# Weak solutions for a bi-fluid model for a mixture of two compressible non
interacting fluids with general boundary data

Stanislav Kračmar S.K. has been supported by the Czech Science Foundation
(GAČR) project GA19-04243S within RVO:67985840. Young-Sam Kwon  The work of
Y-S.K. was supported by the National Research Foundation of Korea
(NRF2020R1F1A1A01049805) Šárka Nečasová  Š. N. has been supported by the Czech
Science Foundation (GAČR) project GA19-04243S. The Institute of Mathematics,
CAS is supported by RVO:67985840.  Antonín Novotný The work of the fourth
author was partially supported by the distinguished Edurad Čech visiting
program at the Institute of Mathematics of the Academy of Sciences of the
Czech Republic.

###### Abstract

We prove global existence of weak solutions for a version of one velocity
Baer-Nunziato system with dissipation describing a mixture of two non
interacting viscous compressible fluids in a piecewise regular Lipschitz
domain with general inflow/outflow boundary conditions. The geometrical
setting is general enough to comply with most current domains important for
applications as, for example, (curved) pipes of picewise regular and axis-
dependent cross sections.

As far as the existence proof is concerned, we adapt to the system the
nowaday’s classical Lions-Feireisl approach to the compressible Navier-Stokes
equations which is combined with a generalization of the theory of
renormalized solutions to the transport equations in the spirit of Vasseur-
Wen-Yu. The results related to the families of transport equations presented
in this paper extend/improve some of statements of the theory of renormalized
solutions, and they are therefore of independent interest.

Department of Mathematics, Czech Technical University, Faculty of Mechanical
Engineering,

Resslova 307, 110 00 Praha, Czech Republic<EMAIL_ADDRESS>
Department of Mathematics, Dong-A University

Busan 49315, Republic of Korea<EMAIL_ADDRESS>
Institute of Mathematics of the Academy of Sciences of the Czech Republic,

Žitná 25, 115 67 Praha, Czech Republic<EMAIL_ADDRESS>
University of Toulon, IMATH, EA 2134, BP 20139

839 57 La Garde, France<EMAIL_ADDRESS>
MSC Classification: 76N10, 35Q30

Keywords: bi-fluid system; Baer-Nunziato system; compressible Navier–Stokes
equations; transport equation; continuity equation; renormalized solutions;
non zero inflow and outflow, weak solutions

##  1 Introduction

One of the acceptable models for description of mixture of several
compressible fluids is the so called two velocity Baer-Nunziato model. The
equations of the Baer–Nunziato model with dissipation read (cf. [5], [16],
[28, Section 1]):

$\partial_{t}{\alpha}_{\pm}+{\bf v}_{I}\cdot\nabla{\alpha}_{\pm}=0,$
$\partial_{t}(\alpha_{\pm}\varrho_{\pm})+{\rm
div}(\alpha_{\pm}\varrho_{\pm}{\bf u}_{\pm})=0,$
$\partial_{t}(\alpha_{\pm}\varrho_{\pm}{\bf u}_{\pm})+{\rm
div}(\alpha_{\pm}\varrho_{\pm}{\bf u}_{\pm}\otimes{\bf
u}_{\pm})+\nabla(\alpha_{\pm}P_{\pm}(\varrho_{\pm}))-P_{I}\nabla(\alpha_{\pm})$
$=\alpha_{\pm}\mu_{\pm}(\Delta{\bf
u}_{\pm})+\alpha_{\pm}(\mu_{\pm}+\lambda_{\pm})\nabla{\rm div}{\bf u}_{\pm}$
$0\leq\alpha_{\pm}\leq 1,\quad\alpha_{+}+\alpha_{-}=1.$

In the above $(\alpha_{\pm},\alpha_{\pm}\varrho_{\pm}\geq 0,{\bf
u}_{\pm}\in\mathbb{R}^{3})$ -concentrations, densities, velocities of the
$\pm$ species - are unknown functions of time-space
$(t,x)=Q_{T}:=I\times\Omega$, $t\in I=(0,T)$, $T>0$, and
$x\in\Omega\subset\mathbb{R}^{3}$, $P_{\pm}$ are two (different) given
functions defined on $[0,\infty)$ and $P_{I}$, ${{\bf v}}_{I}$ are
conveniently chosen quantities - they represent pressure and velocity at the
interface. In the multifluid modeling, there are many possibilities how the
quantities ${{\bf v}}_{I}$, $P_{I}$ could be chosen, and there is no consensus
about this choice.

Our goal in this paper is to prove the existence of weak solutions for the
Baer-Nunziato system with dissipation on an arbitrary large time interval
$(0,T)$ in a Lipschitz bounded domain $\Omega$, under the following
simplifying assumptions:

$\mu_{\pm}:=\mu,\;\lambda_{\pm}:=\lambda,\;{\bf v}_{I}={\bf u}_{\pm}:={\bf u}$
(1) $\alpha P_{\pm}(s)={\cal P}_{\pm}(f_{\pm}(\alpha)s)\;\mbox{for all
$\alpha\in(0,1)$, $s\in[0,\infty)$}$ (2)

with some functions ${\cal P}_{\pm}$ defined on $[0,\infty)$ and functions
$f_{\pm}$ defined on $(0,1)$.

With this simplifications, the two velocity Baer-Nunziato system reduces to
the following system (which we will call the one velocity Baer-Nunziato type
system):

$\partial_{t}\alpha+({\bf u}\cdot\nabla)\alpha=0,\quad 0\leq\alpha\leq 1,$ (3)
$\partial_{t}\varrho+\mbox{div}(\varrho{\bf u})=0,$ (4)
$\partial_{t}z+\mbox{div}(z{\bf u})=0,$ (5) $\partial_{t}((\rho+z){\bf
u})+\mbox{div}((\rho+z){\bf u}\otimes{\bf u})+\nabla
P(f(\alpha)\varrho,g(\alpha)z)={\rm div}\mbox{$\mathbb{S}$}(\nabla{\bf u})$
(6)

Here $P:[0,\infty)^{2}\mapsto[0,\infty)$ as well as
$f,g:(0,1)\mapsto[0,\infty)$ are given functions, and

${\mbox{$\mathbb{S}$}(\mbox{$\mathbb{Z}$})}=\mu(\mbox{$\mathbb{Z}$}+{\mbox{$\mathbb{Z}$}}^{T})+\lambda{\rm
Tr}(\mbox{$\mathbb{Z}$})\mbox{$\mathbb{I}$}.$

($\mathbb{I}$ is the identity tensor, Tr denotes the trace) is the viscous
stress tensor. The constant viscosity coefficients satisfy standard physical
assumptions, $\mu>0$, $\lambda+\frac{2}{3}\mu\geq 0$. The system is endowed
with initial conditions

$\alpha|_{t=0}=\alpha_{0},\;\varrho|_{t=0}=\varrho_{0},\;z|_{t=0}=z_{0},\;(\varrho+z){\bf
u}|_{t=0}=(\varrho_{0}+z_{0}){\bf u}_{0},$ (7)

We consider the general inflow-outflow boundary conditions,

${\bf u}|_{\partial\Omega}={\bf u}_{B},\ \varrho|_{\Gamma^{\rm
in}}=\varrho_{B},\ z|_{\Gamma^{\rm in}}=z_{B},\ \alpha|_{\Gamma^{\rm
in}}=\alpha_{B},$ (8)

where

$\Gamma^{\rm in}=\left\\{x\in\partial\Omega\ \Big{|}\ {{\bf u}_{B}}\cdot{\bf
n}<0\right\\}.$ (9)

For further use, we also define

$\Gamma^{\rm out}=\left\\{x\in\partial\Omega\ \Big{|}\ {{\bf u}_{B}}\cdot{\bf
n}>0\right\\},\;\Gamma^{0}={\rm int}_{2}\left\\{x\in\partial\Omega\ \Big{|}\
{{\bf u}_{B}}\cdot{\bf n}=0\;\mbox{or}\;{\bf n}(x)\;\mbox{does not
exist}\,\right\\}.$ (10)

Here and in the sequel, ${\rm int}_{2}A$ denotes the interior of
$A\subset\partial\Omega$ with respect to the trace topology of
$\mathbb{R}^{3}$ on $\partial\Omega$.

Assumption (2) is certainly true in the classical situation of two isentropic
gases when

${P}_{\pm}(s)=a_{\pm}s^{\gamma^{\pm}},\;\gamma_{\pm}>0;$ (11)

indeed, in this case

$P(R,Z)=a_{+}R^{\gamma^{+}}+a_{-}Z^{\gamma^{-}},\;f(s):=f_{+}(s)={s}^{\frac{1}{\gamma^{+}}-1},\;g(s):=f_{-}(s)={(1-s)}^{\frac{1}{\gamma^{-}}-1}.$
(12)

We shall however be able to treat in system (3–7) more general functions
$P,f,g$ than those being given by (12).

System (3–6) belongs to the family of multi-fluid models with differential
closure, cf. Ishii, Hibiki [33], Drew, Passman [20]. It is not without
interest that it can be viewed as a barotropic counterpart of the so called
five-equation bi-fluid model derived in Allaire, Clerc, Kokh [2], [3] or by
Guillard, Murrone [32] by different considerations.

The mathematical literature dealing with these types of models is in a short
supply and none of it deals with the general boundary data. We quote a few
papers dealing with no-slip or periodic or slip boundary conditions for
related problems: [9], [11], [21], [22], [37], [40], [41], [43], [45]. In
particular, existence of weak solutions for the problem (3–7) under quite
general assumptions on constitutive functions $P$ and $f,g$ is known in the
case of no-slip boundary conditions (${\bf u}|_{\partial\Omega}=0$), see [40].
The main goal and achievement of this work is to treat the general non-zero
inflow-outflow problem which is more adequate than the no-slip or slip cases
for most physical and engineering applications.

Similarly as in [40], the proof will be based on the reformulation of the
original problem via the change of variables

$R:=f(\alpha)\varrho,\;Z=g(\alpha)z$ (13)

as follows:

$\displaystyle\partial_{t}\varrho+{\rm div}\,(\varrho{\bf u})$
$\displaystyle=0,$ (14) $\displaystyle\partial_{t}z+{\rm div}\,(z{\bf u})$
$\displaystyle=0,$ $\displaystyle\partial_{t}R+{\rm div}\,(R{\bf u})$
$\displaystyle=0,$ $\displaystyle\partial_{t}Z+{\rm div}\,(Z{\bf u})$
$\displaystyle=0,$ $\displaystyle\partial_{t}\big{(}(\varrho+z){\bf
u}\big{)}+{\rm div}\,((\varrho+z){\bf u}\otimes{\bf u})+\nabla P(R,Z)$
$\displaystyle={\rm div}\mbox{$\mathbb{S}$}(\nabla{\bf u})$

with boundary and initial conditions

$\begin{array}[]{l}{\bf u}|_{\partial\Omega}={\bf u}_{B},\
\varrho|_{\Gamma^{\rm in}}=\varrho_{B},\ z|_{\Gamma^{\rm in}}=z_{B},\\\ \
R|_{\Gamma^{\rm in}}=R_{B}:=f(\alpha_{B})\varrho_{B},\ Z|_{\Gamma^{\rm
in}}=Z_{B}:=g(\alpha_{B})z_{B},\end{array}$ (15)

$\varrho(0,x)=\varrho_{0}(x),\;z(0,x)=z_{0}(x),$ (16)
$R(0,x)=R_{0}(x):=f(\alpha_{0})\varrho_{0}(x),\;Z(0,x)=Z_{0}(x):=g(\alpha_{0})z_{0}(x),$
$(\varrho+z){\bf u}(0,x)=(\varrho_{0}+z_{0}){\bf u}_{0}$

for unknown quintet $(\varrho,z,R,Z,{\bf u})$ of functions defined on the
space-time cylinder $Q_{T}=I\times\Omega$. In this paper, we will call it an
academic bi-fluid system.

It is to be noticed that, in the family of equations (14), the transport
equation for $\alpha$ is tacitly hidden in the first four continuity
equations: We anticipate here the fact that (formally),
$\alpha=f^{-1}(R/\varrho)$ and $\tilde{\alpha}=g^{-1}(Z/z)$ verify transport
equation with the same initial condition $\alpha_{0}$ and the same boundary
condition $\alpha_{B}$. To pass from the academic system (14 –16) to the
original system (3–9), we shall need to identify $\alpha$ and
$\tilde{\alpha}$. This will be done through the observation that the pure
transport equation enjoys the almost uniqueness property (without condition
${\rm div}{\bf u}\in L^{1}(0,T;L^{\infty}(\Omega))$ and without a slightly
weaker condition of Bianchini, Bonnicatto [6]), see Corollary 6. These notions
and further properties related to families of transport equations which in a
sense generalize and complete the results of the seminal paper by DiPerna-
Lions [19] will be specified and put on rigorous grounds in Section 3 which is
of independent interest.

The statement about the existence of weak solutions for the academic system
(14–16) is formulated in Section 2 in Theorem 1 and similar statement about
the existence of weak solutions to the one velocity Baer-Nunziato type system
(3–7) is available in Theorem 2.

This article is inspired by two main sources: 1) The paper [40], where the
author investigates weak solutions for the bi-fluid systems (both ”academic”
and ”realistic” under the no-slip boundary conditions). 2) Papers [12], [13],
[35] and monograph [44], where the authors construct weak solutions for the
”mono-fluid” compressible Navier-Stokes equations with general inflow-outflow
boundary data. We wish to concentrate on the effects of the non-homogenous
boundary conditions. Therefore, in contrast with [40], we renounce at
accommodation of the most general pressure law, in order to avoid the
unessential technical difficulties. Still, the pressure law considered in this
paper covers most of classical situations including the mixture of two
isentropic gases, cf. example (12).

The main steps in our approach are the following:

  1. 1.

In Section 3 we develop the theory of renormalized solutions to families of
transport equations with non-homogenous boundary data, which is one of the
building blocks of the proofs. These results are new and of independent
interest. It includes in this context:

    1. (a)

Passage (via renormalization) from the (two) continuity equations to a pure
transport equation.

    2. (b)

Passage (via renormalization) from the (two) transport equations and a
continuity equation to a continuity equation

and two consequences of these results

    1. (a)

Almost compactness of the ratio of two solutions of continuity equations.

    2. (b)

Almost uniqueness of the solutions to the pure transport equation.

As mentioned already above, these are delicate issues which are somehow
connected (and as far as the uniqueness is concerned, somehow generalize) the
seminal works of DiPerna-Lions [19], Ambrosio, Crippa, [4], Bianchini,
Bonicatto [6], employing only the Eulerian approach. Almost compactness
generalizes to the uniform in time convergence and to the non homogenous
boundary data the original results of Vasseur, Wen, Yu from their seminal
paper [45]. We refer also to papers by Boyer [8] and Crippa et al [15] for
results related to transport equations with non homogenous boundary data.

  2. 2.

We approximate the ”academic” system similarly as in [12]: The momentum
equation with added artificial pressure term (small parameter $\delta>0$) is
approximated by the Galerkin approximation (of dimension $n$) while each
continuity equation is approximated by a specific parabolic boundary value
problem (small parameter $\varepsilon$) with the Robin type boundary
conditions (for a while, we shall call the solutions of these parabolic
problems ”densities”). Having in view applications in numerical analysis, in
contrast with [12]–where one solves the parabolic problems by using the
maximal parabolic regularity and thus needs at least $C^{2}$ boundary– we
solve the parabolic problem on Lipschitz domains following Crippa et al. [14,
Lemma 3.2].

It is known since [37] that the property of domination of one density by
another one perpetuates for all times if it is in force initially. We shall
prove by using the maximum principle that the parabolic equations in
consideration enjoy this property. Following [40] and [35], we easily derive
for the approximate system the energy inequality and uniform bounds.

The approximation on Lipschitz domains has an evident practical advantage: In
[13], the passage from $C^{2}$-domain required in [12] to a Lipschitz
piecewise $C^{2}$ domains (which can be considered as an reasonable geometry
for the inflow-outflow problems) is effectuated by a laborious approximation
of domains, while in the present approach, this step is for free (and requires
slightly less of the domain than it is required in [13, Theorem 2.4]) – only
at cost of more work at the level of the Galerkin approximation. This part of
the proof is treated in Sections 4–5, see Lemma 8 and Proposition 9.

  3. 3.

In contrast to [12], we derive the renormalized equation for the parabolic
problem for densities at the level of its weak formulation, see Section 5.3.
In particular, we do not need its satisfaction almost everywhere in the time
cylinder. This allows to simplify the approximation of the momentum equation:
compared to [12] or [40], there is no need to consider the
$\varepsilon$-dependent power law dissipation.

  4. 4.

For the remaining limit passages $\varepsilon\to 0$ and then $\delta\to 0$, we
need to improve the estimates of pressure. With the domination principle at
hand, this is done via the Bogovskii operator exactly in the same way as for
the simple mono-fluid case, cf. [26] completed with [41] or [40]. However, due
to the non homogenous data, these estimates are available only on compact
subsets of $\Omega$.

  5. 5.

The main difficulty in both passages $\varepsilon\to 0$ and then $\delta\to 0$
is to pass to the limit in the non-linear pressure term $P(R,Z)$. It was
observed for the first time by Vasseur et al. [45] (and later improved in [41,
Proposition 7]) that the quantity $Z/R$ is ”almost compact” provided $R$
dominates $Z$ and both quantities $R$ and $Z$ satisfy continuity equation with
the same transporting velocity. The almost compactness of $Z/R$ is the
property of the continuity equations and it is completely decoupled from the
remaining equations in the system. With this observation at hand, it is enough
to prove the compactness for the quantity
$\Pi(R_{\varepsilon},t,x):=P(R_{\varepsilon},R_{\varepsilon}\theta)$ (resp.
$\Pi(R_{\delta},t,x):=P(R_{\delta},R_{\delta}\theta)$, where $\theta(t,x)$ is
a given function (a ratio of weak limits of the sequences $Z_{\varepsilon}$
and $R_{\varepsilon}$ resp. of the weak limits of sequences $Z_{\delta}$ and
$R_{\delta}$). The task thus reduces practically to the task to prove the
strong convergence of density in the ”mono-fluid” case (with the pressure
dependent on only “dominating” density and time-space $(t,x)$). This process
is nowadays well understood, cf. Lions [36], Feireisl et al [26].111Since
2018, there exists an alternative approach to [36] due to Bresch, Jabin [10],
which is however not exploited in the present paper. It passes through:

    1. (a)

Derivation of the effective viscous flux identity.

    2. (b)

Eliminating oscillations in the sequence of densities by using the theory of
renormalized solutions due to DiPerna-Lions [19] which must be modified to
accommodate the non homogenous boundary conditions and renormalizing functions
of several variables.

The first point is very similar to the ”mono-fluid” case. It is briefly
explained in Section 6 for $\varepsilon\to 0$ and in Section 7 for $\delta\to
0.$ The second point is more delicate since the theory of renormalized
solutions to the transport equation is not available for the problems with the
non-homogenous boundary conditions. Some elements of it are developed in
Plotnikov, Sokolowski [44] and in [12] but this is not enough for our purpose.
We treat this part in Section 6 ($\varepsilon\to 0$) and Section 7 ($\delta\to
0$) referring abundantly to Section 3.

  6. 6.

In Section 9 we gather all necessary specific results from functional analysis
needed throughout the proofs.

Theorem 2 is the first rigorous result on existence of weak solutions for a
version of the Baer-Nunziato type bi-fluid model with non zero inflow-outflow
boundary conditions.

The Di-Perna, Lions transport theory in conjonction with the absence of
improved estimates of pressure up to the boundary imposes limitations on
adiabatic coefficients $\gamma^{\pm}$ in formula (11) - or an equivalent
limitation on growth conditions of $P$ (see the next Section): at least one of
them has to be greater or equal than $2$. In view of the existing mono-fluid
theory, existence of weak solutions could be possibly hoped to be achieved if
the adiabatic coefficients of constituents were greater than $3/2$. This
remains however a very interesting open problem.

In what follows, the scalar-valued functions will be printed with the usual
font, the vector-valued functions will be printed in bold, and the tensor-
valued functions with a special font, i.e. $\varrho$ stands for the density,
${\bf u}$ for the velocity field and $\mathbb{S}$ for the stress tensor. We
use standard notation for the Lebesgue and Sobolev spaces equipped by the
standard norms $\|\cdot\|_{L^{p}(\Omega)}$ and $\|\cdot\|_{W^{k,p}(\Omega)}$,
respectively. We will sometimes distinguish the scalar-, the vector- and the
tensor-valued functions in the notation, i.e. we use $L^{p}(\Omega)$ for
scalar quantities, $L^{p}(\Omega;\mathbb{R}^{3})$ for vectors and
$L^{p}(\Omega;\mathbb{R}^{3\times 3})$ for tensors. The indication of the $R$
or tensor character of the fields (here $;\mathbb{R}^{3}$ or
$;\mathbb{R}^{3\times 3}$) may be omitted, when there is no lack of confusion.
The Bochner spaces of integrable functions on $I$ with values in a Banach
space $X$ will be denoted $L^{p}(I;X)$; likewise the spaces of continuous
functions on $\overline{I}$ with values in $X$ will be denoted
$C(\overline{I};X)$. The norms in the Bochner spaces will be denoted
$\|\cdot\|_{L^{p}(I;X)}$ and $\|\cdot\|_{C(\overline{I};X)}$, respectively. In
most cases, the Banach space $X$ will be either the Lebesgue or the Sobolev
space. Finally, we use vector spaces $C_{\rm weak}(\overline{I};X)$ which is a
subspace of $L^{\infty}(I;X)$ of continuous functions in $\overline{I}$ with
respect to weak topology of $X$ (meaning that $f\in C_{\rm
weak}(\overline{I};X)$ iff $t\mapsto{\cal F}(f(t))$ belongs for any ${\cal
F}\in X^{*}$ to $C(\overline{I})$).

The generic constants will be denoted by $c$, $\underline{c}$, $\overline{c}$,
$C$, $\underline{C}$, $\overline{C}$ and their value may change even in the
same formula or in the same line. Sometimes, for two quantities $a$, $b$, we
shall write

$a\stackrel{{\scriptstyle<}}{{\sim}}b\;\mbox{if $a\leq cb$},\;c>0\;\mbox{a
constant},\;a\approx
b\;\mbox{if}\,a\stackrel{{\scriptstyle<}}{{\sim}}b\;\mbox{and}\,b\stackrel{{\scriptstyle<}}{{\sim}}a.$

Here ”constant” typically means a generic quantity independent on the
approximating parameters of the problem (as number of Galerkin modes $n$,
artificial diffusion parameter $\varepsilon$ or artificial pressure parameter
$\delta$).

