}|\tilde{q}|^{2}|J|\,d\bm{y}\bigg{)}^{1/2}\leq\sup_{\tilde{\bm{v}}\in\bm{H}^{1}_{0}(K_{+})}\frac{\int_{K_{+}}\tilde{q}\,\tilde{\Phi}_{ki}\partial_{k}\tilde{v}_{i}|J|\,d\bm{y}}{\|\tilde{\bm{v}}\|_{\bm{H}^{1}(K_{+})}}+C\bigg{|}\int_{K_{+}}\tilde{q}|J|\,d\bm{y}\bigg{|}.$

Regarding $\tilde{q}|J|$ as new $\tilde{q}$ and using $C\leq 1/|J|$, we
conclude the desired estimate. ∎

Let $\tilde{\bm{v}}\in\bm{H}^{1}_{0}(K_{+})$ be arbitrary and
$0<|h|<(1/2)\operatorname{dist}(\operatorname{supp}\tilde{\theta},\partial
K)$. We take $D_{-h}(\tilde{\theta}\tilde{\bm{v}})$ as a test function in
(3.2) to have

$\displaystyle\int_{K_{+}}(\tilde{\theta}D_{h}\tilde{p})\,\tilde{\Phi}_{ki}\partial_{k}\tilde{v}_{i}\,d\bm{y}$
$\displaystyle=\int_{K_{+}}\tilde{\theta}\tilde{\Phi}_{kj}(\partial_{k}D_{h}\tilde{u}_{i})\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}\,d\bm{y}+\int_{K_{+}}\tilde{\theta}^{2}\tilde{g}\,\tilde{\Phi}_{kj}D_{h}\Big{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\Big{]}\,\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}\,d\bm{y}$
$\displaystyle\hskip
85.35826pt-\int_{K_{+}}\tilde{F}_{i}\,D_{-h}(\tilde{\theta}\tilde{v}_{i})\,d\bm{y}+N_{1},$

where, as in the proof of Lemma 3.1, only principal parts are explicitly shown
and all the other lower-order terms are represented by $N_{1}$. We see that
$N_{1}$ is bounded as

$|N_{1}|\leq
C(\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})})\|\tilde{\bm{v}}\|_{\bm{H}^{1}(K_{+})}.$

We also notice that

$\left|\tilde{\Phi}_{kj}D_{h}\Big{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\Big{]}\right|\leq
C(1+\epsilon^{-1})|\nabla D_{h}\tilde{\bm{u}}|\quad\text{a.e.\ in
}K_{+}\quad(i,j=1,\dots,d).$

Consequently,

$\displaystyle\frac{\displaystyle\bigg{|}\int_{K_{+}}(\tilde{\theta}D_{h}\tilde{p})\,\tilde{\Phi}_{ki}\partial_{k}\tilde{v}_{i}\,d\bm{y}\bigg{|}}{\|\tilde{\bm{v}}\|_{\bm{H}^{1}(K_{+})}}$
$\displaystyle\leq
C(1+\epsilon^{-1})(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}+\|\tilde{\theta}\nabla
D_{h}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})})$ $\displaystyle\leq
C(1+\epsilon^{-1})(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}),$

where we have used (3.6). Substituting this into Lemma 3.2 with
$\tilde{q}=\tilde{\theta}D_{h}\tilde{p}$ and noting that
$|\int_{K_{+}}\tilde{\theta}\,(D_{h}\tilde{p})\,d\bm{y}|=|\int_{K_{+}}(D_{-h}\tilde{\theta})\,\tilde{p}\,d\bm{y}|\leq
C\|\tilde{p}\|_{L^{2}(K_{+})}$, we deduce

$\|\tilde{\theta}D_{h}\tilde{p}\|_{L^{2}(K_{+})}\leq
C(1+\epsilon^{-1})(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}).$

Restricting to $K_{+}\cap\tilde{W}$ and making $h\to 0$ prove
$\nabla^{\prime}\tilde{p}\in\bm{L}^{2}(K_{+}\cap\tilde{W})$.

######  Remark 3.3.

If $\bm{x}_{0}\in\Omega$ and its neighborhoods $W_{\bm{x}_{0}},U_{\bm{x}_{0}}$
are chosen as in Proposition 3.1(ii), then the above difference-quotient
argument works as well for all the directions $\alpha=1,\dots,d$ (and for
$|h|$ sufficiently small) without transforming the domain to a half-space (in
other words, one may assume $\bm{\varphi}$ is identity). Therefore,
Proposition 3.1(ii), that is, the interior regularities
$\bm{u}\in\bm{H}^{2}_{\text{loc}}(\Omega)$ and $p\in
H^{1}_{\text{loc}}(\Omega)$ are already proved; cf. [15, Theorem 3.3.1]. In
particular, $\nabla^{2}\bm{u}$ and $\nabla p$ make sense as measurable
functions defined in $\Omega$, and the strong form of the PDE holds in the
almost-everywhere sense:

$-\Delta\bm{u}-\operatorname{div}\Big{(}\frac{\nabla\bm{u}}{\sqrt{|\nabla\bm{u}|^{2}+\epsilon^{2}}}\Big{)}+\nabla
p=\bm{f}\quad\text{a.e.\ in }\Omega.$

###  3.3. Regularity in the normal direction

It remains to examine the regularity of $\tilde{u}$ w.r.t. $y_{d}$. To this
end, we re-choose $\tilde{W}$ and $\tilde{\theta}$ in such a way that
$\tilde{W}:=B(\bm{0};1/4)$ and $\tilde{\theta}\in
C_{0}^{\infty}(B(\bm{0};1/2))$, $\theta\equiv 1$ in $\overline{\tilde{W}}$ and
$0\leq\tilde{\theta}\leq 1$ in $\mathbb{R}^{d}$. The following is a key lemma
of this subsection.

######  Lemma 3.3.

We have

$\|\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+}\cap\tilde{W})}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega\cap U_{\bm{x}_{0}})}+\|g\|_{H^{1}(\Omega\cap
U_{\bm{x}_{0}})}+\|\nabla\bm{u}\|_{\bm{L}^{2}(\Omega\cap
U_{\bm{x}_{0}})}+\|p\|_{L^{2}(\Omega\cap U_{\bm{x}_{0}})}).$

Once this is proved, Lemmas 3.1 and 3.3, transformed back to the
$\bm{x}$-coordinate, imply

$\|\nabla^{2}\bm{u}\|_{\bm{L}^{2}(\Omega\cap W_{\bm{x}_{0}})}\leq
C(\|\bm{f}\|_{\bm{L}^{2}(\Omega\cap U_{\bm{x}_{0}})}+\|g\|_{H^{1}(\Omega\cap
U_{\bm{x}_{0}})}+\|\nabla\bm{u}\|_{\bm{L}^{2}(\Omega\cap
U_{\bm{x}_{0}})}+\|p\|_{L^{2}(\Omega\cap U_{\bm{x}_{0}})}),$

which completes the proof of Proposition 3.1(i). Now we show Lemma 3.3 via the
following four steps.

####  3.3.1. Proof of $\partial_{d}^{2}\tilde{\bm{u}}\in\bm{L}^{2}(K_{+})$ and
$\partial_{d}\tilde{p}\in L^{2}(K_{+})$

Let us first prove $\bm{u}\in\bm{H}^{2}(\Omega)$ and $p\in H^{1}(\Omega)$ up
to the boundary, allowing the estimates to depend on $\epsilon$. Observe that
the strong form of (3.2) is as follows:

(3.7)
$\displaystyle-\partial_{l}\big{(}\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\tilde{\Phi}_{lj}\big{)}-\partial_{l}\bigg{[}\tilde{g}\,\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\bigg{]}+\partial_{k}(\tilde{p}\,\tilde{\Phi}_{ki})$
$\displaystyle=\tilde{F}_{i}\quad\text{in}\;K_{+}\quad(i=1,\dots,d),$
$\displaystyle\tilde{\Phi}_{ki}\partial_{k}\tilde{u}_{i}$
$\displaystyle=0\quad\text{in}\;K_{+}.$

which, as noticed in Remark 3.3, holds in the almost-everywhere sense.
Focusing on the principal terms, we have

$-\tilde{\Phi}_{kj}\tilde{\Phi}_{lj}\partial_{k}\partial_{l}\tilde{u}_{i}-\tilde{g}\,\tilde{\Phi}_{kj}\tilde{\Phi}_{lj}\,\partial_{l}\bigg{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}+\tilde{\Phi}_{ki}\partial_{k}\tilde{p}\in
L^{2}(K_{+})\quad(i=1,\dots,d).$

From the tangential regularity
$\nabla\nabla^{\prime}\tilde{\bm{u}}\in\bm{L}^{2}(K_{+}\cap W)$ we see that
(note that $S:=\sum_{j=1}^{d}|\tilde{\Phi}_{dj}|^{2}>0$):

(3.8)
$-\bigg{[}1+\frac{\tilde{g}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}\partial_{d}^{2}\tilde{u}_{i}-\tilde{g}\,\frac{\tilde{\Phi}_{dj}}{S}\,(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i})\,\partial_{d}\bigg{[}\frac{1}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}+\frac{\tilde{\Phi}_{di}}{S}\partial_{d}\tilde{p}\in
L^{2}(K_{+}\cap W)$

for $i=1,\dots,d$. Here notice that the divergence-free condition
$\tilde{\Phi}_{ki}\partial_{k}\tilde{u}_{i}=0$ and the tangential regularity
imply

$\tilde{\Phi}_{di}\partial_{d}^{2}\tilde{u}_{i}\in L^{2}(K_{+}\cap W).$

This together with (3.8), which is multiplied by $\tilde{\Phi}_{di}$ and added
for $i=1,\dots,d$, deduces

(3.9)
$\partial_{d}\tilde{p}\in\tilde{g}\,\frac{\tilde{\Phi}_{di}\tilde{\Phi}_{dj}}{S}\,(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i})\,\partial_{d}\bigg{[}\frac{1}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}+L^{2}(K_{+}\cap
W).$

Multiplying (3.8) by $-\partial_{d}^{2}\tilde{u}_{i}$ and adding the resulting
equations for $i=1,\dots,d$, we obtain

(3.10)
$\displaystyle\bigg{[}1+\tilde{g}\frac{|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}\bigg{]}|\partial_{d}^{2}\tilde{\bm{u}}|^{2}$
$\displaystyle\hskip
28.45274pt+\tilde{g}\,\frac{\tilde{\Phi}_{dj}}{S}\,(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i})\Big{(}1-\frac{|\tilde{\Phi}_{di}|^{2}}{S}\Big{)}\,\partial_{d}\bigg{[}\frac{1}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}\,\partial_{d}^{2}\tilde{u}_{i}=\bm{R}^{(1)}\cdot\partial_{d}^{2}\tilde{\bm{u}},$

for some $\bm{R}^{(1)}\in\bm{L}^{2}(K_{+}\cap W)$. We have

$\displaystyle\partial_{d}\bigg{[}\frac{1}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}$
$\displaystyle=-\frac{(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):\partial_{d}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}})}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}$
(3.11)
$\displaystyle=-\frac{(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}})}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}-\frac{(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):\big{[}(\partial_{d}\tilde{\bm{\Phi}}^{\top})\nabla\otimes\tilde{\bm{u}}\big{]}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}},$

in which the second term in the right-hand side is bounded by
$C(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{-1/2}$.
Therefore, if we set
$\tilde{\bm{\Phi}}_{d}:=(\tilde{\Phi}_{d1},\dots,\tilde{\Phi}_{dd})$, the
second term in the left-hand side of (3.10) is bounded by

$\displaystyle\frac{\tilde{g}}{S}\left|(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i})\,\partial_{d}\bigg{[}\frac{1}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}\,\tilde{\Phi}_{dj}\partial_{d}^{2}\tilde{u}_{i}\right|$
$\displaystyle\leq\;$
$\displaystyle\frac{\tilde{g}}{S}\frac{|(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}})|\,|(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}_{d}^{\top}\otimes\partial_{d}^{2}\tilde{\bm{u}})|}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}+\big{|}\bm{R}^{(2)}\cdot\partial_{d}^{2}\tilde{\bm{u}}\big{|}$
$\displaystyle\leq\;$
$\displaystyle\frac{\tilde{g}}{S}\frac{|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}\,|\tilde{\bm{\Phi}}_{d}^{\top}|^{2}\,|\partial_{d}^{2}\tilde{\bm{u}}|^{2}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}+|R^{(3)}|\,|\partial_{d}^{2}\tilde{\bm{u}}\big{|}=\tilde{g}\frac{|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}|\partial_{d}^{2}\tilde{\bm{u}}|^{2}+|R^{(3)}|\,|\partial_{d}^{2}\tilde{\bm{u}}\big{|}$

for some $\bm{R}^{(2)}\in\bm{L}^{2}(K_{+}\cap W)$ and $R^{(3)}\in
L^{2}(K_{+})$. Here we have used the fact that

$\Big{(}\sum_{k=1}^{d-1}\tilde{\Phi}_{k,j}\partial_{k}\partial_{d}\tilde{u}_{i}\Big{)}(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{-1/2}\in
L^{2}(K_{+}\cap W)\qquad(i,j=1,\dots,d),$

which is a consequence of the tangential regularity, in the third line. Then
it follows from (3.10) that

$\bigg{[}1+\tilde{g}\frac{\epsilon^{2}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}\bigg{]}|\partial_{d}^{2}\tilde{\bm{u}}|^{2}\leq|R^{(4)}|+\frac{1}{2}|\partial_{d}^{2}\tilde{\bm{u}}|^{2}\quad\text{a.e.
in}\quad K_{+}\cap W,$

where $R^{(4)}\in L^{1}(K_{+})$, which implies
$\partial_{d}^{2}\tilde{\bm{u}}\in\bm{L}^{2}(K_{+}\cap W)$. By (3.9) one also
has $\partial_{d}\tilde{p}\in L^{2}(K_{+}\cap W)$. Combined with the
tangential regularity, we have

$\tilde{\bm{u}}\in\bm{H}^{2}(K_{+}\cap W),\quad\tilde{p}\in H^{1}(K_{+}\cap
W),$

and hence $\bm{u}\in\bm{H}^{2}(\Omega\cap W_{\bm{x}_{0}})$, $p\in
H^{1}(\Omega\cap W_{\bm{x}_{0}})$. This combined with the interior regularity
and a standard covering argument yields

$\bm{u}\in\bm{H}^{2}(\Omega),\quad p\in H^{1}(\Omega).$

####  3.3.2. Satisfaction of the slip boundary condition and
$H^{3}$-approximation for $\bm{u}$

Since $\bm{u}\in\bm{H}^{2}(\Omega)$ and $p\in H^{1}(\Omega)$, the traces of
$\nabla\bm{u}$ and $p$ are well defined in $\bm{H}^{1/2}(\Gamma)$ and
$H^{1/2}(\Gamma)$. Let us confirm that $\tilde{\bm{u}}$ satisfies the perfect
slip boundary condition.

