# Weak Reproductive Solutions for a Convection-Diffusion Model Describing a
Binary Alloy Solidification Processes

Blanca Climent-Ezquerra  Mario Durán  Elva Ortega-Torres  Marko Rojas-Medar
Dpto. EDAN, Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Espana.
E-mail<EMAIL_ADDRESS>Departamento de Ingeniería Matemática, Universidad de
Concepción, Concepción, Chile. E-mail<EMAIL_ADDRESS>Departamento de
Matemáticas, Universidad Católica del Norte, Antofagasta, Chile. E-mail:
<EMAIL_ADDRESS>Departamento de Matemática, Universidad de Tarapacá, Arica,
Chile. E-mail<EMAIL_ADDRESS>
###### Abstract

We study the existence of reproductive weak solutions for a system of
equations describing a solidification process of a binary alloy confined into
a bounded and regular domain in $\mathbb{R}^{3}$, having mixed boundary
conditions.

###### keywords:

Reproductive solution, Solidification process, Navier-Stokes equations.
Mathematics Subject Classification: MSC 35Q35 · MSC 35K51 · MSC 76D03.

, ††thanks: Partially supported by Ministerio de Ciencia, Innovación y
Universidades, Grant PGC2018-098308-B-I00, with the collaboration of FEDER. ,
††thanks: Supported by FONDECYT-Chile grant No 1040205 and No 7060025. ,
††thanks: Supported by of MATH-AMSUD project 21-MATH-03 (CTMicrAAPDEs), CAPES-
PRINT 88887.311962/2018-00 (Brazil), Project UTA-Mayor, 4753-20, Universidad
de Tarapacá .

##  1 Introduction

The modeling and mathematical study of the behavior of metal alloys during the
solidification process has attracted the attention of a great number of
researchers and engineers in the last few years. The complex dynamics which
unfold in this process have led to the development of acute theories which try
to explain the advance of the solidification front on a macroscopic level, as
well as a describe on a microscopic level the dendritic growth in the solid
state zone. From a thermodynamic point of view we can point out the rigorous
work of Hills & al. [11], and from the point of view of a simple modeling,
Rappaz & Voller [14], who obtain results based on the properties of
thermodynamic averages. From a numeric stance, some implemented software
packages partially solve the evolving problem of the solidification of a multi
component alloy. We can mention programs by Ahmad [1] and Voller & Prakash
[16] in the area of Finite Elements, and Combeau & Lesoult [4] with respect to
Finite Volumes.

From a theoretic point of view the question of existence, uniqueness,
asymptotic or periodic behavior of solution for the evolving model remains
basically open, since we are dealing with a degenerate parabolic and free
boundary problem of high mathematical complexity. Some preliminaries works are
Bermúdez & Saguez [5], Donnelly [7] and Luckhaus & Visintin [19]. For the
associated stationary model, about the existence of solution, uniqueness and
computing aspects, we can mention the works by Blanc & al. [3], Duran & al.
[6], Duran & al. [5],and Badillo & al. [2], Gasser [10], Gaillard & Rappaz
[9], among others. It is known that certain dynamical systems may not have
periodic solutions because there exist many orbits, or branches of
bifurcations, that can be randomly reached by the solution. However, several
of these systems are still of the reproductive type, in the sense that there
exist at least two different times where the solution takes the same value.

In this paper, we deal with the existence of weak reproductive solutions of
the system modeling the solidification process. It is organized as follows: In
section 2, we state the mathematical model of solidification for a binary
alloy. In section 3, we state some notations and known inequalities, and we
define the notion of reproductive weak solution. In section 4, we study a
regularized problem associated to the solidification model. In section 5, we
prove the existence of a reproductive weak solutions of the regularized
problem studied in Section 4. The results of Section 5 are employed in Section
6, where we prove the main result about the existence of reproductive weak
solution, which is obtained as the limit of a sequence of reproductive weak
solutions of regularized problems.

##  2 Mathematical model for a binary alloy solidification

In this section we explain the mathematical model we will use to study the
solidification process.

###  2.1 The physical problem

The defective solidification of metal parts during their mould is one of the
main reasons for failure before the time envisioned by the industrial design.
The most popular procedure for molding of metal parts consists in pouring a
material (pure or compound) in its liquid state at a high temperature in a
mould. Once the incandescent matter has been poured, it will loose heat and
begin to solidify. Depending on various factors, amongst them the temperature
gradient and the concentration of solute in the solvent, it is possible that,
during the solidification inside the mould, the fluid may tend to separate
into regions with different characteristics, creating solute nodules
surrounded by solvent with a greater level of purity than that in the original
material. An important consequence of this segregation is that the resulting
metallic part will have, in some sectors, metallurgical properties that differ
with those calculated. Amongst these, the most important ones are the tensile
strength and malleability. The binary alloy is poured into the mould in such a
way that the lowest part is in contact with the bottom wall at low
temperature, meanwhile the top part remains at fusion temperature. This can
create a strong temperature gradient, due to which the process of
solidification will produce an heterogeneous material. In the top of domain
part we will see a liquid section, while at the bottom we will have already
solid material, leaving a middle zone in which the process of solidification
is not yet complete. This section of the mixture is characterized by the
coexistence of liquid and solid material.

###  2.2 Solidification model for a binary alloy

In a binary alloy the element called solute is the one that is dissolved in
the other one, called solvent. Without loss of generality, we can assume that
in the liquid state the mixture is homogeneous. Therefore, at the beginning of
the process a solidification front appears, which consists of solid material
dendrites, around which the material in liquid state will adhere until we
reach a completely solid phase. As it known, this process of solidification is
fundamentally controlled by the thermodynamical variables of solute
concentration and temperature in the mould. The variables determine the state
of the matter trough the phase diagram (see Figure 2). As a fundamental
reference we can point out, among others, the texts by Milne-Thomson [13],
Shapiro [15] and the doctoral theses of Ahmad [1], Gaillard [8] and Gasser
[10].

In the domain we can distinguish three regions, depending on the concentration
$c$ and the temperature $\theta$. Denoting with $\Omega$ the mould into which
the melted matter has been poured, we call $\Omega_{l},\Omega_{m}$ and
$\Omega_{s}$ the regions occupied by matter in liquid, mixture (coexistence of
solid and liquid) and solid state, respectively. Furthermore, we identify the
interfaces between these three sub-domains, calling $\Gamma_{ml}$ the mixture-
liquid interface, and $\Gamma_{sm}$ the one corresponding to solid-mixture.
With the purpose of considering borders conditions, we will also fix some
notation for the three regions of the exterior frontier $\Gamma$. This regions
are denoted by $\Gamma_{t},\Gamma_{b}$ and $\Gamma_{v}$ (see Figure 1).

Figure 1. Solidification mould

In presence of thermodynamic equilibrium, we use the phase diagram to
determine the state of the alloy. As a fundamental reference we can point out
the doctoral theses by Ahmad [1], Gasser [10] and Gaillard [8]. In this
diagram (see Figure 2), the horizontal axis corresponds to the concentration
$c$, the vertical one corresponds to the temperature $\theta$ and $\theta_{F}$
and $\theta_{E}$ are the fusion and eutectic temperature respectively. Therein
we represent two regular curves: the liquids $\gamma_{\ell}(\theta)$ and the
solidus $\gamma_{s}(\theta)$, jointing the point $(0,\theta_{F})$ of fusion of
solvent pure with the points $(\gamma_{l}(\theta_{E}),\theta_{E})$ and
$(\gamma_{s}(\theta_{E}),\theta_{E})$, respectively, where
$\gamma_{s}(\theta_{E})=c_{A}$ and $\gamma_{l}(\theta_{E})=c_{E}$. They
represent the temperatures of solidification of start and end in the
equilibrium.

Figure 2. Phase Diagram

In the phase diagram we identify three zones: the liquid $\Theta_{l}$, the
mixture $\Theta_{m}$ and the solid $\Theta_{s}$, which are defined as:

$\displaystyle\Theta_{l}$ $\displaystyle=$
$\displaystyle\\{(c,\theta)\in\mathbb{R}^{2}\,|\
0<c<c_{E},\>\>\theta>\gamma_{l}^{-1}(c)\\},$ $\displaystyle\Theta_{m}$
$\displaystyle=$ $\displaystyle\\{(c,\theta)\in\mathbb{R}^{2}\,|\
\theta_{E}<\theta<\theta_{F},\>\>\gamma_{s}(\theta)<c<\gamma_{l}(\theta)\\},$
$\displaystyle\Theta_{s}$ $\displaystyle=$
$\displaystyle\\{(c,\theta)\in\mathbb{R}^{2}\,|\
0<c<c_{A},\>\>\theta<\gamma_{l}^{-1}(c)\\},$

Moreover, we denote
$\Theta_{\pm}=(\mathbb{R}_{\pm}\times\mathbb{R})\setminus\overline{\Theta_{l}\cup\Theta_{s}\cup\Theta_{m}}$.