##  2 Main results

###  2.1 Definition of weak solutions

We first explain the notion of the weak solution to problem (14–16) and to
problem (3–8). Before starting the definition, we must underline, that,
without loss of generality, the boundary data (15) (resp. (8) ) are considered
as a restriction to the boundary of functions defined on the whole
$\mathbb{R}^{3}$ (their regularity, as well as the regularity of the initial
data will be specified later, in Section 2.2).

######  Definition 1.

A quintet $(\varrho,z,R,Z,{\bf v}={\bf u}-{\bf u}_{B})$ is a bounded energy
weak solution to problem (14–16), if the following holds:

  1. 1.

The quintet belongs to the functional spaces $\varrho,z,R,Z\geq 0$ a.e. in
$I\times\Omega$, $(\varrho,z,R,Z)\in C_{\rm weak}(\overline{I};$
$L^{\gamma}(\Omega))\cap L^{\gamma}(I;L^{\gamma}(\partial\Omega;|{\bf
u}_{B}\cdot{\bf n}|{\rm d}S_{x}))$ with some $\gamma>1$, ${\bf v}\in
L^{2}(I;W^{1,2}_{0}(\Omega;\mathbb{R}^{3}))$, $(\varrho+z)|{\bf v}|^{2}\in
L^{\infty}(I;L^{1}(\Omega))$, $P(R,Z)\in L^{1}(I\times\Omega)$,
$(\varrho+z){\bf u}\in C_{\rm weak}(\overline{I};L^{q}(\Omega))$ with some
$q>1$.

  2. 2.

Continuity equations

$\begin{array}[]{l}\int_{\Omega}r(\tau,\cdot)\varphi(\tau,\cdot){\,\rm
d}x-\int_{\Omega}r_{0}(\cdot)\varphi(0,\cdot){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r{\bf u}_{B}\cdot{\bf n}\varphi\
{\rm d}S_{x}{\rm d}t\\\ \\\
=\int_{0}^{\tau}\int_{\Omega}\big{(}r\partial_{t}\varphi+r{\bf
u}\cdot\nabla\varphi\big{)}{\,\rm d}x\,{\rm
d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}r_{B}{\bf u}_{B}\cdot{\bf n}\varphi\
{\rm d}S_{x}{\rm d}t\end{array}$ (17)

are satisfied for any $\tau\in[0,T]$ and with any $\varphi\in
C^{1}_{c}([0,T]\times\overline{\Omega})$, where $r$ stands for $\varrho$, $z$,
$R$, $Z$.

  3. 3.

Momentum equation

$\displaystyle\int_{\Omega}(\varrho+z){\bf
u}\cdot{\boldsymbol{\varphi}}(\tau,\cdot){\,\rm
d}x-\int_{\Omega}(\varrho_{0}+z_{0}){\bf
u}_{0}\cdot{\boldsymbol{\varphi}}(0,\cdot){\,\rm
d}x=\int_{0}^{\tau}\int_{\Omega}\Big{(}(\varrho+z){\bf
u}\cdot\partial_{t}{\boldsymbol{\varphi}}$ (18) $\displaystyle+(\varrho+z){\bf
u}\otimes{\bf u}:\nabla{\boldsymbol{\varphi}}+P(R,Z){\rm
div}\,{\boldsymbol{\varphi}}-\mbox{$\mathbb{S}$}(\nabla{\bf
u}):\nabla{\boldsymbol{\varphi}}\Big{)}{\,\rm d}x\,{\rm d}t$

holds with any $\tau\in[0,T]$ and ${\boldsymbol{\varphi}}\in
C^{1}_{c}([0,T)\times\Omega;\mathbb{R}^{3})$.

  4. 4.

Finally, the energy inequality

$\displaystyle\int_{\Omega}\Big{(}\frac{1}{2}(\varrho+z)|{\bf
v}|^{2}+H(R,Z)\Big{)}(\tau,\cdot){\,\rm
d}x+\int_{0}^{\tau}\int_{\Omega}\mbox{$\mathbb{S}$}(\nabla{\bf u}):\nabla{\bf
u}{\,\rm d}x\,{\rm d}t$ (19) $\displaystyle+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}H(R,Z){\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm d}t$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}\Big{(}\frac{1}{2}(\varrho_{0}+z_{0}){\bf
v}_{0}^{2}+H(R_{0},Z_{0})\Big{)}{\,\rm d}x-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}H(R_{B},Z_{B}){\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm d}t$
$\displaystyle+$ $\displaystyle\int_{0}^{\tau}\int_{\Omega}\Big{(}-P(R,Z){\rm
div}{\bf u}_{B}-\rho{\bf u}\cdot\nabla{\bf u}_{B}\cdot{\bf
v}+\mbox{$\mathbb{S}$}(\nabla{\bf u}):\nabla{\bf u}_{B}\Big{)}{\,\rm d}x$

is satisfied for a.a. $\tau\in(0,T)$, where ${\bf v}_{0}={\bf u}_{0}-{\bf
u}_{B}$ and

$H(R,Z)=R\int_{1}^{R}\frac{P(s,s\frac{Z}{R})}{s^{2}}{\rm d}s,\;\mbox{if
$R>0$},\;H(0,Z)=0.$ (20)

######  Definition 2.

A quartet $(\alpha,\varrho,z,{\bf u})$ is a bounded energy weak solution to
problem (3–8), if the following holds:

  1. 1.

$\varrho,z\geq 0$ a.e. in $I\times\Omega$, $(\varrho,z)\in C_{\rm
weak}(\overline{I};L^{\gamma}(\Omega))\cap
L^{\gamma}(I;L^{\gamma}(\partial\Omega;|{\bf u}_{B}\cdot{\bf n}|{\rm
d}S_{x}))$ with some $\gamma>1$, $\alpha\in L^{\infty}(Q_{T})\cap C_{\rm
weak}(\overline{I};L^{\gamma}(\Omega)){\cap
L^{\infty}(I\times\partial\Omega)}$, $0\leq\alpha\leq 1$, ${{\bf v}={\bf
u}-{\bf u}_{\infty}}\in L^{2}(I;W^{1,2}_{0}(\Omega;\mathbb{R}^{3}))$,
$(\varrho+z)|{\bf u}|^{2}\in L^{\infty}(I;L^{1}(\Omega))$,
$P(f(\alpha)\varrho,g(\alpha)z)\in L^{1}(I\times\Omega)$, $(\varrho+z){\bf
u}\in C_{\rm weak}(\overline{I};L^{q}(\Omega))$ with some $q>1$.

  2. 2.

Continuity equations

$\begin{array}[]{l}\int_{\Omega}r(\tau,\cdot)\varphi(\tau,\cdot){\,\rm
d}x-\int_{\Omega}r_{0}(\cdot)\varphi(0,\cdot){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r{\bf u}_{B}\cdot{\bf n}\varphi\
{\rm d}S_{x}{\rm d}t\\\
=\int_{0}^{\tau}\int_{\Omega}\big{(}r\partial_{t}\varphi+r{\bf
u}\cdot\nabla\varphi\big{)}{\,\rm d}x\,{\rm
d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}r_{B}{\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t\end{array}$ (21)

are satisfied for all $\tau\in[0,T]$ and with any $\varphi\in
C^{1}_{c}([0,T]\times\overline{\Omega})$, where $r$ stands for $\varrho$, $z$.

  3. 3.

Pure transport equation

$\begin{array}[]{l}\int_{\Omega}\alpha(\tau,\cdot)\varphi(\tau,\cdot){\,\rm
d}x-\int_{\Omega}\alpha_{0}(\cdot)\varphi(0,\cdot){\,\rm
d}x{+\int_{0}^{\tau}\int_{\Gamma{\rm out}}\alpha{\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t}\\\
=\int_{0}^{\tau}\int_{\Omega}\big{(}\alpha\partial_{t}\varphi+\alpha{\bf
u}\cdot\nabla\varphi{+}\varphi\alpha{\rm div}{\bf u}\big{)}{\,\rm d}x\,{\rm
d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}\alpha_{B}{\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t\end{array}$ (22)

holds for all $\tau\in[0,T]$ with any $\varphi\in
C^{1}_{c}([0,T]\times\overline{\Omega})$.

  4. 4.

Momentum equation

$\displaystyle\int_{\Omega}(\varrho+z){\bf
u}\cdot{\boldsymbol{\varphi}}(\tau,\cdot){\,\rm
d}x-\int_{\Omega}(\varrho_{0}+z_{0}){\bf
u}_{0}\cdot{\boldsymbol{\varphi}}(0,\cdot){\,\rm
d}x=\int_{0}^{\tau}\int_{\Omega}\big{(}(\varrho+z){\bf
u}\cdot\partial_{t}{\boldsymbol{\varphi}}$ (23) $\displaystyle+(\varrho+z){\bf
u}\otimes{\bf
u}:\nabla{\boldsymbol{\varphi}}+P(f(\alpha)\varrho,g(\alpha)z){\rm
div}\,{\boldsymbol{\varphi}}-\mbox{$\mathbb{S}$}(\nabla{\bf
u}):\nabla{\boldsymbol{\varphi}}\big{)}{\,\rm d}x\,{\rm d}t$

holds for all $\tau\in[0,T]$ with any ${\boldsymbol{\varphi}}\in
C^{1}_{c}([0,T)\times\Omega;\mathbb{R}^{3})$.

  5. 5.

The energy inequality holds

$\displaystyle\int_{\Omega}\Big{(}\frac{1}{2}(\varrho+z)|{\bf
u}|^{2}+H(f(\alpha)\varrho,g(\alpha)z)\Big{)}(\tau,\cdot){\,\rm
d}x+\int_{0}^{\tau}\int_{\Omega}\mbox{$\mathbb{S}$}(\nabla{\bf u}):\nabla{\bf
u}{\,\rm d}x\,{\rm d}t$ (24) $\displaystyle+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}H(f(\alpha)\varrho,g(\alpha)z){\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm
d}t\leq\int_{\Omega}\Big{(}\frac{1}{2}(\varrho_{0}+z_{0}){\bf
u}_{0}^{2}+H(f(\alpha_{0})\varrho_{0},g(\alpha_{0})z_{0})\Big{)}{\,\rm d}x$
$\displaystyle-\int_{0}^{\tau}\int_{\Gamma_{\rm
in}}H(f(\alpha_{B})\varrho_{B},g(\alpha_{B})z_{B}){\bf u}_{B}\cdot{\bf n}{\rm
d}S_{x}{\rm d}t$
$\displaystyle+\int_{0}^{\tau}\int_{\Omega}\Big{(}-P(f(\alpha)\varrho,g(\alpha)z){\rm
div}{\bf u}_{B}-\rho{\bf u}\cdot\nabla{\bf u}_{B}\cdot{\bf
v}+\mbox{$\mathbb{S}$}(\nabla{\bf u}):\nabla{\bf u}_{B}\Big{)}{\,\rm d}x{\rm
d}t$

for a.a. $\tau\in(0,T)$, where $H$ is the same as in (19).

###  2.2 Assumptions

Motivated by [41, Section 2] and [40, Section 2.2] we shall gather the
hypotheses for Theorems 1 and 2. In this paper, we however concentrate to the
phenomenons due to the the effects of the non-zero inflow-outflow, and we do
not insist on the most general hypotheses concerning constitutive law for
pressure, which would introduce to the problem further unessential technical
difficulties. We refer the reader to the Remark 1 for the possible relaxation
of this part of hypotheses.

To start, we define an admissible inflow-outflow boundary related to ${{\bf
u}}_{B}$ as follows:

######  Definition 3.

We say that $\Omega$ is a domain with admissible inflow-outflow boundary
related to ${{\bf u}}_{B}$ iff:

  1. 1.

$\Omega$ is a bounded Lipschitz domain.

  2. 2.

$\Gamma^{\mathfrak{a}}=\emptyset\ \mbox{or}\
\Gamma^{\mathfrak{a}}=\cup_{k_{\mathfrak{a}}=1}^{\overline{k_{\overline{\mathfrak{a}}}}}\Gamma^{\mathfrak{a}}_{k_{\mathfrak{a}}},\
\mbox{ $\mathfrak{a}$ stands for ``in'', ``out''}$ (25)

and $\Gamma^{\rm in}_{k_{\rm in}}$, $\Gamma^{\rm out}_{k_{\rm out}}$ are
(open) parametrized $C^{2}$-surfaces or $C^{2}$ compact manifolds, cf. Section
3.1.222Condition that each “$\Gamma=\Gamma^{\rm in/out}_{\rm\cdot}$ is a
$C^{2}$-surface or a $C^{2}$ compact manifold” can be relaxed. It is a
sufficient contition to guarantee existence of a projection $P$ to $\Gamma$ on
a neighborghood $U$ of $\Gamma$, which is continuous on $U$. Indeed, this is
the only condition from conclusion of Lemma 3 which is used in the proof.
Also, conditions on $\mathfrak{g}^{\rm bd}$ could be relaxed. Indeed, the only
thing we need in the proofs is that $\partial\Gamma^{\rm
in}\cup\partial\Gamma^{\rm out}$ satisfies (48). These are the least
conditions needed in the proofs and they enter into the game only through the
Proposition 4.

  3. 3.

$\partial\Gamma^{\rm in}\cup\partial\Gamma^{\rm out}=\mathfrak{g}^{\rm bd},$
(26)

where

${\mathfrak{g}}^{\rm bd}=\Big{(}\cup_{k_{\rm bd}}^{\overline{k_{\rm
bd}}}{{\mathfrak{g}}^{\rm bd}_{k_{\rm bd}}}\Big{)}\cup\Big{(}\cup_{j_{\rm
bd}}^{\overline{j_{\rm bd}}}{{\mathfrak{p}}^{\rm bd}_{j_{\rm bd}}}\Big{)}$

with ${\mathfrak{g}}^{\rm bd}_{k_{\rm bd}}$ being bounded parametrized (open)
$C^{1}$-curves and ${\mathfrak{p}}^{\rm bd}_{j_{\rm bd}}$ points in
$\mathbb{R}^{3}$.

  4. 4.

The sets $\Gamma^{\rm in}_{k_{\rm in}}$, $\Gamma^{\rm out}_{k_{\rm out}}$,
${\mathfrak{g}}^{\rm bd}_{k_{\rm bd}}$, ${\mathfrak{p}}^{\rm bd}_{j_{\rm bd}}$
have two by two empty intersection.

We are now at the point to summarize the hypotheses for Theorem 1.

  1. 1.

Boundary and initial conditions:333The strict inequalities $r_{B}>0$ and
$R_{0}>0$ are here for the sake of simplicity. They could be relaxed up to
$r_{B}\geq 0$ and $R_{0}\geq 0$.

$0<r_{B}\in C_{c}(\mathbb{R}^{3}),\;{{\bf u}_{B}\in
C_{c}^{1}(\mathbb{R}^{3})},\;r\;\mbox{stands for $\varrho$, $z$, $R$, $Z$}.$
(27) $0<R_{0}\in L^{\gamma}(\Omega),\,{\gamma\geq 2},\;Z_{0}\in
L^{\beta}(\Omega)\;\text{ if }\beta>\gamma,\;(\varrho_{0}+z_{0})|{\bf
u}_{0}|^{2}\in L^{1}(\Omega).$ (28) $(R_{0},Z_{0})(x)\in\overline{\cal
O},\;\underline{F}R_{0}(x)\leq\varrho_{0}(x)\leq\overline{F}R_{0}(x),\;\underline{G}Z_{0}(x)\leq
z_{0}(x)\leq\overline{G}Z_{0}(x),$ (29) $(R_{B},Z_{B})(x)\in\overline{\cal
O},\underline{F}R_{B}(x)\leq\varrho_{B}(x)\leq\overline{F}R_{B}(x),\;\underline{G}Z_{0}(x)\leq
z_{B}(x)\leq\overline{G}Z_{B}(x),$

In the above $0<\underline{F}<\overline{F}$, $0<\underline{G}<\overline{G}$
and

${\cal O}:=(R,Z)\in R^{2}\,|\,\underline{a}R<Z<\overline{a}R\\}\;\mbox{with
some $0\leq\underline{a}<\overline{a}$}.$ (30)

  2. 2.

Domain

$\mbox{$\Omega$ is a bounded domain with admissible inflow-outflow boundary {
related} to ${{{\bf u}}}_{B}$}.$ (31)

  3. 3.

Regularity and growth of the pressure function $P$:

$P\in C^{1}(\overline{\cal O})\cap C^{2}({\cal O}),\;P(0,0)=0.$ (32)
$R^{\gamma}+Z^{\beta}-1\stackrel{{\scriptstyle<}}{{\sim}}P(R,Z)\stackrel{{\scriptstyle<}}{{\sim}}R^{\gamma}+Z^{\beta}+1\;\mbox{in
${\cal O}$},$ (33)
$0\leq\partial_{Z}P(R,Z)\stackrel{{\scriptstyle<}}{{\sim}}R^{\underline{\gamma}-1}+R^{\overline{\gamma}-1}\;\mbox{in
${\cal O}$ with
some}\;\underline{\gamma}\in(0,1],\;1\leq\overline{\gamma}<\gamma+\gamma_{\rm
Bog}$ (34)

where

${\gamma\geq 2},\;\beta>0,\;\gamma_{\rm
Bog}=\min\\{\frac{2}{3}\gamma-1,\frac{\gamma}{2}\\},$
$R^{\gamma-1}\stackrel{{\scriptstyle<}}{{\sim}}\partial_{R}P(R,Z)\;\mbox{in
${\cal O}$}.$ (35)

Finally,

$H\;\mbox{is convex on}\;{\cal O}.$ (36)

At this stage a few remarks impose.

######  Remark 1.

  1. 1.

The pressure function introduced in (12) with $1\leq\gamma^{-}\leq\gamma^{+}$,
$\gamma^{+}\geq 2$ is an example of a constitutive law which satisfies all
conditions (32–36).

  2. 2.

We introduce

$\forall s\in L^{\infty}(Q_{T}),\,\mbox{ such that}\
s\in[\underline{a},\overline{a}],\;\Pi(R,t,x)=P(R,Rs(t,x)).$ (37)

Due to (34–35), there exists $d>0$ such that

for a.a. $(t,x)\in Q_{T}$, $\Pi(R,t,x)=dR^{\gamma}+\pi(R,t,x)$ (38)

where for a.a. $(t,x)\in Q_{T}$, $R\mapsto\pi(R,t,x)$ is an non-decreasing
function on $(0,\infty)$. This observation is an important point in the proof
of the strong convergence of the dominating density sequence.

  3. 3.

The function $H$ defined in (20) is called Helmholtz function. We easy verify
that

$H\in C^{1}({\cal O})\cap{C(\overline{\cal O})},\;H(1,Z)=0,\,\forall
Z\in[\underline{a},\overline{a}],\;H(R,Z)\geq\underline{H}>-\infty\;\mbox{for
all $(R,Z)\in\overline{\cal O}$}.$ (39)

provided $P\in C^{2}({\cal O})\cap{C^{1}(\overline{\cal O})}$, $P(0,0)=0$ and
it is a solution of the first order partial differential equation

$R\partial_{R}H(R,Z)+Z\partial_{Z}H(R,Z)-H(R,Z)=P(R,Z).$ (40)

  4. 4.

Due to formula (20), the function $H$ inherits the growth conditions of $P$,
in particular,

$R^{\gamma}+Z^{\beta}-1\stackrel{{\scriptstyle<}}{{\sim}}H(R,Z)\stackrel{{\scriptstyle<}}{{\sim}}R^{\gamma}+Z^{\beta}+1\;\mbox{for
all $(R,Z)\in{\cal O}$}.$ (41)

  5. 5.

The conditions (32–36) are not the most general ones to guarantee the
existence of weak solutions, cf. [40]. Nevertheless, they provide a reasonable
compromise between a presentable proof and overhelming technical complexity.
Indeed:

Condition (34) could be replaced by a weaker one,

$-1\stackrel{{\scriptstyle<}}{{\sim}}\partial_{Z}P(R,Z)\stackrel{{\scriptstyle<}}{{\sim}}R^{\underline{\gamma}-1}+R^{\overline{\gamma}-1}\;\mbox{in
${\cal O}$}$ (42)

with the range of $\underline{\gamma}$ and $\overline{\gamma}$ as in (34).

Condition (35) could be replaced by a weaker one,

$R^{\gamma-1}-1\stackrel{{\scriptstyle<}}{{\sim}}\partial_{R}P(R,Z)\;\mbox{in
${\cal O}$ }.$ (43)

The most restrictive hypotheses (36) could be replaced by

$\forall(R,Z)\in{\cal
O},\;|H(R,Z)-H_{0}(R,Z)|\stackrel{{\scriptstyle<}}{{\sim}}R^{\tilde{\gamma}}+1,\
0<\tilde{\gamma}<\gamma,$ (44)

with some $H_{0}\in C^{1}({\cal O})\cap C(\overline{\cal O})$ convex on ${\cal
O}$.

###  2.3 Main results

The first main result of the paper deals with the academic system (14–16) and
reads

######  Theorem 1.

Under Hypotheses (27–36), problem (14–16) admits at least one bounded energy
weak solution in the sense of Definition 1. Moreover, for all
$t\in\overline{I}$, $(R(t,x),Z(t,x))\in\overline{\cal O}$,
$\underline{F}R(t,x)\leq\varrho(t,x)\leq\overline{F}R(t,x)$ and
$\underline{G}Z(t,x)\leq z(t,x)\leq\overline{G}Z(t,x)$ for a.a. $x\in\Omega$,
and further for a.a. $(t,x)\in I\times\partial\Omega$,
$(R(t,x),Z(t,x))\in\overline{\cal O}$,
$\underline{F}R(t,x)\leq\varrho(t,x)\leq\overline{F}R(t,x)$ and
$\underline{G}Z(t,x)\leq z(t,x)\leq\overline{G}Z(t,x)$. Finally,
$\varrho,z,R,Z\in C(\overline{I};L^{1}(\Omega))$, $(\varrho+z){\bf u}\in
C_{weak}([0,T);$ $L^{\frac{2\gamma}{\gamma+1}}(\Omega;\mathbb{R}^{3}))$ and
$P(R,Z)\in L^{q}(I\times\Omega)$ for some $q>1$ and $Z\in
C_{weak}([0,T);L^{\beta}(\Omega)){\cap
L^{\beta}(I,L^{\beta}(\partial\Omega;|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x}))}$
if $\beta>\gamma$.