######  Lemma 3.4.

It holds that $(\nabla\bm{u}\,\bm{n})_{\tau}=\bm{0}$ a.e. on $\Gamma$. This
reads, in the $\bm{y}$-coordinate, $\partial_{d}\tilde{u}_{i}=0$ on $K_{0}$
for $1\leq i\leq d-1$.

###### Proof.

Let $\bm{\eta}\in\bm{H}^{1/2}(\Gamma),\;\bm{\eta}\cdot\bm{n}=0$ be arbitrary.
It follows from integration by parts and (2.5) that

$\displaystyle-\bigg{(}\Big{(}1+\frac{g}{\sqrt{|\nabla\bm{u}|^{2}+\epsilon^{2}}}\Big{)}(\nabla\bm{u}\,\bm{n})_{\tau},\bm{\eta}\bigg{)}_{\Gamma}=(\bm{f},\bm{v})-(\nabla\bm{u},\nabla\bm{v})-\Big{(}g\frac{\nabla\bm{u}}{\sqrt{|\nabla\bm{u}|^{2}+\epsilon^{2}}},\nabla\bm{v}\Big{)}+(p,\operatorname{div}\bm{v})=0,$

where $\bm{v}\in\bm{V}$ is any $\bm{H}^{1}(\Omega)$-extension of $\bm{\eta}$.
Then
$\big{(}1+g/\sqrt{|\nabla\bm{u}|^{2}+\epsilon^{2}}\big{)}(\nabla\bm{u}\,\bm{n})_{\tau}=\bm{0}$,
and thus $(\nabla\bm{u}\,\bm{n})_{\tau}=\bm{0}$ a.e. on $\Gamma$. From this
one has $\nabla\bm{u}\,\bm{n}=[(\nabla\bm{u}\,\bm{n})\cdot\bm{n}]\bm{n}$ on
$\Gamma$, so that

$\tilde{\Phi}_{kj}(\partial_{k}\tilde{u}_{i})\tilde{n}_{j}=\tilde{\Phi}_{kj}(\partial_{k}\tilde{u}_{l})\tilde{n}_{j}\tilde{n}_{l}\tilde{n}_{i}\quad\text{on}\quad
K_{0}\quad(i=1,\dots,d).$

This combined with $\tilde{\bm{n}}=(0,\dots,0,-1)^{\top}$ and
$\tilde{\Phi}_{id}=\tilde{\Phi}_{di}=0$ for $i\neq d$, $\tilde{\Phi}_{dd}=-1$
on $K_{0}$ leads to the conclusion. ∎

In order to estimate $\|\partial_{d}^{2}\bm{\tilde{u}}\|_{\bm{L}^{2}(K_{+})}$,
it is tempting to multiply (3.7)1 by
$-\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})$ and integrate
over $K_{+}$. However, for the integration by parts to be valid, we need—at
least formally—third order derivatives of $\tilde{\bm{u}}$. To address this
issue, let us introduce an approximation $\bm{u}^{(m)}\in\bm{H}^{3}(\Omega)$
by solving, for $m=1,2,\dots$,

(3.12) $\displaystyle-\Delta\bm{u}^{(m)}+\nabla q^{(m)}$
$\displaystyle=\bm{h}^{(m)}$ $\displaystyle\text{in }\;\Omega,$
$\displaystyle\operatorname{div}\bm{u}^{(m)}$ $\displaystyle=0$
$\displaystyle\text{in }\;\Omega,$
$\displaystyle\bm{u}^{(m)}\cdot\bm{n}=0,\quad(\nabla\bm{u}^{(m)}\bm{n})_{\tau}$
$\displaystyle=0$ $\displaystyle\text{on }\;\Gamma,$

where $\bm{h}^{(m)}\in\bm{H}^{1}(\Omega)$ is chosen in such a way that

$\bm{h}^{(m)}\to\bm{h}:=-\Delta\bm{u}+\nabla q\quad\text{ strongly in
$\bm{L}^{2}(\Omega)$ as $m\to\infty$},\quad q:=0.$

By virtue of the $H^{2}$-regularity result for the linear Stokes equations
under the perfect slip boundary condition (which may be proved similarly as in
[5, Theorem 1.2]), we find that

(3.13) $\|\bm{u}^{(m)}-\bm{u}\|_{\bm{H}^{2}(\Omega)}\leq
C\|\bm{h}^{(m)}-\bm{h}\|_{\bm{L}^{2}(\Omega)}\to 0\quad(m\to\infty).$

The traces of the second-order derivatives of $\bm{u}^{(m)}$ are now well
defined in $\bm{H}^{1/2}(\Gamma)$, to which we obtain the following estimate.
We emphasize that the perfect slip boundary condition plays an essential role
here.

######  Lemma 3.5.

Let $\bm{u}^{(m)}\in\bm{H}^{3}(\Omega)\cap\bm{V}_{\sigma}$ be as above. Then
we have

(3.14)
$|\partial_{d}\tilde{u}_{d}^{(m)}|+|\partial_{d}^{2}\tilde{u}_{d}^{(m)}|\leq
C|\nabla^{\prime}\tilde{\bm{u}}^{(m)}|\text{ a.e.\ on }K_{0}.$

###### Proof.

From the divergence-free condition
$\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}=0$ we see that the
equalities

$\displaystyle\tilde{\Phi}_{dd}\,\partial_{d}\tilde{u}_{d}^{(m)}$
$\displaystyle=-\sum_{(i,k)\neq(d,d)}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)},$
$\displaystyle\tilde{\Phi}_{dd}\,\partial_{d}^{2}\tilde{u}_{d}^{(m)}$
$\displaystyle=-\partial_{d}\tilde{\Phi}_{dd}\,\partial_{d}\tilde{u}_{d}^{(m)}-\sum_{(i,k)\neq(d,d)}\Big{(}\partial_{d}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}+\tilde{\Phi}_{ki}\,\partial_{k}\partial_{d}\tilde{u}_{i}^{(m)}\Big{)}$

hold in $H^{1}(K_{+})$. Taking the trace at $y_{d}=0$, we notice that
$\tilde{\Phi}_{ki}\,\partial_{k}\partial_{d}\tilde{u}_{i}^{(m)}=0$ for
$(i,k)\neq(d,d)$111We do not apply the summation convention here. and that
$\partial_{k}\tilde{u}_{i}^{(m)}=0$ if only one of $i,k$ equals $d$. This
leads to

(3.15) $\displaystyle\partial_{d}\tilde{u}_{d}^{(m)}$
$\displaystyle=\sum_{i,k<d}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}\quad\text{a.e.\
on}\;K_{0},$ (3.16) $\displaystyle\partial_{d}^{2}\tilde{u}_{d}^{(m)}$
$\displaystyle=\partial_{d}\tilde{\Phi}_{dd}\,\sum_{i,k<d}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}+\sum_{i,k<d}\partial_{d}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}\quad\text{a.e.\
on}\;K_{0},$

which concludes (3.14). ∎

######  Remark 3.4.

A similar argument without relying on the $H^{3}$-approximation shows
$|\partial_{d}\tilde{u}_{d}|\leq C|\nabla^{\prime}\tilde{\bm{u}}|$ a.e. on
$K_{0}$.

####  3.3.3. Estimates for $\nabla\partial_{d}\tilde{\bm{u}}$ uniform in
$\epsilon$

We now multiply (3.7)1 by
$-\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})$, integrate over
$K_{+}$, take the summation for $i=1,\dots,d$. The resulting equation reads

(3.17)
$I_{1}+I_{2}+I_{3}=\int_{K_{+}}\tilde{F}_{i}\big{[}-\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})\big{]}\,d\bm{y},$

where

$\displaystyle I_{1}$
$\displaystyle=\int_{K_{+}}\partial_{l}\big{(}\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\tilde{\Phi}_{lj}\big{)}\,\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})\,d\bm{y},$
$\displaystyle I_{2}$
$\displaystyle=\int_{K_{+}}\partial_{l}\bigg{[}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\bigg{]}\,\partial_{d}(\tilde{\theta}\partial_{d}\tilde{u}_{i})\,d\bm{y},$
$\displaystyle I_{3}$
$\displaystyle=\int_{K_{+}}\partial_{k}(\tilde{p}\,\tilde{\Phi}_{ki})\big{[}-\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})\big{]}\,d\bm{y}.$

Let us show that the places of $\partial_{l}$ and $\partial_{d}$ in $I_{1}$
and $I_{2}$ above can be interchanged.

######  Lemma 3.6.

We have

$\displaystyle
I_{1}=\int_{K_{+}}\partial_{d}\big{(}\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\tilde{\Phi}_{lj}\big{)}\,\partial_{l}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})\,d\bm{y},\qquad
I_{2}=\int_{K_{+}}\partial_{d}\bigg{[}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\bigg{]}\,\partial_{l}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})\,d\bm{y}.$

###### Proof.

Since the proof for $I_{1}$ is similar, we only deal with $I_{2}$. Let
$\bm{u}^{(m)}$ be the approximation of $\bm{u}$ introduced in (3.12). We find
from integration by parts that

$\displaystyle\int_{K_{+}}\partial_{l}\Big{(}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\Big{)}\,\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\,d\bm{y}$
$\displaystyle=\int_{K_{+}}\partial_{d}\Big{(}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\Big{)}\,\partial_{l}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\,d\bm{y}$
$\displaystyle\qquad+\int_{K_{0}}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\tilde{n}_{l}\,\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\,d\bm{y}^{\prime}$
$\displaystyle\qquad-\int_{K_{0}}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\tilde{n}_{d}\,\partial_{l}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\,d\bm{y}^{\prime}$
$\displaystyle=:I_{21}^{(m)}+I_{22}^{(m)}+I_{23}^{(m)}.$

It suffices to prove $I_{22}^{(m)}+I_{23}^{(m)}=0$, because then the desired
equality follows by making $m\to\infty$ above (recall (3.13)). We claim that
non-zero contribution from $I_{23}^{(m)}$ owes only to the case $l=d$. In
fact, suppose $l\neq d$; then

  * •

$i\neq d$: we have $\partial_{d}\tilde{u}_{i}^{(m)}=0$ (and hence
$\partial_{l}\partial_{d}\tilde{u}_{i}^{(m)}=0$) on $K_{0}$.

  * •

$i=d$ and $k\neq d$: we have $\partial_{k}\tilde{u}_{i}^{(m)}=0$ on $K_{0}$.

  * •

$i=k=d$: we see that $\Phi_{kj}\neq 0$ only when $j=d$, in which case
$\Phi_{lj}=0$ on $K_{0}$.

Therefore, we may assume $l=d$ in $I_{23}^{(m)}$, which implies
$I_{22}^{(m)}+I_{23}^{(m)}=0$. ∎

We split $I_{1}$ into a principal part and lower-order terms $L_{1}$ to derive

$I_{1}=\int_{K_{+}}\tilde{\theta}^{2}|\bm{\Phi}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}}|^{2}\,d\bm{y}+L_{1}\geq
C\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}+L_{1},$

where

$|L_{1}|\leq
C\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}\big{(}\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}\big{)}.$

Next observe that $I_{2}$ is rewritten as

$\displaystyle I_{2}$
$\displaystyle=\int_{K_{+}}\tilde{\theta}^{2}\tilde{g}\,\tilde{\Phi}_{kj}\partial_{d}\bigg{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}\,\tilde{\Phi}_{lj}\partial_{l}\partial_{d}\tilde{u}_{i}\,d\bm{y}\\!+\\!\int_{K_{+}}\tilde{g}\tilde{\Phi}_{kj}\partial_{d}\bigg{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}\,\tilde{\Phi}_{lj}\partial_{l}(\tilde{\theta}^{2})\partial_{d}\tilde{u}_{i}\,d\bm{y}+L_{2}$
$\displaystyle=:I_{21}+I_{22}+L_{2},$

where the lower-order terms $L_{2}$ can be bounded by

$\displaystyle|L_{2}|$ $\displaystyle\leq
C\|\tilde{g}\|_{H^{1}(K_{+})}\bigg{\|}\frac{\nabla\tilde{\bm{u}}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{\|}_{\bm{L}^{\infty}(K_{+})}(\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})})$
$\displaystyle\leq
C\|\tilde{g}\|_{H^{1}(K_{+})}(\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}).$

Since (cf. (3.11))

$\displaystyle\tilde{\Phi}_{kj}\partial_{d}\bigg{[}\frac{\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\bigg{]}$
$\displaystyle=\frac{\tilde{\Phi}_{kj}\partial_{k}\partial_{d}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}-\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\frac{(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}})}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}$
$\displaystyle\quad-\underbrace{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\frac{(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):\big{[}(\partial_{d}\tilde{\bm{\Phi}}^{\top})\nabla\otimes\tilde{\bm{u}}\big{]}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}}_{|\cdot|\leq
C}\qquad(i,j=1,\dots,d),$

we obtain, by the Cauchy–Schwarz inequality,

$I_{21}=\int_{K_{+}}\tilde{\theta}^{2}\tilde{g}\frac{|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}}|^{2}\,(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})-\big{|}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}})\big{|}^{2}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}d\bm{y}+L_{21}\geq
0+L_{21},$

where $|L_{21}|\leq
C\|\tilde{g}\|_{L^{2}(K_{+})}\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}$.
Similarly,

$\displaystyle I_{22}$
$\displaystyle=\int_{K_{+}}\tilde{g}\frac{(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}}):\big{(}\tilde{\bm{\Phi}}^{\top}\nabla(\tilde{\theta}^{2})\otimes\partial_{d}\tilde{\bm{u}}\big{)}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}d\bm{y}$
$\displaystyle\hskip
56.9055pt-\int_{K_{+}}\tilde{g}\frac{\big{[}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}^{\top}\nabla(\tilde{\theta}^{2})\otimes\partial_{d}\tilde{\bm{u}})\big{]}\,\big{[}(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}):(\tilde{\bm{\Phi}}^{\top}\nabla\otimes\partial_{d}\tilde{\bm{u}})\big{]}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{3/2}}d\bm{y}+L_{22},$

where $|L_{22}|\leq
C\|\tilde{g}\|_{L^{2}(K_{+})}\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}$.
Consequently,

$|I_{2}|\leq
C\|\tilde{g}\|_{H^{1}(K_{+})}(\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}).$

We now focus on $I_{3}$. It is represented as

$I_{3}=\lim_{m\to\infty}I_{3}^{(m)},\qquad
I_{3}^{(m)}:=\int_{K_{+}}\partial_{k}(\tilde{p}\,\tilde{\Phi}_{ki})\big{[}-\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\big{]}\,d\bm{y}.$

Let us admit for the moment that the following key estimate holds:

(3.18) $|I_{3}^{(m)}|\leq
C\big{(}\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})})(\|\nabla\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla^{2}\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}),$

which will be shown in Subsection 3.3.4. Taking the limit $m\to\infty$, we
find from (3.6) and the arithmetic-geometric mean inequality that

$|I_{3}|\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}^{2}+\|\tilde{g}\|_{H^{1}(K_{+})}^{2}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}+\|\tilde{p}\|_{L^{2}(K_{+})}^{2})+\frac{1}{4}\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2}.$

Combining the estimates for $I_{1}$–$I_{3}$ with

$\Big{|}\int_{K_{+}}\tilde{F}_{i}\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i})\,d\bm{y}\Big{|}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}^{2}+\|\tilde{g}\|_{L^{2}(K_{+})}^{2}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2})+\frac{1}{4}\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}^{2},$

we deduce from (3.17) that

$\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}\leq
C(\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}),$

which completes the proof of Lemma 3.3 except (3.18).