In a stationary case, in thermodynamic equilibrium, the sub-domains
$\Omega_{l},\Omega_{m}$ and $\Omega_{s}$ can be described by the
thermodynamics variables of concentration and temperature, $(c,\theta)$ as
follows:

$\displaystyle\Omega_{a}$ $\displaystyle=$ $\displaystyle\\{x\in\Omega\,|\
(c(x),\theta(x))\in\Theta_{a}\\},\,a=l,m,s,$ $\displaystyle\Gamma_{ml}$
$\displaystyle=$ $\displaystyle\\{x\in\Omega\,|\
c(x)=\gamma_{l}(\theta(x))\\}=\partial\Omega_{m}\cap\partial\Omega_{l},$
$\displaystyle\Gamma_{sm}$ $\displaystyle=$ $\displaystyle\\{x\in\Omega\,|\
c(x)=\gamma_{s}(\theta(x))\\}=\partial\Omega_{s}\cap\partial\Omega_{m},$
$\displaystyle\Omega_{ml}$ $\displaystyle=$
$\displaystyle\Omega_{m}\cup\Gamma_{ml}\cup\Omega_{l}.$

In order to model the transport phenomenon in the mixture region, we denote by
$c_{l}$ and $c_{s}$ the concentration of solute in liquid and solid phase,
respectively, which can be precisely determined through the phase diagram (see
e.g. [6], [7], [8]), then, $c_{l}=c$ in $\Theta_{l}$ and
$c_{l}=\gamma_{\ell}(\theta)$ if
$(c,\theta)\in\overline{\Theta_{m}}\setminus(\partial\Theta_{-}\cup\partial\Theta_{+})$.
Artificially, we extend the definition of $c_{l}$ to whole $\mathbb{R}^{2}$
as:

$c_{l}(c,\theta)=\left\\{\begin{array}[]{ll}c&\ \mbox{ if }\
(c,\theta)\in\Theta_{l},\\\ \gamma_{\ell}(\theta)&\ \mbox{ if }\
(c,\theta)\in\overline{\Theta_{m}}\setminus(\partial\Theta_{-}\cup\partial\Theta_{+}),\\\
\gamma_{\ell}(\gamma_{s}^{-1}(c))&\ \mbox{ if }\ (c,\theta)\in\Theta_{s},\\\
0&\ \mbox{ if }\ (c,\theta)\in\overline{\Theta_{-}},\\\ c_{E}&\ \mbox{ if }\
(c,\theta)\in\overline{\Theta_{+}}.\end{array}\right.$ (1)

In the solid-liquid mixture region it makes sense to define a function that
indicate the fraction of matter in solid state $f_{s}(c,\theta)$ and in liquid
state

$f_{l}(c,\theta)=1-f_{s}(c,\theta),$

per unit of volume. Thus, $f_{s}$ can be calculated by (see Figure 2):

$f_{s}(c,\theta)=\left\\{\begin{array}[]{ll}0&\ \mbox{ if }\
(c,\theta)\in\Theta_{l},\\\ \displaystyle\frac{a}{a+b}&\ \mbox{ if }\
(c,\theta)\in\Theta_{m},\\\ \,1&\ \mbox{ if
}\,(c,\theta)\in\Theta_{s},\end{array}\right.$

where $a=\gamma_{l}(\theta)-c$ and $b=c-\gamma_{s}(\theta)$. Analogously, we
extend this definition of $f_{s}$ to $\mathbb{R}^{2}$ by 1 if
$(c,\theta)\in\overline{\Theta_{s}\cup\Theta_{+}}$ and by 0 in other case.

Moreover, the unknown function of concentration $c$ that intervenes in the
equations is a weighted average between $c_{l}$ and $c_{s}$, that is,

$c=c_{l}\,(1-f_{s})+c_{s}\,f_{s}.$

Taking these expressions into account, we propose to model the dynamics of the
mixture from the moment it is poured into the mould in liquid phase until it
has finished its solidification. It is clear that, as time passes, the zones
$\Omega_{a},\,a\in\\{l,m,s\\}$, will appear, evolve, and disappear within
$\Omega$. This way, we will see the appearance of a concentration and a
temperature gradient which, added to the action of the force of gravity which
acts over the alloy, can develop a movement of velocity $\textit{{v}}(x,t)$
and pressure field $p(x,t)$ in the interior of the mixture
$\Omega_{ml}\times(0,T)$ (see [9]). It is important to observe that the set
$\Omega_{ml}$ and the interfaces $\Gamma_{sm},\Gamma_{ml}$ are unknown a
priori, because its dependence on the concentration $c$ and temperature
$\theta$. This is an additional difficult of the problem.

We model the transport during the solidification process using the
incompressible Navier-Stokes equations, coupled with terms of force (external
and internal) as follows (see [8])

$\displaystyle\rho\frac{\partial\textit{{v}}}{\partial t}-2\nu\,{\rm
div}\,e(\textit{{v}})+\rho(\textit{{v}}\cdot\nabla)\,\textit{{v}}+F_{i}(c,\theta)\,\textit{{v}}+\nabla
p$ $\displaystyle=$ $\displaystyle\mathbf{F}_{e}(c,\theta)\ \ {\rm in}\
\mathcal{Q}_{ml},$ (2) $\displaystyle{\rm div}\,\textit{{v}}$ $\displaystyle=$
$\displaystyle 0\ \ {\rm in}\ \mathcal{Q}_{ml},$ (3)

where $\mathcal{Q}_{ml}=\Omega_{ml}\times(0,T)$, $\nu$ is the dynamical
viscosity, $\rho$ is the mean density in the mixture, both considered
constants, and
$e(\textit{{v}})=(\nabla\textit{{v}}+(\nabla\textit{{v}})^{t})/2$ is the
linearized stress tensor of v. The external force
${\mathbf{F}}_{e}(\cdot,\cdot)$ is given by the Boussinesq approximation:

$\mathbf{F}_{e}(c,\theta)=\rho\,\mathbf{g}\,(\alpha\,(\theta-\theta_{r})+\beta\,(c_{l}(c,\theta)-c_{r})),$
(4)

where $\alpha,\beta$ are known real constants, $\theta_{r},c_{r}$ are
respectively a temperature and a concentration of reference, and
$\mathbf{g}=(0,0,-g)^{t}$ is the gravitational force.

The term $F_{i}(\cdot,\cdot)$ represents the internal force within a porous
material which opposes the movement of the fluid on $\Omega_{ml}$. It is known
as the Carman-Kozeny law, and it is given by

$F_{i}(c,\theta)=C_{0}\,\frac{f_{s}^{2}(c,\theta)}{(1-f_{s}(c,\theta))^{3}}\quad\mbox{
with }C_{0}>0.$ (5)

In the liquid region $F_{i}(\cdot,\cdot)=0$ and is hypersingular in
$\Omega_{ml}$. On the other hand, we suppose that the velocity field v is $0$
on $\Omega_{s}$ and continuous over the whole domain $\Omega$.

The diffusion process of the concentration and the heat transfer in a multi-
component mixture is described by a non-linear system of parabolic equations.
The concentration, the temperature and the velocity are the variables involve.
These equations take into account the phenomena of diffusion and transport,
and are deduced considering the laws of solute conservation in the mixture and
conservation of energy. Simplifying these equations, using the laws of Fourier
and Fick, we obtain

$\displaystyle\frac{\partial c}{\partial
t}-\eta\,\Delta\,c+\textit{{v}}\cdot\nabla c_{l}(c,\theta)$ $\displaystyle=$
$\displaystyle 0\quad\mbox{ in }\ \mathcal{Q},$ (6)
$\displaystyle\frac{\partial}{\partial
t}\,H(c,\theta)-\kappa\,\Delta\,\theta+\rho\,C_{p}\,\textit{{v}}\cdot\nabla\theta$
$\displaystyle=$ $\displaystyle 0\quad\mbox{ in }\mathcal{Q},$ (7)

where $\mathcal{Q}=\Omega\times(0,T)$, $\kappa,\eta$ are two strictly positive
real constants that represent the physical diffusions, $C_{p}$ is the caloric
capacity, $c_{l}(\cdot,\cdot)$ is the liquid concentration and
$H(\cdot,\cdot)$ is the enthalpy, defined as

$H(\cdot,\cdot)=\rho\,C_{p}\,\theta+\rho\,L_{f}\,f_{l}(c,\theta),$

where $L_{f}$ is the latent heat of fusion. There follows, we approximate
$H(\cdot,\cdot)$ by $\rho\,C_{p}\,\theta$ because we do not know how to treat
the non-linear term
$\partial_{t}\frac{\gamma_{l}(\theta)-c}{\gamma_{l}(\theta)-\gamma_{s}(\theta)}$
that appears in (7) in $\Omega_{m}$.