The second main result of the paper deals with the one velocity Baer-Nunziato
type system (3–8) and reads:

######  Theorem 2.

Suppose that $f,g\in{C^{1}(0,1)}$ are two strictly monotone and strictly
positive functions on interval $(0,1)$ and that the boundary conditions
$\varrho_{B}$, $z_{B}$, ${\bf u}_{B}$ satisfy conditions (27). Let $\gamma\geq
2$, $\beta>0$ and, in addition,

$\alpha_{B}\in
C(\overline{\Omega}),\;0<\underline{\alpha}\leq\alpha_{B}\leq\overline{\alpha}<1,\;(f(\alpha_{B})\varrho_{B},g(\alpha_{B})z_{B})(x)\in\overline{\cal
O},$ $\alpha_{0}\in
L^{\infty}(\Omega),\;0<\underline{\alpha}\leq\alpha_{0}\leq\overline{\alpha}<1,\;(f(\alpha_{0})\varrho_{0},g(\alpha_{0})z_{0})(x)\in\overline{\cal
O}$ ${0<\varrho_{0}}\in L^{\gamma}(\Omega),\;z_{0}\in
L^{\beta}(\Omega)\;\text{ if }\beta>\gamma,\;(\varrho_{0}+z_{0})|{\bf
u}_{0}|^{2}\in L^{1}(\Omega).$ (45)

Suppose that the domain $\Omega$ is a bounded Lipschitz domain with the
admissible inflow-outflow boundary with respect to ${\bf u}_{B}$, cf. (31).
Finally suppose that the pressure $P$ and its Helmholtz function $H$ verify
hypotheses (32–36). Then the problem (3–8) admits at least one bounded energy
weak solution in the sense of Definition 2. Moreover, for all
$t\in\overline{I}$, $(f(\alpha)\varrho(t,x),g(\alpha)z(t,x))\in\overline{\cal
O}$ and $\underline{\alpha}\leq\alpha(t,x)\leq\overline{\alpha}$ for a.a.
$x\in\Omega$, and further for a.a. $(t,x)\in I\times\partial\Omega$,
$(f(\alpha)\varrho(t,x),g(\alpha)z(t,x))\in\overline{\cal O}$ and
$\underline{\alpha}\leq\alpha(t,x)\leq\overline{\alpha}$. Finally,
$\alpha,\varrho,z\in C(\overline{I};L^{1}(\Omega))$, $z\in
C_{weak}(\overline{I};$ $L^{{\beta}}(\Omega)){\cap
L^{\beta}(I,L^{\beta}(\partial\Omega;|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x}))}$
if $\beta>\gamma$, $(\varrho+z){\bf u}\in C_{weak}([0,T);$
$L^{\frac{2\gamma}{\gamma+1}}(\Omega;\mathbb{R}^{3}))$ and
$P(f(\alpha)\varrho,g(\alpha)z)\in L^{q}(I\times\Omega)$ with some $q>1$.

######  Remark 2.

  1. 1.

As already mentioned in Remark 1, Theorems 1, 2 remain still valid if we
replace the hypotheses (34–36) for the constitutive law for pressure by weaker
hypotheses (42–44).

  2. 2.

The growth corresponding to $\beta\in(0,1]$ is covered by Theorems 1–2 only
provided $0<\underline{a}$ and provided one considers the version of the
theorems with weaker hypotheses (42–44). Indeed, $\beta\in(0,1]$ is in
contradiction with (36) whatever is the value of $\underline{a}$ and it is in
contradiction with (42) if $\underline{a}=0$.

Theorem 1 will be proved through Sections 4–7. Theorem 2 is proved in Section
8. Without loss of generality, we shall concentrate on the case when
$\Gamma^{\rm in}$ and $\Gamma^{\rm out}$ are not empty. For the sake of
simplicity, we shall also discard the situation, when $\Gamma^{\rm in}$ or
$\Gamma^{\rm out}$ contain a compact manifold. Finally, also for the sake of
simplicity, we shall limit ourselves to the case $\beta\leq\gamma$.

##  3 Transport equations with non homogenous boudary data

The properties of solutions to the continuity equation and an interplay
between the solutions of continuity and transport equations play essential
role in the proofs in this paper. These results are well known in the case,
when the transporting velocity ${\bf u}$ is zero at the boundary, essentially
due to seminal paper of Di-Perna, Lions [19], see Vasseur et al. [45], and
[41, Section 3], [40, Section 3] for further extension, still with the
condition ${\bf u}=0$ at $\partial\Omega$. The purpose of this section is to
extend them to non-homogenous boundary data. This section is therefore of
independent interest.

###  3.1 Some elements of differential geometry

Let $k=1,2,\ldots$. A set $\Gamma\subset\mathbb{R}^{3}$ (resp.
$\mathfrak{g}\subset\mathbb{R}^{3}$) is a $C^{k}$-parametrized bounded surface
(resp. $C^{k}$-parametrized bounded curve) iff there exists a bounded domain
${\cal O}\subset\mathbb{R}^{2}$ (resp. ${\cal O}\subset\mathbb{R}$) a
bijection ${\cal G}\in C^{k}(\overline{\cal O};\overline{\Gamma})$ (resp.
${\cal G}\in C^{k}(\overline{\cal O};\overline{\mathfrak{g}})$) such that
${\cal G}$ is an $C^{1}$-differomorphism from ${\cal O}$ onto $\Gamma$ (resp.
$\mathfrak{g}$).444 Meaning that ${\cal G}$ is a bijection of regularity
$C^{1}({\cal O})$ and the differential $\nabla{\cal G}(x)$ is a bijection of
$R^{2}$ to $R^{2}$ (resp. $R$ to $R$) for any $x\in{\cal O}$. For a set
$A\subset\mathbb{R}^{3}$ we denote $d_{A}(x)={\rm dist}(x;A):=\inf_{y\in
A}|x-y|$ the distance function to $A$ and ${\bf
n}=\frac{\partial_{1}G\times\partial_{2}G}{|{\partial_{1}G\times\partial_{2}G}|}$
the (outer) normal vector to the surface $\Gamma\subset\partial\Omega$ with
respect to $\Omega$.

We define

$T(\Gamma;\varepsilon):=\\{x\in\mathbb{R}^{3}\,|\,x=x_{\Gamma}+s{\bf
n}(x_{\Gamma}),\,x_{\Gamma}\in\Gamma,\,s\in{(-\varepsilon,\varepsilon)}\\},$
${T^{\pm}(\Gamma;\varepsilon)}:=\\{x\in\mathbb{R}^{3}\,|\,x=x_{\Gamma}+s{\bf
n}(x_{\Gamma}),\,x_{\Gamma}\in\Gamma,\,\pm s\in{(0,\varepsilon)}\\},$ (46)
$T(\mathfrak{g};\varepsilon):=B(\mathfrak{g};\varepsilon)\setminus\overline{B(\partial\mathfrak{g};\varepsilon)},$

where, for a subset $A$ of $\mathbb{R}^{3}$,

$B(A;\varepsilon):=\\{x\in\mathbb{R}^{3}|{\rm dist}(A;x)<\varepsilon\\}.$

It is well known that $d_{A}$ is 1-Lipschitz function on $\mathbb{R}^{3}$ and
if $A$ is closed, for almost all $x\notin A$, there exists a unique point
$P_{A}(x)\in A$ nearest to $A$ such that

$\nabla d_{A}(x)=\frac{x-P_{A}(x)}{d_{A}(x)},$ (47)

cf. Ziemer [46, Exercice 1.15]. We also recall that if $k\geq 1$,

$|T(\Gamma;\varepsilon)|\stackrel{{\scriptstyle<}}{{\sim}}\varepsilon,\quad|T(\mathfrak{g};\varepsilon)|\stackrel{{\scriptstyle<}}{{\sim}}\varepsilon^{2},\
|B(\partial\mathfrak{g};\varepsilon)|\stackrel{{\scriptstyle<}}{{\sim}}\varepsilon^{3}$
(48)

cf. Gray [31].

The following Lemma resumes the properties of distance and projection to
$\Gamma$ which can be deduced from Theorems 1,2 in Foote [27].

######  Lemma 3.

Let $k\geq 2$. Then there exists $\overline{\varepsilon}>0$ such that for all
$0<\varepsilon\leq\overline{\varepsilon}$, we have the following:

  1. 1.

$T_{\varepsilon}:=T(\Gamma;\varepsilon)$,
$T^{\pm}_{\varepsilon}:=T^{\pm}(\Gamma;\varepsilon)$ are open sets in
$\mathbb{R}^{3}$.

  2. 2.

$\forall x\in
T_{\varepsilon},\;\exists!x_{\Gamma}\in\Gamma,\;|x_{\Gamma}-x|=d_{\Gamma}(x)$.
We denote $P_{\Gamma}(x):=x_{\Gamma}$. Then $P_{\Gamma}\in
C^{k-1}(T_{\varepsilon})$.

  3. 3.

$d_{\Gamma}\in C^{k}({T_{\varepsilon}^{+}\cup\Gamma})\cup
C^{k}(T_{\varepsilon}^{-}\cup\Gamma)$ and for all $x\in
T^{\pm}_{\varepsilon}$, $\nabla
d_{\Gamma}(x)=\frac{x-P_{\Gamma}(x)}{|x-P_{\Gamma}(x)|}=\pm{\bf
n}(P_{\Gamma}(x))$.

###  3.2 Renormalized solutions to families of transport equations

We consider the general transport equations on the time-space cylinder
$Q_{T}=I\times\Omega$, $\Omega$ a bounded Lipschitz domain in $\mathbb{R}^{3}$
and $I=(0,T)$, $T>0$ a time interval. The equations read:

${\partial_{t}r+{\rm div}\,(r{\bf u})+rv=0\;\mbox{in $(0,T)\times\Omega$}}$
(49)

with initial and boundary conditions

$r(0,\cdot)=r_{0}(\cdot)\;\mbox{in $\Omega$},\;r|_{I\times\Gamma_{\rm
in}}=r_{B}.$

Equation (49) is called continuity equation if $v=0$ and pure transport
equation if $v=-{\rm div}\,{\bf u}$.

We shall consider the following regularity of transporting coefficients

${\bf u}={\bf v}+{\bf u}_{B},\;{\bf v}\in
L^{2}(I;W_{0}^{1,2}(\Omega;\mathbb{R}^{3})),\;v\in L^{2}(Q_{T}).$ (50)

A function $0\leq r\in L^{2}(Q)\cap L^{\infty}(I,L^{\gamma}(\Omega))\cap
L^{\gamma}(I;L^{\gamma}(\partial\Omega;|{\bf u}_{B}\cdot{\bf n}|{\rm
d}S_{x}))$ is a weak solution of the transport equation (49) with boundary
data $r_{B}$ and initial data $r_{0}$ iff

$\int_{0}^{T}\int_{\Gamma^{\rm in}}r_{B}{\bf u}_{B}\cdot{\bf n}{\rm d}S{\rm
d}t+\int_{0}^{T}\int_{\Gamma^{\rm out}}r{\bf u}_{B}\cdot{\bf n}{\rm d}S{\rm
d}t-\int_{\Omega}r_{0}\varphi(0,\cdot){\,\rm d}x$ (51)
$=\int_{0}^{T}\int_{\Omega}\Big{(}r\partial_{t}\varphi+r{\bf
u}\cdot\nabla\varphi-rv\varphi\Big{)}{\,\rm d}x{\rm d}t$

holds with any $\varphi\in C^{1}_{c}([0,T)\times\overline{\Omega})$.

We shall need the following result on the renormalized solutions to the
transport equation, which is of independent interest. It is formulated in the
specific functional setting needed for the purpose of this paper and its
formulation could be easily generalized to the $L^{p}-L^{q}$ setting.

######  Proposition 4.

[Renormalized solutions to families of transport equations] Let
$\Omega\subset\mathbb{R}^{3}$ be a bounded Lipschitz domain and $\gamma>1$.
Let ${\bf u}_{B}$ belong to class (27), let each component of
${\mathfrak{r}}_{B}=(r_{B}^{(1)},\ldots,r_{B}^{(M)})$,
$\mathfrak{s}_{B}=(s_{B}^{(1)},\ldots,s_{B}^{(N)})$ be non negative function
in $C(\overline{\Omega})$ and let each component of
${\mathfrak{r}}_{0}=(r_{0}^{(1)},\ldots,r_{0}^{(M)})$,
${\mathfrak{s}}_{0}=(s_{0}^{(1)},\ldots,s_{0}^{(N)})$ be non negative and
belong to $L^{\gamma}(\Omega)$ (for the components of $\mathfrak{r}$) resp. to
$L^{\infty}(\Omega)$ (for the components of $\mathfrak{s}$). Suppose that
$\Omega$ has an admissible inflow-outflow boundary relative to ${{\bf u}}_{B}$
in the sense of Definition 3.555 As we already mentioned in Definition 3,
condition that each “$\Gamma=\Gamma^{\rm in/out}_{\rm\cdot}$ is a
$C^{2}$-surface or a $C^{2}$ compact manifold” can be relaxed. It is a
sufficient contition to guarantee existence of a projection $P$ to $\Gamma$ on
a neighborghood $U$ of $\Gamma$, which is continuous on $U$. Indeed, this is
the only condition from conclusion of Lemma 3 which is used in the proof.
Also, conditions on $\mathfrak{g}^{\rm bd}$ in Definition 3 could be relaxed.
The only thing we need in the proofs is that $\partial\Gamma^{\rm
in}\cup\partial\Gamma^{\rm out}$ satisfies (48). Assume further that ${\bf u}$
belongs to the class (50) and that

$0\leq\mathfrak{r}=[(r^{(1)},\ldots,r^{(M)}]\in L^{2}(I;L^{2}(\Omega))\cap
L^{\infty}(I,L^{\gamma}(\Omega))\cap{L^{\gamma}(I;L^{\gamma}(\Gamma^{\rm
out};|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x})),}$
$0\leq\mathfrak{s}=[s^{(1)},\ldots,s^{(N)})\in
L^{\infty}((0,T)\times\Omega){\cap L^{\infty}(I;L^{\infty}({\Gamma^{\rm
out}};|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x}))}$

are such that each component of $\mathfrak{r}$ is a weak solution of the
continuity equation (49) (i.e., it satisfies (51) with ${\bf u}$ and $v=0$),
while each component of $\mathfrak{s}$ is a weak solution the pure transport
equation (49) (i.e., it satisfies (51) with ${\bf u}$ and $v=-{\rm div}\,{\bf
u}$). In the above the sign ”$\leq$” means that each component of the vector
is a non negative number.

Then there holds:

  1. 1.

The quantities $\mathfrak{r},B(\mathfrak{r})\in C_{\rm
weak}(\overline{I};L^{\gamma}(\Omega))\cap C(\overline{I};L^{p}(\Omega))$,
$1\leq p<\gamma$ and we have

$\int_{\Omega}B(\mathfrak{r})\varphi{\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}B(\mathfrak{r}){{\bf u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm in}}B(\mathfrak{r}_{B}){{\bf
u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm d}t$ (52)
$=\int_{0}^{\tau}\int_{\Omega}\Big{(}{B({\mathfrak{r}})\partial_{t}\varphi}+B(\mathfrak{r}){{\bf
u}}\cdot\nabla\varphi-\varphi\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{{\bf u}}\Big{)}{\,\rm d}x{\rm d}t,\
B(\mathfrak{r}(0))=B(\mathfrak{r}_{0})$

with any $\tau\in\overline{I}$, for any $\varphi\in
C_{c}^{1}([0,T]\times\overline{\Omega}$) and any

$B\in C([0,\infty)^{M}),\;\nabla_{\mathfrak{r}}B\in
L^{\infty}((0,\infty)^{M}).$ (53)

  2. 2.

The quantities $\mathfrak{s},B(\mathfrak{s})\in C([0,T];L^{p}(\Omega))$,
$1\leq p<\infty$ and we have

$\int_{\Omega}B(\mathfrak{s})\varphi{\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}B(\mathfrak{s}){{\bf u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm in}}B(\mathfrak{s}_{B}){{\bf
u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm d}t$ (54)
$=\int_{0}^{\tau}\int_{\Omega}\Big{(}{B({\mathfrak{s}})\partial_{t}\varphi}+B(\mathfrak{s}){{\bf
u}}\cdot\nabla\varphi+\varphi B(\mathfrak{s}){\rm div}\,{{\bf u}}\Big{)}{\,\rm
d}x{\rm d}t,\ B(\mathfrak{s}(0))=B(\mathfrak{s}_{0})$

with any $\tau\in\overline{I}$, for any $\varphi\in
C_{c}^{1}(\overline{I}\times\overline{\Omega})$ and any $B$ in class

$B\in C([0,\infty)^{N})\cap C^{1}((0,\infty)^{N}).$ (55)

  3. 3.

Let now $r$ be one component of $\mathfrak{r}$. Then $rB(\mathfrak{s})\in
C_{\rm weak}([0,T];L^{\gamma}(\Omega))\cap C([0,T];L^{p}(\Omega))$, $1\leq
p<\gamma$ and we have:

$\int_{\Omega}rB(\mathfrak{s})\varphi{\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}rB(\mathfrak{s}){{\bf u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm in}}r_{B}B(\mathfrak{s}_{B}){{\bf
u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm d}t$ (56)
$=\int_{0}^{\tau}\int_{\Omega}\Big{(}{rB({\mathfrak{s}})\partial_{t}\varphi}+rB(\mathfrak{s}){{\bf
u}}\cdot\nabla\varphi\Big{)}{\,\rm d}x,\
\mathfrak{r}(0)B(\mathfrak{s}(0))=\mathfrak{r}_{0}B(\mathfrak{s}_{0})$

with any $\tau\in\overline{I}$, for any $\varphi\in
C_{c}^{1}(\overline{I}\times\overline{\Omega}))$, and any $B$ in the class
(55).

######  Remark 3.

  1. 1.

The condition (53) on the renormalizing function $B$ in Lemma 4 can be relaxed
by using the Lebesgue dominated convergence theorem: one can take, e.g., $B\in
C([0,\infty)^{M})$ $\cap$ $C^{1}((0,\infty)^{M})$,
$\mathfrak{r}\cdot\nabla_{{\mathfrak{r}}}B-B\in C[0,\infty)$ and
$|B(\mathfrak{r})|\leq c(1+r)^{q_{1}}$,
$|\mathfrak{r}\cdot\nabla_{{\mathfrak{r}}}B(\mathfrak{r})-B(\mathfrak{r})|\stackrel{{\scriptstyle<}}{{\sim}}c(1+|\mathfrak{r}|)^{q_{2}}$,
$0\leq q_{1}\leq\frac{5}{6}\gamma$, $0\leq q_{2}\leq\min\\{1,\gamma/2\\}$. In
this case, relation (52) remains still valid and $B(\mathfrak{r})\in C_{\rm
weak}(\overline{I};L^{\gamma/q_{1}}(\Omega))$ $\cap$
$C(\overline{I};L^{p}(\Omega))$, $1\leq p<\gamma/q_{1}$.

  2. 2.

Equation (52) implies

$-\int_{0}^{T}\int_{\Omega}B({\mathfrak{r}})\eta\partial_{t}\varphi{\,\rm
d}x{\rm d}t=<{\cal F},\varphi>,\;\forall\varphi\in
C^{1}_{c}((0,T)\times\overline{\Omega}),\;\forall\eta\in C^{1}_{c}(\Omega).$

where

$<{\cal F},\varphi>=\int_{\Omega}B(\mathfrak{r})\eta{{\bf
u}}\cdot\nabla\varphi+B(\mathfrak{r})\varphi{{\bf
u}}\cdot\nabla\eta-\varphi\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{{\bf u}}\Big{)}{\,\rm d}x{\rm d}t$ $-\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}B(\mathfrak{r}){{\bf u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm
d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}B(\mathfrak{r}_{B}){{\bf
u}}_{B}\cdot{\bf n}\varphi\ {\rm d}S_{x}{\rm d}t.$

We verify by using the Hölder inequality and the trace theorem that

$\partial_{t}[B(\mathfrak{r})\eta]\in
L^{2}(I;[W^{1,p}(\Omega)]^{*})\,\mbox{with
$\frac{1}{p}+\frac{q}{\gamma}=\frac{5}{6}$, $q=\max\\{q_{1},q_{2}\\}$}.$

Proof of Proposition 4
We shall limit ourselves to show Item 1. (i.e. (52)) in the ”simple
situation”$M=1$. To this end we set $\mathfrak{r}=r^{(1)}$,
$\mathfrak{r}_{0}=r^{(1)}_{0}$, $\mathfrak{r}_{B}=r^{(1)}_{B}$ and consider
$B\in C([0,\infty))$, $B^{\prime}\in L^{\infty}(0,\infty)$. The proof is
divided into seven steps. Extension from $M=1$ to $M>1$ is briefly described
in Step 8. The general case and the proof of Items 2.-3. follow the same
strategy.

In Step 1 we construct convenient outer neighborhoods ${\cal U}^{+}$ of
$\Gamma^{0}$ and of each component $\Gamma\subset\Gamma^{\rm out/in}$. In
particular, the projection operator$P$ (cf. Lemma 3) must be sufficiently
regular on the outer neighborhoods of components of $\Gamma^{\rm out/in}$. In
Step 2, we construct an outer neighborhood ${\cal V}^{+}\subset{\cal U}^{+}$
of any component $\Gamma\subset\Gamma^{\rm in}$ to which the density field can
be extended via the characteristics of the vector field $-{\bf u}_{B}$ in such
a way that the couple $(\mbox{density},{\bf u}_{B})$ satisfies the continuity
equation on $(0,T)\times{\cal V}^{+}$–see (70). In Step 3, we do the same for
any component $\Gamma\subset\Gamma^{\rm out}$– see (73), and, in Step 4., for
$\Gamma^{0}$. The global extension is then defined in Step 5.; it satisfies
the continuity equation in the sense of distributions on a domain
$\tilde{\Omega}$ created as union of $\Omega$ with $\Gamma$’s and their outer
neighboghoods. It provides enough space to apply the DiPerna-Lions
regularization procedure (cf. [19]) in Step 6.. The result of Steps 1-6, is
the renormalized continuity equation satisfied with test functions up to
$\overline{\Omega}$ but with compact support in $I$. The extension to its time
integrated form is discussed in Step 7.

Step 1: Construction of a particular outer neighborhoods of $\Gamma^{\rm in}$,
$\Gamma^{\rm out}$ and $\Gamma^{0}$–cf. (9), (10) and Definition 3. We denote,

$B^{+}(\xi,\varepsilon)=B(\xi,\varepsilon)\cap\mathbb{R}^{3}\setminus\overline{\Omega},\;B^{-}(\xi,\varepsilon)=B(\xi,\varepsilon)\cap\Omega,$
(57)

  1. 1.

Let $\Gamma=\Gamma^{0}$. Then for any $\xi\in\Gamma$ there exists
$\epsilon(\xi)$ such that
$B^{+}(\xi,\epsilon)\cap\partial\Omega\subset\Gamma$. We then set

${\cal U}^{+}(\Gamma)=\cup_{\xi\in\Gamma}B^{+}(\xi;\epsilon).$ (58)

  2. 2.