####  3.3.4. Proof of (3.18)

Integration by parts gives

$\displaystyle I_{3}^{(m)}$
$\displaystyle=\int_{K_{+}}\tilde{p}\tilde{\Phi}_{ki}\,\partial_{k}\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\,d\bm{y}-\int_{K_{0}}\tilde{n}_{k}\tilde{p}\,\tilde{\Phi}_{ki}\,\partial_{d}(\tilde{\theta}^{2}\partial_{d}\tilde{u}_{i}^{(m)})\,d\bm{y}^{\prime}=:I_{31}^{(m)}+I_{32}^{(m)}.$

If we divide $I_{31}^{(m)}$ into a principal part and other lower-order terms
represented by $L_{31}^{(m)}$, we get

(3.19)
$I_{31}^{(m)}=\int_{K_{+}}\tilde{\theta}^{2}\tilde{p}\,\partial_{d}^{2}(\underbrace{\tilde{\Phi}_{ki}\partial_{k}\tilde{u}_{i}^{(m)}}_{=0})\,d\bm{y}+L_{31}^{(m)},\qquad|L_{31}^{(m)}|\leq
C\|\tilde{p}\|_{L^{2}(K_{+})}\big{(}\|\nabla\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\partial_{d}\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}\big{)}.$

######  Remark 3.5.

We see that $\nabla^{2}\tilde{\bm{\Phi}}$, i.e., third-order derivatives of
$\bm{\varphi}$ appear in $L_{31}^{(m)}$, which accounts for why we need
$C^{3,1}$-regularity of $\Omega$.

Since both $n_{k}$ and $\Phi_{ki}$ become non-zero only when $k=i=d$ and
$n_{d}=\Phi_{dd}=-1$, we see that

$I_{32}^{(m)}=-\int_{K_{0}}\tilde{p}\,\partial_{d}(\tilde{\theta}^{2}\,\partial_{d}\tilde{u}_{d}^{(m)})\,d\bm{y}^{\prime}.$

To address this term, let us estimate the pressure trace $\tilde{p}|_{K_{0}}$
in $H^{-1/2}(K_{0})+L^{2}(K_{0})$ as follows.

######  Lemma 3.7.

For arbitrary $\tilde{\eta}\in H^{1/2}(K_{0})$ with
$\operatorname{supp}\tilde{\eta}\subset K_{0}\cap B(\bm{0};1/2)$ we have

$\displaystyle\left|\int_{K_{0}}\tilde{p}\tilde{\eta}\,d\bm{y}^{\prime}\right|$
$\displaystyle\leq
C\big{(}\|\tilde{\bm{F}}\|_{\bm{H}^{1}(K_{+})^{\prime}}+\|\tilde{g}\|_{L^{2}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})})\|\tilde{\eta}\|_{H^{1/2}(K_{0})}$
$\displaystyle\hskip
56.9055pt+C(\|\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{0}\cap
B(\bm{0};1/2))}+\|\tilde{g}\|_{L^{2}(K_{0})}\big{)}\|\tilde{\eta}\|_{L^{2}(K_{0})}.$

###### Proof.

There exists some $\tilde{\bm{v}}\in\bm{H}^{1}(K_{+})$ satisfying
$\tilde{\bm{v}}=\tilde{\eta}\tilde{\bm{n}}$ on $K_{0}$,
$\tilde{\bm{v}}=\bm{0}$ on $\partial K_{+}\setminus K_{0}$, and
$\|\tilde{\bm{v}}\|_{\bm{H}^{1}(K_{+})}\leq
C\|\tilde{\eta}\|_{H^{1/2}(K_{0})}$. We multiply (3.7)1 by $\tilde{v}_{i}$,
integrate over $K_{+}$, perform integration by parts, and add for
$i=1,\dots,d$ to have

$\displaystyle\int_{K_{+}}\Big{[}(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i})(\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i})+\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\;\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}-\tilde{p}\,\tilde{\Phi}_{kj}\partial_{k}\tilde{v}_{i}\Big{]}\,d\bm{y}$
$\displaystyle\qquad-\int_{K_{0}}\tilde{n}_{l}\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\,\tilde{\Phi}_{lj}\tilde{v}_{i}\,d\bm{y}^{\prime}-\int_{K_{0}}\tilde{n}_{l}\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\Phi}_{lj}\tilde{v}_{i}\,d\bm{y}^{\prime}+\int_{K_{0}}\tilde{n}_{k}\tilde{p}\tilde{\Phi}_{ki}\tilde{v}_{i}\,d\bm{y}^{\prime}=\int_{K_{+}}\tilde{F}_{i}\tilde{v}_{i}\,d\bm{y}.$

It follows that, by the property of $\tilde{\bm{n}}$ and $\tilde{\bm{\Phi}}$
on $K_{0}$ (recall Subsection 3.1.1) and by $\tilde{v}_{d}=-\tilde{\eta}$ on
$K_{0}$,

$\displaystyle\int_{K_{0}}\tilde{p}\tilde{\eta}\,d\bm{y}^{\prime}$
$\displaystyle=\int_{K_{+}}\Big{[}(\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i})(\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i})+\tilde{g}\frac{\tilde{\Phi}_{kj}\partial_{k}\tilde{u}_{i}\;\tilde{\Phi}_{lj}\partial_{l}\tilde{v}_{i}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}-\tilde{p}\,\tilde{\Phi}_{kj}\partial_{k}\tilde{v}_{i}-\tilde{F}_{i}\tilde{v}_{i}\Big{]}\,d\bm{y}$
$\displaystyle\qquad-\int_{K_{0}}\partial_{d}\tilde{u}_{d}\,\tilde{\eta}\,d\bm{y}^{\prime}-\int_{K_{0}}\tilde{g}\frac{\partial_{d}\tilde{u}_{d}}{(|\tilde{\bm{\Phi}}^{\top}\nabla\otimes\tilde{\bm{u}}|^{2}+\epsilon^{2})^{1/2}}\tilde{\eta}\,d\bm{y}^{\prime}.$

Since $|\partial_{d}\tilde{u}_{d}|\leq C|\nabla^{\prime}\tilde{\bm{u}}|$ a.e.
on $K_{0}$ by Remark 3.4, we obtain

$\displaystyle\left|\int_{K_{0}}\tilde{p}\tilde{\eta}\,d\bm{y}^{\prime}\right|$
$\displaystyle\leq
C\big{(}\|\tilde{\bm{F}}\|_{\bm{H}^{1}(K_{+})^{\prime}}+\|\tilde{g}\|_{L^{2}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})})\|\tilde{\bm{v}}\|_{\bm{H}^{1}(K_{+})}$
$\displaystyle\hskip
56.9055pt+C(\|\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{0}\cap
B(\bm{0};1/2)))}+\|\tilde{g}\|_{L^{2}(K_{0})}\big{)}\|\tilde{\eta}\|_{L^{2}(K_{0})},$

which proves the lemma. ∎

Setting
$\tilde{\eta}:=\partial_{d}(\tilde{\theta}^{2}\,\partial_{d}\tilde{u}_{d}^{(m)})\in
H^{1}(K_{+})$, we see from (3.15) and (3.16) that

$\tilde{\eta}=\partial_{d}(\tilde{\theta}^{2})\sum_{i,k<d}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}+\tilde{\theta}^{2}\sum_{i,k<d}\big{(}\partial_{d}\tilde{\Phi}_{dd}\,\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}+\partial_{d}\tilde{\Phi}_{ki}\,\partial_{k}\tilde{u}_{i}^{(m)}\big{)}\quad\text{on}\quad
K_{0}.$

If the function defined by the right-hand side above is denoted as
$\tilde{\chi}\in H^{1}(K_{+})$, we find from a trace inequality that

$\displaystyle\|\tilde{\eta}\|_{H^{1/2}(K_{0})}=\|\tilde{\chi}\|_{H^{1/2}(K_{0})}\leq
C\|\tilde{\chi}\|_{H^{1}(K_{+})}\leq
C(\|\nabla\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\nabla^{\prime}\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}).$

Applying Lemma 3.7 to this $\tilde{\eta}$ and using (3.3) and Remark 3.2, we
deduce that

$\displaystyle|I_{32}^{(m)}|$ $\displaystyle\leq
C\big{(}\|\tilde{\bm{F}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})}+\|\nabla^{\prime}\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{0}\cap
B(\bm{0};1/2)))})$ $\displaystyle\hskip
56.9055pt\times(\|\nabla\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\nabla^{\prime}\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})})$
$\displaystyle\leq
C\big{(}\|\tilde{\bm{f}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{g}\|_{H^{1}(K_{+})}+\|\nabla\tilde{\bm{u}}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{p}\|_{L^{2}(K_{+})})(\|\nabla\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}+\|\tilde{\theta}\nabla\nabla^{\prime}\tilde{\bm{u}}^{(m)}\|_{\bm{L}^{2}(K_{+})}).$

This combined with (3.19) proves (3.18), establishing Lemma 3.3 and hence
Proposition 3.1.

##  4\. $H^{2}$-regularity for the non-stationary problem

In this section, we focus on the non-stationary Bingham–Navier–Stokes problem
(1.4)–(1.7), retrieving the notation $\nu$ to represent the positive viscosity
constant. Similarly to Proposition 2.1(i), one can interpret (1.4)–(1.6) as a
variational inequality (cf. [13, Section VI.3.2]) as follows:

(4.1)
$(\bm{u}^{\prime}(t),\bm{v}-\bm{u}(t))-\nu(\Delta\bm{u}(t),\bm{v}-\bm{u}(t))+b(\bm{u}(t),\bm{u}(t),\bm{v}-\bm{u}(t)){}+(g(t),|\nabla\bm{v}|-|\nabla\bm{u}(t)|)\geq(\bm{f}(t),\bm{v}-\bm{u}(t))\quad\forall\bm{v}\in\bm{V}_{\sigma}$

and for a.a. $t\in(0,T)$ with the initial condition $\bm{u}(0)=\bm{u}_{0}$ in
$\bm{H}_{\sigma}$. Here, $\bm{u}^{\prime}$ means the time derivative
$\partial\bm{u}/\partial t$, and the trilinear form $b$ is given by

$b(\bm{u},\bm{v},\bm{w})=\int_{\Omega}u_{i}\frac{\partial v_{j}}{\partial
x_{i}}w_{j}dx\qquad\forall\bm{u},\bm{v},\bm{w}\in\bm{V}_{\sigma}.$

We also define $B:\bm{V}_{\sigma}\to\bm{V}_{\sigma}^{\prime}$ by
$B\bm{u}:=P[(\bm{u}\cdot\nabla)\bm{u}]$, where
$P:\bm{L}^{2}(\Omega)\to\bm{H}_{\sigma}$ is the Helmholtz–Leray projection,
which can be naturally extended to
$\bm{V}_{\sigma}^{\prime}\to\bm{V}_{\sigma}^{\prime}$. Then we have $\langle
B\bm{u},\bm{v}\rangle=b(\bm{u},\bm{u},\bm{v})$ (recall
$\langle\cdot,\cdot\rangle$ denotes the duality pairing between
$\bm{V}_{\sigma}^{\prime}$ and $\bm{V}_{\sigma}$) and

(4.2) $\|B\bm{v}-B\bm{w}\|_{\bm{V}_{\sigma}^{\prime}}\leq C_{\rm
L}\|\bm{v}-\bm{w}\|_{\bm{V}_{\sigma}}\\{\|\bm{v}\|_{\bm{V}_{\sigma}}+\|\bm{w}\|_{\bm{V}_{\sigma}}\\}$

for some constant $C_{\rm L}=C(\Omega)>0$, which implies the continuity of $B$
(cf. [3, pp. 251–253]).

Under these settings we obtain the following extension of [3, Theorem 5.9] to
the Bingham–Navier–Stokes problem.

######  Theorem 4.1.

Let $\Omega$ be of $C^{3,1}$-class, $\bm{f}\in
W^{1,1}(0,T;\bm{L}^{2}(\Omega))$, and $g\in H^{1}(0,T;L^{2}(\Omega))\cap
L^{\infty}(0,T;H^{1}(\Omega))$, $g\geq 0$. Assume that there exists
$\bm{f}^{*}\in\bm{L}^{2}(\Omega)$ such that
$\bm{u}_{0}\in\bm{V}_{\sigma}\cap\bm{H}^{2}(\Omega)$ satisfies

(4.3) $\displaystyle(\nabla\bm{u}_{0}\bm{n})_{\tau}=\bm{0}\quad\text{a.e.\ on
}\;\Gamma,$ (4.4)
$\displaystyle(-\nu\Delta\bm{u}_{0}+B\bm{u}_{0}-\bm{f}(0),\bm{v}-\bm{u}_{0})+(g(0),|\nabla\bm{v}|-|\nabla\bm{u}_{0}|)\geq(\bm{f}^{*},\bm{v}-\bm{u}_{0})\qquad\forall\bm{v}\in\bm{V}_{\sigma}.$

Then, there exist $T_{*}\in(0,T]$ and a unique function $\bm{u}\in
W^{1,\infty}(0,T_{*};\bm{H}_{\sigma})\cap H^{1}(0,T_{*};\bm{V}_{\sigma})\cap
L^{\infty}(0,T_{*};\bm{H}^{2}(\Omega))$ such that $\bm{u}$ satisfies
$\bm{u}(0)=\bm{u}_{0}$ and (4.1) for a.a. $t\in(0,T_{*})$. If $d=2$, $T_{*}$
can be taken as $T$.

######  Remark 4.1.

(i) Assumption (4.4) is a kind of a compatibility condition for initial data
and can be expressed as
$\partial\varphi^{0}(\bm{u}_{0})+B\bm{u}_{0}-P\bm{f}(0)\cap\bm{H}_{\sigma}\neq\emptyset$
($\varphi^{t}$ is defined in Subsection 4.3 below).

(ii) If $g\in H^{1}((0,T)\times\Omega)$ only, we still have a solution such
that $\bm{u}\in L^{2}(0,T;\bm{H}^{2}(\Omega))$.

We can obtain another strong solution with less regular $\bm{f}$, $g$, and
$\bm{u}_{0}$ as follows. Note that the compatibility condition (4.4) is not
necessary to assume in this case.