The initial conditions are obtained from the moment in which the melted
material is poured into the solidification mould and are, in general, regular
functions. The limit conditions considered are:

  * •

Adherence of the fluid over $\Gamma_{bt}=\Gamma_{b}\cup\Gamma_{t}$.

  * •

The normal flux and the tangential stress are zero over $\Gamma_{v}$.

  * •

The normal concentration flux is zero over $\Gamma$ and a fixed total
concentration of solute in the domain. The temperature is known over
$\Gamma_{bt}$ and the wall $\Gamma_{v}$ is adiabatic.

###  2.3 Setting the evolutive mathematical model

Let be $\Omega$ an open domain subset of $\mathbb{R}^{3}$ with regular
boundary $\Gamma=\partial\Omega$. Then, from (2)-(7) and the boundary
conditions stated above, the problem of our present study reads: Find
functions $(c,\theta,\textit{{v}},p):\mathcal{Q}\rightarrow\mathbb{R}^{6}$,
satisfying

$\displaystyle\frac{\partial c}{\partial t}-\eta\,\Delta
c+\textit{{v}}\cdot\nabla c_{l}(c,\theta)=0\quad{\rm in}\ \mathcal{Q},$ (8)
$\displaystyle\frac{\partial c}{\partial n}=0\quad{\rm on}\
\Gamma\times(0,T),\qquad\int\limits_{\Omega}c\,(x,t)\,dx=c_{g}\quad{\rm in}\
(0,T),$ (9) $\displaystyle\ c(x,0)=c_{0}(x)\quad{\rm in}\ \Omega,$ (10)
$\displaystyle\frac{\partial\theta}{\partial
t}-\kappa\Delta\theta+\rho\,C_{p}\,\textit{{v}}\cdot\nabla\theta=0\quad{\rm
in}\ \mathcal{Q},$ (11) $\displaystyle\theta=\theta_{\delta}\quad{\rm on}\
\Gamma_{bt}\times(0,T),\qquad\frac{\partial\theta}{\partial\textit{{n}}}=0\quad{\rm
on}\ \Gamma_{v}\times(0,T),$ (12)
$\displaystyle\theta(x,0)=\theta_{0}(x)\quad{\rm in}\ \Omega,$ (13)
$\displaystyle\frac{\partial\textit{{v}}}{\partial t}-2\nu\,{\rm
div}\,e(\textit{{v}})+\rho\,(\textit{{v}}\cdot\nabla)\textit{{v}}+F_{i}(c,\theta)\,\textit{{v}}+\nabla
p=\mathbf{F}_{e}(c,\theta)\ {\rm in}\ \mathcal{Q}_{ml},$ (14)
$\displaystyle{\rm div}\,\textit{{v}}=0\quad{\rm in}\ \mathcal{Q}_{ml},\
\hskip 79.6678pt\textit{{v}}=\mathbf{0}\quad{\rm in}\ \Omega_{s}\times(0,T),$
(15) $\displaystyle\textit{{v}}=\mathbf{0}\quad{\rm on}\
\Gamma_{bt}\times(0,T),\ \hskip 59.75095pt\textit{{v}}\cdot\textit{{n}}=0\
{\rm on}\quad\Gamma_{v}\times(0,T),$ (16)
$\displaystyle\sigma(\textit{{v}},p)\,\textit{{n}}\wedge\textit{{n}}=0\quad{\rm
on}\ \Gamma_{v}\times(0,T),\ \
\quad\textit{{v}}(x,0)=\textit{{v}}_{0}(x)\quad{\rm in}\ \Omega.$ (17)

Here $c_{g}$ is the total quantity of solute in the mixture and is a strictly
positive real constant that satisfies the compatibility relation

$0\leq c_{g}\leq\gamma_{\ell}(\theta_{E})|\Omega|.$ (18)

Furthermore, $\theta_{\delta}$ is a known distribution of temperature on
$\Gamma_{bt}$, $n$ denote the vector unit outward to the surface $\Gamma$ and
$\sigma(\textit{{v}},p)$ represents the stress tensor given by
$\sigma(\textit{{v}},p)=-p\,I+2\,\nu\,e(\textit{{v}})$.

##  3 Function spaces and preliminaries

The vector-valued functions in $\mathbb{R}^{3}$ are denoted by
$\\{\,\textit{{u}},\,\textit{{v}},\,\textit{{w}}\,\\}$, while the scalar ones
are written solely by $\\{\,p,\,\varphi,\,\psi,\,\phi,\,c,\,\theta\\}$. For
simplicity, we denote $\mathbf{L}^{2}$ and $\mathbf{H}^{1}$ instead of
$L^{2}(\Omega)^{3}$ and $H^{1}(\Omega)^{3}$, $L^{p}(X)$ instead of
$L^{p}(0,T;X)$, for $1\leq p\leq\infty$, etc. Also, the scalar product in
$L^{2}$ will be denoted by $(\cdot,\cdot)$, and $\langle\cdot,\cdot\rangle$
will denote some duality products. For all integer $m\geq 0$, we denote the
norm of the scalar and vector-valued Hilbert spaces $H^{m}$ and $L^{2}$ as
$\|\cdot\|_{m}$ and $\|\cdot\|$ respectively. The $L^{p}$ norma will be
denoted by $\|\cdot\|_{L^{p}}$. Other notation that we use are: $H^{1}(H^{1})$
to denote the space $H^{1}(0,T;H^{1}(\Omega))$ and $W^{1,\infty}(\Omega)$
denote the usual Sobolev space which consists of functions in $L^{\infty}$
such that their derivatives in the sense of the distributions are in
$L^{\infty}$.

Let us consider the following function spaces:

H $\displaystyle=$ $\displaystyle\\{\textit{{u}}\in\mathbf{L}^{2}:{\rm
div}\,\textit{{u}}=0\mbox{ in }\Omega,\textit{{u}}\cdot\textit{{n}}=0\mbox{ on
}\partial\Omega\\},$ V $\displaystyle=$
$\displaystyle\\{\textit{{u}}\in\mathbf{H}^{1}_{0}:{\rm
div}\,\textit{{u}}=0\mbox{ in }\Omega\\},$ $\displaystyle\textit{H}_{\theta}$
$\displaystyle=$ $\displaystyle\\{\,\psi\in\textit{H}^{1}:\psi=0\ {\rm on}\
\Gamma_{bt}\\},$ $\displaystyle\textit{{H}}_{v}$ $\displaystyle=$
$\displaystyle\\{\,\textit{{w}}\in\mathbf{H}^{1}:{\rm
div}\,\textit{{w}}=0\mbox{ in }\Omega,\ \textit{{w}}=0\mbox{ on
}\Gamma_{bt}\mbox{ and }\textit{{w}}\cdot\textit{{n}}=0\mbox{ on
}\Gamma_{v}\\}.$

From now on, $C>0$ will denote different constants, depending only on the
fixed data of the problem.