Let $\Gamma$ be any parametrized surface in the decomposition (25). We deduce
that

$\forall\xi\in\Gamma,\;\exists\epsilon=\epsilon(\xi)\in(0,\varepsilon),\;B(\xi,\epsilon){\subset{T}}(\Gamma;\varepsilon),$
(59)

where $T(\Gamma;\varepsilon)$ is defined in Lemma 3. We define open sets

${\cal U}={\cal
U}_{\varepsilon}(\Gamma):=\cup_{\xi\in\Gamma}B(\xi;\epsilon(\xi)),\quad{\cal
U}^{\pm}={\cal
U}_{\varepsilon}^{\pm}(\Gamma):=\cup_{\xi\in\Gamma}B^{\pm}(\xi;\epsilon(\xi)).$
(60)

Taking into account Lemma 3, we deduce that for any $\varepsilon>0$
sufficiently small, we have, in particular,

${\cal U}={\cal U}^{+}\cup{\cal U}^{-}\cup\Gamma,\ {\cal
U}^{+}\subset\mathbb{R}^{3}\setminus\overline{\Omega},\ {\cal
U}^{-}\subset\Omega,$

and

$P_{\Gamma}\in{C^{1}({\overline{\cal
U}})},\;d_{\Gamma}\in{C^{2}(\overline{{\cal U}^{+}})\cap C^{2}(\overline{{\cal
U}^{-}}),\ \forall x\in{\cal U}^{\pm},\ \nabla d_{\Gamma}(x)=\pm{\bf
n}(P(x))}$ (61)

  3. 3.

We realize that construction (58), (60) can be done in such a way that

${\cal U}^{+}(\Gamma)\cap{\cal U}^{+}(\tilde{\Gamma})=\emptyset,\;\mbox{for
any couple $\Gamma\neq\tilde{\Gamma}$ in the decomposition (\ref{Gin0}) }$
(62) $\mbox{or for $\Gamma$ in the decomposition (\ref{Gin0}) and
$\tilde{\Gamma}=\Gamma^{0}$}.$

The goal now is to extend the density and velocity fields $(\mathfrak{r},{\bf
u})$ from $\Omega$ to an outer neighborhood of $\Gamma$ (which is a subset of
${\cal U}^{+}$ and which we will denote for a moment ${\cal V}^{+}(\Gamma)$)
in such a way that the extended fields will satisfy the continuity equation in
the sense of distributions on the open set $\Omega\cup{{\cal
V}^{+}(\Gamma)}\cup\Gamma)$. The construction will depend on the fact whether
$\Gamma\subset\Gamma^{\rm in}$ or $\Gamma\subset\Gamma^{\rm out}$ or
$\Gamma=\Gamma^{0}$.

Step 2: Extension of the density beyond the inflow boundary:

  1. 1.

${\cal V}^{+}$ in the case $\Gamma=\Gamma^{\rm in}$.

    1. (a)

Flux of $-{\bf u}_{B}$. We denote by $\mathfrak{X}$ the flux of the vector
field $-{\bf u}_{B}$, i.e. solution of the following family of Cauchy problems
for ODE,

$\frac{\rm d}{{\rm d}s}\mathfrak{X}(s;x)=-{\bf
u}_{B}(\mathfrak{X}),\;\mathfrak{X}(0;x)=x,\;s\in\mathbb{R},\;x\in\mathbb{R}^{3}.$
(63)

It is well known, cf. e.g. [17, Chapter XI], that,

$\mathfrak{X}\in
C^{1}(\mathbb{R}\times\mathbb{R}^{3}),\;\mathfrak{X}(t,\cdot)\;\mbox{is
$C^{1}$ diffeomorphism of $\mathbb{R}^{3}$ onto $\mathbb{R}^{3}$},$

in particular,

$\forall(s,x)\in\mathbb{R}\times\mathbb{R}^{3},\;\mathfrak{X}(-s,\mathfrak{X}(s;x))=x,\;{\rm
det}\Big{[}\nabla\mathfrak{X}(s;x)\Big{]}={\rm exp}\Big{(}-\int_{0}^{s}{\rm
div}\,{\bf u}_{B}(z,\mathfrak{X}(z,x)){\rm d}z\Big{)}>0.$

    2. (b)

Construction of ${\cal V}$, ${\cal V}^{\pm}$.

      1. i.

Let $K\subset\Gamma$ be a compact set (with respect to the trace topology of
$\mathbb{R}^{3}$ on $\partial\Omega$) where $\Gamma\subset\Gamma^{\rm in}$ is
any component in decomposition (25). We want to prove that
$\forall\varepsilon>0$, $\exists\delta=\delta_{K}>0,$

$\forall(s,\xi)\in(-\delta,\delta)\times K,\;\mathfrak{X}(s,\xi)\subset{\cal
U}_{\varepsilon}(\Gamma)\;\mbox{and}\;\forall(s,\xi)\in(0,\delta)\times
K,\;\mathfrak{X}(s,\xi)\subset{\cal U}^{+}_{\varepsilon}(\Gamma).$ (64)

Indeed:

      2. ii.

By the uniform continuity of $\mathfrak{X}$ on compacts of
$\mathbb{R}\times\mathbb{R}^{3}$ we easily get

$\exists\delta>0,\;\mathfrak{X}((-\delta,\delta);K)\subset{\cal
U}_{\varepsilon}(\Gamma).$

      3. iii.

Moreover, due to (63),

$\forall x_{B}\in\Gamma^{\rm in},\;\exists\delta>0,\;\forall
s\in(0,\delta),\;(\mathfrak{X}(s,x_{B})-P(\mathfrak{X}(0,x_{B})))\cdot{\bf
n}(x_{B})>0.$

By the uniform continuity of ${\bf n}$ on compacts of $\Gamma$, $P$ on
compacts of ${T}(\Gamma)$ and $\mathfrak{X}$ on compacts of
$\mathbb{R}\times\mathbb{R}^{3}$, we deduce, in particular, that

$\exists\delta>0,\;\forall(s,\xi)\in(0,\delta)\times
K,\;\Big{(}\mathfrak{X}(s,\xi)-P(\mathfrak{X}(s,\xi))\Big{)}\cdot{\bf
n}(P(\mathfrak{X}(s,\xi))>0.$

This means that $\mathfrak{X}((0,\delta);K)\subset{\cal
U}^{+}_{\varepsilon}(\Gamma)$ which finishes the proof of (64)

      4. iv.

Recalling Definition 3 and the definition of the parametrized surface in
Section 3.1, we know that $\overline{\Gamma}={\cal G}(\overline{\cal O})$
where ${\cal G}\in C^{2}(\overline{\cal O};\mathbb{R}^{3})$ is a
$C^{2}$-diffeomorhism of a domain ${\cal O}\in\mathbb{R}^{2}$ onto $\Gamma$.
Let $L_{n}$ be an exhaustive sequence of compacts of ${\cal O}$,

$L_{n}\subset{\rm int}_{2}L_{n+1},\;\cup_{n=1}^{\infty}L_{n}={\cal O}$ (65)

so that $K_{n}:={\cal G}(L_{n})$ is an exhaustive sequence of compacts in
$\Gamma$ (one can take $L_{n}=\\{x\in{\cal O}\,|\,{\rm
dist}(x,R^{2}\setminus\overline{{\cal O}})\geq 1/n\\}$).

We define

${\cal W}={\cal W}_{\varepsilon}={\cal W}_{\varepsilon}(\Gamma),\;{\cal
W}^{+}={\cal W}^{+}_{\varepsilon}={\cal
W}_{\varepsilon}^{+}(\Gamma)=\cup_{n\in N}{\cal W}^{+}_{n}$ (66) ${\cal
V}={\cal V}_{\varepsilon}={\cal V}_{\varepsilon}(\Gamma)=\cup_{n\in N}{\cal
V}_{n},\;{\cal V}^{+}={\cal V}^{+}_{\varepsilon}={\cal
V}_{\varepsilon}^{+}(\Gamma)=\cup_{n\in N}{\cal V}^{+}_{n},$

where

${\cal W}_{n}=(-\delta_{K_{n}},\delta_{K_{n}})\times{\rm int}_{2}L_{n},\;{\cal
W}_{n}^{+}=(0,\delta_{K_{n}})\times{\rm int}_{2}L_{n},$ ${\cal
V}_{n}:=\mathfrak{X}((-\delta_{K_{n}},\delta_{K_{n}});{\cal G}({\rm
int}_{2}L_{n}))\subset{\cal U}_{\varepsilon}(\Gamma),\;{\cal
V}^{+}_{n}:=\mathfrak{X}((0,\delta_{K_{n}});{\cal G}({\rm
int}_{2}L_{n}))\subset{\cal U}_{\varepsilon}^{+}(\Gamma).$

    3. (c)

Construction of a local diffeomorphism

      1. i.

Now, we define a map,

$\Phi:(-\infty,\infty)\times{\cal O}\ni(s,\zeta)\mapsto\mathfrak{X}(s;{\cal
G}(\zeta))\in\mathbb{R}^{3}.$ (67)

Clearly, $\Phi$ is contiunous and $\Phi(0,\cdot)$ is a bijection from $L_{n}$
to $K_{n}$. We shall prove that $\Phi|_{{\cal W}_{n}}$ is a $C^{1}$ local
diffeomorphisms of ${\cal W}_{n}$ onto ${\cal V}_{n}$. Likewise, $\Phi|_{{\cal
W}_{n}^{+}}$ is a $C^{1}$ local diffeomorphisms of ${\cal W}_{n}^{+}$ onto
${\cal V}_{n}^{+}$. In particular, ${\cal V}$, ${\cal V}^{+}$ and ${\cal
V}^{+}\cup\Gamma\cup\Omega$ are open.

      2. ii.

Indeed, in view of the theorem of local inversion, it is enough to show that

$\forall\zeta\in{\cal O},\;s\in R,\;{\rm
det}\Big{[}\partial_{s}\Phi,\nabla_{\zeta}\Phi\Big{]}(s,\zeta)\neq 0.$

Seeing that, $\mathfrak{X}(s;\mathfrak{X}(-s;\xi))=\xi$, we infer

$\forall\xi\in\mathbb{R}^{3},\;s\in\mathbb{R},\quad\partial_{s}\mathfrak{X}(s;\mathfrak{X}(-s;\xi))+{\bf
u}_{B}(\mathfrak{X}(-s;\xi))\cdot\nabla_{x}\mathfrak{X}(s;\mathfrak{X}(-s;\xi))=0,$

i.e., equivalently,

$\partial_{s}\Phi(s,\zeta)=-{\bf u}_{B}({\cal
G}(\zeta))\cdot\nabla_{\xi}\mathfrak{X}(s;{\cal G}(\zeta)),\;\mbox{in
particular, for all $s>0$, $\zeta\in{\cal O}$,}$

we easily find that

$\Big{[}\partial_{s}\Phi,\partial_{\zeta_{1}}\Phi,\partial_{\zeta_{2}}\Phi\Big{]}(s,\zeta)=-\begin{bmatrix}[{\bf
u}_{B}({\cal G}(\zeta))]^{T}\\\ [\partial_{\zeta_{1}}{\cal G}(\zeta)]^{T}\\\
[\partial_{\zeta_{1}}{\cal G}(\zeta)]^{T}\\\
\end{bmatrix}\;\begin{bmatrix}\nabla\mathfrak{X}_{1}(s;{\cal
G}(\zeta)),\nabla\mathfrak{X}_{2}(s;{\cal
G}(\zeta)),\nabla\mathfrak{X}_{3}(s;{\cal G}(\zeta))\end{bmatrix},$

where vectors and $\nabla$ are columns. Whence,

${\rm det}\Big{[}\partial_{s}\Phi,\nabla_{\zeta}\Phi\Big{]}(s,\zeta)=-{\bf
u}_{B}\cdot{\bf n}({\cal G}(\zeta)){\rm exp}\Big{(}-\int_{0}^{s}{\rm
div}\,{\bf u}_{B}(z,\mathfrak{X}(z;{\cal G}(\zeta))){\rm d}z\Big{)}>0$

for all $\zeta\in{\cal O}$ and $s\in R$.

    4. (d)

A diffeomorphism induced by $\Phi$. Finally, we observe (employing the uniform
continuity and the fact that $\Phi$ is a local diffeomorphism), that
$\delta_{K_{n}}$ can be chosen so small that

$\forall s_{1},s_{2}\in(-\delta_{K_{n}},\delta_{K_{n}}),\
\Phi(s_{1},L_{n})\cap\Phi(s_{2},L_{n})=\emptyset.$

Consequently, in particular,

$\forall\xi\in{\cal
V}^{+},\;\exists!(s,x_{B})\in{\mathfrak{W}}^{+}\;\mbox{such that}\
\xi=\mathfrak{X}(s,x_{B}),$ (68)

where $\mathfrak{W}^{+}=\cup_{n\in N}(0,\delta_{K_{n}})\times{\cal G}({\cal
O}_{n})$.

  2. 2.

Extension of the density beyond the inflow boundary. We may therefore extend
the boundary data to ${\cal V}^{+}$ by setting

$\tilde{\mathfrak{r}}(t,\xi)=\mathfrak{r}_{B}({x}_{B}){\rm
exp}\Big{(}\int_{0}^{s}{\rm div}{\bf u}_{B}(\mathfrak{X}(z;{x}_{B})){\rm
d}z\Big{)},\ \mbox{where $\xi=\mathfrak{X}(s,x_{B})$}.$ (69)

Clearly, $\tilde{\mathfrak{r}}\in C^{1}(\overline{I}\times\overline{{\cal
V^{+}}})$ and

$\partial_{t}\tilde{\mathfrak{r}}+{\rm div}\,(\tilde{\mathfrak{r}}{\bf
u}_{B})=0\ \mbox{in}\ (0,T)\times{\cal V}^{+},$ (70)

and

$\partial_{t}B(\tilde{\mathfrak{r}})+{\rm div}(B(\tilde{\mathfrak{r}}){\bf
u}_{B})+[\tilde{{\mathfrak{r}}}B^{\prime}(\tilde{\mathfrak{r}})-B(\tilde{\mathfrak{r}})]{\rm
div}{\bf u}_{B}=0\ \mbox{in $(0,T)\times\mathcal{V}^{+}$.}$ (71)

Step 3: Extension of the density beyond the outflow boundary

  1. 1.

We construct the open set $\mathcal{V}$, $\mathcal{V}^{\pm}$ as in the Step
3.2 using the flow determined by EDO (63), where we replace $-{\bf u}_{B}$ by
${\bf u}_{B}$. We denote this flow again by $\mathfrak{X}$ and the
corresponding diffeomorphisme by $\Phi$. In particular, for any
$\xi\in\mathcal{V}^{+}$ there exists unique $(s,x_{B})\in\mathfrak{W}^{+}$
such that $\mathfrak{X}(s,x_{B})=\xi$.

  2. 2.

We take a sequence

$C_{c}^{1}(\Gamma)\ni\mathfrak{r}^{n}\to\mathfrak{r}\ \mbox{in
$L^{\gamma}(I;L^{\gamma}(\Gamma;|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x}))$}.$
(72)

  3. 3.

We extend $\mathfrak{r}^{n}$ from $\Gamma$ to ${\cal V}^{+}$:
$\forall(t,\xi)\in[0,T]\times\mathcal{V}^{+}$,

$\tilde{\mathfrak{r}^{n}}(t,\xi)=\left\\{\begin{array}[]{c}{\mathfrak{r}}^{n}(t-s,x_{B}){\rm
exp}\Big{(}-\int_{0}^{s}{\rm div}{\bf u}_{B}(\mathfrak{X}(\zeta,x_{B})){\rm
d}\zeta\Big{)},\;\xi=\mathfrak{X}(s,x_{B})\;\mbox{if $t\geq s$}\\\
{\mathfrak{r}^{n}}(0,x_{B}){\rm exp}\Big{(}-\int_{0}^{s}{\rm div}{\bf
u}_{B}(\mathfrak{X}(\zeta,x_{B})){\rm
d}\zeta\Big{)},\;\xi=\mathfrak{X}(s,x_{B})\;\mbox{if $0\leq
t<s$}\end{array}\right\\}$ (73)

and easily verify that that the couple $(\tilde{{\mathfrak{r}}^{n}},{\bf
u}_{B})$ verifies equations (70)–(71).

  4. 4.

Seeing (72) an using change of variables $\xi\to\Phi(s,x_{B})$ we infer

$\tilde{\mathfrak{r}^{n}}\to\tilde{\mathfrak{r}}\ \mbox{in
$L^{\gamma}(I;L^{\gamma}({\cal V}^{+}_{\mathfrak{h}}))$},\ {\cal
V}^{+}_{\mathfrak{h}}=({\cal V}^{+}\cup\Gamma)\setminus
B(\partial\Gamma;\mathfrak{h})$ (74)

with any $0<\mathfrak{h}$ sufficiently small. Consequently, we deduce from
(73),

$\int_{0}^{T}\int_{{\cal
V}^{+}}{\Big{(}\tilde{\mathfrak{r}}\partial_{t}\varphi+\tilde{\mathfrak{r}}{\bf
u}\cdot\nabla{\bf
u}_{B}\Big{)}}-\int_{0}^{T}\int_{\Gamma}\tilde{\mathfrak{r}}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t=0,\ \varphi\in C^{1}_{c}((0,T)\times({\cal
V}^{+}\cup\Gamma))$ (75)

and

$\int_{0}^{T}\int_{{\cal
V}^{+}}\Big{(}B(\tilde{\mathfrak{r}})\partial_{t}\varphi+B(\tilde{\mathfrak{r}})\cdot{\bf
u}_{B}\cdot\nabla\varphi-[\tilde{\mathfrak{r}}B^{\prime}(\tilde{\mathfrak{r}})-B(\mathfrak{r})]{\rm
div}{\bf u}\Big{)}{\rm d}x{\rm d}t$ (76)
$-\int_{0}^{T}\int_{\Gamma}B(\tilde{\mathfrak{r}})\cdot{\bf n}\varphi{\rm
d}S_{x}{\rm d}t=0,\ \varphi\in C^{1}_{c}((0,T)\times({\cal
V}^{+}\cup\Gamma)),$

where, here, ${\bf n}$ is the outer normal to $\mathcal{V}^{+}$ (at $\Gamma$).

Step 4: Extension of the density field beyond the slip boundary.
Let now $\Gamma=\Gamma^{0}$. We take ${\cal V}^{+}={\cal
V}_{\varepsilon}^{+}(\Gamma)={\cal U}^{+}(\Gamma)$, cf. (58), and we set in
this case simply

${\tilde{\mathfrak{r}}}(t,x)=0,\;(t,x)\in(0,T)\times{\cal V}^{+}.$ (77)

Clearly, equations (70) and (71) hold in this case.

Step 5: Continuity equation extended.
Referring to the decomposition (25), we construct ${\cal V}^{+}(\Gamma^{\rm
in}_{{\mathfrak{k}}_{\rm in}})$ according to (66), ${\cal V}^{+}(\Gamma^{\rm
out}_{{\mathfrak{k}}_{\rm out}})$ according to Item 1 in Step 3, and ${\cal
V}^{+}(\Gamma^{0})$ according to (58), cf. (77). These open sets are mutually
disjoint by virtue of (62). Finally, we set,

$\tilde{\cal V}^{+}:=\Big{[}\cup_{{\mathfrak{k}}_{\rm
in}}^{\overline{{\mathfrak{k}}_{\rm in}}}\Big{(}{\cal V}^{+}(\Gamma^{\rm
in}_{{\mathfrak{k}}_{\rm in}})\cup\Gamma^{\rm in}_{{\mathfrak{k}}_{\rm
in}}\Big{)}\Big{]}\cup\Big{[}\cup_{{\mathfrak{k}}_{\rm
in}}^{\overline{{\mathfrak{k}}_{\rm in}}}\Big{(}{\cal V}^{+}(\Gamma^{\rm
out}_{{\mathfrak{k}}_{\rm in}})\cup\Gamma^{\rm out}_{{\mathfrak{k}}_{\rm
in}}\Big{)}\Big{]}\cup\Big{[}{\cal
V}^{+}(\Gamma^{0})\cup\Gamma^{0}\Big{]},\quad\tilde{\Omega}=\tilde{\cal
V}^{+}\cup\Omega$ (78)

and extend $[\mathfrak{r},{\bf u}]$ from $(0,T)\times\Omega$ to
$(0,T)\times\tilde{\Omega}$ as follows

$(\mathfrak{r},{\bf u})(t,x)=\left\\{\begin{array}[]{c}(\mathfrak{r},{\bf
u})(t,x)\;\mbox{if $(t,x)\in(0,T)\times\Omega$},\\\
(\tilde{\mathfrak{r}}(t,x),{\bf u}_{B}(x))\;\mbox{if
$(t,x)\in(0,T)\times\tilde{\cal V}^{+}$}\end{array}\right\\},$ (79)

where $\tilde{\mathfrak{r}}$ in $\tilde{\cal V}^{+}$ is defined through (69)
or (73)–(74) or (77), according to the case.

By virtue of (51)v=0 and (70), (77), (75) we easily deduce, that the new
couple $[\mathfrak{r},{\bf u}]$ satisfies continuity equation (49)v=0 in the
sense of distributions on ${\cal D}((0,T)\times\tilde{\Omega})$.

Step 6: Application of the DiPerna-Lions regularization, proof of equation
(52). Next, we use the regularization procedure due to DiPerna and Lions [19]
applying convolution with a family of regularizing kernels obtaining for the
regularized function $[\mathfrak{r}]_{\mathfrak{e}}$,

$\partial_{t}[\mathfrak{r}]_{\mathfrak{e}}+{\rm
div}\,([\mathfrak{r}]_{\mathfrak{e}}{\bf u})=R_{\mathfrak{e}}\ \mbox{a.e. in}\
(0,T)\times\tilde{\Omega}_{{\mathfrak{e}}},$ (80)

where

$\tilde{\Omega}_{{\mathfrak{e}}}=\left\\{x\in\tilde{\Omega}\ \Big{|}\ {\rm
dist}(x,\partial\tilde{\Omega})>{\mathfrak{e}}\right\\},R_{\mathfrak{e}}:={\rm
div}\,([\mathfrak{r}]_{\mathfrak{e}}{\bf u})-{\rm div}\,([\mathfrak{r}{\bf
u}]_{\mathfrak{e}})\to 0\ \mbox{in}\ L_{\rm
loc}^{1}((0,T)\times\tilde{\Omega})\ \mbox{as}\ {\mathfrak{e}}\to 0.$

The convergence of $R_{\mathfrak{e}}$ evoked above results from the
application of the refined version of the Friedrichs lemma on commutators, see
e.g. [19] or [25, Lemma 10.12 and Corollary 10.3].