######  Theorem 4.2.

Let $\Omega$ be of $C^{3,1}$-class, $\bm{f}\in L^{2}(0,T;\bm{L}^{2}(\Omega))$,
$g\in W^{1,1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{1}(\Omega))$, $g\geq 0$, and
$\bm{u}_{0}\in\bm{V}_{\sigma}$. Then, there exist $T_{*}\in(0,T]$ and a unique
function $\bm{u}\in H^{1}(0,T_{*};\bm{H}_{\sigma})\cap
C([0,T_{*}];\bm{V}_{\sigma})\cap L^{2}(0,T_{*};\bm{H}^{2}(\Omega))$ such that
$\bm{u}$ satisfies $\bm{u}(0)=\bm{u}_{0}$ and (4.1) for a.a. $t\in(0,T_{*})$.
If $d=2$, $T_{*}$ can be taken as $T$.

######  Remark 4.2.

(i) Since the embedding $H^{1}(0,T_{*};\bm{L}^{2}(\Omega))\cap
L^{2}(0,T_{*};\bm{H}^{2}(\Omega))\hookrightarrow
C([0,T_{*}];\bm{H}^{1}(\Omega))$ holds, it suffices to construct a solution
$\bm{u}\in H^{1}(0,T_{*};\bm{H}_{\sigma})\cap
L^{2}(0,T_{*};\bm{H}^{2}(\Omega))$.

(ii) The well-posedness in this class may be discussed also from the abstract
theory of evolution inclusion governed by the subdifferential with nonmonotone
perturbation [29]. However, in order to apply this abstract theory we need
stronger assumption on $g$ with respect to $t$, more precisely the Lipschitz
continuity (see [29, p. 273]). In this sense, our assumption of Theorem 4.2
with respect to $g$ is weaker than [29].

###  4.1. Truncation of $B$

The proof of the above theorems is based on the truncation method. To this
end, for $M>0$ we introduce a truncated operator
$B_{M}:\bm{V}_{\sigma}\to\bm{V}_{\sigma}^{\prime}$ by

$B_{M}\bm{v}=\begin{cases}B\bm{v}&{\rm if~{}}\|\bm{v}\|_{\bm{V}_{\sigma}}\leq
M,\\\ \displaystyle\frac{M^{2}}{\|\bm{v}\|_{\bm{V}_{\sigma}}^{2}}B\bm{v}&{\rm
if~{}}\|\bm{v}\|_{\bm{V}_{\sigma}}>M.\\\ \end{cases}$

The essential idea of this truncation goes back to [2, 3, 4], see also a
similar method in [10, 12]. It follows from [3, Chapter 5] that if
$\bm{v}\in\bm{V}\cap\bm{H}^{2}(\Omega)$ then $B_{M}\bm{v}$ makes sense in
$\bm{H}_{\sigma}$ and

(4.5) $\displaystyle\|B_{M}\bm{v}\|_{\bm{H}_{\sigma}}$ $\displaystyle\leq
C_{\rm L}M^{\frac{3}{2}}\|\bm{v}\|_{\bm{H}^{2}(\Omega)}^{\frac{1}{2}},$ (4.6)
$\displaystyle\|B_{M}\bm{v}\|_{\bm{H}_{\sigma}}$ $\displaystyle\leq C_{\rm
L}\|\bm{v}\|_{\bm{H}_{\sigma}}^{\frac{1}{2}}\|\bm{v}\|_{\bm{V}_{\sigma}}\|\bm{v}\|_{\bm{H}^{2}(\Omega)}^{\frac{1}{2}}\quad$
$\displaystyle{\rm if~{}}d=2,$ (4.7)
$\displaystyle\|B_{M}\bm{v}\|_{\bm{H}_{\sigma}}$ $\displaystyle\leq C_{\rm
L}\|\bm{v}\|_{\bm{V}_{\sigma}}^{\frac{3}{2}}\|\bm{v}\|_{\bm{H}^{2}(\Omega)}^{\frac{1}{2}}\quad$
$\displaystyle{\rm if~{}}d=3,$

for some constant $C_{\mathrm{L}}=C(\Omega)>0$ independent of $M$. Moreover,
similarly to (4.2), for all $\bm{v},\bm{w}\in\bm{V}_{\sigma}$ we have

(4.8) $\|B_{M}\bm{v}-B_{M}\bm{w}\|_{\bm{V}_{\sigma}^{\prime}}\leq 2C_{\rm
L}\|\bm{v}-\bm{w}\|_{\bm{V}_{\sigma}}\\{\|\bm{v}\|_{\bm{V}_{\sigma}}+\|\bm{w}\|_{\bm{V}_{\sigma}}\\}.$

Given a nonnegative function $g\in H^{1}(\Omega)$, define a proper, lower
semi-continuous, and convex functional $j_{g}:\bm{H}_{\sigma}\to[0,+\infty]$
by

$j_{g}(\bm{v})=\frac{\nu}{2}\int_{\Omega}|\nabla\bm{v}|^{2}\,d\bm{x}+\int_{\Omega}g|\nabla\bm{v}|\,d\bm{x}\quad\text{if
}\;\bm{v}\in D(j_{g}):=\bm{V}_{\sigma},\qquad
j_{g}(\bm{v})=+\infty\quad\text{if
}\;\bm{v}\in\bm{H}_{\sigma}\setminus\bm{V}_{\sigma}.$

By the definition of a subdifferential, we have $\bm{f}\in\partial
j_{g}(\bm{u})$ in $\bm{H}_{\sigma}$ if and only if $\bm{u}\in\bm{V}_{\sigma}$
satisfies

(4.9)
$\nu(\nabla\bm{u},\nabla(\bm{v}-\bm{u}))+(g,|\nabla\bm{v}|-|\nabla\bm{u}|)\geq(\bm{f},\bm{v}-\bm{u})\qquad\forall\bm{v}\in\bm{V}_{\sigma}.$

Then Corollary 3.1 implies that
$\bm{u}\in\\{\bm{v}\in\bm{V}_{\sigma}\cap\bm{H}^{2}(\Omega)\mid(\nabla\bm{v}\,\bm{n})_{\tau}=\bm{0}\;\text{
a.e.\ on }\Gamma\\}=:\bm{W}$ and that

(4.10) $\|\bm{u}\|_{\bm{H}^{2}(\Omega)}\leq C_{\rm
K}(\|\bm{f}\|_{\bm{L}^{2}(\Omega)}+\|g\|_{H^{1}(\Omega)}),$

where $C_{\rm K}=C(\Omega,\nu)$ is independent of $\bm{f}$ and $g$. In
particular, we have $D(\partial j_{g})\subset\bm{W}$.

Under these settings, according to [3] we define an operator
$\Gamma_{g,M}:\bm{H}_{\sigma}\to 2^{\bm{H}_{\sigma}}$ by

$\Gamma_{g,M}=\partial j_{g}+B_{M},\quad D(\Gamma_{g,M})=D(\partial j_{g}).$

######  Lemma 4.1.

Let $g\in H^{1}(\Omega)$ and $M\in(0,\infty)$. Then there exists a constant
$\alpha_{M}=C(\Omega,\nu,M)>0$ independent of $g$ such that
$\Gamma_{g,M}+\alpha I$ is maximal monotone for all $\alpha\geq\alpha_{M}$,
where $I$ denotes the identity operator.

###### Proof.

We closely follow the arguments of [3, Lemma 5.2]. We remark that, for
operators in a Hilbert space, the concepts of maximal monotonicity (where its
dual is identified with itself) and $m$-accretivity coincide.

Step 1. By [3, pp. 254–255] there exists a positive constant
$c_{M}=C(\Omega,\nu,M)$ such that

(4.11) $|\langle
B_{M}\bm{w}-B_{M}\bm{z},\bm{w}-\bm{z}\rangle|\leq\frac{\nu}{8}\|\bm{w}-\bm{z}\|_{\bm{V}_{\sigma}}^{2}+c_{M}\|\bm{w}-\bm{z}\|_{\bm{H}_{\sigma}}^{2}\qquad\forall\bm{w},\bm{z}\in\bm{V}_{\sigma}.$

Therefore, we see from (4.9), (4.11), and (2.2) that for all
$\bm{w}^{*}\in((1/8)\partial j_{g}+B_{M}+\alpha I)\bm{w}$,
$\bm{z}^{*}\in((1/8)\partial j_{g}+B_{M}+\alpha I)\bm{z}$, and
$\alpha\geq\alpha_{M}:=c_{M}$, there exist $\bm{f}\in\partial j_{g}(\bm{w})$
and $\bm{h}\in\partial j_{g}(\bm{z})$ such that

$\displaystyle(\bm{w}^{*}-\bm{z}^{*},\bm{w}-\bm{z})$
$\displaystyle=(\bm{f}-\bm{h},\bm{w}-\bm{z})+\langle
B_{M}\bm{w}-B_{M}\bm{z},\bm{w}-\bm{z}\rangle+\alpha\|\bm{w}-\bm{z}\|_{\bm{H}_{\sigma}}^{2}$
$\displaystyle\geq\frac{\nu}{8}(\nabla\bm{w},\nabla(\bm{w}-\bm{z}))+(g,|\nabla\bm{w}|-|\nabla\bm{z}|)+\frac{\nu}{8}(\nabla\bm{z},\nabla(\bm{z}-\bm{w}))+(g,|\nabla\bm{z}|-|\nabla\bm{w}|)$
$\displaystyle\qquad-\frac{\nu}{8}\|\bm{w}-\bm{z}\|_{\bm{V}_{\sigma}}^{2}+(\alpha-
c_{M})\|\bm{w}-\bm{z}\|_{\bm{H}_{\sigma}}^{2}$ $\displaystyle=(\alpha-
c_{M})\|\bm{w}-\bm{z}\|_{\bm{H}_{\sigma}}^{2}\geq 0.$

This implies the monotonicity of $(1/8)\partial j_{g}+B_{M}+\alpha I$, and
hence that of $\Gamma_{g,M}+\alpha I$.

Step 2. Let us consider $(1/4)\partial_{*}j_{g}+B_{M}:\bm{V}_{\sigma}\to
2^{\bm{V}_{\sigma}^{\prime}}$ where $\partial_{*}$ stands for the
subdifferential between $\bm{V}_{\sigma}$ and $\bm{V}_{\sigma}^{\prime}$.
Recall that $\partial j_{g}(\bm{u})\subset\partial_{*}j_{g}(\bm{u})$ in
$\bm{V}_{\sigma}^{\prime}$ and that if $\bm{f}\in\bm{H}_{\sigma}$ and
$\bm{u}\in\bm{V}_{\sigma}$ satisfy $\bm{f}\in\partial_{*}j_{g}(\bm{u})$ then
$\partial j_{g}(\bm{u})=\partial_{*}j_{g}(\bm{u})$ in $\bm{H}_{\sigma}$. Now
we prove that $(1/4)\partial_{*}j_{g}+B_{M}+\alpha I:\bm{V}_{\sigma}\to
2^{\bm{V}^{\prime}_{\sigma}}$ is maximal monotone. For this we define

$j_{1}(\bm{v})=\frac{\nu}{4}\int_{\Omega}|\nabla\bm{v}|^{2}\,d\bm{x}+\int_{\Omega}g|\nabla\bm{v}|\,d\bm{x},\quad
j_{2}(\bm{v})=\frac{\nu}{4}\int_{\Omega}|\nabla\bm{v}|^{2}\,d\bm{x},\quad
D(j_{1})=D(j_{2})=\bm{V}_{\sigma}.$

Then $(1/4)\partial_{*}j_{1}:\bm{V}_{\sigma}\to 2^{\bm{V}_{\sigma}^{\prime}}$
is maximal monotone with $D((1/4)\partial_{*}j_{1})=\bm{V}_{\sigma}$.
Moreover, $(1/4)\partial_{*}j_{2}+B_{M}+\alpha
I:\bm{V}_{\sigma}\to\bm{V}_{\sigma}^{\prime}$ is single-valued, monotone (by a
similar computation to Step 1), and demi-continuous. Therefore, it follows
from the Rockafellar theorem [3, Corollary 2.6] that the sum

$\frac{1}{4}\partial_{*}j_{g}+B_{M}+\alpha
I=\frac{1}{4}\partial_{*}j_{1}+\Big{(}\frac{1}{4}\partial_{*}j_{2}+B_{M}+\alpha
I\Big{)}:\bm{V}_{\sigma}\to 2^{\bm{V}_{\sigma}^{\prime}}$

is maximal monotone. Furthermore, again by a similar computation to Step 1, it
is coercive. Hence we can apply the Minty theorem [3, Corollary 2.2] to find
that it is surjective, i.e., $R((1/4)\partial_{*}j_{g}+B_{M}+\alpha
I)=\bm{V}^{\prime}_{\sigma}$.

Step 3. Let us define $F_{1}:\bm{H}_{\sigma}\to 2^{\bm{H}_{\sigma}}$ and
$F_{2}:\bm{H}_{\sigma}\to 2^{\bm{H}_{\sigma}}$ by

$\displaystyle F_{1}$ $\displaystyle=\frac{3}{4}\partial j_{g},\qquad$
$\displaystyle D(F_{1})=D(\partial
j_{g})\subset\bm{W}\subset\bm{H}^{2}(\Omega),$ $\displaystyle F_{2}$
$\displaystyle=\frac{1}{4}\partial_{*}j_{g}+B_{M}+\alpha I,\qquad$
$\displaystyle
D(F_{2})=\Big{\\{}\bm{v}\in\bm{V}_{\sigma}\,\Big{|}\,\frac{1}{4}\partial_{*}j_{g}(\bm{v})+B_{M}\bm{v}\subset\bm{H}_{\sigma}\Big{\\}}.$

Note that $F_{2}$ is the restriction of $(1/4)\partial_{*}j_{g}+B_{M}+\alpha
I$ to $\bm{H}_{\sigma}$. It is surjective as a result of Step 2, and so is
$F_{2}+I$. Therefore, again by the Minty theorem, it is maximal monotone (cf.
[8, Exemple 2.3.7]).