In forthcoming section of the paper we use some inequalities, whose proofs can
be found in [12], and that we state in what follows:

###### Lemma 3.1

(Young’s Inequality) Let $A\geq 0,\,B\geq 0$ be real constants. Suppose that
$p,\,q\in\mathbb{R}$ such that $\,1\leq p<\infty\,$ and
$\,\displaystyle\frac{1}{p}+\frac{1}{q}=1$. Then, for all $\,\xi>0\,$, the
following inequality hold:

$A\,B\leq\frac{\xi}{p}\,A^{p}+\frac{(p-1)\,\xi^{1-p}}{p}\,B^{q}.$ (19)

###### Lemma 3.2

(Gronwall’s Inequality) Let functions $g(t)$ and $h(t)$ nonnegative,
continuous with $\,g^{{}^{\prime}}(t)\geq 0\,$ and $h(t)$ sumable on $[0,T]$.
Assume that it is satisfied

$\varphi\,(t)+\int_{0}^{t}\varphi^{*}\,(\tau)\,d\,\tau\leq\frac{g(t)}{\lambda}+\int_{0}^{t}h\,(\tau)\,\varphi\,(\tau)\,d\,\tau,$

for two positive continuous functions $\,\varphi\,$ and $\,\varphi^{*}\,$ on
$\,[0,T]\,$ and the constant $\,\lambda>0$. Then it holds

$\varphi\,(t)+\int_{0}^{t}\varphi^{*}\,(\tau)\,d\,\tau\leq\frac{g(t)}{\lambda}\,(1+\int_{0}^{t}h\,(\tau)\,d\,\tau\,)\,\exp\,(\int_{0}^{t}h\,(\tau)\,d\,\tau\,).$
(20)

###### Lemma 3.3

(Friedrich’s Lemma) Let $\Omega$ be a fixed bounded domain of $\mathbb{R}^{3}$
with boundary $\,\partial\Omega\,$ of class $\,C^{2}$. For any $\omega>0$,
$N_{\omega}$ basis functions ${\textit{{w}}}^{i}$, $i=1,2,\dots,N_{\omega}$,
can be constructed satisfying the following inequality

$\|\textit{{u}}\|^{2}\leq\sum_{i=1}^{N_{\omega}}\,(\textit{{u}},{\textit{{w}}}^{i})^{2}+\omega\,\|\nabla\textit{{u}}\|^{2}\quad\forall\,\textit{{u}}\in\mathbf{H}^{1}_{0}.$
(21)

We denote the classical bilinear and trilinear forms by

$(\nabla\textit{{u}},\nabla\textit{{v}})=\sum_{i,j=1}^{3}\int_{\Omega}\frac{\partial
u_{j}}{\partial x_{i}}\frac{\partial v_{j}}{\partial
x_{i}}\,dx,\quad(\textit{{u}}\cdot\nabla\textit{{v}},\textit{{w}})=\sum_{i,j=1}^{3}\int_{\Omega}u_{j}\,\frac{\partial
v_{i}}{\partial x_{j}}\,w_{i}\,dx,$

for all vector-valued functions $\,\textit{{u}},\textit{{v}},\textit{{w}}$ for
which the integrals make sense. Moreover, the trilinear form has the
properties

$(\textit{{u}}\cdot\nabla\textit{{v}},\textit{{w}})=-(\textit{{u}}\cdot\nabla\textit{{w}},\textit{{v}}),\quad(\textit{{u}}\cdot\nabla\textit{{v}},\textit{{v}})=0\quad\forall\,\textit{{u}}\in\textit{{H}}_{v},\
\forall\,\textit{{v}},\,\textit{{w}}\,\in\mathbf{H}^{1}.$

Hereinafter, we will use the following classical interpolation and Sobolev
inequalities (for $3D$ domains):

$\|v\|_{L^{6}}\leq
C\|v\|_{1},\quad\|v\|_{L^{3}}\leq\|v\|^{1/2}\|v\|_{1}^{1/2}\quad\forall v\in
H^{1}$

and

$\|v\|_{L^{\infty}}\leq C\|v\|_{1}^{1/2}\|v\|_{2}^{1/2}\quad\forall v\in
H^{2}.$

We state now our problem rigorously establishing regularity assumptions on the
boundary $\Gamma$ and some technical assumption on the external data.

Hypothesis:

  * (S1)

Let $O_{0}$ be a neighborhood of the origin. Thus $O_{0}\subseteq{\rm int\
}\Omega$, $\Omega$ is connected and $\,\Omega\subseteq B(0,R),\,R>0$. The
boundary $\Gamma\in C^{2}$;

  * (S2)

The given external field of temperature $\theta_{\delta}$ belongs to
$H^{1}(L^{\infty}(\Gamma_{bt}))$ and there exists a continuous extension in
$H^{1}(H^{1})\cap L^{2}(H^{1})$, satisfying
$\theta_{\delta}(x,0)=\theta_{\delta}(x,T)$ in $\Omega$;

  * (S3)

The function of concentration in liquid phase $c_{l}(\cdot,\cdot)$ belongs to
$W^{1,\infty}(\mathbb{R}^{2})$.

There follows, we define precisely the notion of a weak reproductive solution
for the system (8)-(17).

###### Definition 3.4

We say that the triplet of functions
$(c(x,t),\,\theta(x,t),\,\textit{{v}}(x,t))$ is a weak reproductive solution
of the evolutive problem (8)-(17) at time $T$ if and only if there exists
$c_{0},\theta_{0}\in\,L^{2}$ and $\textit{{v}}_{0}\in\,\textit{{H}}$ such that
the following relations hold:

$\displaystyle
c(x,T)=c_{0}(x),\quad\theta(x,T)=\theta_{0}(x),\quad\textit{{v}}(x,T)=\textit{{v}}_{0}(x)\mbox{
a.e. in }\Omega,$ (22) $\displaystyle c\in L^{2}(H^{1})\cap
L^{\infty}(L^{2}),\quad\ c(x,0)=c(x,T)\mbox{ in }\Omega,$ (23)
$\displaystyle\theta\in L^{2}(H^{1})\cap L^{\infty}(L^{2}),\quad\
\theta(x,0)=\theta(x,T)\mbox{ in }\Omega,$ (24) $\displaystyle\textit{{v}}\in
L^{2}(\textit{{H}}_{v})\cap L^{\infty}(\textit{{H}}),\quad\
\textit{{v}}(x,0)=\textit{{v}}(x,T)\mbox{ in }\Omega,$ (25) $\displaystyle
0\leq c\leq\gamma_{\ell}\,(\theta_{E})\ \mbox{ in }\mathcal{Q},$ (26)
$\displaystyle\int\limits_{\Omega}c\,(x,t)\,dx=c_{g}\ \mbox{ in }(0,T),$ (27)
$\displaystyle\theta(\cdot,t)-\theta_{\delta}(\cdot,t)\in H_{\theta}\ \mbox{
in }(0,T),$ (28) $\displaystyle\textit{{v}}=0\quad\mbox{ in
}\Omega_{s}\times(0,T),$ (29)

and the equations are satisfies in a distributional sense:

$\displaystyle(\frac{\partial c}{\partial t},\varphi)+\eta\,(\nabla
c,\nabla\varphi)+(\textit{{v}}\cdot\nabla c_{l}(c,\theta),\varphi)=0,$ (30)
$\displaystyle(\frac{\partial\theta}{\partial
t},\psi)+\kappa\,(\nabla\theta,\nabla\psi)+\rho\,C_{p}\,(\textit{{v}}\cdot\nabla\theta,\psi)=0,$
(31) $\displaystyle(\frac{\partial\textit{{v}}}{\partial
t},\,\textit{{w}})+\nu\,(\nabla\textit{{v}},\,\nabla\textit{{w}})+\rho\,((\textit{{v}}\cdot\nabla)\textit{{v}},\,\textit{{w}})+(F_{i}(c,\theta)\,\textit{{v}},\,\textit{{w}})=(\mathbf{F}_{e}(c,\theta),\,\textit{{w}}),$
(32)

for all $\varphi\in H^{1},\psi\in H_{\theta},\textit{{w}}\in\textit{{H}}_{v}$
with ${\rm supp}\,\textit{{w}}\subset{\Omega_{ml}}$.

It is important to note that any reproductive weak solution of the problem
(8)-(17) satisfy the following result.

###### Proposition 3.5

Let $(c,\theta,\textit{{v}})$ be a weak reproductive solution of the problem
$(\ref{eq8})-(\ref{eq10c})$. Then the following inequalities are satisfied

$\displaystyle\hskip 71.13188pt0\leq$ $\displaystyle c(x,t)$
$\displaystyle\leq\gamma_{\ell}(\theta_{E})\ \mbox{ a.e. in }\mathcal{Q},$
(33) $\displaystyle\theta_{\delta}^{\inf}\leq\,$ $\displaystyle\theta(x,t)$
$\displaystyle\,\leq\,\theta_{\delta}^{\sup}\ \mbox{ a.e. in }\mathcal{Q},$
(34)

where
$\theta_{\delta}^{\inf}=\min\\{\mathop{\inf}\limits_{\Omega}\theta_{0}(x),\,\mathop{\inf}\limits_{\Lambda}\theta_{\delta}(x,t)\\},\
\theta_{\delta}^{\sup}=\max\\{\mathop{\sup}\limits_{\Omega}\theta_{0}(x),\,\mathop{\sup}\limits_{\Lambda}\theta_{\delta}(x,t)\\}$
and $\Lambda=\Gamma_{bt}\times(0,T)$.