Multiplying equation (80) on $B^{\prime}([\mathfrak{r}]_{\mathfrak{e}})$, we
get

$\partial_{t}B([\mathfrak{r}]_{\mathfrak{e}})+{\rm
div}\,(B([\mathfrak{r}]_{\mathfrak{e}}){\bf
u})+\left(B^{\prime}([\mathfrak{r}]_{\mathfrak{e}})[\mathfrak{r}]_{\mathfrak{e}}-B([\mathfrak{r}]_{\mathfrak{e}})\right){\rm
div}\,{\bf u}=B^{\prime}([\mathfrak{r}]_{\mathfrak{e}})R_{\mathfrak{e}}$ (81)

or

$\int_{0}^{T}\int_{\tilde{\Omega}}\Big{(}B([\mathfrak{r}]_{\mathfrak{e}})\partial_{t}\varphi+B([\mathfrak{r}]_{\mathfrak{e}}){\bf
u}\cdot\nabla\varphi-\varphi\left(B^{\prime}([\mathfrak{r}]_{\mathfrak{e}})[\mathfrak{r}]_{\mathfrak{e}}-B([\mathfrak{r}]_{\mathfrak{e}})\right){\rm
div}\,{\bf u}\Big{)}\ {\,\rm d}x{\rm d}t$
$-\int_{0}^{T}\int_{\tilde{\Omega}}{\varphi
B^{\prime}([\mathfrak{r}]_{\mathfrak{e}})R_{\mathfrak{e}}}{\rm d}x{\ d}t=0$

for any $\varphi\in C^{1}_{c}(I\times\tilde{\Omega})$, $0<{\mathfrak{e}}<{\rm
dist}({\rm supp}(\varphi),\partial\tilde{\Omega})$. Since the last term at the
right hand side converges to $0$ by virtue of the Friedrichs commutator lemma
(cf. Lemma 22), letting ${\mathfrak{e}}\to 0$, we get

$\int_{0}^{T}\int_{\Omega}\Big{(}B(\mathfrak{r})\partial_{t}\varphi+B(\mathfrak{r}){\bf
u}\cdot\nabla\varphi-\varphi\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{\bf u}\Big{)}\ {\,\rm d}x{\rm d}t$ (82) $+\int_{0}^{T}\int_{\tilde{\cal
V}^{+}}\Big{(}B(\mathfrak{r})\partial_{t}\varphi+B(\mathfrak{r}){\bf
u}\cdot\nabla\varphi-\varphi\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{\bf u}\Big{)}\ {\,\rm d}x{\rm d}t=0$

for any $\varphi\in C^{1}_{c}(I\times\tilde{\Omega})$, where, by virtue of
(71), (76), (77),

$\int_{0}^{T}\int_{\tilde{\cal
V}^{+}}\Big{(}B(\mathfrak{r})\partial_{t}\varphi+B(\mathfrak{r}){\bf
u}\cdot\nabla\varphi-\varphi\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{\bf u}\Big{)}\ {\,\rm d}x{\rm d}t$
$=-\int_{0}^{T}\int_{\partial\Omega}B(\mathfrak{r}){\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t,\ \mathfrak{r}=\mathfrak{r}_{B}\;\mbox{on
$\Gamma^{\rm in}$}.$

Using in (82) test function

$\varphi(t,x)=\phi(1-\chi_{\mathfrak{h}}),\ \phi\in
C^{1}_{c}(I\times\overline{\Omega}),\
\chi_{\mathfrak{h}}(x)=\chi\Big{(}\frac{d_{{\mathfrak{g}}^{\rm
bd}}(x)}{\mathfrak{h}}\Big{)}\Big{)},$

where

$\chi\in C^{1}[0,\infty),\;|\chi^{\prime}(x)|\leq
3,\;\chi(x)\left\\{\begin{array}[]{c}\in[0,1]\\\ =1\;\mbox{if
$x\in[0,1/2]$}\\\ =0\;\mbox{if $x>1$}\\\ \end{array}\right\\},$

and $\mathfrak{h}$ is a positive sufficiently small number, we get

$\int_{0}^{T}\int_{\Omega}\Big{(}B(\mathfrak{r})\partial_{t}\phi(1-\chi_{\mathfrak{h}})+B(\mathfrak{r}){\bf
u}\cdot\nabla\phi(1-\chi_{\mathfrak{h}})-\phi(1-\chi_{\mathfrak{h}})\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{\bf u}\Big{)}\chi_{\mathfrak{h}}\ {\,\rm d}x{\rm d}t$
$+\int_{0}^{T}\int_{\Gamma^{\rm out}}B(\mathfrak{r}){\bf u}_{B}\cdot{\bf
n}\varphi\chi_{\mathfrak{h}}{\rm d}S_{x}{\rm d}t+\int_{0}^{T}\int_{\Gamma^{\rm
in}}B(\mathfrak{r}){\bf u}_{B}\cdot{\bf n}\phi\chi_{\mathfrak{h}}{\rm
d}S_{x}{\rm d}t$ (83) $-\int_{0}^{T}\int_{\Omega\cap B({\mathfrak{g}}^{\rm
bd};\mathfrak{h})}\phi B(\mathfrak{r}){\bf
u}\cdot\nabla\chi_{\mathfrak{h}}{\,\rm d}x\,{\rm d}t=0,$

where the last term at the left hand side can be written as

$\int_{0}^{T}\int_{\Omega\cap B({\mathfrak{g}}^{\rm bd};\mathfrak{h})}\phi
B(\mathfrak{r})({\bf u}-{\bf u}_{B})\cdot\nabla\chi_{\mathfrak{h}}{\,\rm
d}x\,{\rm d}t+\int_{0}^{T}\int_{\Omega\cap B({\mathfrak{g}}^{\rm
bd};\mathfrak{h})}\phi B(\mathfrak{r}){\bf
u}_{B}\cdot\nabla\chi_{\mathfrak{h}}{\,\rm d}x\,{\rm d}t.$

The first term in the latter expression tends to $0$ as $\mathfrak{h}\to 0$ by
virtue of the Hardy inequlity and the second one tends to $0$ due to the
Hölder inequality and (27), cf. (47) and (48).

Thus letting $\mathfrak{h}\to 0$ we obtain the desired result, namely

$\int_{0}^{T}\int_{\Omega}\Big{(}B(\mathfrak{r})\partial_{t}\phi+B(\mathfrak{r}){\bf
u}\cdot\nabla\phi-\phi\left(B^{\prime}(\mathfrak{r})\mathfrak{r}-B(\mathfrak{r})\right){\rm
div}\,{\bf u}\Big{)}\ {\,\rm d}x{\rm d}t$ $-\int_{0}^{T}\int_{\Gamma^{\rm
out}}B(\mathfrak{r}){\bf u}_{B}\cdot{\bf n}\phi{\rm d}S_{x}{\rm
d}t-\int_{0}^{T}\int_{\Gamma^{\rm in}}B(\mathfrak{r}){\bf u}_{B}\cdot{\bf
n}\phi{\rm d}S_{x}{\rm d}t=0,\ \phi\in C^{1}_{c}(I\times\overline{\Omega}).$

Step 7: Time integrated renormalized continuity equation
Since the couple $(\mathfrak{r},{\bf u})$ satisfies (51)v=0, and since
$\mathfrak{r}\in L^{\infty}(I;L^{\gamma}(\Omega))$, it is standard to see that
${\mathfrak{r}}\in C_{\rm weak}(\overline{I};L^{\gamma}(\Omega))$. Moreover,
since the renormalized equation holds in the sense of distributions, one can
infer that ${\mathfrak{r}}\in C(\overline{I};L^{1}(\Omega))$ and the
renormalized time integrated equation (52) holds. This can be deduced from Di-
Perna, Lions [19], see e.g. [42, Theorems 3, 5] for more details.

Step 8: The case $M>1$:
In this case, we obtain instead of (80) $M$ identities,

$\partial_{t}[\mathfrak{r}^{i}]_{\mathfrak{e}})+{\rm
div}\,([{r}^{i}]_{\mathfrak{e}}){\bf u})=R^{i}_{\mathfrak{e}}:={\rm
div}([r^{i}]_{\mathfrak{e}}{\bf u})-{\rm div}[r^{i}{\bf
u}]_{\mathfrak{e}},\;i=1,\ldots,M$

We obtain the required result by multiplying $i$-th equation by
$\partial_{i}B([\mathfrak{r}^{1}]_{\varepsilon},\ldots,[\mathfrak{r}^{M}]_{\varepsilon})$,
summing the resulting equations and then proceeding in the same way as in Step
6.

This finishes the proof of Proposition 4.

Proposition 4 gives rise to several useful corollaries. Before stating them we
define

$\forall\tau\in\overline{I},\ \mbox{for a.a.
$x\in\Omega$},\quad\mathfrak{s}_{\mathfrak{d}}(\tau,x)=[Z/_{\mathfrak{d}}R](\tau,x):=\left\\{\begin{array}[]{c}\frac{Z(\tau,x)}{R(\tau,x}\;\mbox{if
$R(\tau,x)\neq 0$},\\\ \mathfrak{d}\;\mbox{if
$R(\tau,x)=0$}\end{array}\right\\},\ \mathfrak{d}\in\mathbb{R},$ (84)
$\mbox{for a.a. $(\tau,x)\in
I\times\partial\Omega$},\quad\mathfrak{s}_{\mathfrak{d}}(\tau,x)=[Z/_{\mathfrak{d}}R](\tau,x):=\left\\{\begin{array}[]{c}\frac{Z(\tau,x)}{R(\tau,x}\;\mbox{if
$R(\tau,x)\neq 0$},\\\ \mathfrak{d}\;\mbox{if
$R(\tau,x)=0$}\end{array}\right\\}.$

The first of the corollaries is the following:

######  Corollary 5.

[From continuity to pure transport equation] Let $\gamma$, $\Omega$, ${\bf
u}_{B}$ and ${\bf u}$ be the same as in Proposition 4. Suppose that

$0\leq Z\leq\overline{a}R,\;R\in L^{2}(I,L^{2}(\Omega))\cap
L^{\infty}(I;L^{\gamma}(\Omega))\cap L^{\gamma}(I;L^{\gamma}(\Gamma^{\rm
out};|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x}))$

and that both $Z$ and $R$ are weak solutions of the continuity equation (49)
with initial conditions

$0\leq Z_{0}\leq\overline{a}R_{0},\;R_{0}\in L^{\gamma}(\Omega)$

and boundary conditions

$0\leq Z_{B}\leq\overline{a}R_{B},\;R_{B}\in C(\overline{\Omega}).$

Then we have: The quantities

$Z,R\in C_{\rm weak}(\overline{I};L^{\gamma}(\Omega))\cap
C(\overline{I};L^{p}(\Omega)),\;1\leq p<\gamma,$

and for any $\mathfrak{d}\in R$ the quantity
$\mathfrak{s}_{\mathfrak{d}}:=Z/_{\mathfrak{d}}R$ (cf. (84) belongs to
$C(\overline{I};L^{p}(\Omega))\cap L^{\infty}(I,L^{\infty}(\Gamma^{\rm
out}))$, and it satisfies the pure transport equation

$\int_{\Omega}{\mathfrak{s}}_{\mathfrak{d}}\varphi(\tau){\,\rm
d}x-\int_{\Omega}{\mathfrak{s}}_{0,\mathfrak{d}}\varphi(0){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}{\mathfrak{s}}_{\mathfrak{d}}{\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}{\mathfrak{s}}_{B,\mathfrak{d}}{\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm
d}t$
$=\int_{0}^{\tau}\int_{\Omega}\Big{(}{\mathfrak{s}}_{\mathfrak{d}}\partial_{t}\varphi+{\mathfrak{s}}_{\mathfrak{d}}{\bf
u}\cdot\nabla\varphi+\varphi{\mathfrak{s}}_{\mathfrak{d}}{\rm div}\,{\bf
u}\Big{)}{\,\rm d}x{\rm d}t$ (85)

with any $\tau\in\overline{I}$ and any $\varphi\in
C^{1}_{c}([0,T]\times\overline{\Omega}).$ In the above,
$\mathfrak{s}_{0,\mathfrak{d}}=Z_{0}/_{\mathfrak{d}}R_{0}$,
$\mathfrak{s}_{B,\mathfrak{d}}=Z_{B}/_{\mathfrak{d}}R_{B}$.

Indeed, the identity (85) can be obtained from (52) with $M=2$, $r^{(1)}=Z$,
$r^{(2)}=R$, $B(R,Z)=\frac{Z}{R+\mathfrak{a}}$ after letting $\mathfrak{a}\to
0$ with help of the Lebesgue dominated convergence theorem.

The last item of Proposition 4, namely identity (56) yields readily the
following corollary about the ”almost uniqueness” for the transport equation.

######  Corollary 6.

[Almost uniqueness to the pure transport equation] Let $\Omega$, ${\bf u}$,
${\bf u}_{B}$ be the same as in Proposition 4. Let $0\leq s^{(i)}\in
L^{\infty}(Q_{T})\cap C(\overline{I},L^{1}(\Omega)\cap
L^{\infty}(I;L^{\infty}(\Gamma^{\rm out}))$, $i=1,2$ be two weak solutions of
the pure transport equation (49) (i.e. they satisfy (51) with $v=-{\rm
div}{\bf u}$).

If $s^{(1)}(0,\cdot)=s^{(2)}(0,\cdot)$, $s_{B}^{(1)}=s_{B}^{(2)}$ then

$\mbox{for all
$\tau\in\overline{I}$}\;s^{(1)}(\tau,\cdot)=s^{(2)}(\tau,\cdot)\;\mbox{for
a.a. $x\in\\{\varrho(\tau,\cdot)>0\\}$},$ (86) $s^{(1)}=s^{(2)}\ \mbox{ a.e.
in $\\{(t,x)\in I\times\Gamma^{\rm out}|\varrho>0\\}$},$

where $\varrho$ is any weak solution to the continuity equation (49) (i.e.
satisfying (51) with $v=0$) in the class $0\leq\varrho\in
C(\overline{I},L^{1}(\Omega))\cap L^{2}(I;L^{2}(\Omega))\cap
L^{\infty}(I;L^{p}(\Omega))\cap L^{\gamma}(I;L^{p}(\Gamma^{\rm out}(|{\bf
u}_{B}\cdot{\bf n}|{\rm d}S_{x}))$, $p>1$.

To prove Corollary 6, it is enough to take in formula (56) in Proposition 4
$N=2$, $r=\varrho$, $B(s^{(1)},s^{(2)})=(s^{(1)}-s^{(2)})^{2}$.

This result generalizes [40, Proposition 5] from the case of zero transporting
velocity to the case of general boundary data. It also generalizes the
uniqueness results from seminal paper of DiPerna-Lions [19, Theorem II.2] and
its improvement which can be deduced from Bianchini-Bonicatto [6].

The next corollary is one of the crucial point of the compactness argument in
the existence proof. The case with zero velocity at the boundary has been
treated in Vasseur et al. [45], and improved in [41, Proposition 7]. The
generalization of [41, Proposition 7] to the general boundary data reads as
follows.

######  Corollary 7.

We suppose that $\Omega$, ${\bf u}_{B}$, $R_{B},Z_{B}$, $R_{0},Z_{0}$ satisfy
assumptions of Corollary 5. Let

${\bf u}_{n}\in L^{2}(I,W^{1,2}(\Omega;\mathbb{R}^{3})),\;{\bf
u}_{n}|_{I\times\partial\Omega}={\bf u}_{B},\ 0\leq
Z^{n}\leq\overline{a}R^{n},$ $(R_{n},Z_{n})\in
L^{\infty}(I;L^{\gamma}(\Omega))\cap L^{2}(I;L^{2}(\Omega))\cap
L^{\gamma}(I;L^{\gamma}(\Gamma^{\rm out};|{\bf u}_{B}\cdot{\bf n}|{\rm
d}S_{x})),$

where $\gamma>1$. Suppose that

$\sup_{n\in
N}\Big{(}\|R_{n}\|_{L^{2}(Q_{T})}+\|R_{n}\|_{L^{\infty}(I;L^{\gamma}(\Omega))}+\|{\bf
u}_{n}\|_{L^{2}(I;W^{1,2}(\Omega))}\Big{)}<\infty,$ (87)

and that both couples $(R_{n},{\bf u}_{n})$, $(Z_{n},{\bf u}_{n})$ satisfy
continuity equation in the weak sense (i.e. (51) with $v=0$, ${\bf u}={\bf
u}_{n}$ holds). Then:

  1. 1.

$R_{n},B(R_{n})$, $Z_{n},B(Z_{n})$ $\in C_{\rm
weak}(\overline{I};L^{\gamma}(\Omega))\cap C(\overline{I};L^{p}(\Omega))$,
$1\leq p<\gamma$ and each of $R_{n}$, $Z_{n}$ is a renormalized solution of
the continuity equation, i.e. it satisfies equation (52)${}_{{\bf u}={\bf
u}_{n},M=1}$ with $B$ specified in (53), see also Remark 3.

  2. 2.

Up to a subsequence (not relabeled)

$\displaystyle(R_{n},Z_{n})$ $\displaystyle\to(R,Z)\;\mbox{in $C_{\rm
weak}(\overline{I};L^{\gamma}(\Omega))$ and weakly in
$L^{\gamma}(I;L^{\gamma}(\Gamma^{\rm out};|{\bf u}_{B}\cdot{\bf n}|{\rm
d}S_{x}))$},$ (88) $\displaystyle{\bf u}_{n}$
$\displaystyle\rightharpoonup{\bf u}\;\mbox{in
$L^{2}(I;W^{1,2}(\Omega;\mathbb{R}^{3}))$,}$

where $(R,Z)$, $0\leq Z\leq R$ belongs to spaces

$L^{2}(I;L^{2}(\Omega))\cap L^{\infty}(I,L^{\gamma}(I,\Omega))\cap
C(\overline{I};L^{p}(\Omega)),\;1\leq p<\gamma$

and $(R,{\bf u})$ as well as $(Z,{\bf u})$ verify continuity equation in the
renormalized sense (i.e., integral identity (52)u,M=1 with $B$ specified in
(53), see also Remark 3), is satisfied.

  3. 3.

Let $\mathfrak{d}\in[0,\overline{a}]$. We define in agreement with convention
(84) for all $t\in\overline{I}$,

$s_{B}(x)={Z_{B}(x)/_{\mathfrak{d}}R_{B}(x)},\;s_{n}(t,x)={Z_{n}(t,x)/_{\mathfrak{d}}R_{n}(t,x)},\;s(t,x)={Z(t,x)}/_{\mathfrak{d}}{R(t,x)}.$
(89)

Then $s_{n},s\in C(\overline{I};L^{q}(\Omega))$, $1\leq q<\infty$ and for all
$t\in\overline{I}$, $0\leq s_{n}(t,x)\leq\overline{a}$, $0\leq
s(t,x)\leq\overline{a}$ for a.a. $x\in\Omega$, for a.a. $(t,x)\in
I\times\partial\Omega$, $0\leq s_{n}(t,x)\leq\overline{a}$, $0\leq
s(t,x)\leq\overline{a}$. Moreover, both $(s_{n},{\bf u}_{n})$ and $(s,{\bf
u})$ satisfy the pure transport equation in the renormalized sense (i.e.,
integral identity (54)u,N=1 with $B$ specified in (53), see also Remark 3), is
satisfied.

  4. 4.

Finally,

$\int_{\Omega}(R_{n}|s_{n}-s|^{2})(\tau,\cdot)\,{\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}R_{n}|s_{n}-s|^{2}{\rm d}S_{x}{\rm
d}t\to 0\;\mbox{for all $\tau\in\overline{I}$.}$ (90)

The first statement is statement 1 of Proposition 4. The second statement is
nowadays a mathematical folklore (see (152–155) for the reasoning). The third
statement follows from statement 3 in Proposition 4.

We prove (90). We realize, employing (88) and (89),

$\forall\tau\in[0,T],\;\lim_{n\to\infty}\Big{(}\int_{\Omega}R_{n}({s}_{n}-{s})^{2}(\tau){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}R_{n}{s_{n}}^{2}|{\bf
u}_{B}\cdot{\bf n}|{\rm d}S_{x}{\rm d}t\Big{)}$ (91)
$=\lim_{n\to\infty}\Big{(}\int_{\Omega}R_{n}\mathfrak{s}_{n}^{2}(\tau){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}R_{n}{s_{n}}^{2}|{\bf
u}_{B}\cdot{\bf n}|{\rm d}S_{x}{\rm d}t\Big{)}$
$-\int_{\Omega}R{s}^{2}(\tau){\,\rm d}x-\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}R{s}^{2}|{\bf u}_{B}\cdot{\bf n}|{\rm d}S_{x}{\rm d}t,$

where $R_{n}s_{n}^{2}$ satisfies continuity equation, in particular, with test
function $\varphi=1$,

$\int_{\Omega}R_{n}{s}^{2}_{n}(\tau,x){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}R_{n}{s_{n}}^{2}|{\bf
u}_{B}\cdot{\bf n}|{\rm d}S_{x}{\rm d}t$
$=\int_{\Omega}R_{0}(x){s}^{2}(0,x){\,\rm d}x-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}R_{B}{s}^{2}_{B}{\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm d}t,$

while $Rs^{2}$ satisfies the same continuity equation with ${\bf u}_{n}$
replaced by ${\bf u}$. Inserting both latter identities into the right hand
side of (91) yields the statement.

##  4 Approximations

Starting from now, we shall suppose, without loss of generality,

$0<r_{0}\in C^{1}(\overline{\Omega}),\;{\bf u}_{0}={\bf v}_{0}+{\bf
u}_{B},\;{\bf v}_{0}\in C^{1}(\overline{\Omega}).$ (92)

The proof of Theorem 1 is based on a multilevel approximation scheme that
shares certain common features with the approximation of the compressible
Navier–Stokes in [26], see also monographs [24] or [39]. First, we introduce a
sequence of finite–dimensional spaces $X_{n}\subset
L^{2}(\Omega;\mathbb{R}^{3})$,

$X_{n}={\rm span}\left\\{{\bf w}_{i}\ \Big{|}\ {\bf w}_{i}\in
C^{\infty}_{c}(\Omega;\mathbb{R}^{3}),\ i=1,\dots,n\right\\}.$

Without loss of generality, we may assume that ${\bf w}_{i}$ are orthonormal
with respect to the standard scalar product in $L^{2}(\Omega)$.