Step 4. It remains to show the maximal monotonicity of $F_{1}+F_{2}$ with the
domain $D(\partial j_{g})$. Recall that the minimal section of $F_{2}$ is
given by

$F_{2}^{\circ}\bm{v}:=\left\\{\bm{v}^{*}_{\rm min}\in
F_{2}\bm{v}\,\middle|\,\|\bm{v}^{*}_{\rm
min}\|_{\bm{H}_{\sigma}}=\min_{\bm{v}^{*}\in
F_{2}\bm{v}}\|\bm{v}^{*}\|_{\bm{H}_{\sigma}}\right\\}.$

Let $\bm{u}\in D(\partial j_{g})\subset\bm{W}$ and
$\bm{f}\in\partial_{*}j_{g}(\bm{u})=\partial j_{g}(\bm{u})$ be arbitrary.
Then, from (4.5) and (4.10) we obtain

$\displaystyle\|F_{2}^{\circ}\bm{u}\|_{\bm{H}_{\sigma}}$
$\displaystyle\leq\frac{1}{4}\|\bm{f}\|_{\bm{H}_{\sigma}}+\|B_{M}\bm{u}\|_{\bm{H}_{\sigma}}+\alpha\|\bm{u}\|_{\bm{H}_{\sigma}}\leq\frac{1}{4}\|\bm{f}\|_{\bm{H}_{\sigma}}+C_{\rm
L}M^{\frac{3}{2}}\|\bm{u}\|_{\bm{H}^{2}(\Omega)}^{\frac{1}{2}}+\alpha\|\bm{u}\|_{\bm{H}_{\sigma}}$
$\displaystyle\leq\frac{1}{4}\|\bm{f}\|_{\bm{H}_{\sigma}}+C_{\rm
L}M^{\frac{3}{2}}C_{\rm
K}^{\frac{1}{2}}\|\bm{f}\|_{\bm{H}_{\sigma}}^{\frac{1}{2}}+C_{\rm
L}M^{\frac{3}{2}}C_{\rm
K}^{\frac{1}{2}}\|g\|_{H^{1}(\Omega)}^{\frac{1}{2}}+\alpha\|\bm{u}\|_{\bm{H}_{\sigma}}$
$\displaystyle\leq\frac{1}{2}\|\bm{f}\|_{\bm{H}_{\sigma}}+C_{\rm
L}^{2}M^{3}C_{\rm K}+C_{\rm L}M^{\frac{3}{2}}C_{\rm
K}^{\frac{1}{2}}\|g\|_{H^{1}(\Omega)}^{\frac{1}{2}}+\alpha\|\bm{u}\|_{\bm{H}_{\sigma}},$

which implies, with some constant $C=C(M,C_{\rm L},C_{\rm K},\alpha,g)>0$,

$\|F_{2}^{\circ}\bm{u}\|_{\bm{H}_{\sigma}}\leq\frac{2}{3}\|F_{1}^{\circ}\bm{u}\|_{\bm{H}_{\sigma}}+C(\|\bm{u}\|_{\bm{H}_{\sigma}}+1).$

Therefore, we conclude from a perturbation result for $m$-accretive operators
(see [3, Proposition 3.9] or [8, Corollaire 2.6]) that $F_{1}+F_{2}$ is
maximal monotone in $\bm{H}_{\sigma}$, especially, $D(F_{1}+F_{2})=D(\partial
j_{g})$. ∎

###  4.2. Discretization in time and a priori estimates

To show Theorems 4.1 and 4.2, we construct an approximate solution of (4.1) by
the method of discretization in time. For this purpose, fix $M>0$, choose a
division number $N\in\mathbb{N}$ of the time interval $[0,T]$ to be
sufficiently large (at least $N/T\geq 4\alpha_{M}$), and define the time
increment $h_{N}=T/N$. We introduce discretizations of $\bm{f}\in
L^{2}(0,T;\bm{L}^{2}(\Omega))$ and $g\in W^{1,1}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;H^{1}(\Omega))$ by

$\bm{f}_{k}=\frac{1}{h_{N}}\int_{(k-1)h_{N}}^{kh_{N}}\bm{f}(s)\,ds\in\bm{L}^{2}(\Omega)\quad\text{and}\quad
g_{k}=\frac{1}{h_{N}}\int_{(k-1)h_{N}}^{kh_{N}}g(s)\,ds\in
H^{1}(\Omega)\qquad(k=1,\dots,N),$

together with $g_{0}:=g(0)\in L^{2}(\Omega)$. We also use
$\bm{f}_{0}:=\bm{f}(0)\in\bm{L}^{2}(\Omega)$, which makes sense in the setting
of Theorem 4.1. Then we set $\varphi_{k}:=j_{g_{k}}$, that is,

$\varphi_{k}(\bm{v})=\frac{\nu}{4}\int_{\Omega}|\nabla\bm{v}|^{2}\,d\bm{x}+\int_{\Omega}g_{k}|\nabla\bm{v}|\,d\bm{x},\quad
D(\varphi_{k})=\bm{V}_{\sigma}\qquad(k=1,\dots,N).$

Recalling that $\bm{u}_{0}\in\bm{V}_{\sigma}$ is given as an initial datum, we
consider the following differential inclusions:

(4.12)
$\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}+\partial\varphi_{k}(\bm{u}_{k})+B_{M}\bm{u}_{k}\ni
P\bm{f}_{k}\quad\text{in }\;\bm{H}_{\sigma}\qquad(k=1,\dots,N),$

which is equivalent to

(4.13)
$\Big{(}\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}},\bm{u}_{k}-\bm{v}\Big{)}+\nu(\nabla\bm{u}_{k},\nabla(\bm{u}_{k}-\bm{v}))+(B_{M}\bm{u}_{k},\bm{u}_{k}-\bm{v})+(g_{k},|\nabla\bm{u}_{k}|-|\nabla\bm{v}|)\leq(\bm{f}_{k},\bm{u}_{k}-\bm{v})\quad\forall\bm{v}\in\bm{V}_{\sigma}.$

Since $\partial\varphi_{k}+B_{M}+(1/h_{N})I:\bm{H}_{\sigma}\to
2^{\bm{H}_{\sigma}}$ is maximal monotone by Lemma 4.1, there exists
$\bm{u}_{k}\in D(\partial\varphi_{k})\subset\bm{W}\subset\bm{H}^{2}(\Omega)$
satisfying (4.12) for all $k=1,\dots,N$.

######  Lemma 4.2.

Let $\\{\bm{u}_{k}\\}_{k=1}^{N}$ be defined as above. Then, for
$m=1,2,\ldots,N$ we have

$\displaystyle\|\bm{u}_{m}\|_{\bm{H}_{\sigma}}^{2}+\sum_{k=1}^{m}\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}+\nu\sum_{k=1}^{m}\|\bm{u}_{k}\|_{\bm{V}_{\sigma}}^{2}h_{N}+\sum_{k=1}^{m}(g_{k},|\nabla\bm{u}_{k}|)h_{N}\leq
C(\|\bm{u}_{0}\|_{\bm{H}_{\sigma}}^{2}+\|\bm{f}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}),$

where the constant $C=C(\Omega,\nu)$ is independent of $m$, $N$, and $M$.

###### Proof.

Taking $\bm{v}=\bm{0}$ in (4.13), using the relations (recall (2.2) as well)

$\displaystyle\left(\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}},\bm{u}_{k}\right)=\frac{1}{2h_{N}}(\|\bm{u}_{k}\|_{\bm{H}_{\sigma}}^{2}-\|\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}+\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}),$
$\displaystyle(B_{M}\bm{u}_{k},\bm{u}_{k})=0,\quad(\bm{f}_{k-1},\bm{u}_{k})\leq\frac{\nu}{2}\|\nabla\bm{u}_{k}\|_{\bm{L}^{2}(\Omega)}^{2}+C\|\bm{f}_{k-1}\|_{\bm{L}^{2}(\Omega)}^{2},$

and adding the resulting inequalities for $k=1,\dots,m$, we obtain the desired
estimate. ∎

Next we obtain an $\ell^{\infty}(0,T;\bm{H}^{1}(\Omega))$-a priori estimate
for $\\{\bm{u}_{k}\\}_{k=1}^{N}$. To this end we rewrite (4.12) into the
equation

(4.14)
$\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}+\bm{u}_{k}^{*}+B_{M}\bm{u}_{k}=P\bm{f}_{k}\quad\text{with
some }\bm{u}_{k}^{*}\in\partial\varphi_{k}(\bm{u}_{k})\qquad(k=1,\dots,N).$

######  Lemma 4.3.

For $m=1,\dots,N$ we have

(4.15)
$\displaystyle\|\nabla\bm{u}_{m}\|_{\bm{L}^{2}(\Omega)}^{2}+(g_{m},|\nabla\bm{u}_{m}|)+\sum_{k=1}^{m}\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}+\sum_{k=1}^{m}\|\bm{u}_{m}\|_{\bm{H}^{2}(\Omega)}^{2}h_{N}+\sum_{k=1}^{m}\left\|\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\right\|_{\bm{H}_{\sigma}}^{2}h_{N}$
$\displaystyle\qquad\leq
C_{M}(\|\nabla\bm{u}_{0}\|_{\bm{L}^{2}(\Omega)}^{2}+\|\bm{f}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}+\|g\|_{L^{2}(0,T;H^{1}(\Omega))}^{2}+\|g\|_{W^{1,1}(0,T;L^{2}(\Omega))}^{2}),$

where the constant $C_{M}=C(\Omega,\nu,T,M)$ is independent of $m$ and $N$.

###### Proof.

By the definition of a subdifferential,
$\varphi_{k}(\bm{u}_{k})+(\bm{u}_{k-1}-\bm{u}_{k},\bm{u}_{k}^{*})\leq\varphi_{k}(\bm{u}_{k-1})$.
Substituting (4.14) gives

$\varphi_{k}(\bm{u}_{k})+\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}+(B_{M}\bm{u}_{k}-\bm{f}_{k},\bm{u}_{k}^{*})h_{N}\leq\varphi_{k}(\bm{u}_{k-1}),$

so that, for $k=1,\dots,N$,

(4.16)
$\displaystyle\frac{\nu}{2}(\|\nabla\bm{u}_{k}\|_{L^{2}(\Omega)}^{2}-\|\nabla\bm{u}_{k-1}\|_{L^{2}(\Omega)}^{2})+(g_{k},|\nabla\bm{u}_{k}|)-(g_{k-1},|\nabla\bm{u}_{k-1}|)+\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}$
$\displaystyle\leq$
$\displaystyle(\bm{f}_{k}-B_{M}\bm{u}_{k},\bm{u}_{k}^{*})h_{N}+(g_{k}-g_{k-1},|\nabla\bm{u}_{k-1}|)$
$\displaystyle\leq$
$\displaystyle\frac{1}{4}\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}+\|\bm{f}\|_{L^{2}((k-1)h_{N},kh_{N};\bm{L}^{2}(\Omega))}^{2}-(B_{M}\bm{u}_{k},\bm{u}_{k}^{*})h_{N}+(g_{k}-g_{k-1},|\nabla\bm{u}_{k-1}|),$

where we have used
$\|\bm{f}_{k}\|_{\bm{L}^{2}(\Omega)}\leq(1/\sqrt{h_{N}})\|\bm{f}\|_{L^{2}((k-1)h_{N},kh_{N};\bm{L}^{2}(\Omega))}$.
Since $\bm{u}_{k}^{*}\in\partial\varphi_{k}(\bm{u}_{k})$ implies
$\|\bm{u}_{k}\|_{\bm{H}^{2}(\Omega)}\leq
C(\|\bm{u}_{k}^{*}\|_{\bm{H}_{\sigma}}+\|g_{k}\|_{H^{1}(\Omega)})$, cf.
(4.10), we find from (4.5) that

$\displaystyle|(B_{M}\bm{u}_{k},\bm{u}_{k}^{*})|$ $\displaystyle\leq
C_{M}\|\bm{u}_{k}\|_{\bm{H}^{2}(\Omega)}^{1/2}\|\bm{u}_{k}^{*}\|_{\bm{H}_{\sigma}}\leq
C_{M}(\|\bm{u}_{k}^{*}\|_{\bm{H}_{\sigma}}^{3/2}+\|g_{k}\|_{H^{1}(\Omega)}^{1/2}\|\bm{u}_{k}^{*}\|_{\bm{H}_{\sigma}})$
$\displaystyle\leq\frac{1}{4}\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}+C_{M}\Big{(}\frac{1}{\sqrt{h_{N}}}\|g\|_{L^{2}((k-1)h_{N},kh_{N};H^{1}(\Omega))}+1\Big{)},$

where we have used
$\|g_{k}\|_{H^{1}(\Omega)}\leq(1/\sqrt{h_{N}})\|g\|_{L^{2}((k-1)h_{N},kh_{N};H^{1}(\Omega))}$.
Finally, by direct computation one has

$\displaystyle(g_{k}-g_{k-1},|\nabla\bm{u}_{k-1}|)$
$\displaystyle=\frac{1}{h_{N}}\int_{(k-1)h_{N}}^{kh_{N}}\int_{\min\\{0,s-h_{N}\\}}^{s}(g^{\prime}(r),|\nabla\bm{u}_{k-1}|)\,dr\,ds$
$\displaystyle\leq\|g^{\prime}\|_{L^{1}(\min\\{(k-2)h_{N},0\\},kh_{N};L^{2}(\Omega))}\|\nabla\bm{u}_{k-1}\|_{\bm{L}^{2}(\Omega)}.$

Substituting the two estimates above into (4.16) and absorbing
$\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}$ to the left-hand side, we see
that

$\displaystyle\frac{\nu}{2}(\|\nabla\bm{u}_{k}\|_{L^{2}(\Omega)}^{2}-\|\nabla\bm{u}_{k-1}\|_{L^{2}(\Omega)}^{2})+(g_{k},|\nabla\bm{u}_{k}|)-(g_{k-1},|\nabla\bm{u}_{k-1}|)+\frac{1}{2}\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}$
$\displaystyle\leq\;$ $\displaystyle
C_{M}(\|\bm{f}\|_{L^{2}((k-1)h_{N},kh_{N};\bm{L}^{2}(\Omega))}^{2}+\|g\|_{L^{2}((k-1)h_{N},kh_{N};H^{1}(\Omega))}h_{N}^{1/2}+h_{N})$
$\displaystyle\qquad+\|g^{\prime}\|_{L^{1}(\min\\{(k-2)h_{N},0\\},kh_{N};L^{2}(\Omega))}\|\nabla\bm{u}_{k-1}\|_{\bm{L}^{2}(\Omega)}.$

Now, if
$m:=\operatorname{argmax}_{k=0,\dots,N}\|\nabla\bm{u}_{k}\|_{\bm{L}^{2}(\Omega)}\geq
1$, then we add the above estimate for $k=1,\dots,m$ to obtain

(4.17)
$\displaystyle\frac{\nu}{2}(\|\nabla\bm{u}_{m}\|_{L^{2}(\Omega)}^{2}-\|\nabla\bm{u}_{0}\|_{L^{2}(\Omega)}^{2})+(g_{m},|\nabla\bm{u}_{m}|)-(g_{0},|\nabla\bm{u}_{0}|)+\frac{1}{2}\sum_{k=1}^{m}\|\bm{u}^{*}_{k}\|_{\bm{H}_{\sigma}}^{2}h_{N}$
$\displaystyle\leq$ $\displaystyle
C_{M}(\|\bm{f}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}+\|g\|_{L^{2}(0,T;H^{1}(\Omega))}^{2}+T)+\|g^{\prime}\|_{L^{1}(0,T;L^{2}(\Omega))}\|\nabla\bm{u}_{m}\|_{\bm{L}^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C_{M}(\|\bm{f}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}+\|g\|_{L^{2}(0,T;H^{1}(\Omega))}^{2}+\|g^{\prime}\|_{L^{1}(0,T;L^{2}(\Omega))}^{2}+T)+\frac{\nu}{4}\|\nabla\bm{u}_{m}\|_{\bm{L}^{2}(\Omega)}^{2},$

which gives us a bound of
$\max_{k=0,\dots,N}\|\nabla\bm{u}_{k}\|_{\bm{L}^{2}(\Omega)}$ independent of
$N$ (note: $\|g_{0}\|_{L^{2}(\Omega)}\leq
C(\Omega,T)\|g\|_{W^{1,1}(0,T;L^{2}(\Omega))}$). A similar computation to
(4.17), in which $m$ is replaced by an arbitrary integer, proves (4.15) for
the first 3 terms on the left-hand side. The fourth and fifth terms can be
addressed by

$\displaystyle\|\bm{u}_{k}\|_{\bm{H}^{2}(\Omega)}^{2}$ $\displaystyle\leq
C\Big{(}\|\bm{u}_{k}^{*}\|_{\bm{H}_{\sigma}}^{2}+\frac{1}{h_{N}}\|g\|_{L^{2}((k-1)h_{N},kh_{N};H^{1}(\Omega))}^{2}\Big{)},$
$\displaystyle\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}$
$\displaystyle=P\bm{f}_{k}-\bm{u}_{k}^{*}-B_{M}\bm{u}_{k},\quad\|B_{M}\bm{u}_{k}\|_{\bm{H}_{\sigma}}^{2}\leq
C_{M}(\|\bm{u}_{k}^{*}\|_{\bm{H}_{\sigma}}+\|g_{k}\|_{H^{1}(\Omega)}),$

which completes the proof of the lemma. ∎

Finally we derive an $H^{1}(0,T;\bm{H}^{1}(\Omega))$-like a priori estimate
for $\\{\bm{u}_{k}\\}_{k=1}^{N}$ to show Theorem 4.1.