Proof: By definition $f_{+}=\max\\{0,f\\}$ and $f_{-}=\max\\{0,-f\\}$. To
prove the inequality on the right side of (33), we take
$\varphi=[c(\cdot,t)-\gamma_{\ell}(\theta_{E})]_{+}\in H^{1}$ as test function
in (30), by a simple calculation, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|[c(t)-\gamma_{\ell}(\theta_{E})]_{+}\|^{2}+\eta\,\|\nabla[c(t)-\gamma_{\ell}(\theta_{E})]_{+}\|^{2}$
(35) $\displaystyle\hskip 28.45274pt+(\textit{{v}}\cdot\nabla
c_{l}(c,\theta),\,[c\,(t)-\gamma_{\ell}(\theta_{E})]_{+})=0.$

From (1), if $c(t)\geq\gamma_{\ell}(\theta_{E})$, then
$(c,\theta)\in\Theta_{+}$ and $c_{l}$ is constant, therefore

$(\textit{{v}}\cdot\nabla
c_{l}(c,\theta),\,[c(t)-\gamma_{\ell}(\theta_{E})]_{+})=0.$

By integrating (35) with respect to $t$ from $0$ to $T$ and since $c(T)=c(0)$,
we get

$\eta\int_{0}^{T}\|\nabla[c(t)-\gamma_{\ell}(\theta_{E})]_{+}\|^{2}dt=0,$ (36)

and we conclude that $[c(x,t)-\gamma_{\ell}(\theta_{E})]_{+}=M$, being $M$ a
non-negative constant. By definition, $M=0$ or
$M=c(x,t)-\gamma_{\ell}(\theta_{E})$, then if
$M=c(x,t)-\gamma_{\ell}(\theta_{E})$ we have
$c(x,t)=M+\gamma_{\ell}(\theta_{E})$, so from (27) and by integrating, we
deduce that $c_{g}-\gamma_{\ell}(\theta_{E})|\Omega|=M\,|\Omega|>0$, thus
$c_{g}>\gamma_{\ell}(\theta_{E})|\Omega|$ in contradiction with (18).
Therefore, $[c(t)-\gamma_{\ell}(\theta_{E})]_{+}=0$ which implies that
$c(t)-\gamma_{\ell}(\theta_{E})\leq 0$, obtaining the upper bound of (33).

To prove that $c$ is not negative we use the test function
$\varphi=[c(\cdot,t)]_{-}\in H^{1}$, and from (30) similarly as to (36), we
can obtain

$\eta\int_{0}^{T}\|\nabla[c(t)]_{-}\|^{2}dt=0,$

which implies $[c(x,t)]_{-}=M_{1}$ being $M_{1}$ a non-negative constant. By
definition $M_{1}=0$ or $M_{1}=-c(x,t)$, then if $M_{1}=-c(x,t)$ from (27), we
get $c_{g}=-M_{1}\,|\Omega|<0$ in contradiction with (18). Thus,
$[c(x,t)]_{-}=0$ and the lower bound of (33) is proved.

To prove the left side inequality of (34), we choose
$\psi=[\theta(\cdot,t)-\theta_{\delta}^{sup}]_{+}\in H_{\theta}$ as function
test in (31) and taking into account that

$(\textit{{v}}\cdot\nabla\theta,[\theta-\theta_{\delta}^{sup}]_{+})=(\textit{{v}}\cdot\nabla[\theta-\theta_{\delta}^{sup}]_{+},[\theta-\theta_{\delta}^{sup}]_{+})=0,$

we obtain

$\frac{d}{dt}\|[\theta\,(t)-\theta_{\delta}^{sup}]_{+}\|^{2}+2\,\kappa\|\nabla[\theta\,(t)-\theta_{\delta}^{sup}]_{+}\|^{2}=0.$
(37)

Since $\theta(T)=\theta(0)$ a.e. in $\Omega$, by integrating (37) with respect
to $t$ from $0$ to $T$, we get

$\int_{0}^{T}\|\nabla[\theta(t)-\theta_{\delta}^{sup}]_{+}\|^{2}dt=0.$

Hence, taking into account (28), we can conclude that
$[\theta(x,t)-\theta_{\delta}^{sup}]_{+}=0$ and the upper bound of (34) is
obtained. The lower bound of (34) is proved similarly, using the test function
$\psi=[\theta(\cdot,t)-\theta_{\delta}^{inf}]_{-}\in H_{\theta}$.

##  4 The regularized mathematical problem

Observe that the Carman-Kozeny term given in (5), becomes discontinuous when
$x\in\Omega_{s}$ since $f_{s}(c(x,t),\theta(x,t))=1$. Then, to avoid this
singularity and to consider the system defined in whole the domain $\Omega$,
we define a family of regularized problems, which are obtained by modifying
the Carman-Kozeny term, as follows:

$F_{i}^{\epsilon}\,(c_{\epsilon},\theta_{\epsilon})=C_{0}\frac{f_{s}^{2}(c_{\epsilon},\theta_{\epsilon})}{(1-f_{s}(c_{\epsilon},\theta_{\epsilon})+\epsilon)^{3}}\quad\forall\,\epsilon\in(0,1]\,.$
(38)

More precisely, the family of the regularized associated problems is defined
as: Find the functions
$(\,c_{\epsilon},\,\theta_{\epsilon},\,p_{\epsilon},\,\textit{{v}}_{\epsilon}\,):\,\mathcal{Q}\rightarrow\mathbb{R}^{6}$,
such that

$\displaystyle\frac{\partial c_{\epsilon}}{\partial t}-\eta\,\Delta
c_{\epsilon}+\textit{{v}}_{\epsilon}\cdot\nabla
c_{l}(c_{\epsilon},\theta_{\epsilon})=0\quad{\rm in}\ \mathcal{Q},$ (39)
$\displaystyle\ \frac{\partial c_{\epsilon}}{\partial\textit{{n}}}=0\quad{\rm
on}\ \Gamma\times(0,T),\qquad\int_{\Omega}c_{\epsilon}(x,t)\,dx=c_{g}\quad{\rm
in}\ (0,T),$ (40) $\displaystyle\ c_{\epsilon}(x,0)=c_{0}(x)\quad{\rm in}\
\Omega,$ (41) $\displaystyle\frac{\partial\theta_{\epsilon}}{\partial
t}-\kappa\Delta\theta_{\epsilon}+\rho\,C_{p}\,\textit{{v}}_{\epsilon}\cdot\nabla\theta_{\epsilon}=0\quad{\rm
in}\ \mathcal{Q},$ (42) $\displaystyle\
\theta_{\epsilon}=\theta_{\delta}\quad{\rm on}\
\Gamma_{bt}\times(0,T),\qquad\frac{\partial\theta_{\epsilon}}{\partial\textit{{n}}}=0\quad{\rm
on}\ \Gamma_{v}\times(0,T),$ (43) $\displaystyle\
\theta_{\epsilon}(x,0)=\theta_{0}(x)\quad{\rm in}\ \Omega,$ (44)
$\displaystyle\frac{\partial\textit{{v}}_{\epsilon}}{\partial t}-2\nu\,{\rm
div}\,e(\textit{{v}}_{\epsilon})+\rho\,(\textit{{v}}_{\epsilon}\cdot\nabla)\textit{{v}}_{\epsilon}+F_{i}^{\epsilon}(c_{\epsilon},\theta_{\epsilon})\,\textit{{v}}_{\epsilon}+\nabla
p_{\epsilon}=\mathbf{F}_{e}(c_{\epsilon},\theta_{\epsilon})\ {\rm in}\
\mathcal{Q},$ (45) $\displaystyle\ {\rm
div}\,\textit{{v}}_{\epsilon}=0\quad{\rm in}\
\mathcal{Q},\qquad\qquad\qquad\textit{{v}}_{\epsilon}=\mathbf{0}\quad{\rm on}\
\Gamma_{bt}\times(0,T),$ (46) $\displaystyle\
\textit{{v}}_{\epsilon}\cdot\textit{{n}}=0\quad{\rm on}\
\Gamma_{v}\times(0,T),\qquad\sigma(\textit{{v}}_{\epsilon},p_{\epsilon})\,\textit{{{}}{{n}}}\wedge\textit{{n}}=0\quad{\rm
on}\ \Gamma_{v}\times(0,T),$ (47) $\displaystyle\
\textit{{v}}_{\epsilon}(x,0)=\textit{{v}}_{0}(x)\quad{\rm in}\ \Omega.$ (48)

In the same way as Definition 3.4, we give the following definition of
reproductive regularized weak solution of the problem (8)-(17).