Following [12], [35] and [41], [40] we use the following parabolic
approximation of the continuity equations,

$\partial_{t}r+{\rm div}\,(r{\bf u})=\varepsilon\Delta r\ \mbox{in}\
(0,T)\times\Omega,\ \varepsilon>0,$ (93)

supplemented with the boundary conditions

$\varepsilon\nabla r\cdot{\bf n}+(r_{B}-r)[{\bf u}_{B}\cdot{\bf n}]^{-}=0\
\mbox{in}\ [0,T]\times\partial\Omega,$ (94)

and the initial condition

$r(0,\cdot)=r_{0}.$ (95)

Here, ${\bf u}={\bf v}+{\bf u}_{B}$, with ${\bf v}\in C([0,T];X_{n})$, in
particular, ${\bf u}|_{\partial\Omega}={\bf u}_{B}$ and $r$ stands for
$\varrho,z,R,Z$, according to the case. Note that for given ${\bf u}$,
$r_{B}$, ${\bf u}_{B}$, this is a linear parabolic problem with the Robin
boundary conditions for the unknown $r$.

Following [35], [40], we use the Galerkin approximation of the momentum
equation: we look for the approximate velocity field in the form

${\bf u}={\bf v}+{\bf u}_{B},\ {\bf v}\in C([0,T];X_{n}).$

Accordingly, the approximate momentum balance reads

$\int_{\Omega}(\varrho+z){\bf u}\cdot\boldsymbol{\varphi}{\,\rm
d}x\Big{|}_{t=0}^{t=\tau}=\int_{0}^{\tau}\int_{\Omega}\Big{[}(\varrho+z){\bf
u}\cdot\partial_{t}\boldsymbol{\varphi}+(\varrho+z){\bf u}\otimes{\bf
u}:\nabla\boldsymbol{\varphi}+P_{\delta}(R,Z){\rm div}\,\boldsymbol{\varphi}$
(96) $-\mathbb{S}(\nabla{\bf
u}):\nabla\boldsymbol{\varphi}-\varepsilon\nabla(\varrho+z)\cdot\nabla{\bf
u}\cdot\boldsymbol{\varphi}\Big{]}{\,\rm d}x\,{\rm d}t$

for any $\boldsymbol{\varphi}\in C^{1}([0,T];X_{n})$, with the initial
condition

$(\varrho+z){\bf u}(0,\cdot)=(\varrho_{0}+z_{0}){\bf u}_{0},\ {\bf u}_{0}={\bf
v}_{0}+{\bf u}_{B},\ {\bf v}_{0}\in X_{n},$ (97)

where

$P_{\delta}(R,Z)=P(R,Z)+\delta(R^{\mathfrak{c}}+Z^{\mathfrak{c}}),\;\mathfrak{c}>\max\\{\frac{9}{2},\beta,\gamma\\}.$
(98)

For fixed parameters $n$, $\delta>0$, $\varepsilon>0$, the first level
approximation is a solution $[\varrho,z,R,Z,{\bf u}]$ 666Here in the sequel,
we skip the indexes $n$, $\varepsilon$, $\delta$ and write e.g. $\xi$ instead
of $\xi_{n,\varepsilon,\delta}$, etc. and will use eventually only one of them
in the situations when it will be useful to underline the corresponding limit
passage. of the parabolic problem (93)–(95), and the Galerkin approximation
(96), (97).

###  4.1 Parabolic problem (93–95)

In contrast with [12], [35], we do not want to use the maximal regularity
theory of parabolic equations (which requires at least $C^{2}$ boundary, see
Denk, Hüber, Prüss [18]), but, we shall employ rather the theory from Crippa,
Donadello, Spignolo [14] which allows merely the Lipschitz boundaries.

For Lipschitz domains the usual parabolic estimates fail at the level of the
spatial derivatives and we are forced to use the weak formulation:

$\begin{split}\left[\int_{\Omega}r\varphi{\,\rm
d}x\right]_{t=0}^{t=\tau}&=\int_{0}^{\tau}\int_{\Omega}\left[r\partial_{t}\varphi+r{\bf
u}\cdot\nabla\varphi-\varepsilon\nabla r\cdot\nabla\varphi\right]{\,\rm
d}x\,{\rm d}t\\\ &-\int_{0}^{\tau}\int_{\partial\Omega}\varphi r{\bf
u}_{B}\cdot{\bf n}\ {\rm d}S_{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\partial\Omega}\varphi(r-r_{B})[{\bf u}_{B}\cdot{\bf
n}]^{-}\ {\rm d}S_{x}\,{\rm d}t,\ r(0,\cdot)=r_{0},\end{split}$ (99)

for any test function

$\varphi\in L^{2}(0,T;W^{1,2}(\Omega)),\ \partial_{t}\varphi\in
L^{1}(0,T;L^{2}(\Omega)).$

######  Lemma 8.

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded Lipschitz domain and ${\bf
u}={\bf v}+{\bf u}_{B}$, ${\bf v}\in C(\overline{I};X_{n})$. Suppose that
$(r_{B},{\bf u}_{B})$ belongs to the class (27) while $(r_{0},{\bf u}_{0})$
belongs to the class (92). Then we have:

  1. 1.

The initial–boundary value problem (93–95) admits a weak solution $r$
specified in (99), unique in the class

$r\in L^{2}(I;W^{1,2}(\Omega))\cap C(\overline{I};L^{2}(\Omega)).$

The norm in the aforementioned spaces is bounded only in terms of the data
$r_{B}$, $r_{0}$, ${\bf u}_{B}$ and $\|{\bf v},{\rm div}{\bf
v}\|_{L^{\infty}(I;L^{\infty}(\Omega))}$.

  2. 2.

Moreover, $\partial_{t}r\in L^{2}(I\times\Omega)$ and
$\sqrt{\varepsilon}\nabla r\in L^{\infty}(I;L^{2}(\Omega))$ are bounded in
terms of the data $r_{B}$, $r_{0}$, ${\bf u}_{B}$ and $\|{\bf v},{\rm div}{\bf
v}\|_{L^{\infty}(I;L^{\infty}(\Omega))}$ and $\nabla^{2}r\in
L^{2}(I;L^{2}_{\rm loc}(\Omega)$ is bounded in the same way on any compact set
$K$ of $\Omega$ with the constant dependent in addition on $K$.

  3. 3.

Strong maximum principle: The solution satisfies,

$\begin{array}[]{c}{\forall\tau\in\overline{I},\;\|r(\tau)\|_{L^{\infty}(\Omega)}}\leq
M\exp\left(T\|{\rm div}\,{\bf
u}\|_{L^{\infty}((0,\tau)\times\Omega)}\right),\\\ \\\ \mbox{for a.a. $\tau\in
I$},\;r(\tau,x)\leq M\exp\left(T\|{\rm div}\,{\bf
u}\|_{L^{\infty}((0,\tau)\times\Omega)}\right)\;\mbox{for a.a.
$x\in\partial\Omega$},\end{array}$ (100)

where

$M=\max\left\\{\max_{\Omega}r_{0},\max_{\Gamma^{\rm in}}r_{B},\|{\bf
u}_{B}\|_{L^{\infty}((0,T)\times\Omega)}\right\\}.$

  4. 4.

Renormalization: For any $B\in C^{2}(R)$,

$\int_{\Omega}B(r){\,\rm
d}x\Big{|}_{0}^{\tau}+\varepsilon\int_{0}^{\tau}\int_{\Omega}|\nabla
r|^{2}B^{\prime\prime}(r){\,\rm d}x{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\Big{(}rB^{\prime}(r)-B(r)\Big{)}{\rm
div}\,{\bf u}{\,\rm d}x{\rm d}t$ (101)
$-\int_{0}^{\tau}\int_{\partial\Omega}[{\bf u}_{B}\cdot{\bf
n}]^{-}E_{B}(r_{B}|r){\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\partial\Omega}[{\bf u}_{B}\cdot{\bf n}]^{+}B(r){\rm
d}S_{x}{\rm d}t=-\int_{0}^{\tau}\int_{\partial\Omega}[{\bf u}_{B}\cdot{\bf
n}]^{-}B(r_{B}){\rm d}S_{x}{\rm d}t,$

where $\tau\in\overline{I}$ and

$E_{B}(r|\tilde{r})=B(r)-B^{\prime}(\tilde{r})(r-\tilde{r})-B^{\prime}(\tilde{r}).$

  5. 5.

Strong minimum principle: The solution satisfies,

$\begin{array}[]{c}\forall\tau\in\overline{I},\;{\rm
ess}\inf_{x\in\Omega}r(\tau,x)\geq m\exp\left(-T\|{\rm div}\,{\bf
u}\|_{L^{\infty}((0,T)\times\Omega)}\right),\\\ \\\ \mbox{for a.a. $\tau\in
I$},\;r(\tau,x)\geq m\exp\left(-T\|{\rm div}\,{\bf
u}\|_{L^{\infty}((0,\tau)\times\Omega)}\right)\;\mbox{for a.a.
$x\in\partial\Omega$},\end{array}$ (102)

where

$m=\min\left\\{\min_{\Omega}r_{0},\min_{\Gamma^{\rm in}}r_{B}\right\\}.$

The first item is a particular case of Lemma 3.2 in Crippa et al. [14], see
also [1, Lemma 3.1]. The latter reference contains also the proof of the
second item. The third item, maximum principle, is proved Crippa et al. [14,
Lemma 3.4], see also [1, Lemma 3.2]. Renormalization is proved in [1, Lemma
3.3]. Finally, the minimum principle is shown in the latter reference in
Corollary 3.4.777 Inequalities (100) and (102) are proved in [1] with ${\rm
ess}\sup_{t,x}r(t,x)$ and ${\rm ess}\inf_{t,x}r(t,x)$, respectively. They
however hold for all $\tau\in\overline{I}$ provided, in addition, $r\in C_{\rm
weak}(\overline{I};L^{\gamma}(\Omega))$, $\gamma\geq 1$. Indeed, suppose for
example for the quantity $r$ in addition to the latter regularity, $r(t,x)\geq
0$ for a.a. $(t,x)\in I\times\Omega$. Then for any $M\subset\Omega$, $|M|>0$,
any $\tau_{0}\in I$ and $\delta>0$ ”small”,
$\int_{\tau_{0}-\delta}^{\tau_{0}+\delta}\int_{M}r(t,x){\rm d}x{\rm d}t\geq
0$; whence $\int_{M}r(\tau_{0},x){\rm d}x\geq 0$, and $r(\tau_{0},\cdot)\geq
0$ a. a. in $\Omega$ by the theorem on Lebesgue points. Since $r\in
L^{2}(I;W^{1,2}(\Omega))$, the second inequality in (100) resp. in (102)
follows from the first one and the trace theorem.

###  4.2 Existence of approximations at level I

The _existence_ of the approximate solutions at the level of the parabolic
problem (93–95) coupled with the Galerkin approximation (96–97) can be proved
in the same way as in [12, Section 4] (mono-fluid case with non zero inflow-
outflow) combined with [40, Section 3], eventually with [41, Section 4]
(multi-fluid with zero boundary conditions). Specifically, for ${\bf u}={\bf
u}_{B}+{\bf v}$, ${\bf v}\in C([0,T];X_{n})$, we identify the unique solutions
$r=r[{\bf u}]$ of (93–95), where $r$ stands for $\varrho,z,R,Z$ and plug them
as $\varrho$, $z$, $R$, $Z$ in (96). The unique solution ${\bf u}={\bf
u}[\varrho,z,R,Z]$ of (96) defines a mapping

$\mathcal{T}:{\bf v}\in C([0,T];X_{n})\mapsto\mathcal{T}[{\bf v}]=({\bf
u}[\varrho,z,R,Z]-{\bf u}_{B})\in C([0,T];X_{n}).$

The first level approximate solutions $r=r_{\delta,\varepsilon,n}$, ${\bf
u}={\bf u}_{\delta,\varepsilon,n}$ – here, $r$ stands for $\varrho$, $z$, $R$,
$Z$–are obtained via a fixed point through the mapping $\mathcal{T}$. This
procedure is detailed in [12] and in [35] for the mono-fluid case with the non
zero inflow-outflow and in [40] for the multi-fluid case with the no-slip
boundary conditions. Combinnig [35, Section 4] with [40, Section 4], we easily
deduce the following result.888 The energy inequality (104) in [35, Lemma 4.2]
and in [40, Section 4] is derived under assumption $\Omega\in C^{2}$. This
assumption is needed due to the treatment of the parabolic problem (93–95) via
the classical maximal regularity methods. With Lemma 8 at hand, the same proof
can be carried out without modifications also in Lipschitz domains.

######  Proposition 9.

[Approximate solutions, level I] Let $\Omega\subset\mathbb{R}^{3}$ be a
bounded Lipschitz domain. Let the data $(\varrho_{B},z_{B},R_{B},Z_{B},{\bf
u}_{B})$, $(\varrho_{0},z_{0},R_{0},Z_{0},{\bf u}_{0})$ belong to the class
(27–30), (92). Suppose that assumptions (32–36) are satisfied.

Then for each fixed $\delta>0$, $\varepsilon>0$, $n>0$, there exists a
solution $(\varrho_{n},z_{n},R_{n},Z_{n},{\bf u}_{n}={\bf v}_{n}+{\bf u}_{B})$
of the approximate problem (99) and (96), (97). Moreover, the following holds:

  1. 1.

Lower and upper bounds of ”densities”:

$\begin{array}[]{c}\forall
t\in\overline{I},\;R_{n}(t,x)\geq\underline{c}(\delta,n)>0,\;Z_{n}(t,x)\geq\underline{c}(\delta,n)>0,\;\underline{a}R_{n}(t,x)\leq
Z_{n}(t,x)\leq\overline{a}R_{n}(t,x),\\\
\underline{F}R_{n}(t,x)\leq\varrho_{n}(t,x)\leq\overline{F}R_{n}(t,x),\;\underline{G}R_{n}(t,x)\leq
z_{n}(t,x)\leq\overline{F}Z_{n}(t,x)\;\mbox{ for a.a. $x\in\Omega$,}\\\ \\\
\mbox{for a.a. $t\in
I$},\;R_{n}(t,x)\geq\underline{c}(\delta,n)>0,\;Z_{n}(t,x)\geq\underline{c}(\delta,n)>0,\;\underline{a}R_{n}(t,x)\leq
Z_{n}(t,x)\leq\overline{a}R_{n}(t,x),\\\
\underline{F}R_{n}(t,x)\leq\varrho_{n}(t,x)\leq\overline{F}R_{n}(t,x),\;\underline{G}R_{n}(t,x)\leq
z_{n}(t,x)\leq\overline{F}Z_{n}(t,x)\;\mbox{ for a.a.
$x\in\partial\Omega$.}\end{array}$ (103)

  2. 2.

The approximate energy inequality

$\begin{split}&\left[\int_{\Omega}\left[\frac{1}{2}(\varrho_{n}+z_{n})|{\bf
v}_{n}|^{2}+{H}_{\delta}(R_{n},Z_{n})\right]{\,\rm
d}x\right]_{t=0}^{t=\tau}+\int_{0}^{\tau}\int_{\Omega}\mathbb{S}(\nabla{\bf
u}_{n}):\nabla{\bf u}_{n}{\,\rm d}x\,{\rm d}t\\\
&+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}{H}_{\delta}(R_{n},Z_{n}){\bf
u}_{B}\cdot{\bf n}\ {\rm d}S_{x}\,{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}E_{{H}_{\delta}}(R_{B},Z_{B}|R_{n},Z_{n}){\bf u}_{B}\cdot{\bf n}\ {\rm
d}S_{x}\ \,{\rm d}t\\\
&+\varepsilon\int_{0}^{\tau}\int_{\Omega}{\nabla_{R,Z}^{2}{H}_{\delta}(R_{n},Z_{n})}[\nabla
R_{n},\nabla Z_{n}]{\,\rm d}x\,{\rm d}t\\\
&\leq-\int_{0}^{\tau}\int_{\Omega}\left[(\varrho_{n}+z_{n}){\bf
u}_{n}\otimes{\bf u}_{n}+P_{\delta}(R_{n},Z_{n})\mathbb{I}\right]:\nabla{\bf
u}_{B}{\,\rm d}x\,{\rm d}t\\\
&+\int_{0}^{\tau}{\int_{\Omega}(\varrho_{n}+z_{n}){\bf u}_{n}\cdot\nabla{\bf
u}_{B}\cdot{\bf u}_{B}{\,\rm d}x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\mathbb{S}(\nabla{\bf u}_{n}):\nabla{\bf
u}_{B}{\,\rm d}x\,{\rm d}t\\\ &-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}{H}_{\delta}(R_{B},Z_{B}){\bf u}_{B}\cdot{\bf n}\ {\rm d}S_{x}\,{\rm
d}t\end{split}$ (104)

holds for any $0\leq\tau\leq T$, where

${H}_{\delta}(R,Z)=H(R,Z)+\frac{\delta}{\mathfrak{c}-1}\Big{(}R^{\mathfrak{c}}+Z^{\mathfrak{c}}\Big{)},$
(105)

and

$\nabla_{R,Z}^{2}{H}_{\delta}(R,Z)[\nabla R,\nabla
Z]=\partial^{2}_{R}H_{\delta}(R,Z)|\nabla
R|^{2}+2\partial_{R}\partial_{Z}H_{\delta}\nabla R\cdot\nabla
Z+\partial^{2}_{Z}H_{\delta}(R,Z)|\nabla Z|^{2}.$

  3. 3.

Renormalized identity

$\|r_{n}\|^{2}_{L^{2}(\Omega)}\Big{|}_{0}^{\tau}+\varepsilon\int_{0}^{\tau}\|\nabla
r_{n}\|_{L^{2}(\Omega)}^{2}{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}{\bf
u}_{B}\cdot{\bf n}(r_{n}-r_{B})^{2}{\rm d}S_{x}{\rm d}t$ (106)
$+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r_{n}^{2}{\bf u}_{B}\cdot{\bf n}{\rm
d}S_{x}{\rm d}t=-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}r_{B}^{2}{\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm
d}t-\int_{0}^{\tau}\int_{\Omega}r_{n}^{2}{\rm div}\,{\bf u}_{n}{\,\rm d}x{\rm
d}t$

holds, where $r_{n}$ stands for $\varrho_{n},z_{n},R_{n},Z_{n}$.

This is level I of approximations (with three parameters $n$, $\varepsilon$,
$\delta$). We shall pass first to the limit $n\to\infty$ in order to obtain
level II of approximations (with two parameters $\varepsilon$, $\delta$). Then
we obtain level III of approximations (with one parameter $\delta$) by letting
$\varepsilon\to 0+$. Finally, we effectuate limit $\delta$ to $0+$ in order to
obtain a weak solution of the (academic) problem (14–16).

##  5 Limit passage from level I to level II (limit $n\to\infty$)

The goal of this section is to pass to the limit $n\to\infty$ in Proposition
9. The result is formulated in Proposition 10 at the end of the section.

###  5.1 Limit in the parabolic equations and in the momentum equation (start)

In view of (29–36), (27), (92), (39–41) and (98), (105) we deduce from
(103–106) the following bounds:

$\begin{array}[]{c}\mbox{$\forall t\in\overline{I}$},\;R_{n}(t,x)\geq
0,\;Z_{n}(t,x)\geq 0,\;\underline{a}R_{n}(t,x)\leq
Z_{n}(t,x)\leq\overline{a}R_{n}(t,x),\\\
\underline{F}R_{n}(t,x)\leq\varrho_{n}(t,x)\leq\overline{F}R_{n}(t,x),\;\underline{G}R_{n}(t,x)\leq
z_{n}(t,x)\leq\overline{F}Z_{n}(t,x)\;\mbox{ for a.a. $x\in\Omega$,}\\\ \\\
\mbox{for a.a. $(t,x)\in I\times\partial\Omega$},\;R_{n}(t,x)\geq
0,\;Z_{n}(t,x)\geq 0,\;\underline{a}R_{n}(t,x)\leq
Z_{n}(t,x)\leq\overline{a}R_{n}(t,x),\\\
\underline{F}R_{n}(t,x)\leq\varrho_{n}(t,x)\leq\overline{F}R_{n}(t,x),\;\underline{G}R_{n}(t,x)\leq
z_{n}(t,x)\leq\overline{F}Z_{n}(t,x).\end{array}$ (107)

and

$\displaystyle\|(\varrho_{n}+z_{n})|{\bf
v}_{n}|^{2}\|_{L^{\infty}(I,L^{1}(\Omega))}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),$
(108) $\displaystyle\|{\bf v}_{n}\|_{L^{2}(I,W^{1,2}(\Omega))}\leq C(\delta),$
(109)
$\displaystyle\|r_{n}\|_{L^{\infty}(I,L^{\mathfrak{c}}(\Omega))}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),$
(110)
$\displaystyle\varepsilon\|\nabla{r_{n}}\|^{2}_{L^{2}(Q_{T})}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),$
(111) $\displaystyle\|r_{n}|{\bf u}_{B}\cdot{\bf
n}|^{1/{\mathfrak{c}}}\|_{L^{\mathfrak{c}}((0,T)\times\partial\Omega)}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),$
(112)

where $r_{n}$ stands for $\varrho_{n}$, $z_{n}$, $R_{n}$, $Z_{n}$, and

$\varepsilon\|\nabla(r_{n}^{{\mathfrak{c}/2}})\|^{2}_{L^{2}(Q_{T})}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),\;\mbox{where
$r_{n}=Z_{n}$ or $r_{n}=R_{n}$.}$ (113)

By (109),

${\bf v}_{n}\rightharpoonup{\bf v}\;\mbox{in $L^{2}(I;W_{0}^{1,2}(\Omega))$}.$
(114)

By virtue of (110), Arzela-Ascoli theorem combined with equation (99),
estimates (to verify equi-continuity) and density of $C^{\infty}_{c}(\Omega)$
in $L^{{\mathfrak{c}}^{\prime}}(\Omega)$,

$r_{n}\to r\;\mbox{in $C_{\rm weak}(\overline{I};L^{\mathfrak{c}}(\Omega))$,
$r$ states for $\varrho,z,R,Z$.}$

Consequently in particular- cf. (108),

$(\varrho_{n}+z_{n}){\bf u}_{n}\rightharpoonup_{*}(\varrho+z){\bf u}\;\mbox{in
$L^{\infty}(I;L^{\frac{2\mathfrak{c}}{\mathfrak{c}+1}}(\Omega))$},\;{\bf
u}={\bf v}+{\bf u}_{B}.$

By the similar Arzela-Ascoli argument as above, but now with the momentum
equation (96) one gets first,

$(\varrho_{n}+z_{n}){\bf u}_{n}\to(\varrho+z){\bf u}\;\mbox{in $C_{\rm
weak}(I;L^{\frac{2\mathfrak{c}}{\mathfrak{c}+1}}(\Omega))$}$

and then

$(\varrho_{n}+z_{n}){\bf u}_{n}\otimes{\bf u}_{n}\to(\varrho+z){\bf
u}\otimes{\bf u}\;\mbox{in
$L^{2}(\overline{I};L^{\frac{6\mathfrak{c}}{4\mathfrak{c}+3}}(\Omega))$}.$