######  Lemma 4.4.

Under the assumptions of Theorem 4.1, let
$M\geq\|\bm{u}_{0}\|_{\bm{V}_{\sigma}}$. Then, for $m=1,\dots,N$ we have

$\displaystyle\left\|\frac{\bm{u}_{m}-\bm{u}_{m-1}}{h_{N}}\right\|_{\bm{H}_{\sigma}}^{2}+\sum_{k=1}^{m}\left\|\nabla\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\right\|_{\bm{L}^{2}(\Omega)}^{2}h_{N}$
$\displaystyle\leq\;C_{M}(\|\bm{u}_{0}\|_{\bm{V}_{\sigma}}^{2}+\|\bm{f}\|_{W^{1,1}(0,T;\bm{L}^{2}(\Omega))}^{2}+\|g\|_{H^{1}((0,T)\times\Omega)}^{2}+\|\bm{f}^{*}\|_{\bm{H}_{\sigma}}^{2}),$

where the constant $C_{M}=C(\Omega,\nu,T,M)$ is independent of $m$ and $N$.

###### Proof.

For the functional $j_{2}$ defined in the Step 2 of the proof of Lemma 4.1, we
see that $\partial j_{2}=-(\nu/2)P\Delta$ and that $F:=(1/4)\partial
j_{2}+B_{M}+\alpha_{M}I$ is monotone. In addition, observe that (4.13) is
rewritten as

(4.18)
$\displaystyle\Big{(}\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}},\bm{u}_{k}-\bm{v}\Big{)}+\frac{7\nu}{8}(\nabla\bm{u}_{k},\nabla(\bm{u}_{k}-\bm{v}))+(F\bm{u}_{k},\bm{u}_{k}-\bm{v})+(g_{k},|\nabla\bm{u}_{k}|-|\nabla\bm{v}|)$
$\displaystyle\leq$
$\displaystyle(\bm{f}_{k},\bm{u}_{k}-\bm{v})+\alpha_{M}(\bm{u}_{k},\bm{u}_{k}-\bm{v})\qquad\forall\bm{v}\in\bm{V}_{\sigma},\quad
k=1,\dots,N.$

Taking $\bm{v}=\bm{u}_{k-1}$ in (4.18), taking $\bm{v}=\bm{u}_{k}$ in (4.18)
with $k$ replaced by $k-1$, adding the two inequalities, and applying the
monotonicity of $F$, we obtain for $k=2,\dots,N$

$\displaystyle\frac{\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}-\|\bm{u}_{k-1}-\bm{u}_{k-2}\|_{\bm{H}_{\sigma}}^{2}}{2h_{N}}+\frac{\|\bm{u}_{k}-\bm{u}_{k-2}\|_{\bm{H}_{\sigma}}^{2}}{2h_{N}}+\frac{7\nu}{8}\|\nabla(\bm{u}_{k}-\bm{u}_{k-1})\|_{\bm{L}^{2}(\Omega)}^{2}$
$\displaystyle\leq\;$
$\displaystyle(\bm{f}_{k}-\bm{f}_{k-1},\bm{u}_{k}-\bm{u}_{k-1})+\alpha_{M}\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}-(g_{k}-g_{k-1},|\nabla\bm{u}_{k}|-|\nabla\bm{u}_{k-1}|)$
$\displaystyle\leq\;$
$\displaystyle\|\bm{f}_{k}-\bm{f}_{k-1}\|_{\bm{L}^{2}(\Omega)}\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}+\alpha_{M}\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}+C\|g_{k}-g_{k-1}\|_{L^{2}(\Omega)}^{2}+\frac{3\nu}{8}\|\nabla(\bm{u}_{k}-\bm{u}_{k-1})\|_{\bm{L}^{2}(\Omega)}^{2}.$

Therefore,

(4.19)
$\displaystyle\Big{\|}\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}-\Big{\|}\frac{\bm{u}_{k-1}-\bm{u}_{k-2}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}+\nu\Big{\|}\nabla\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{L}^{2}(\Omega)}^{2}h_{N}$
$\displaystyle\leq$
$\displaystyle\|\bm{f}^{\prime}\|_{L^{1}((k-2)h_{N},kh_{N};\bm{L}^{2}(\Omega))}\Big{\|}\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}+2\alpha_{M}\Big{\|}\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}h_{N}+2C\|g^{\prime}\|_{L^{2}((k-2)h_{N},kh_{N};L^{2}(\Omega))}^{2},$

where
$\|\bm{f}_{k}-\bm{f}_{k-1}\|_{\bm{L}^{2}(\Omega)}\leq\|\bm{f}^{\prime}\|_{L^{1}((k-2)h_{N},kh_{N};\bm{L}^{2}(\Omega))}$
and
$\|g_{k}-g_{k-1}\|_{\bm{L}^{2}(\Omega)}\leq\sqrt{h_{N}}\|g^{\prime}\|_{L^{2}((k-2)h_{N},kh_{N};\bm{L}^{2}(\Omega))}$
are used. Now we add (4.19) for $k=2,\dots,m$, argue similarly as in (4.17),
and then substitute (4.15) to arrive at

(4.20)
$\displaystyle\Big{\|}\frac{\bm{u}_{m}-\bm{u}_{m-1}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}+\nu\sum_{k=2}^{m}\Big{\|}\nabla\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{L}^{2}(\Omega)}^{2}h_{N}$
$\displaystyle\leq$
$\displaystyle\Big{\|}\frac{\bm{u}_{1}-\bm{u}_{0}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}+C_{M}(\|\bm{u}_{0}\|_{\bm{V}_{\sigma}}^{2}+\|\bm{f}\|_{W^{1,1}(0,T;\bm{L}^{2}(\Omega))}^{2}+\|g\|_{H^{1}((0,T)\times\Omega)}^{2})\qquad(m=2,\dots,N).$

On the other hand, recall that (4.4) holds, in which we have $B=B_{M}$ since
$M\geq\|\bm{u}_{0}\|_{\bm{V}_{\sigma}}$. Then it follows that

$\frac{7\nu}{8}(\nabla\bm{u}_{0},\nabla(\bm{u}_{0}-\bm{v}))+(F\bm{u}_{0},\bm{u}_{0}-\bm{v})+(g_{0},|\nabla\bm{u}_{0}|-|\nabla\bm{v}|)\leq(\bm{f}_{0}+\bm{f}^{*}+\alpha_{M}\bm{u}_{0},\bm{u}_{0}-\bm{v})\quad\forall\bm{v}\in\bm{V}_{\sigma}.$

Taking $\bm{v}=\bm{u}_{1}$ above, taking $\bm{v}=\bm{u}_{0}$ in (4.18) for
$k=1$, adding the two inequalities, and dividing the both sides by $h_{N}$, we
derive (recall $N$ is chosen so large that $h_{N}\alpha_{M}\leq 1/4$)

(4.21)
$\displaystyle\Big{\|}\frac{\bm{u}_{1}-\bm{u}_{0}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}+\frac{7\nu}{8}\Big{\|}\nabla\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{L}^{2}(\Omega)}^{2}h_{N}$
$\displaystyle\leq$
$\displaystyle\Big{(}\bm{f}_{1}-\bm{f}_{0}-\bm{f}^{*},\frac{\bm{u}_{1}-\bm{u}_{0}}{h_{N}}\Big{)}+h_{N}\alpha_{M}\Big{\|}\frac{\bm{u}_{1}-\bm{u}_{0}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}-\frac{1}{h_{N}}(g_{1}-g_{0},|\nabla\bm{u}_{1}|-|\nabla\bm{u}_{0}|)$
$\displaystyle\leq$ $\displaystyle
C(\|\bm{f}^{\prime}\|_{L^{1}(0,h_{N};\bm{L}^{2}(\Omega))}^{2}+\|\bm{f}^{*}\|_{\bm{H}_{\sigma}}^{2})+\frac{1}{2}\Big{\|}\frac{\bm{u}_{1}-\bm{u}_{0}}{h_{N}}\Big{\|}_{\bm{H}_{\sigma}}^{2}+C\|g^{\prime}\|_{L^{2}(0,h_{N};\bm{L}^{2}(\Omega))}^{2}+\frac{3\nu}{8}\Big{\|}\nabla\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{L}^{2}(\Omega)}^{2}h_{N}.$

Combining (4.20) and (4.21) concludes the desired estimate. ∎

###  4.3. Limiting procedure as $N\to\infty$

We continue to deal with $\\{\bm{u}_{k}\\}_{k=0}^{N}$ constructed in the
previous subsection. Let us introduce a piecewise-linear interpolation
$\hat{\bm{u}}_{N}:[0,T]\to\bm{H}_{\sigma}$ and piecewise-constant
interpolations $\bar{\bm{u}}_{N}:(0,T]\to\bm{H}_{\sigma}$ and
$\bar{\bm{f}}_{N}:(0,T]\to\bm{H}_{\sigma}$ as follows: for $k=1,\dots,N$,

$\displaystyle\hat{\bm{u}}_{N}(t)$
$\displaystyle=\frac{kh_{N}-t}{h_{N}}\bm{u}_{k-1}+\frac{t-(k-1)h_{N}}{h_{N}}\bm{u}_{k}\qquad$
$\displaystyle{\rm for~{}}t\in[(k-1)h_{N},kh_{N}],$
$\displaystyle\bar{\bm{u}}_{N}(t)$
$\displaystyle=\bm{u}_{k},\quad\bar{\bm{f}}_{N}(t)=\bm{f}_{k}\qquad$
$\displaystyle{\rm for~{}}t\in((k-1)h_{N},kh_{N}].$

Define $\Phi_{N},\Phi:L^{2}(0,T;\bm{V}_{\sigma})\to\mathbb{R}$ by

$\Phi_{N}(\bm{\upsilon})=\sum_{k=1}^{N}\int_{(k-1)h_{N}}^{kh_{N}}\varphi_{k}(\bm{\upsilon}(t))\,dt,\qquad\Phi(\bm{\upsilon})=\int_{0}^{T}\varphi^{t}(\bm{\upsilon}(t))\,dt,$

where $\varphi^{t}:=j_{g(t)}$, that is,
$\varphi^{t}(\bm{v})=(\nu/2)\|\nabla\bm{v}\|_{\bm{L}^{2}(\Omega)}^{2}+(g(t),|\nabla\bm{v}|)$
for each $t\in[0,T]$. By direct computation, we see that
$\Phi_{N}(\bm{\upsilon})\to\Phi(\bm{\upsilon})$ as $N\to\infty$ for
$\bm{\upsilon}\in L^{2}(0,T;\bm{V}_{\sigma})$ and that
$|\Phi_{N}(\bm{\upsilon}_{1})-\Phi_{N}(\bm{\upsilon}_{2})|\leq\|g\|_{L^{2}(0,T;L^{2}(\Omega))}\|\bm{\upsilon}_{1}-\bm{\upsilon}_{2}\|_{L^{2}(0,T;\bm{V}_{\sigma})}$.

Now we observe that (4.13) can be reformulated as the following space-time
variational inequality:

(4.22)
$\displaystyle\int_{0}^{T}(\hat{\bm{u}}_{N}^{\prime}(t),\bm{\upsilon}(t)-\bar{\bm{u}}_{N}(t))dt+\int_{0}^{T}(B_{M}\bar{\bm{u}}_{N}(t),\bm{\upsilon}(t)-\bar{\bm{u}}_{N}(t))dt+\Phi_{N}(\bm{\upsilon})-\Phi_{N}(\bar{\bm{v}}_{N})\geq\int_{0}^{T}(\bar{\bm{f}}_{N}(t),\bm{\upsilon}(t)-\bar{\bm{u}}_{N}(t))dt$

for all $\bm{\upsilon}\in L^{2}(0,T;\bm{V}_{\sigma})$. From this we construct
a solution of (4.1) in which $B$ is replaced with its truncation $B_{M}$. The
following result is an extension of [3, Proposition 5.15] to a time-dependent
nonlinear multivalued operator $\partial\varphi^{t}$.

######  Proposition 4.1.

Under the assumptions in Theorem 4.1, there exists at least one $\bm{u}\in
W^{1,\infty}(0,T;\bm{H}_{\sigma})\cap H^{1}(0,T;\bm{V}_{\sigma})\cap
L^{\infty}(0,T;\bm{W})$ such that $\bm{u}(0)=\bm{u}_{0}$ and

(4.23)
$(\bm{u}^{\prime}(t),\bm{v}-\bm{u}(t))-\nu(\Delta\bm{u}(t),\bm{v}-\bm{u}(t))+(B_{M}\bm{u}(t),\bm{v}-\bm{u}(t))+(g(t),|\nabla\bm{v}|-|\nabla\bm{u}(t)|)\geq(\bm{f}(t),\bm{v}-\bm{u}(t))$

for all $\bm{v}\in\bm{V}_{\sigma}$ and a.a. $t\in(0,T)$.