###### Definition 4.1

We say that the triplet of functions $(c_{\epsilon}(x,t),$
$\,\theta_{\epsilon}(x,t),\,\textit{{v}}_{\epsilon}(x,t))$ is a regularized
reproductive weak solution of the evolutive problem (8)-(17) at time $T$ if
and only if there exist $c_{0},\theta_{0}\in\,L^{2}$ and
$\textit{{v}}_{0}\in\,\textit{{H}}$ such that the following relations hold:

$\displaystyle
c_{\epsilon}(x,T)=c_{0}(x),\,\theta_{\epsilon}(x,T)=\theta_{0}(x),\,\textit{{v}}_{\epsilon}(x,T)=\textit{{v}}_{0}(x)\mbox{
a.e. in }\Omega,$ (49) $\displaystyle c_{\epsilon}\in L^{2}(H^{1})\cap
L^{\infty}(L^{2}),\quad c_{\epsilon}(x,0)=c_{\epsilon}(x,T)\mbox{ in }\Omega,$
(50) $\displaystyle\theta_{\epsilon}\in L^{2}(H^{1})\cap
L^{\infty}(L^{2}),\quad\theta_{\epsilon}(x,0)=\theta_{\epsilon}(x,T)\mbox{ in
}\Omega,$ (51) $\displaystyle\textit{{v}}_{\epsilon}\in
L^{2}(\textit{{H}}_{v})\cap
L^{\infty}(\textit{{H}}),\quad\textit{{v}}_{\epsilon}(x,0)=\textit{{v}}_{\epsilon}(x,T)\mbox{
in }\Omega,$ (52) $\displaystyle\int_{\Omega}c_{\epsilon}(x,t)\,dx=c_{g}\
\mbox{ in }(0,T),\quad\theta_{\epsilon}(\cdot,t)-\theta_{\delta}(\cdot,t)\in
H_{\theta}\ \mbox{ in }(0,T),$ (53)

and $c_{\epsilon},\,\theta_{\epsilon}$ and $\textit{{v}}_{\epsilon}$ satisfy
the variational equations a.e. in (0,T):

$\displaystyle(\frac{\partial c_{\epsilon}}{\partial
t},\,\varphi)+\eta\,(\nabla\,c_{\epsilon},\,\nabla\varphi)+(\textit{{v}}_{\epsilon}\cdot\nabla
c_{l}(c_{\epsilon},\theta_{\epsilon}),\,\varphi)=0,$ (54)
$\displaystyle(\frac{\partial\theta_{\epsilon}}{\partial
t},\,\psi)+\kappa\,(\nabla\theta_{\epsilon},\,\nabla\psi)+\rho\,C_{p}\,(\textit{{v}}_{\epsilon}\cdot\nabla\theta_{\epsilon},\,\psi)=0,$
(55) $\displaystyle(\frac{\partial\textit{{v}}_{\epsilon}}{\partial
t},\,\textit{{w}})+\nu\,(\nabla\textit{{v}}_{\epsilon},\,\nabla\textit{{w}})+\rho\,(\textit{{v}}_{\epsilon}\cdot\nabla\textit{{v}}_{\epsilon},\,\textit{{w}})+(F_{i}^{\epsilon}(c_{\epsilon},\theta_{\epsilon})\,\textit{{v}}_{\epsilon},\textit{{w}})\hskip
42.67912pt$ $\displaystyle\hskip
42.67912pt=(\mathbf{F}_{e}(c_{\epsilon},\theta_{\epsilon}),\,\textit{{w}}),$
(56)

for all $\ \varphi\in H^{1},\ \psi\in H_{\theta},\
\textit{{w}}\in\textit{{H}}_{v}$.

###### Remark 4.2

A weak reproductive solution of the regularized problem (39)-(48) is a
reproductive regularized weak solution of the problem (8)-(17). Thus, in what
follows, we refer to reproductive regularized weak solution of the problem
(8)-(17) as reproductive weak solution of the problem (39)-(48).

###### Remark 4.3

Applying the Maximal Principle to the diffusion and heat equations, we obtain
the inequalities

$\displaystyle\hskip 71.13188pt0\,\leq$ $\displaystyle c_{\epsilon}(x,t)$
$\displaystyle\leq\,\gamma_{\ell}(\theta_{E})\quad\mbox{ a.e. in
}\mathcal{Q}\,,$ (57) $\displaystyle\theta_{\delta}^{\inf}\ \leq\ $
$\displaystyle\theta_{\epsilon}(x,t)$ $\displaystyle\ \leq\
\theta_{\delta}^{\sup}\quad\mbox{a.e. in }\mathcal{Q}.$ (58)

The proof of (57) and (58) is similar to Proposition 3.5.

##  5 Weak reproductive solution of the regularized problem

In this section, we prove the existence of at least one weak reproductive
solution of the problem (39)-(48), by characterizing it as the limit of a
sequence of approximated solutions defined on finite dimensional spaces. There
follows, we state this result and the proof is done at the end of this
section.

###### Theorem 5.1

Under the hypotheses (S1), (S2) and (S3) the regularized problem (39)-(48)
admits at least one weak reproductive solution.

For the sake of simplicity we denote
$\tilde{\theta}_{\epsilon}=\theta_{\epsilon}-\theta_{\delta}$ and
$\tilde{c}_{\epsilon}=c_{\epsilon}-c_{g}|\Omega|^{-1}$, then to find a weak
reproductive solution of the problem (39)-(48) is equivalent to: Find the
functions
$(\,\tilde{c}_{\epsilon},\,\tilde{\theta}_{\epsilon},\,\textit{{v}}_{\epsilon}\,)\in
L^{2}(H^{1})\times L^{2}(H_{\theta})\times L^{2}(\textit{{H}}_{v}\,)$, such
that the following equations are satisfied in the distribution sense in
$(0,T)$,

$\displaystyle(\frac{\partial\tilde{c}_{\epsilon}}{\partial
t},\varphi)+\eta\,(\nabla\tilde{c}_{\epsilon},\nabla\varphi)+(\textit{{v}}_{\epsilon}\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}),\varphi)=0,$ (59)
$\displaystyle(\frac{\partial\tilde{\theta}_{\epsilon}}{\partial
t},\psi)+\kappa\,(\nabla(\tilde{\theta}_{\epsilon}+\theta_{\delta}),\nabla\psi)+\rho\,C_{p}\,(\textit{{v}}_{\epsilon}\cdot\nabla(\tilde{\theta}_{\epsilon}+\theta_{\delta}),\psi)=-(\frac{\partial\theta_{\delta}}{\partial
t},\psi),$ (60) $\displaystyle(\frac{\partial\textit{{v}}_{\epsilon}}{\partial
t},\textit{{w}})+\nu\,(\nabla\textit{{v}}_{\epsilon},\nabla\textit{{w}})+\rho\,(\textit{{v}}_{\epsilon}\cdot\nabla\textit{{v}}_{\epsilon},\,\textit{{w}})+(F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\,\textit{{v}}_{\epsilon},\textit{{w}})=(\mathbf{F}_{e}(\textit{{z}}_{\epsilon}),\textit{{w}}),$
(61)

for all $\ \varphi\in H^{1},\ \psi\in H_{\theta},\
\textit{{w}}\in\textit{{H}}_{v}$, where
$\textit{{z}}_{\epsilon}=(\tilde{c}_{\epsilon}+c_{g}|\Omega|^{-1},\tilde{\theta}_{\epsilon}+\theta_{\delta})$
and

$\int_{\Omega}\tilde{c}_{\epsilon}(x,t)\,dx=0.$ (62)

Thus, if we find
$(\,\tilde{c}_{\epsilon},\,\tilde{\theta}_{\epsilon},\,\textit{{v}}_{\epsilon}\,)$
reproductive solution of (59)-(62), we have that
$(c_{\epsilon}=\tilde{c}_{\epsilon}+c_{g}|\Omega|^{-1},\,\theta_{\epsilon}=\tilde{\theta}_{\epsilon}+\theta_{\delta},\,\textit{{v}}_{\epsilon})$
is a weak reproductive solution of (39)-(48).