By (111),

$\nabla r_{n}\rightharpoonup\nabla r\;\mbox{ in $L^{2}(I\times\Omega)$,
$r=\varrho$ or $z$ or $R$ or $Z$.}$

and by (112), for the traces of the same $r$’s,

$r_{n}\rightharpoonup r\;\mbox{in
$L^{\mathfrak{c}}(I;L^{\mathfrak{c}}(\partial\Omega;|{\bf u}_{B}\cdot{{\bf
n}}|{\rm d}S_{x}))$}$

We realize that (99) rewrites in the form

$\partial_{t}r_{n}={\cal F}(r_{n},{\bf u}_{n})\;\mbox{in
$L^{2}(I;[W^{1,2}(\Omega)]^{*}))$},$ (115)

where

$<{\cal F}(r,{\bf u}),\varphi>=\int_{0}^{T}\int_{\Omega}\left[r{\bf
u}\cdot\nabla\varphi-\varepsilon\nabla r\cdot\nabla\varphi\right]{\,\rm
d}x\,{\rm d}t$ $-\int_{0}^{\tau}\int_{\partial\Omega}\varphi r{\bf
u}_{B}\cdot{\bf n}\ {\rm d}S_{x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\partial\Omega}\varphi(r-r_{B})[{\bf u}_{B}\cdot{\bf
n}]^{-}\ {\rm d}S_{x}\,{\rm d}t.$

Consequently, we deduce from the estimates, that

$\|\partial_{t}r_{n}\|_{L^{2}(I;[W^{1,2}(\Omega)]^{*})}\stackrel{{\scriptstyle<}}{{\sim}}C(\varepsilon,\delta).$
(116)

In view of (115) and (116), we find in the limit

$\partial_{t}r={\cal F}(r,{\bf u})\;\mbox{in
$L^{2}(I;[W^{1,2}(\Omega)]^{*}))$}.$ (117)

By virtue of (111), (116) and Lions-Aubin lemma, for $r=R$ or $Z$,

$r_{n}\to r\;\mbox{a.e. in $I\times\Omega$};\;\mbox{whence}\;r_{n}\to r\in
L^{q}(I\times\Omega)\;\mbox{with some $q>\mathfrak{c}$},$

where we have used (110) and (113) and interpolation. Consequently,

$P_{\delta}(R_{n},Z_{n})\to P_{\delta}(R,Z)\;\mbox{in $L^{q}(I\times\Omega)$
with some $q>1$}.$

With what we derived so far in this section, we are at the point to be able to
pass to the limit $n\to\infty$ in the parabolic equations (93–95). In order to
pass to the limit in the momentum equation (96), it remains to prove strong
convergence of $\nabla(\varrho_{n}+z_{n})$ and consequently,

$\nabla(\varrho_{n}+z_{n})\cdot\nabla{\bf
u}_{n}\rightharpoonup\nabla(\varrho+z)\cdot\nabla{\bf u}\;\mbox{in ${\cal
D}^{\prime}(I\times\Omega)$}.$ (118)

We shall postpone this point to Section 5.3. Indeed, to do this, we shall need
to derive first the renormalized equation (130) for the limiting couple
$(r,{\bf u})$.999 In [12], the authors obtained the strong convergence of
$\nabla\Sigma_{n}$, $\Sigma_{n}=\varrho_{n}+z_{n}$ directly by deriving the
$L^{4/3}$-bound for the second derivatives of $\Sigma_{n}$ via the maximal
parabolic regularity – at cost of adding the additional dissipation
$\varepsilon{\rm div}(|\nabla{\bf v}|^{2}\nabla{\bf v})$ to the approximated
momentum equation (99). This bound was used also for the derivation of the
renormalized identities of type (131). In addition to the non homogenous
boundary data, the present approach generalizes [12] in several directions: 1)
There is no need of the additional dissipation in the approximations of the
momentum equation. 2)There is no need of $C^{2}$ boundary in order to derive
the renormalized equations.

###  5.2 Limit in the energy inequality

We have,

$\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int_{\Omega}(\varrho_{n}+z_{n}){\bf
v}_{n}^{2}{\,\rm d}x{\rm
d}t\to\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int_{\Omega}(\varrho+z){\bf
v}^{2}{\,\rm d}x{\rm d}t$ with $\lim_{\delta\to
0+}\frac{1}{\delta}\int_{\tau}^{\tau+\delta}\int_{\Omega}(\varrho+z){\bf
v}^{2}{\,\rm d}x{\rm d}t=\int_{\Omega}(\varrho+z){\bf v}^{2}(\tau){\,\rm d}x$
for a.a. $\tau\in(0,T)$

where the latter conclusion is a consequence of the theorem on Lebesgue
points.

We further use lower weak semi-continuity of convex functionals, as follows:

$\forall\tau\in\overline{I},\;\int_{\Omega}H_{\delta}(R,Z)(\tau){\,\rm
d}x\leq\liminf_{n\to\infty}\int_{\Omega}H_{\delta}(R_{n},Z_{n})(\tau){\,\rm
d}x,$ $\int_{0}^{\tau}\int_{\Gamma^{\rm out}}H_{\delta}(R,Z){\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm
d}t\leq\liminf_{n\to\infty}\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}H_{\delta}(R_{n},Z_{n}){\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm d}t,$
$\int_{0}^{\tau}\int_{\Omega}|\nabla{\bf v}|^{2}{\,\rm d}x{\rm
d}t\leq\liminf_{n\to\infty}\int_{0}^{\tau}\int_{\Omega}|\nabla{\bf
v}_{n}|^{2}{\,\rm d}x{\rm d}t.$

At this point, we can deduce from the energy inequality (104) the energy
inequality (129) in Proposition 10.

###  5.3 Renormalization

In the sequel, we shall work with equation (117). To begin, we take $B\in
C^{2}_{c}((0,\infty))$. As $\partial_{t}r\in
L^{2}(0,T;[W^{1,2}(\Omega)]^{*})$, $B^{\prime}(r)\in
L^{2}(0,T;W^{1,2}(\Omega))$ we deduce from this formulation

$\begin{split}\int_{\tau_{1}}^{\tau_{2}}&\left<\partial_{t}r,B^{\prime}(r)\varphi\right>_{[W^{1,2}]^{*};W^{1,2}}\,{\rm
d}t\\\ &=\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}r{\bf
u}\cdot\nabla(B^{\prime}(r)\varphi){\,\rm d}x\,{\rm
d}t-\varepsilon\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}\Big{[}|\nabla
r|^{2}B^{\prime\prime}(r)\varphi+B^{\prime}(r)\nabla
r\cdot\nabla\varphi\Big{]}{\,\rm d}x\,{\rm d}t\\\
&-\int_{\tau_{1}}^{\tau_{2}}\int_{\partial\Omega}\varphi rB^{\prime}(r){\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}\ \,{\rm
d}t+\int_{\tau_{1}}^{\tau_{2}}\int_{\partial\Omega}\varphi
B^{\prime}(r)(r-r_{B})[{\bf u}_{B}\cdot{\bf n}]^{-}\ {\rm d}S_{x}\ \,{\rm
d}t,\end{split}$ (119)

$0<\tau_{1}<\tau_{2}<T$, where $\varphi\in C^{1}(\overline{\Omega})$.

Using the standard temporal regularization via a family of $t-$dependent
convolution kernels, we find a sequence of functions

$r_{n}\in C^{1}([\tau_{1},\tau_{2}];W^{1,2}(\Omega)),\ r_{n}\to r\ \mbox{in}\
L^{2}(\tau_{1},\tau_{2};W^{1,2}(\Omega)),\ \partial_{t}r_{n}\to\partial_{t}r\
\mbox{in}\ L^{2}(\tau_{1},\tau_{2};[W^{1,2}(\Omega)]^{*})$

for any $0<\tau_{1}<\tau_{2}$, and as $r\in C([0,T];L^{p}(\Omega))$, $1\leq
p<\mathfrak{c}$,

$B(r_{n})(\tau)\to B(r(\tau))\ \mbox{in $L^{p}(\Omega)$ for any}\
\tau\in(0,T).$

Thus, we obtain,

$\int_{\Omega}B(r)\varphi{\,\rm
d}x\Big{|}_{\tau_{1}}^{\tau_{2}}=\lim_{n\to\infty}\left[\int_{\Omega}B(r_{n})\varphi{\,\rm
d}x\right]_{t=\tau_{1}}^{t=\tau_{2}}=\lim_{n\to\infty}\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}B^{\prime}(r_{n})\partial_{t}r_{n}\varphi{\,\rm
d}x\,{\rm
d}t=\int_{\tau_{1}}^{\tau_{2}}\left<{\partial_{t}r},B^{\prime}(r)\varphi\right>\,{\rm
d}t$

and

$\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}r{\bf
u}\cdot\nabla(B^{\prime}(r)\varphi){\,\rm d}x\,{\rm d}t=\lim_{n\to
0}\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}r_{n}{\bf
u}\cdot\nabla(B^{\prime}(r_{n})\varphi){\,\rm d}x\,{\rm d}t$ ${=-\lim_{n\to
0}\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}\Big{[}\varphi{\rm
div}\,(B(r_{n}){\bf u})+(r_{n}B^{\prime}(r_{n})-B(r_{n})){\rm div}\,{\bf
u}\Big{]}{\,\rm d}x\,{\rm
d}t+\int_{\tau_{1}}^{\tau_{2}}\int_{\partial\Omega}\varphi
r_{n}B^{\prime}(r_{n}){\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}\,{\rm d}t}$
$={\lim_{n\to
0}\Big{(}\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}\Big{[}B(r_{n}){\bf
u}\cdot\nabla\varphi-\varphi(r_{n}B^{\prime}(r_{n})-B(r_{n})){\rm div}\,{\bf
u}\Big{]}{\,\rm d}x\,{\rm d}t}$
$+\int_{\tau_{1}}^{\tau_{2}}\int_{\partial\Omega}\varphi\Big{[}-B(r_{n})+r_{n}B^{\prime}(r_{n})\Big{]}{\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}\,{\rm d}t$
$=\int_{\tau_{1}}^{\tau_{2}}\int_{\Omega}\Big{[}B(r){\bf
u}\cdot\nabla\varphi-\varphi(rB^{\prime}(r)-B(r)){\rm div}\,{\bf
u}\Big{]}{\,\rm d}x\,{\rm
d}t+\int_{\tau_{1}}^{\tau_{2}}\int_{\partial\Omega}\varphi\Big{[}-B(r)+rB^{\prime}(r)\Big{]}{\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}\,{\rm d}t$

for any $0<\tau_{1}<\tau_{2}<T$.

Inserting the last two formulas to (119) yields,

${\int_{\Omega}B(r)\varphi{\,\rm
d}x\Big{|}_{0}^{\tau}=}\int_{0}^{\tau}\int_{\Omega}\Big{[}\Big{(}B(r){\bf
u}-\varepsilon B^{\prime}(r)\nabla
r\Big{)}\cdot\nabla\varphi-\varphi\Big{(}\varepsilon
B^{\prime\prime}(r)|\nabla r|^{2}+(rB^{\prime}(r)-B(r)){\rm div}{\bf
u}\Big{)}\Big{]}{\,\rm d}x{\rm d}t$
${+}\int_{0}^{\tau}\int_{\partial\Omega}\varphi[{\bf u}_{B}\cdot{\bf
n}]^{-}\Big{(}B(r_{B})-B^{\prime}(r)(r_{B}-r)-B(r)\Big{)}{\rm d}S_{x}{\rm d}t$
(120) $-\int_{0}^{\tau}\int_{\partial\Omega}\varphi[{\bf u}_{B}\cdot{\bf
n}]^{-}B(r_{B}){\rm d}S_{x}{\rm
d}t-\int_{0}^{\tau}\int_{\partial\Omega}\varphi[{\bf u}_{B}\cdot{\bf
n}]^{+}B(r){\rm d}S_{x}{\rm d}t.$

Identity (120) can be extended to $B\in C^{2}([0,\infty))$ with growth

$|B(r)|\stackrel{{\scriptstyle<}}{{\sim}}(1+r)^{\frac{1}{2}\mathfrak{c}},\;|B^{\prime}(r)|\stackrel{{\scriptstyle<}}{{\sim}}(1+r)^{\frac{1}{2}\mathfrak{c}-1},\;B^{\prime\prime}(r)\stackrel{{\scriptstyle<}}{{\sim}}1.$
(121)

Under these conditions, and if moreover $B$ is convex on $[0,\infty)$, we
deduce

$\int_{\Omega}B(r)\varphi{\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\partial\Omega}\varphi[{\bf
u}_{B}\cdot{\bf n}]^{+}B(r){\rm d}S_{x}{\rm d}t$ (122)
$\leq\int_{0}^{\tau}\int_{\Omega}\Big{(}B(r){\bf
u}\cdot\nabla\varphi-\varphi(rB^{\prime}(r)-B(r)){\rm div}{\bf u}\Big{)}{\,\rm
d}x{\rm d}t$ $-\int_{0}^{\tau}\int_{\partial\Omega}\varphi[{\bf
u}_{B}\cdot{\bf n}]^{-}B(r_{B}){\rm d}S_{x}{\rm
d}t-\varepsilon\int_{0}^{\tau}\int_{\Omega}B^{\prime}(r)\nabla
r\cdot\nabla\varphi{\,\rm d}x{\rm d}t$

with any $\tau\in[0,T]$ and any non negative $\varphi\in
C^{1}(\overline{\Omega})$.

Inequality (122) is generalizable to renormalizing functions of several
variables and several parabolic equations (93–95) with the same ${\bf u}$. In
particular, if $B\in C^{2}([0,\infty)^{2})$ is convex with the growth

$|B(R,Z)|\stackrel{{\scriptstyle<}}{{\sim}}(1+\sqrt{R^{2}+Z^{2}})^{\frac{1}{2}\mathfrak{c}},\;|\nabla_{R,Z}B(R,Z)|\stackrel{{\scriptstyle<}}{{\sim}}(1+\sqrt{R^{2}+Z^{2}})^{\frac{1}{2}\mathfrak{c}-1},\;|\nabla^{2}_{R,Z}(R,Z)|\stackrel{{\scriptstyle<}}{{\sim}}1,$
(123)

then

$\int_{\Omega}B(R,Z){\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\partial\Omega}\varphi[{\bf
u}_{B}\cdot{\bf n}]^{+}B(R,Z){\rm d}S_{x}{\rm d}t$ (124)
$\leq\int_{0}^{\tau}\int_{\Omega}\Big{[}B(R,Z){\bf
u}\cdot\nabla\varphi-\varphi{{\rm div}\,{\bf
u}}\Big{(}R\partial_{R}B+Z\partial_{Z}B-B\Big{)}(R,Z)\Big{]}{\,\rm d}x{\rm
d}t$ $-\int_{0}^{\tau}\int_{\partial\Omega}[{\bf u}_{B}\cdot{\bf
n}]^{-}B(R_{B},Z_{B}){\rm d}S_{x}{\rm
d}t-\varepsilon\int_{0}^{\tau}\int_{\Omega}\Big{(}\partial_{R}B(R,Z)\nabla
R+\partial_{Z}B(R,Z)\nabla Z\Big{)}\cdot\nabla\varphi{\,\rm d}x{\rm d}t$

with any $\tau\in\overline{I}$ and with any non negative $\varphi\in
C^{2}(\overline{\Omega})$.

We shall use (120–124) in the following situations

  1. 1.

If $B(r)=r^{2}$ and $\varphi=1$, we obtain the renoramlized identity (130).

  2. 2.

If $B(r)=r\log(r+\mathfrak{a})$, $\mathfrak{a}>0$ we obtain the renormalized
inequality (131).

  3. 3.

If $B(R,Z)=\frac{Z^{2}}{R+\mathfrak{a}}$, $\mathfrak{a}>0$ in (124), we obtain
(132) after letting $\mathfrak{a}\to 0+$.

###  5.4 Limit in the momentum equation (end)

Combining (106) with (130) together with the lower weak semicontinuity of
norms, we deduce

$\|r(\tau)\|^{2}_{L^{2}(\Omega)}+2\varepsilon\int_{0}^{\tau}\|\nabla
r\|_{L^{2}(\Omega)}^{2}{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}{\bf
u}_{B}\cdot{\bf n}(r-r_{B})^{2}{\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r^{2}{\bf u}_{B}\cdot{\bf n}{\rm
d}S_{x}{\rm d}t$
$\leq\liminf_{n\to\infty}\Big{(}\|r_{n}(\tau)\|^{2}_{L^{2}(\Omega)}+2\varepsilon\int_{0}^{\tau}\|\nabla
r_{n}\|_{L^{2}(\Omega)}^{2}{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}{\bf
u}_{B}\cdot{\bf n}(r-r_{B})^{2}{\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r^{2}{\bf u}_{B}\cdot{\bf n}{\rm
d}S_{x}{\rm d}t\Big{)}$
$\leq\|r(\tau)\|^{2}_{L^{2}(\Omega)}+2\varepsilon\int_{0}^{\tau}\|\nabla
r\|_{L^{2}(\Omega)}^{2}{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}{\bf
u}_{B}\cdot{\bf n}(r-r_{B})^{2}{\rm d}S_{x}{\rm
d}t+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r^{2}{\bf u}_{B}\cdot{\bf n}{\rm
d}S_{x}{\rm d}t,$

where $r$ stands for $\varrho$, $z$, $R$, $Z$, and any of its linear
combinations; whence, in particular,

$\nabla(\varrho_{n}+z_{n})\to\nabla(\varrho+z)\;\mbox{in
$L^{2}(I\times\Omega)$}.$

This in combination with (114) justifies convergence (118) and ends the proof
of the convergence in the momentum equation, yielding (128).

###  5.5 Summary of limit passage from level I to level II

We resume the results of Section 5.

######  Proposition 10.

[Approximate solutions, level II] Let $\Omega\subset\mathbb{R}^{3}$ be a
bounded Lipschitz domain. Let the data $(\varrho_{B},z_{B},R_{B},Z_{B},{\bf
u}_{B})$, $(\varrho_{0},z_{0},R_{0},Z_{0},{\bf u}_{0})$ belong to the class
(27–30), (92). Suppose that $P$ satisfies assumptions (32–36).

Then for each fixed $\delta>0$, $\varepsilon>0$, there exists a quintet
$(\varrho,z,R,Z,{\bf u}={\bf v}+{\bf
u}_{B})=(\varrho_{\varepsilon},z_{\varepsilon},R_{\varepsilon},Z_{\varepsilon},{\bf
u}_{\varepsilon}={\bf v}_{\varepsilon}+{\bf u}_{B},)$ with the following
properties:

  1. 1.

It belongs to the function spaces

$r\in C_{\rm weak}(\overline{I};L^{\mathfrak{c}}(\Omega))\cap
L^{2}(0,T;W^{1,2}(\Omega))\cap
L^{\mathfrak{c}}(I;L^{\mathfrak{c}}(\partial\Omega,|{\bf u}_{B}\cdot{\bf
n}|{\rm d}S_{x})),$ (125) ${{\bf v}:={\bf u}-{\bf u}_{B}\in
L^{2}(0,T;W_{0}^{1,2}(\Omega;\mathbb{R}^{3})),}$ $(\varrho+z){\bf v}\in C_{\rm
weak}(\overline{I};L^{\frac{2{\mathfrak{c}}}{\mathfrak{c}+1}}(\Omega)),\;(\varrho+z){\bf
v}^{2},P(R,Z)\in L^{\infty}(I;L^{1}(\Omega)).$

  2. 2.

It obeys the following domination inequalities

$\begin{array}[]{c}\mbox{$\forall
t\in\overline{I}$},\;R_{\varepsilon}(t,x)\geq 0,\;Z_{\varepsilon}(t,x)\geq
0,\;\underline{a}R_{\varepsilon}(t,x)\leq
Z_{\varepsilon}(t,x)\leq\overline{a}R_{\varepsilon}(t,x),\\\
\underline{F}R_{\varepsilon}(t,x)\leq\varrho_{\varepsilon}(t,x)\leq\overline{F}R_{\varepsilon}(t,x),\;\underline{G}R_{\varepsilon}(t,x)\leq
z_{\varepsilon}(t,x)\leq\overline{F}Z_{\varepsilon}(t,x)\;\mbox{ for a.a.
$x\in\Omega$,}\\\ \\\ \mbox{for a.a. $(t,x)\in
I\times\partial\Omega$},\;R_{\varepsilon}(t,x)\geq
0,\;Z_{\varepsilon}(t,x)\geq 0,\;\underline{a}R_{\varepsilon}(t,x)\leq
Z_{\varepsilon}(t,x)\leq\overline{a}R_{\varepsilon}(t,x),\\\
\underline{F}R_{\varepsilon}(t,x)\leq\varrho_{\varepsilon}(t,x)\leq\overline{F}R_{\varepsilon}(t,x),\;\underline{G}R_{\varepsilon}(t,x)\leq
z_{\varepsilon}(t,x)\leq\overline{F}Z_{\varepsilon}(t,x).\end{array}$ (126)

  3. 3.

Each of

$\varrho_{\varepsilon},z_{\varepsilon},Z_{\varepsilon},R_{\varepsilon}$
satisfies the weak formulations (99) and (117) of parabolic equation with
${\bf u}={\bf u}_{\varepsilon}$.  (127)

  4. 4.

The momentum equation

$\int_{\Omega}(\varrho_{\varepsilon}+z_{\varepsilon}){\bf
u}_{\varepsilon}\cdot\boldsymbol{\varphi}{\,\rm
d}x\Big{|}_{0}^{\tau}=\int_{0}^{\tau}\int_{\Omega}\Big{[}(\varrho_{\varepsilon}+z_{\varepsilon}){\bf
u}_{\varepsilon}\cdot\partial_{t}\boldsymbol{\varphi}+(\varrho_{\varepsilon}+z_{\varepsilon}){\bf
u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon}:\nabla\boldsymbol{\varphi}+P_{\delta}(R_{\varepsilon},Z_{\varepsilon}){\rm
div}\,\boldsymbol{\varphi}$ (128) $-\mathbb{S}(\nabla{\bf
u}_{\varepsilon}):\nabla\boldsymbol{\varphi}-\varepsilon\nabla(\varrho_{\varepsilon}+z_{\varepsilon})\cdot\nabla{\bf
u}_{\varepsilon}\cdot\boldsymbol{\varphi}\Big{]}{\rm d}x\,{\rm d}t$

for any $\boldsymbol{\varphi}\in C_{c}^{1}([0,T]\times\Omega)$.