###### Proof.

Collecting the estimates in Lemmas 4.2, 4.3, and 4.4, we deduce that, for
sufficiently large $N$,

(4.24)
$\displaystyle\|\hat{\bm{u}}_{N}^{\prime}\|_{L^{\infty}(0,T;\bm{H}_{\sigma})}+\|\hat{\bm{u}}_{N}^{\prime}\|_{L^{2}(0,T;\bm{V}_{\sigma})}+\|\hat{\bm{u}}_{N}\|_{L^{\infty}(0,T;\bm{V}_{\sigma})}+\|\hat{\bm{u}}_{N}\|_{L^{2}(0,T;\bm{H}^{2}(\Omega))}$
$\displaystyle\leq C_{M},$ (4.25)
$\displaystyle\|\bar{\bm{u}}_{N}\|_{L^{\infty}(0,T;\bm{V}_{\sigma})}+\|\bar{\bm{u}}_{N}\|_{L^{2}(0,T;\bm{H}^{2}(\Omega))}$
$\displaystyle\leq C_{M},$ (4.26)
$\displaystyle\|\hat{\bm{u}}_{N}-\bar{\bm{u}}_{N}\|_{L^{2}(0,T;\bm{V}_{\sigma})}^{2}=\frac{h_{N}}{3}\sum_{k=1}^{N}\Big{\|}\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\Big{\|}_{\bm{V}_{\sigma}}^{2}h_{N}\leq\frac{h_{N}}{3}C_{M}\to
0\quad\text{as $N\to\infty$},$

where $C_{M}=C(\Omega,\nu,T,\bm{f},g,\bm{u}_{0},\bm{f}^{*},M)$ is independent
of $N$. As a result of the weak compactness and of the Aubin–Lions compactness
theorem [33, Corollary 4], we can find some $\bm{u}\in
W^{1,\infty}(0,T;\bm{H}_{\sigma})\cap H^{1}(0,T;\bm{V}_{\sigma})\cap
L^{2}(0,T;\bm{W})$, $\bar{\bm{u}}\in L^{\infty}(0,T;\bm{V}_{\sigma})\cap
L^{2}(0,T;\bm{W})$, and subsequence $\\{N_{n}\\}_{n\in\mathbb{N}}$ such that

$\displaystyle\hat{\bm{u}}_{N_{n}}^{\prime}\to\bm{u}^{\prime}$
$\displaystyle\quad{\rm
weakly~{}star~{}in~{}}L^{\infty}(0,T;\bm{H}_{\sigma})\cap
L^{2}(0,T;\bm{V}_{\sigma}),$ $\displaystyle\hat{\bm{u}}_{N_{n}}\to\bm{u}$
$\displaystyle\quad{\rm
weakly~{}star~{}in~{}}L^{\infty}(0,T;\bm{V}_{\sigma})\cap L^{2}(0,T;\bm{W}),$
(4.27) $\displaystyle\hat{\bm{u}}_{N_{n}}\to\bm{u}$ $\displaystyle\quad{\rm
strongly~{}in~{}}C([0,T];\bm{H}_{\sigma})\cap L^{2}(0,T;\bm{V}_{\sigma}),$
$\displaystyle\bar{\bm{u}}_{N_{n}}\to\bar{\bm{u}}$ $\displaystyle\quad{\rm
weakly~{}star~{}in~{}}L^{\infty}(0,T;\bm{V}_{\sigma})\cap L^{2}(0,T;\bm{W}),$

as $n\to\infty$. By virtue of (4.26) and (4.27), one has

$\|\bar{\bm{u}}_{N_{n}}-\bm{u}\|_{L^{2}(0,T;\bm{V}_{\sigma})}\leq\|\bar{\bm{u}}_{N_{n}}-\hat{\bm{u}}_{N_{n}}\|_{L^{2}(0,T;\bm{V}_{\sigma})}+\|\hat{\bm{u}}_{N_{n}}-\bm{u}\|_{L^{2}(0,T;\bm{V}_{\sigma})}\to
0\quad\text{as $n\to\infty$},$

so that $\bar{\bm{u}}_{N_{n}}\to\bm{u}$ strongly in
$L^{2}(0,T;\bm{V}_{\sigma})$ and $\bm{u}=\bar{\bm{u}}$. This implies that
$B_{M}\bar{\bm{u}}_{N_{n}}\to B_{M}\bm{u}$ strongly in
$L^{2}(0,T;\bm{V}_{\sigma}^{\prime})$ in view of (4.8) and that

$|\Phi_{N_{n}}(\bar{\bm{u}}_{N_{n}})-\Phi(\bm{u})|\leq|\Phi_{N_{n}}(\bar{\bm{u}}_{N_{n}})-\Phi_{N_{n}}(\bm{u})|+|\Phi_{N_{n}}(\bm{u})-\Phi(\bm{u})|\\\
\to 0\quad\text{as $n\to\infty$}.$

Therefore, setting $N=N_{n}$ and letting $n\to+\infty$ in (4.22), we obtain,
for all $\bm{\upsilon}\in L^{2}(0,T;\bm{V}_{\sigma})$,

$\displaystyle\int_{0}^{T}(\bm{u}^{\prime}(t),\bm{\upsilon}(t)-\bm{u}(t))dt+\int_{0}^{T}(B_{M}\bm{u}(t),\bm{\upsilon}(t)-\bm{u}(t))dt+\Phi(\bm{\upsilon})-\Phi(\bm{u})\geq\int_{0}^{T}(\bm{f}(t),\bm{\upsilon}(t)-\bm{u}(t))dt.$

In particular, $\bm{u}(t)\in D(\partial\varphi^{t})\subset\bm{W}$ for a.a.
$t\in(0,T)$ and the following evolution equation holds:

(4.28)
$\displaystyle\begin{cases}\bm{u}^{\prime}(t)+\partial\varphi^{t}(\bm{u}(t))+B_{M}\bm{u}(t)\ni
P\bm{f}(t)\quad\mbox{in~{}}\bm{H}_{\sigma},\quad\mbox{for~{}a.a.~{}}t\in(0,T),\\\
\bm{u}(0)=\bm{u}_{0}\quad\mbox{in~{}}\bm{H}_{\sigma}.\end{cases}$

This $\bm{u}$ satisfies the equivalent variational formulation (4.23).
Finally, using (4.5), (4.28), and (4.10), we have

$\displaystyle\|\bm{u}(t)\|_{\bm{H}^{2}(\Omega)}$ $\displaystyle\leq
C(\|P\bm{f}(t)-\bm{u}^{\prime}(t)-B_{M}\bm{u}(t)\|_{\bm{H}_{\sigma}}+\|g(t)\|_{H^{1}(\Omega)})$
$\displaystyle\leq
C_{M}(\|\bm{f}(t)\|_{\bm{L}^{2}(\Omega)}+\|\bm{u}^{\prime}(t)\|_{\bm{H}_{\sigma}}+\|g(t)\|_{H^{1}(\Omega)}+1)+\frac{1}{2}\|\bm{u}(t)\|_{H^{2}(\Omega)}\quad\text{for
a.a.\ $t\in(0,T)$},$

which yields $\bm{u}\in L^{\infty}(0,T;\bm{W})$. This completes the proof of
the proposition. ∎

###  4.4. Proof of Theorem 4.1

Let $\bm{u}$ be a solution obtained by Proposition 4.1, which means the
existence of some $\bm{u}^{*}\in L^{2}(0,T;\bm{H}_{\sigma})$ such that

$\bm{u}^{\prime}(t)+\bm{u}^{*}(t)+B_{M}\bm{u}(t)=P\bm{f}(t)\text{ in
}\bm{H}_{\sigma}\quad\text{and}\quad\bm{u}^{*}(t)\in\partial\varphi^{t}(\bm{u}(t))\text{
in }\bm{H}_{\sigma}\quad\text{for a.a.\ }t\in(0,T).$

Let us recall a chain rule for a time-dependent convex functional, which is
essentially due to [1, Lemma 2.10], [22, Theorem 2.1.1], [29, Proposition
3.4], [30, Lemma 2.4], and [31, Lemma 3.6].

######  Lemma 4.5.

The mapping $t\mapsto\varphi^{t}(\bm{u}(t))$ is differentiable a.e. on
$(0,T)$, and it satisfies

(4.29)
$\frac{d}{dt}\varphi^{t}(\bm{u}(t))-(\bm{u}^{*}(t),\bm{u}^{\prime}(t))\leq\|g^{\prime}(t)\|_{L^{2}(\Omega)}\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}\quad\mbox{for
a.a.\ }t\in(0,T).$

###### Proof.

An argument based on the Moreau–Yosida regularization (see e.g. [22, p. 29])
shows that $t\mapsto\varphi^{t}(\bm{u}(t))$ is absolutely continuous on
$[0,T]$, which implies its differentiability. Since $\bm{u}^{\prime}\in
L^{2}(0,T;\bm{H}_{\sigma})$, we have

$\frac{\bm{u}(t+h)-\bm{u}(t)}{h}=\frac{1}{h}\int_{t}^{t+h}\bm{u}^{\prime}(\tau)d\tau\to\bm{u}^{\prime}(t)\quad{\rm
in~{}}\bm{H}_{\sigma}\text{ as $h\to 0$, for a.a.\ $t\in(0,T)$}.$

Next, since $\bm{u}^{*}\in L^{2}(0,T;\bm{H}_{\sigma})$, for a.a. $t\in(0,T)$
there exists $\\{h_{n}\\}$, $h_{n}\searrow 0$ such that

$\bm{u}^{*}(t+h_{n})\in\partial\varphi^{t+h_{n}}(\bm{u}(t+h_{n})),\quad\bm{u}^{*}(t+h_{n})\to\bm{u}^{*}(t)\quad{\rm
in~{}}\bm{H}_{\sigma}\quad\text{as $n\to\infty$}.$

By the definition of the subdifferential of $\varphi^{t+h_{n}}$,

$\displaystyle\varphi^{t+h_{n}}(\bm{u}(t+h_{n}))-\varphi^{t}(\bm{u}(t))-(\bm{u}(t+h_{n})-\bm{u}(t),\bm{u}^{*}(t+h_{n}))$
$\displaystyle\leq\varphi^{t+h_{n}}(\bm{u}(t))-\varphi^{t}(\bm{u}(t))$
$\displaystyle\leq\int_{t}^{t+h_{n}}\|g^{\prime}(\tau)\|_{L^{2}(\Omega)}d\tau\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)},$

which proves (4.29) after dividing by $h_{n}$ and taking $n\to\infty$. ∎

Let us start the proof of the existence part in Theorem 4.1. For this purpose,
it suffices to establish an upper bound of $\|\bm{u}(t)\|_{\bm{V}_{\sigma}}$
that is uniform in $t$ and independent of $M$. By Lemma 4.2, we have
$\|\bm{u}\|_{L^{\infty}(0,T;\bm{H}_{\sigma})\cap
L^{2}(0,T;\bm{V}_{\sigma})}\leq C(\Omega,\nu,\bm{u}_{0},\bm{f})$. Next we
apply Lemma 4.5 to obtain

$\frac{d}{dt}\varphi^{t}(\bm{u}(t))+\|\bm{u}^{*}(t)\|_{\bm{H}_{\sigma}}^{2}\leq(\bm{u}^{*}(t),\bm{f}(t)-B_{M}\bm{u}(t))+\|g^{\prime}(t)\|_{L^{2}(\Omega)}\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}\quad\text{for
a.a.\ $t\in(0,T)$,}$

where we observe from (4.7), (2.2), (4.10), and Young’s inequality that

$\displaystyle(\bm{u}^{*}(t),\bm{f}(t)-B_{M}\bm{u}(t))$
$\displaystyle\leq\|\bm{u}^{*}(t)\|_{\bm{H}_{\sigma}}\big{[}\|\bm{f}(t)\|_{\bm{L}^{2}(\Omega)}+C\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}^{3/2}(\|\bm{u}^{*}(t)\|_{\bm{L}^{2}(\Omega)}^{1/2}+\|g(t)\|_{H^{1}(\Omega)}^{1/2})\big{]}$
$\displaystyle\leq\frac{1}{2}\|\bm{u}^{*}(t)\|_{\bm{H}_{\sigma}}^{2}+C(\|\bm{f}(t)\|_{\bm{L}^{2}(\Omega)}^{2}+\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}^{6}+\|g(t)\|_{H^{1}(\Omega)}^{2}).$

Consequently, for a.a. $t\in(0,T)$,

$\frac{d}{ds}\varphi^{s}(\bm{u}(t))\leq
C(\|\bm{f}(t)\|_{\bm{L}^{2}(\Omega)}^{2}+\|g(t)\|_{H^{1}(\Omega)}^{2})+\|g^{\prime}(t)\|_{L^{2}(\Omega)}\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}+C\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}^{6}.$

If we set $y(t):=\varphi^{t}(\bm{u}(t))+1\geq 1$, this yields

(4.30)
$y^{\prime}(t)\leq\underbrace{C(\|\bm{f}(t)\|_{\bm{L}^{2}(\Omega)}^{2}+\|g(t)\|_{H^{1}(\Omega)}^{2}+\|g^{\prime}(t)\|_{L^{2}(\Omega)}+\|\nabla\bm{u}(t)\|_{\bm{L}^{2}(\Omega)}^{2})}_{=:a(t)}y(t)^{2},\quad
C=C(\Omega,\nu),$

which implies, for suitably small $t>0$,

$y(t)\leq\frac{y(0)}{1-y(0)\int_{0}^{t}a(s)\,ds}.$

It is possible to choose $T^{*}=C(\Omega,\nu,\bm{f},g,\bm{u}_{0})$ in such a
way that $y(0)\int_{0}^{T^{*}}a(t)\,dt\leq 1/2$, which gives us a bound
$\|\bm{u}\|_{L^{\infty}(0,T^{*};\bm{V}_{\sigma})}\leq
C(\Omega,\nu,\bm{f},g,\bm{u}_{0})$ independently of $M$. We remark that in
case $d=2$ we need not restrict $T$ to $T^{*}$. In fact, use (4.6) in place of
(4.7) to arrive at a similar estimate to (4.30), where the power of $y(t)$ in
the right-hand side is replaced by 1. Then Gronwall’s inequality enables us to
bound $\bm{u}\in L^{\infty}(0,T;\bm{V}_{\sigma})$ globally in $(0,T)$.

Now fix $M$ to be greater than the $L^{\infty}(0,T^{*};\bm{V}_{\sigma})$-bound
above, and consider $\bm{u}$ constructed in Proposition 4.1. Since
$B_{M}\bm{u}(t)=B\bm{u}(t)$ for a.a. $t\in(0,T^{*})$, this $\bm{u}$ indeed
solves (4.1) on $(0,T^{*})$, which proves the existence part.