To prove the existence of at least a weak reproductive solution of the problem
(59)-(62), we consider the Hilbert orthogonal basis
$\\{\varphi^{i}(x)\\}^{\infty}_{i=1}$ of $H^{1}$,
$\\{\psi^{i}(x)\\}^{\infty}_{i=1}$ of $H_{\theta}$ and
$\\{\textit{{w}}^{i}(x)\\}^{\infty}_{i=1}$ of $\textit{{H}}_{v}$, which are
assumed orthonormal in $L^{2}$ (only to introduce certain simplification in
the subsequent treatment).

Let $H^{k}$ be the subspace of $H^{1}$ spanned by
$\\{\varphi^{1}(x),\ldots,\varphi^{k}(x)\\}$, $H_{\theta}^{k}$ be the subspace
of $H_{\theta}$ spanned by $\\{\psi^{1}(x),\ldots,\psi^{k}(x)\\}$ and
$\textit{{H}}_{v}^{k}$ be the subspace of $\textit{{H}}_{v}$ spanned by
$\\{\textit{{w}}^{1}(x),\ldots,\textit{{w}}^{k}(x)\\}$, respectively. For
every $k\geq 1$, we define approximations $\tilde{c}^{k}_{\epsilon}(x,t)$,
$\tilde{\theta}^{k}_{\epsilon}(x,t)$ and $\textit{{v}}^{k}_{\epsilon}(x,t)$ of
$\tilde{c}_{\epsilon}(x,t)$, $\tilde{\theta}_{\epsilon}(x,t)$ and
$\textit{{v}}_{\epsilon}(x,t)$ respectively, by means of the following finite
expansion:

$\displaystyle\tilde{c}^{k}_{\epsilon}=\sum^{k}_{i=1}c_{ki}^{\epsilon}(t)\,\varphi^{i}(x),\quad\tilde{\theta}^{k}_{\epsilon}=\sum^{k}_{i=1}d_{ki}^{\epsilon}(t)\,\psi^{i}(x),\quad\textit{{v}}^{k}_{\epsilon}=\sum^{k}_{i=1}e_{ki}^{\epsilon}(t)\,\textit{{w}}^{i}(x),\qquad$
(63)

for $t\in(0,T)$, where the coefficients $c_{ik}^{\epsilon}(t),\
d_{ik}^{\epsilon}(t)$ and $e_{ik}^{\epsilon}(t)$ will be calculated in such
way that $\tilde{c}^{k}_{\epsilon},\tilde{\theta}^{k}_{\epsilon}$ and
$\textit{{v}}^{k}_{\epsilon}$ solve the following approximations of system
(59)-(62):

$\displaystyle(\frac{\partial\tilde{c}^{k}_{\epsilon}}{\partial
t},\varphi^{j})+\eta\,(\nabla\tilde{c}^{k}_{\epsilon},\nabla\varphi^{j})+(\textit{{v}}^{k}\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}^{k}),\varphi^{j})=0,$ (64)
$\displaystyle(\frac{\partial\,\tilde{\theta}^{k}_{\epsilon}}{\partial
t},\psi^{j})+\kappa\,(\nabla\tilde{\theta}^{k}_{\epsilon},\nabla\psi^{j})+\rho\,C_{p}\,(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\tilde{\theta}^{k}_{\epsilon},\psi^{j})+\rho\,C_{p}\,(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\theta_{\delta},\psi^{j})$
$\displaystyle\hskip
56.9055pt=-\kappa\,(\nabla\theta_{\delta},\nabla\psi^{j})-(\frac{\partial\theta_{\delta}}{\partial
t},\psi^{j}),$ (65)
$\displaystyle(\frac{\partial\,\textit{{v}}^{k}_{\epsilon}}{\partial
t},\textit{{w}}^{j})+\nu\,(\nabla\textit{{v}}^{k}_{\epsilon},\nabla\textit{{w}}^{j})+\rho\,(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\textit{{v}}^{k}_{\epsilon},\textit{{w}}^{j})+(F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\,\textit{{v}}^{k}_{\epsilon},\textit{{w}}^{j})$
$\displaystyle\hskip 56.9055pt=({\bf
F}_{e}(\textit{{z}}_{\epsilon}^{k}),\textit{{w}}^{j}),$ (66)

for all $\,\varphi^{j}\in H^{k},\ \psi^{j}\in H_{\theta}^{k},\
\textit{{w}}^{j}\in\textit{{H}}_{v}^{k}$, with
$\textit{{z}}_{\epsilon}^{k}=(\tilde{c}^{k}_{\epsilon}+c_{g}|\Omega|^{-1},\tilde{\theta}^{k}_{\epsilon}+\theta_{\delta})$
and

$\displaystyle\int_{\Omega}\tilde{c}^{k}_{\epsilon}(x,t)\,dx=0,$ (67)
$\displaystyle\tilde{c}^{k}_{\epsilon}(x,0)=\tilde{c}_{0}^{k}(x),\quad\tilde{\theta}^{k}_{\epsilon}(x,0)=\tilde{\theta}_{0}^{k}(x),\quad\textit{{v}}^{k}_{\epsilon}(x,0)=\textit{{v}}_{0}^{k}(x)\qquad{\rm
in}\ \Omega,\qquad$ (68)

where

$\tilde{c}_{0}^{k}(x)\rightarrow\tilde{c}_{0}(x)\mbox{ in }\
L^{2},\quad\tilde{\theta}_{0}^{k}(x)\rightarrow\tilde{\theta}_{0}(x)\mbox{ in
}\ L^{2},\quad\textit{{v}}_{0}^{k}(x)\rightarrow\textit{{v}}_{0}(x)\ \mbox{ in
}\ \textit{{H}},$

as $k\rightarrow\infty$ and $\tilde{c}_{0}(x)=c_{0}(x)-c_{g}|\Omega|^{-1}$ and
$\tilde{\theta}_{0}(x)=\theta_{0}(x)-\theta_{\delta}(x,0)$.

###### Remark 5.2

If we denote
$Z^{k}(x,t)=(\tilde{c}^{k}_{\epsilon}(x,t),\tilde{\theta}^{k}_{\epsilon}(x,t),\textit{{v}}^{k}_{\epsilon}(x,t))$,
because the orthonormality of $\varphi^{i},\psi^{i}$ and $\textit{{w}}^{i}$ in
${\textbf{L}}^{2}$, then

$\|Z^{k}(t)\|^{2}=\sum_{i=1}^{k}(\,|c^{\epsilon}_{ki}(t)|^{2}+|d^{\epsilon}_{ki}(t)|^{2}+|e^{\epsilon}_{ki}(t)|^{2}\,).$

Since the system (64)-(66) with the conditions (68) depends analytically on

${\bf
G}(t)=(\,c^{\epsilon}_{k1}(t),\,\cdots,\,c^{\epsilon}_{kk}(t),\,d^{\epsilon}_{k1}(t),\,\cdots,\,d^{\epsilon}_{kk}(t),\,e^{\epsilon}_{k1}(t),\,\cdots,\,e^{\epsilon}_{kk}(t)\,),$

to prove that the approximating solutions
$(\tilde{c}^{k}_{\epsilon},\tilde{\theta}^{k}_{\epsilon},\textit{{v}}^{k}_{\epsilon})$
are defined uniquely for all $t\geq 0$, it is sufficient to prove that the
nonlinear differential system (64)-(66) with the conditions (68) has a maximal
solution ${\bf G}(t)$ defined on some interval $[0,t_{k}]$. If $t_{k}<T$ then
$\|Z^{k}(x,t)\|^{2}$ must tend to $+\infty$ as $t\rightarrow t_{k}$, but the a
priori estimates we shall prove below (see Proposition 5.3) show that this is
not possible and therefore, $t_{k}=T$.

Now, in order to prove that the system (64)-(68) has at least one reproductive
solution, previously we get some estimates in the following prepositions.

###### Proposition 5.3

Let
$(\tilde{c}_{\epsilon}^{k},\tilde{\theta}_{\epsilon}^{k},\textit{{v}}_{\epsilon}^{k})$
be a solution of (64)-(68). Then, for $t\in[0,T]$ the following estimates
holds

$\|Z^{k}(t)\|^{2}+\int_{0}^{t}(\,\eta\,\|\nabla\tilde{c}^{k}_{\epsilon}(s)\|^{2}+\kappa\,\|\nabla\tilde{\theta}^{k}_{\epsilon}(s)\|^{2}+\nu\,\|\nabla\textit{{v}}^{k}_{\epsilon}(s)\|^{2}\,)\,ds\leq
C.$ (69)

where $C$ does not depend on $\epsilon$ and $k$.