  5. 5.

The energy inequality

$\begin{split}&\int_{\Omega}\left[\frac{1}{2}(\varrho_{\varepsilon}+z_{\varepsilon})|{\bf
v}_{\varepsilon}|^{2}+{H}_{\delta}(R_{\varepsilon},Z_{\varepsilon})\right]{\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\Omega}\mathbb{S}(\nabla{\bf
u}_{\varepsilon}):\nabla{\bf u}_{\varepsilon}{\,\rm d}x\,{\rm d}t\\\
&{+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}{H}_{\delta}(R_{\varepsilon},Z_{\varepsilon}){\bf u}_{B}\cdot{\bf n}\
{\rm d}S_{x}\,{\rm
d}t}\leq-\int_{0}^{\tau}\int_{\Omega}\left[(\varrho_{\varepsilon}+z_{\varepsilon}){\bf
u}_{\varepsilon}\otimes{\bf
u}_{\varepsilon}+P_{\delta}(R_{\varepsilon},Z_{\varepsilon})\mathbb{I}\right]:\nabla{\bf
u}_{B}{\,\rm d}x\,{\rm d}t\\\
&+\int_{0}^{\tau}{\int_{\Omega}(\varrho_{\varepsilon}+z_{\varepsilon}){\bf
u}_{\varepsilon}\cdot\nabla{\bf u}_{B}\cdot{\bf u}_{B}{\,\rm d}x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\mathbb{S}(\nabla{\bf
u}_{\varepsilon}):\nabla{\bf u}_{B}{\,\rm d}x\,{\rm d}t\\\
&-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}{H}_{\delta}(R_{B},Z_{B}){\bf
u}_{B}\cdot{\bf n}\ {\rm d}S_{x}\,{\rm d}t\end{split}$ (129)

holds for a.a. $\tau\in I$.

  6. 6.

Renormalized identity

$\|r_{\varepsilon}(\tau)\|^{2}_{L^{2}(\Omega)}-\|r_{0}\|^{2}_{L^{2}(\Omega)}+2\varepsilon\int_{0}^{\tau}\|\nabla
r_{\varepsilon}\|_{L^{2}(\Omega)}^{2}{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}{\bf u}_{B}\cdot{\bf n}(r_{\varepsilon}-r_{B})^{2}{\rm d}S_{x}{\rm d}t$
(130) $+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}r_{\varepsilon}^{2}{\bf
u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm d}t=-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}r_{B}^{2}{\bf u}_{B}\cdot{\bf n}{\rm d}S_{x}{\rm
d}t-\int_{\Omega}r_{\varepsilon}^{2}{\rm div}\,{\bf u}_{\varepsilon}{\,\rm
d}x,$

where $r_{\varepsilon}$ stands for any linear combination of
$\varrho_{\varepsilon},z_{\varepsilon},R_{\varepsilon},Z_{\varepsilon}$

  7. 7.

Renormalized inequality

$\int_{\Omega}r_{\varepsilon}\log(r_{\varepsilon}+\mathfrak{a})(\tau,x)\varphi(x){\,\rm
d}x-\int_{\Omega}r_{0}\log(r_{0}+\mathfrak{a})(x)\varphi(x){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}\varphi
r_{\varepsilon}\log(r_{\varepsilon}+\mathfrak{a}){\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t$ (131)
$\leq\int_{0}^{\tau}\int_{\Omega}\Big{(}r_{\varepsilon}\log(r_{\varepsilon}+\mathfrak{a}){\bf
u}_{\varepsilon}\cdot\nabla\varphi-\varphi\frac{r^{2}_{\varepsilon}}{r_{\varepsilon}+\mathfrak{a}}{\rm
div}{\bf u}_{\varepsilon}\Big{)}{\,\rm d}x{\rm d}t$
$-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}\varphi
r_{B}\log(r_{B}+\mathfrak{a}){\bf u}_{B}\cdot{\bf n}\varphi{\rm d}S_{x}{\rm
d}t-\varepsilon\int_{0}^{\tau}\int_{\Omega}\Big{(}\log(r_{\varepsilon}+\mathfrak{a})+\frac{r_{\varepsilon}}{r_{\varepsilon}+\mathfrak{a}}\Big{)}\nabla
r_{\varepsilon}\cdot\nabla\varphi{\,\rm d}x{\rm d}t$

holds for all $\tau\in\overline{I}$, $\mathfrak{a}>0$, and all
$0\leq\varphi\in C^{1}(\overline{\Omega})$, where $r_{\varepsilon}$ stands for
$\varrho_{\varepsilon},z_{\varepsilon},R_{\varepsilon},Z_{\varepsilon}$.

  8. 8.

Renormalized inequality

$\int_{\Omega}R_{\varepsilon}\mathfrak{s}^{2}_{\varepsilon,\mathfrak{d}}(\tau,x)\varphi(\tau,x){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}R_{\varepsilon}s^{2}_{\varepsilon,\mathfrak{d}}{\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t$ (132)
$\leq\int_{\Omega}R_{0}(x)\mathfrak{s}^{2}_{\mathfrak{d}}(0,x)\varphi(x){\,\rm
d}x-\int_{0}^{\tau}\int_{\Gamma^{\rm in}}R_{B}\mathfrak{s}^{2}_{B}{\bf
u}_{B}\cdot{\bf n}\varphi{\rm d}S_{x}{\rm d}t$
$+\int_{0}^{\tau}\int_{\Omega}R_{\varepsilon}\mathfrak{s}_{\varepsilon,\mathfrak{d}}{\bf
u}_{\varepsilon}\cdot\nabla\varphi{\,\rm d}x{\rm
d}t-\varepsilon\int_{0}^{\tau}\int_{\Omega}\Big{(}2\mathfrak{s}_{\varepsilon,\mathfrak{d}}\nabla
Z_{\varepsilon}-\mathfrak{s}^{2}_{\varepsilon,\mathfrak{d}}\nabla
R_{\varepsilon}\Big{)}\cdot\nabla\varphi{\,\rm d}x{\rm d}t$

holds for all $\tau\in\overline{I}$ with any non negative $\varphi\in
C^{1}(\overline{\Omega}))$ and any $\mathfrak{d}\in\mathbb{R}$, where
$\mathfrak{s}_{B}(x)=\frac{Z_{B}(x)}{R_{B}(x)}$ and
$\mathfrak{s}_{\mathfrak{d}}$ is defined in (84).

##  6 Limit from level II to level III (limit $\varepsilon\to 0$)

We deduce from (126), (129) and (130) readily the following uniform bounds
with respect to $\varepsilon\in(0,1)$:

$\|{\bf
v}_{\varepsilon}\|_{L^{2}(I,W^{1,2}(\Omega))}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),\;\|(\varrho_{\varepsilon}+z_{\varepsilon})|{\bf
u}_{\epsilon}|^{2}\|_{L^{\infty}(I,L^{1}(\Omega))}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta),$
(133) $\|r_{\varepsilon}\|_{L^{\infty}(I,L^{\mathfrak{c}}(\Omega))}\leq
C(\delta),\;\|r_{\varepsilon}|{\bf u}_{B}\cdot{\bf
n}|^{1/{\mathfrak{c}}}\|_{L^{\mathfrak{c}}(I,L^{\mathfrak{c}}(\Omega))}\leq
C(\delta),$ (134)

and $\varepsilon$-dependent bounds

$\sqrt{\varepsilon}\|\nabla
r_{\epsilon}\|_{L^{2}(Q_{T})}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta).$
(135)

In the above, $r$ stands for $\varrho,z,Z,R$.

Estimate (134) yields the pressure bounded solely in
$L^{\infty}(I;L^{1}(\Omega))$. we can improve this estimate by taking

$\phi(t,x)=\psi(t)\mathcal{B}\left[\eta
R(t)-\frac{1}{|\Omega|}\int_{\Omega}\eta R(t){\,\rm d}x\right](x),\
0\leq\psi\in C^{\infty}_{c}((I)),\ 0\leq\eta\in C^{1}_{c}(\Omega)$ (136)

as test function the momentum equation (128), where ${\cal B}$ is the
Bogovskii operator, cf. Lemma 17,

$\int_{0}^{T}\psi\int_{\Omega}P_{\delta}(R,Z)R\eta{\,\rm d}x\,{\rm
d}t=\int_{0}^{T}\Big{[}\psi\frac{1}{|\Omega|}\int_{\Omega}R\eta{\,\rm
d}x\int_{\Omega}P_{\delta}(R,Z){\,\rm d}x\Big{]}{\,{\rm d}t}$ (137)
$-\int_{0}^{T}\psi\int_{\Omega}(\varrho(t)+z(t)){{\bf
u}}(t)\cdot\mathcal{B}\Big{[}\partial_{t}\Big{(}\eta
R(t)-\frac{1}{|\Omega|}\int_{\Omega}\eta R(t){\,\rm d}x\Big{)}\Big{]}{\,\rm
d}x\ \,{\rm d}t$ $-\int_{0}^{T}\psi^{\prime}\int_{\Omega}(\varrho+z){{\bf
u}}(t)\cdot\mathcal{B}\Big{[}\eta R(t)-\frac{1}{|\Omega|}\int_{\Omega}\eta
R(t){\,\rm d}x\Big{)}\Big{]}{\,\rm d}x\ \,{\rm d}t$
$-\int_{0}^{T}\psi\int_{\Omega}((\varrho+z){{\bf u}}\otimes{\bf
u}):\nabla\mathcal{B}\left[R-\frac{1}{|\Omega|}\int_{\Omega}R{\,\rm
d}x\right]{\,\rm d}x\ \,{\rm d}t$
$+\int_{0}^{T}\psi\int_{\Omega}\mathbb{S}(\nabla{\bf u}):\nabla\
\mathcal{B}\left[\eta R-\frac{1}{|\Omega|}\int_{\Omega}\eta R{\,\rm
d}x\right]{\,\rm d}x\,{\rm d}t.$

In the above,101010 Strictly speaking we should take in (136) the time (or
space) mollification of $\eta R$, in order to be able to justify the indentity
$\partial_{t}{\mathcal{B}}\Big{[}\Big{(}\eta
R(t)-\frac{1}{|\Omega|}\int_{\Omega}\eta R(t){\,\rm d}x\Big{)}\Big{]}={\cal
B}\Big{[}\partial_{t}\Big{(}\eta R(t)-\frac{1}{|\Omega|}\int_{\Omega}\eta
R(t){\,\rm d}x\Big{)}\Big{]}$ in $L^{p}(Q_{T})$. This is however standard, see
[25, Section 2.2.5].

$\partial_{t}(R\eta)={\mathcal{F}}_{\eta}\in L^{2}(I;[W^{1,2}(\Omega)]^{*})\
\;<{\cal F}_{\eta};\varphi>=<{\cal F}(R,{\bf u}),\phi\eta>,\ \mbox{cf.
(\ref{A0})}.$

Seeing (133-135), we easily verify the bounds (chosing properly the sequence
of $\psi$’s),

$\|\partial_{t}R_{\varepsilon}\|_{L^{2}(I;[W^{1,2}(\Omega)]^{*}))}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta,{\rm
supp}\eta)),\;\mbox{whence}\;\|\mathcal{B}[\partial_{t}{R_{\varepsilon}(t)}]\|_{L^{2}(Q_{T})}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta,{\rm
supp\eta}),$

where we have used (199) in Lemma 17. In view of the last estimate, estimates
(133–135) and in view of Lemma 17, we easily verify that the right hand side
of the identity (137) is bounded.

Seeing the structural hypotheses (33) we deduce from the above and the
domination inequalities (126) that,

$\|r_{\varepsilon}\|_{L^{\mathfrak{c}+1}(I\times
K)}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta,K),\;r\;\mbox{stands
for}\;\varrho,z,R,Z.$ (138)

and

$\|P_{\delta}(R_{\varepsilon},Z_{\varepsilon})\|_{L^{\frac{\mathfrak{c}+1}{\mathfrak{c}}}(I\times
K)}\stackrel{{\scriptstyle<}}{{\sim}}C(\delta,K)$ (139)

with any compact $K\subset\Omega$.

In absence of improved estimates of the pressure up to the boundary, we shall
need at least its equi-integrability near the boundary, in order to pass to
the limit in the energy inequality in the term containing
$P(R_{\varepsilon},Z_{\varepsilon}){\rm div}{\bf u}_{B}$. To this end we
recall the following lemma (see [35, Lemma 6.1]),

######  Lemma 11.

Let $\Omega$ be a bounded Lipschitz domain. Denote by
$U_{-,h}:=\\{x\in\Omega|{\rm dist}(x,\mathbb{R}^{3}\setminus\Omega\\}$ its
inner neighborhood. Consider a sequence $(p_{\varepsilon},{\bf
z}_{\varepsilon},{\bf
F}_{\varepsilon},\mbox{$\mathbb{G}$}_{\varepsilon})_{\varepsilon>0}$ of
functions which satisfy equation

$\partial_{t}{\bf z}_{\varepsilon}+{\bf F}_{\varepsilon}+{\rm
div}\mbox{$\mathbb{G}$}_{\varepsilon}+\nabla p_{\varepsilon}=0\;\mbox{in
${\cal D}^{\prime}(Q_{T};\mathbb{R}^{d})$}.$ (140)

Suppose finally that

$(z_{\varepsilon},{\bf
F}_{\varepsilon},\mbox{$\mathbb{G}$}_{\varepsilon},p_{\varepsilon})_{\varepsilon>0}\;\mbox{is
bounded in $L^{\infty}(0,T;L^{\alpha}(\Omega))\times L^{\kappa}(Q_{T})\times
L^{\kappa}(Q_{T})\times L^{1}(Q_{T})$ by $k>0$}$

uniformly with respect to $\varepsilon$.

Then there exists $h_{0}>0$ and $c=c(k,T,\Omega)>0$ such that

$\int_{0}^{T}\int_{U-,h}{p_{\varepsilon}}{\rm d}x{\rm d}t\leq
ch^{\Gamma},\;\Gamma=\min\\{1/\alpha^{\prime},1/\kappa^{\prime}\\},$ (141)

uniformly with respect to $\varepsilon$.

We apply this lemma to the momentum equation (128) and obtain, 111111
Alternatively, this estimate can be achieved directly by testing the momentum
equation (140) using test function $\psi{\cal B}\Big{(}1_{\Omega\setminus
U_{-,h}}(x)-\frac{|\Omega|-|U_{-,h}|}{|\Omega|}\Big{)}$ and estimates
(133)–(135), in particular, the bound of $P(R_{\varepsilon},Z_{\varepsilon})$
in $L^{\infty}(I;L^{1}(\Omega))$.

$\|P_{\delta}(R_{\varepsilon},Z_{\varepsilon})\|_{L^{1}(I\times
U_{-,h})}\stackrel{{\scriptstyle<}}{{\sim}}h^{\Gamma},\ \mbox{with some
$\Gamma>0$}$ (142)

uniformly with respect to $\varepsilon$.

The goal now is to pass to the limit $\varepsilon\to 0+$ in Proposition 10 and
to get the following result.

######  Proposition 12.

[Approximate solutions, level III]

Let $\Omega\subset\mathbb{R}^{3}$ be a bounded Lipschitz domain satisfying
(31). Let the data $\varrho_{B},z_{B},R_{B},Z_{B},{\bf u}_{B})$,
$(\varrho_{0},z_{0},R_{0},Z_{0},{\bf u}_{0})$ belong to the class (27–30),
(92). Suppose that $P$ satisfies assumptions (32–36).

Then for each fixed $\delta>0$ there exists a quinted $(\varrho,z,R,Z,{\bf
u}={\bf v}+{\bf
u}_{B})=(\varrho_{\delta},z_{\delta},R_{\delta},Z_{\delta},{\bf
u}_{\delta}={\bf v}_{\delta}+{\bf u}_{B})$ with the following properties:

  1. 1.

It belongs to the function spaces

$r\in C_{\rm weak}(\overline{I};L^{\mathfrak{c}}(\Omega))\cap
L^{\mathfrak{c}}(I;L^{\mathfrak{c}}(\Gamma^{\rm out},|{\bf u}_{B}\cdot{\bf
n}|{\rm d}S_{x})),$ (143) ${\bf v}:={\bf u}-{\bf u}_{B}\in
L^{2}(0,T;W_{0}^{1,2}(\Omega;\mathbb{R}^{3})),\;$ $(\varrho+z){\bf v}\in
C_{\rm
weak}(I;L^{\frac{2{\mathfrak{c}}}{\mathfrak{c}+1}}(\Omega)),\;(\varrho+z){\bf
v}^{2},P(R,Z)\in L^{\infty}(I;L^{1}(\Omega)).$

  2. 2.

Domination inequalities:

${\mbox{$\varrho_{\delta},z_{\delta},R_{\delta},Z_{\delta}$ satisfy domination
inequalities (\ref{eq5.1!})}.}$ (144)

  3. 3.

Each of

$\varrho_{\delta},z_{\delta},Z_{\delta},R_{\delta}$ satisfies the weak
formulation of the continuity equation (17) with ${\bf u}={\bf u}_{\delta}$.
(145)

  4. 4.

The momentum equation

$\int_{\Omega}(\varrho_{\delta}+z_{\delta}){\bf
u}_{\delta}\cdot\boldsymbol{\varphi}{\,\rm d}x\Big{|}_{0}^{\tau}$ (146)
$=\int_{0}^{\tau}\int_{\Omega}\Big{[}(\varrho_{\delta}+z_{\delta}){\bf
u}_{\delta}\cdot\partial_{t}\boldsymbol{\varphi}+(\varrho_{\delta}+z_{\delta}){\bf
u}_{\delta}\otimes{\bf
u}_{\delta}:\nabla\boldsymbol{\varphi}+P_{\delta}(R_{\delta},Z_{\delta}){\rm
div}\,\boldsymbol{\varphi}-\mathbb{S}(\nabla{\bf
u}_{\delta}):\nabla\boldsymbol{\varphi}\Big{]}{\,\rm d}x{\rm d}t$

holds with any $\tau\in\overline{I}$, and any $\boldsymbol{\varphi}\in
C^{1}([0,T]\times\Omega)$.

  5. 5.

The energy inequality

$\begin{split}&\int_{\Omega}\Big{(}\frac{1}{2}(\varrho_{\delta}+z_{\delta})|{\bf
v}_{\delta}|^{2}+{H}_{\delta}(R_{\delta},Z_{\delta})\Big{)}{\,\rm
d}x\Big{|}_{0}^{\tau}+\int_{0}^{\tau}\int_{\Omega}\mathbb{S}(\nabla{\bf
u}_{\delta}):\nabla{\bf u}_{\delta}{\,\rm d}x\,{\rm d}t+\\\
&+\int_{0}^{\tau}\int_{\Gamma^{\rm
out}}{H}_{\delta}(R_{\delta},Z_{\delta}){\bf u}_{B}\cdot{\bf n}\ {\rm
d}S_{x}\,{\rm
d}t-\int_{0}^{\tau}\int_{\Omega}\left[(\varrho_{\delta}+z_{\delta}){\bf
u}_{\delta}\otimes{\bf
u}_{\delta}+P_{\delta}(R_{\delta},Z_{\delta})\mathbb{I}\right]:\nabla{\bf
u}_{B}{\,\rm d}x\,{\rm d}t\\\
&+\int_{0}^{\tau}{\int_{\Omega}({\varrho}_{\delta}+z_{\delta}){\bf
u}_{\delta}\cdot\nabla{\bf u}_{B}\cdot{\bf u}_{B}{\,\rm d}x}\,{\rm
d}t+\int_{0}^{\tau}\int_{\Omega}\mathbb{S}(\nabla{\bf u}_{\delta}):\nabla{\bf
u}_{B}{\,\rm d}x\,{\rm d}t\\\ &-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}{H}_{\delta}(R_{B},Z_{B}){\bf u}_{B}\cdot{\bf n}\ {\rm d}S_{x}\,{\rm
d}t.\end{split}$ (147)

  6. 6.

Renormalized equations

$\int_{\Omega}B(r_{\delta}(\tau))\varphi(x){\,\rm
d}x-\int_{\Omega}B(r_{0})\varphi(x){\,\rm
d}x+{\int_{0}^{\tau}\int_{\Gamma^{\rm out}}B(r_{\delta}){\bf u}_{B}\cdot{\bf
n}\varphi{\rm d}S_{x}{\rm d}t}$ (148)
$=\int_{0}^{\tau}\int_{\Omega}\Big{(}B(r_{\delta}){\bf
u}_{\delta}\cdot\nabla\varphi-{\cal Y}(r_{\delta}){\rm div}{\bf
u}\varphi\Big{)}{\,\rm d}x{\rm d}t-\int_{0}^{\tau}\int_{\Gamma^{\rm
in}}B(r_{B}){\bf u}_{B}\cdot{\bf n}\varphi{\rm d}S_{x}{\rm d}t,$

hold for all $\tau\in\overline{I}$ and all $\varphi\in
C^{1}_{c}(\overline{\Omega})$, where $r_{\delta}$ stands for
$\varrho_{\delta},z_{\delta},R_{\delta},Z_{\delta}$. In the above $(B,{\cal
Y})$ are, in particular, the following couples

$\Big{(}B(s)=s^{\theta},{\cal
Y}(s)=(\theta-1)s^{\theta}\Big{)},\;\Big{(}B(s)=s\log s,{\cal
Y}(s)=s\Big{)},\;\Big{(}B(s)={\cal L}_{k}(s),{\cal Y}(s)={\cal
T}_{k}(s)\Big{)},$

where $\theta\in(0,\gamma/2]$, $k\geq 1$ and

${\cal T}_{k}(s)=k{\cal T}(s/k),\;{\cal T}\in C^{1}([0,\infty))\;{\cal
T}(s)=\left\\{\begin{array}[]{c}s\,\mbox{if $s\in[0,1]$}\\\ 2\,\mbox{if $s\geq
3$}\end{array}\right\\},\;\mbox{concave}$ (149)

is a truncation of the map $s\mapsto s$, while

${\cal L}_{k}(s)=s\int_{1}^{s}\frac{{\cal T}_{k}(t)}{t^{2}}{\rm d}t,\;\mbox{in
particular, $s{\cal L}_{k}^{\prime}(s)-{\cal L}_{k}(s)={\cal T}_{k}(s)$}$
(150)

  7. 7.

Renormalized equation

$\int_{\Omega}R_{\delta}(\tau,x)\mathfrak{s}^{2}_{\delta,\mathfrak{d}}(\tau,x)\varphi(x){\,\rm
d}x+\int_{0}^{\tau}\int_{\Gamma^{\rm out}}R\mathfrak{s}_{\mathfrak{d}}^{2}{\bf