The uniqueness part of Theorem 4.1 is easy. In fact, let $\bm{v}$ and $\bm{u}$
be two solutions of (4.1). Then, we take $\bm{v}(t)$ (resp. $\bm{u}(t)$) in
the variational inequality that $\bm{u}(t)$ (resp. $\bm{v}(t)$) satisfies, and
add the resulting inequalities to get

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\bm{v}(t)-\bm{u}(t)\|_{\bm{H}_{\sigma}}^{2}+\nu\|\nabla(\bm{v}(t)-\bm{u}(t))\|^{2}_{\bm{L}^{2}(\Omega)}$
$\displaystyle\leq|b(\bm{v}(t),\bm{v}(t),\bm{v}(t)-\bm{u}(t))-b(\bm{u}(t),\bm{u}(t),\bm{v}(t)-\bm{u}(t))|$
$\displaystyle=|b(\bm{v}(t)-\bm{u}(t),\bm{v}(t)-\bm{u}(t),\bm{u}(t))|.$

Then a standard argument, e.g. [34, Chapter 3], leads to $\bm{v}=\bm{u}$. This
completes the proof of Theorem 4.1.

###  4.5. Proof of Theorem 4.2

We exploit the same strategy as in Theorem 4.1, however without Lemma 4.4 this
time (Lemmas 4.2 and 4.3 remain to hold in the setting of Theorem 4.2). The
difficulty is two folded: 1) the strong convergence of $\hat{\bm{u}}_{N}$ in
$L^{2}(0,T;\bm{V}_{\sigma})$ is not trivial if we do not know the boundedness
of $\hat{\bm{u}}_{N}^{\prime}\in L^{2}(0,T;\bm{V}_{\sigma})$; 2)
$\hat{\bm{u}}_{N}$ may not belong to $L^{2}(0,h_{N};\bm{H}^{2}(\Omega))$
because we only assume $\bm{u}_{0}\in\bm{V}_{\sigma}$.

To address issue 1), we devise an auxiliary estimate for Lemma 4.3.

######  Lemma 4.6.

We have

(4.31)
$\displaystyle\sum_{k=1}^{N}\|\nabla(\bm{u}_{k}-\bm{u}_{k-1})\|_{\bm{L}^{2}(\Omega)}^{2}\leq
C_{M}(\|\nabla\bm{u}_{0}\|_{\bm{L}^{2}(\Omega)}^{2}+\|\bm{f}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}+\|g\|_{L^{2}(0,T;H^{1}(\Omega))}^{2}+\|g\|_{W^{1,1}(0,T;L^{2}(\Omega))}^{2}),$

where the constant $C_{M}=C(\Omega,\nu,T,M)$ is independent of $N$.

###### Proof.

For $k=1,\dots,N$, we take $\bm{v}=\bm{u}_{k-1}$ in (4.13) to get

$\displaystyle\left\|\frac{\bm{u}_{k}-\bm{u}_{k-1}}{h_{N}}\right\|_{\bm{H}_{\sigma}}^{2}h_{N}+\frac{\nu}{2}(\|\nabla\bm{u}_{k}\|_{\bm{L}^{2}(\Omega)}-\|\nabla\bm{u}_{k-1}\|_{\bm{L}^{2}(\Omega)}^{2}+\|\nabla(\bm{u}_{k}-\bm{u}_{k-1})\|_{\bm{L}^{2}(\Omega)}^{2})+(g_{k},|\nabla\bm{u}_{k}|)-(g_{k-1},|\nabla\bm{u}_{k-1}|)$
$\displaystyle\leq\;$
$\displaystyle-(B_{M}\bm{u}_{k}-\bm{f}_{k-1},\bm{u}_{k}-\bm{u}_{k-1})+(g_{k}-g_{k-1},|\nabla\bm{u}_{k-1}|).$

Adding this for $k=1,\dots,N$ and applying the estimates known from Lemma 4.3,
we obtain (4.31). ∎

Next, to resolve issue 2), we introduce another piecewise linear interpolation
of $\\{\bm{u}_{k}\\}$ by

$\bm{w}_{N}(t)=\begin{cases}\bm{u}_{1}&\quad{\rm if~{}}t\in[0,h_{N}],\\\
\hat{\bm{u}}_{N}(t)&\quad{\rm if~{}}t\in[h_{N},T].\end{cases}$

Then we have $\bm{w}_{N}(t)\in\bm{H}^{2}(\Omega)$ for all $t\in[0,T]$ and

$\displaystyle\|\bm{w}_{N}^{\prime}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}$
$\displaystyle\leq\|\hat{\bm{u}}_{N}^{\prime}\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2},$
$\displaystyle\|\bm{w}_{N}\|_{L^{2}(0,T;\bm{H}^{2}(\Omega))}^{2}$
$\displaystyle\leq\frac{h_{N}}{2}\|\bm{u}_{1}\|_{\bm{H}^{2}(\Omega)}^{2}+\|\bar{\bm{u}}_{N}\|_{L^{2}(0,T;\bm{H}^{2}(\Omega))}^{2},$

which implies $\|\bm{w}_{N}\|_{H^{1}(0,T;\bm{H}_{\sigma})\cap
L^{2}(0,T;\bm{W})}\leq C(\Omega,\nu,T,\bm{f},g,\bm{u}_{0},M)$ by Lemma 4.3.
Combining this with the boundedness of $\\{\hat{\bm{u}}_{N}\\}$ and
$\\{\bar{\bm{u}}_{N}\\}$ already known, we can find some $\bm{u}\in
H^{1}(0,T;\bm{H}_{\sigma})\cap L^{\infty}(0,T;\bm{V}_{\sigma})$,
$\bar{\bm{u}}\in L^{\infty}(0,T;\bm{V}_{\sigma})\cap L^{2}(0,T;\bm{W})$,
$\bm{w}\in H^{1}(0,T;\bm{H}_{\sigma})\cap L^{2}(0,T;\bm{W})$ and subsequence
$\\{N_{n}\\}_{n\in\mathbb{N}}$ such that

$\displaystyle\hat{\bm{u}}_{N_{n}}\to\bm{u}$ $\displaystyle\quad{\rm
weakly~{}star~{}in~{}}H^{1}(0,T;\bm{H}_{\sigma})\cap
L^{\infty}(0,T;\bm{V}_{\sigma}),$ $\displaystyle\hat{\bm{u}}_{N_{n}}\to\bm{u}$
$\displaystyle\quad{\rm strongly~{}in~{}}L^{2}(0,T;\bm{H}_{\sigma}),$
$\displaystyle\bar{\bm{u}}_{N_{n}}\to\bar{\bm{u}}$ $\displaystyle\quad{\rm
weakly~{}star~{}in~{}}L^{\infty}(0,T;\bm{V}_{\sigma})\cap L^{2}(0,T;\bm{W}),$
$\displaystyle\bm{w}_{N_{n}}\to\bm{w}$ $\displaystyle\quad\text{weakly in
}H^{1}(0,T;\bm{H}_{\sigma})\cap L^{2}(0,T;\bm{W}),$
$\displaystyle\bm{w}_{N_{n}}\to\bm{w}$ $\displaystyle\quad\text{strongly in
}C([0,T];\bm{H}_{\sigma})\cap L^{2}(0,T;\bm{V}_{\sigma}),$

when $n\to\infty$, as a result of weak compactness and the Aubin–Lions
theorem.

Now we claim $\bm{u}=\bar{\bm{u}}=\bm{w}$. In fact, since

$\|\bar{\bm{u}}_{N}-\hat{\bm{u}}_{N}\|_{L^{2}(0,T;\bm{H}_{\sigma})}^{2}=\frac{h_{N}}{3}\sum_{k=1}^{N}\|\bm{u}_{k}-\bm{u}_{k-1}\|_{\bm{H}_{\sigma}}^{2}\to
0\quad(N\to\infty)$

by direct computation and Lemma 4.2, we have

$\|\bar{\bm{u}}_{N_{n}}-{\bm{u}}\|_{L^{2}(0,T;\bm{H}_{\sigma})}\leq\|\bar{\bm{u}}_{N_{n}}-\hat{\bm{u}}_{N_{n}}\|_{L^{2}(0,T;\bm{H}_{\sigma})}+\|\hat{\bm{u}}_{N_{n}}-\bm{u}\|_{L^{2}(0,T;\bm{H}_{\sigma})}\to
0\quad(n\to\infty),$

hence $\bm{u}=\bar{\bm{u}}$. A similar computation based on
$\|{\bm{w}}_{N}-\hat{\bm{u}}_{N}\|_{L^{2}(0,T;\bm{H}_{\sigma})}^{2}=(h_{N}/3)\|\bm{u}_{1}-\bm{u}_{0}\|_{\bm{H}_{\sigma}}^{2}$
shows $\bm{u}=\bm{w}$.

We note that these counterparts for the $\bm{H}^{1}(\Omega)$-norm also hold,
that is,

$\|\nabla(\bar{\bm{u}}_{N}-\hat{\bm{u}}_{N})\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}=\frac{h_{N}}{3}\sum_{k=1}^{N}\|\nabla(\bm{u}_{k}-\bm{u}_{k-1})\|_{\bm{L}^{2}(\Omega)}^{2}\to
0,\quad\|\nabla({\bm{w}}_{N}-\hat{\bm{u}}_{N})\|_{L^{2}(0,T;\bm{L}^{2}(\Omega))}^{2}\to
0\quad(N\to\infty)$

by Lemma 4.3. Therefore, we conclude

$\|\bar{\bm{u}}_{N_{n}}-\bm{u}\|_{L^{2}(0,T;\bm{V}_{\sigma})}\leq\|\bar{\bm{u}}_{N_{n}}-\hat{\bm{u}}_{N_{n}}\|_{L^{2}(0,T;\bm{V}_{\sigma})}+\|\hat{\bm{u}}_{N_{n}}-{\bm{w}}_{N_{n}}\|_{L^{2}(0,T;\bm{V}_{\sigma})}+\|{\bm{w}}_{N_{n}}-\bm{u}\|_{L^{2}(0,T;\bm{V}_{\sigma})}\to
0\quad(n\to\infty),$

which establishes the strong convergence of $\\{\bar{\bm{u}}_{N_{n}}\\}$ in
$L^{2}(0,T;\bm{V}_{\sigma})$.

Now we can proceed completely in the same way as in Proposition 4.1 and
Theorem 4.1, finishing the proof of Theorem 4.2.

## References

  * [1] G. Akagi and M. Ôtani, Time-dependent constraint problems arising from macroscopic critical-state models for type-II superconductivity and their approximations, Adv. Math. Sci. Appl., 14 (2004), pp. 683–712.
  * [2] V. Barbu, Time optimal control of Navier–Stokes equations, Systems Control Lett., 30 (1997), pp. 93–100.
  * [3] , Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010.
  * [4] V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier–Stokes equation, J. Math. Anal. Appl., 255 (2001), pp. 281–307.
  * [5] H. Beirão da Veiga, Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions, Adv. Differential Equations, 9 (2004), pp. 1079–1114.
  * [6] M. Boukrouche and G. L. ukaszewicz, On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition, Discrete Contin. Dyn. Syst. Ser. S, 34 (2014), pp. 3969–3983.
  * [7] H. Brezis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis, Academic Press, 1971, pp. 101–156.
  * [8] , Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, 1973.
  * [9] , Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.
  * [10] T. Caraballo, J. Real, and P. E. Kloeden, Unique strong solutions and $v$-attractors of a three dimensional system of globally modified Navier–Stokes equations, J. Math. Anal. Appl., 6 (2006), pp. 411–436.
  * [11] P. Constantin and C. Foias, Navier–Stokes equations, University of Chicago Press, 1989.
  * [12] G. Deugoue and J. K. Djoko, On the time discretization for the globally modified three dimensional Navier–Stokes equations, J. Comput. Appl. Math., 235 (2011), pp. 2015–2029.
  * [13] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer, 1976.
  * [14] M. Fuchs and G. Seregin, Regularity results for the quasi-static Bingham variational inequality in dimensions two and three, Math. Z., 227 (1998), pp. 525–541.
  * [15] , Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Springer, 2000.
  * [16] T. Fukao, On a variational inequality of Bingham and Navier–Stokes type in three dimensional space, in Nonlinear Analysis in Iinterdisciplinary Sciences, vol. 36 of GAKUTO Internat. Ser. Math. Sci. Appl., 2013, pp. 57–71.
  * [17] V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Springer, 1986.
  * [18] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, 1984.
  * [19] R. Glowinski, J. L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, 1981.
  * [20] S. González-Andrade, A BDF2-semismooth Newton algorithm for the numerical solution of the Bingham flow with temperature dependent paramerters, J. Non-Newtonian Fluid Mech., 284 (2020), p. 104380.
  * [21] Y. Kato, Variational inequalities of Bingham type in three dimensions, Nagoya Math. J., 129 (1993), pp. 53–95.
  * [22] N. Kenmochi, Solvability of nonlinear parabolic equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), pp. 1–87.
  * [23] J. U. Kim, On the Cauchy problem associated with the motion of a Bingham fluid in the plane, Trans. Amer. Math. Soc., 298 (1986), pp. 371–400.
  * [24] , On the initial-boundary value problem for a Bingham fluid in a three-dimensional domain, Trans. Amer. Math. Soc., 304 (1987), pp. 751–770.
  * [25] , A finite element approximation of three dimensional motion of a Bingham fluid, Math. Mod. Numer. Anal., 23 (1989), pp. 293–333.
  * [26] B. Merouani, F. Messelmi, and S. Drabla, Dynamical flow of a Bingham fluid with subdifferential boundary condition, Anal. Univ. Oradea, fasc. Math., 16 (2009), pp. 5–30.
  * [27] F. Messelmi, Diffusion-reaction problem for the Bingham fluid with Lipschitz source, Int. J. Adv. Appl. Math. and Mech., 3 (2014), pp. 37–55.
  * [28] J. Naumann and M. Wulst, On evolution inequalities of Bingham type in three dimensions I, J. Math. Anal. Appl., 68 (1979), pp. 211–227.
  * [29] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), pp. 268–299.
  * [30] , Nonlinear evolution equations with time-dependent constraints, Adv. Math. Sci. Appl., 3 (1993/94), pp. 383–399.
  * [31] M. Ôtani and Y. Yamada, On the Navier–Stokes equations in noncylindrical domains: an approach by the subdifferential operator theory, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 25 (1978), pp. 185–204.
  * [32] N. Saito, On the Stokes equation with leak and slip boundary conditions of friction type: regularity of solutions, Publ. RIMS. Kyoto Univ., 40 (2004), pp. 345–383.
  * [33] J. Simon, Compact sets in the spaces ${L}^{p}(0,{T};{B})$, Ann. Mat. Pura. Appl. (4), 146 (1987), pp. 65–96.
  * [34] R. Temam, Navier–Stokes Equations, AMS Chelsea Publishing, 2001.
  * [35] S. Tsubouchi, Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity, arXiv:2208.14640, (2022).
  * [36] J. Wloka, Partial Differential Equations, Cambridge University Press, 1987.