Proof: Considering in (64)-(66),
$\varphi^{j}=\tilde{c}^{k}_{\epsilon}\,(t),\,\psi^{j}=\tilde{\theta}^{k}_{\epsilon}\,(t)$
and $\textit{{w}}^{j}=\textit{{v}}^{k}_{\epsilon}\,(t)$, we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\tilde{c}^{k}_{\epsilon}(t)\|^{2}+\eta\,\|\nabla\tilde{c}^{k}_{\epsilon}(t)\|^{2}+(\textit{{v}}^{k}_{\epsilon}(t)\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}^{k}),c^{k}_{\epsilon}(t))=0,$ (70)
$\displaystyle\frac{1}{2}\frac{d}{dt}\|\tilde{\theta}^{k}_{\epsilon}(t)\|^{2}+\kappa\,\|\nabla\tilde{\theta}^{k}_{\epsilon}(t)\|^{2}+\rho\,C_{p}\,(\textit{{v}}^{k}_{\epsilon}(t)\cdot\nabla\theta_{\delta}(t),\tilde{\theta}^{k}_{\epsilon}(t))$
$\displaystyle\hskip
62.59596pt=-\kappa\,(\nabla\theta_{\delta}(t),\nabla\tilde{\theta}^{k}_{\epsilon}(t))-(\frac{\partial}{\partial
t}\theta_{\delta}(t),\tilde{\theta}^{k}_{\epsilon}(t)),$ (71)
$\displaystyle\frac{1}{2}\frac{d}{dt}\|\textit{{v}}^{k}_{\epsilon}(t)\|^{2}+\nu\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}+(F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\,\textit{{v}}^{k}_{\epsilon}(t),\textit{{v}}^{k}_{\epsilon}(t))=({\bf
F}_{e}(\textit{{z}}_{\epsilon}^{k}),\textit{{v}}^{k}_{\epsilon}(t)).\quad$
(72)

Since $\,F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\geq 0$, adding
(70)-(72), we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|Z^{k}(t)\|^{2}+\eta\,\|\nabla\tilde{c}^{k}_{\epsilon}(t)\|^{2}+\kappa\,\|\nabla\tilde{\theta}^{k}_{\epsilon}(t)\|^{2}+\nu\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}$
$\displaystyle\hskip 14.22636pt\leq|\,({\bf v}^{k}_{\epsilon}(t)\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}^{k}),\tilde{c}^{k}_{\epsilon}(t))\,|+\rho\,C_{p}\,|\,(\textit{{v}}^{k}_{\epsilon}(t)\cdot\nabla\theta_{\delta}(t),\tilde{\theta}^{k}_{\epsilon}(t))\,|$
$\displaystyle\hskip
28.45274pt+\kappa\,|\,(\nabla\theta_{\delta}(t),\nabla\tilde{\theta}^{k}_{\epsilon}(t))\,|+|\,(\frac{\partial}{\partial
t}\theta_{\delta}(t),\tilde{\theta}^{k}_{\epsilon}(t))\,|+|\,({\bf
F}_{e}(\textit{{z}}_{\epsilon}^{k}),\textit{{v}}^{k}_{\epsilon}(t))\,|.\quad$
(73)

Now, by considering (S3) together Hölder and Young inequalities, for the first
term on the right side of inequality (5), we get

$\displaystyle|(\textit{{v}}^{k}_{\epsilon}(t)\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}^{k}),\tilde{c}^{k}_{\epsilon}(t))|$
$\displaystyle=$
$\displaystyle|(\textit{{v}}^{k}_{\epsilon}(t)\cdot\nabla\tilde{c}^{k}_{\epsilon}(t),c_{l}(\textit{{z}}_{\epsilon}^{k}))|$
(74) $\displaystyle\leq$ $\displaystyle
C\|\textit{{v}}^{k}_{\epsilon}(t)\|\|c_{l}(\textit{{z}}_{\epsilon}^{k})\|_{L^{\infty}}\|\nabla\tilde{c}^{k}_{\epsilon}(t)\|$
$\displaystyle\leq$
$\displaystyle\frac{\eta}{2}\,\|\nabla\tilde{c}^{k}_{\epsilon}(t)\|^{2}+C\,\|\textit{{v}}^{k}_{\epsilon}(t)\|^{2}.$

By taking into account (58) and, using the Hölder and Young inequalities, for
the others terms on the right side of (5) we have

$\displaystyle\rho\,C_{p}\,|\,(\textit{{v}}^{k}_{\epsilon}(t)\cdot\nabla\,\theta_{\delta}(t),\tilde{\theta}^{k}_{\epsilon}(t))\,|$
$\displaystyle\leq$ $\displaystyle
C\,\|\textit{{v}}^{k}_{\epsilon}(t)\|_{L^{6}}\|\nabla\theta_{\delta}(t)\|\,\|\tilde{\theta}^{k}_{\epsilon}(t)\|_{L^{3}}$
(75) $\displaystyle\leq$
$\displaystyle\,\frac{\nu}{2}\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}+C\,\|\nabla\theta_{\delta}(t)\|^{2},$
$\displaystyle\kappa\,|\,(\nabla\theta_{\delta}(t),\nabla\tilde{\theta}^{k}_{\epsilon}(t))\,|$
$\displaystyle\leq$
$\displaystyle\,C\,\|\nabla\theta_{\delta}(t)\|^{2}+\frac{\kappa}{2}\,\|\nabla\,\tilde{\theta}^{k}_{\epsilon}(t)\|^{2},$
(76) $\displaystyle|\,(\frac{\partial}{\partial
t}\theta_{\delta}(t),\tilde{\theta}^{k}_{\epsilon}(t))\,|$ $\displaystyle\leq$
$\displaystyle C\,\|\frac{\partial}{\partial
t}\theta_{\delta}(t)\|^{2}+C\,\|\tilde{\theta}^{k}_{\epsilon}(t)\|^{2},$ (77)

considering (58), (S3) and since ${\bf g}\in L^{\infty}$, we get

$\displaystyle|({\bf
F}_{e}(\textit{{z}}_{\epsilon}^{k}),\textit{{v}}^{k}_{\epsilon}(t))|$
$\displaystyle\leq$ $\displaystyle C\,\|{\bf
g}\|\,\|\tilde{\theta}^{k}_{\epsilon}(t)+\theta_{\delta}(t)\|_{L^{3}}\,\|\textit{{v}}^{k}_{\epsilon}(t)\|_{L^{6}}+C\,\|{\bf
g}\|\,\|\textit{{v}}^{k}_{\epsilon}(t)\|$ (78) $\displaystyle+C\,\|{\bf
g}\|\,\|c_{l}(\textit{{z}}_{\epsilon}^{k})\|_{L^{\infty}}\|\textit{{v}}^{k}_{\epsilon}(t)\|$
$\displaystyle\leq$ $\displaystyle
C+\frac{\nu}{4}\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}.$

Then, carrying inequalities (74)-(78) in (5), we have the inequality

$\displaystyle\frac{1}{2}\frac{d}{dt}\|Z^{k}(t)\|^{2}+\frac{\eta}{2}\,\|\nabla\tilde{c}^{k}_{\epsilon}(t)\|^{2}+\frac{\kappa}{2}\,\|\nabla\tilde{\theta}^{k}_{\epsilon}(t)\|^{2}+\frac{\nu}{2}\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}$
$\displaystyle\hskip 14.22636pt\leq
C\,\|Z^{k}(t)\|^{2}+C\,\|\theta_{\delta}(t)\|^{2}_{H^{1}}+C\,\|\frac{\partial}{\partial
t}\theta_{\delta}(t)\|^{2}+C.$ (79)

Now, taking into account (S2), and by integrating (5) from $0$ to $t\
(t\in[0,T])$, we obtain

$\displaystyle\|Z^{k}(t)\|^{2}+\int_{0}^{t}(\eta\,\|\nabla\tilde{c}^{k}_{\epsilon}(s)\|^{2}+\kappa\,\|\nabla\tilde{\theta}^{k}_{\epsilon}(s)\|^{2}+\nu\,\|\nabla\textit{{v}}^{k}_{\epsilon}(s)\|^{2})\,ds$
$\displaystyle\qquad\leq C+\|Z^{k}(0)\|^{2}+C\int_{0}^{t}\|Z^{k}(s)\|^{2}ds,$

thus the Gronwall’s inequality (Lemma 3.2), implies

$\|Z^{k}(t)\|^{2}+\int_{0}^{t}(\eta\,\|\nabla\tilde{c}^{k}_{\epsilon}(s)\|^{2}+\kappa\,\|\nabla\tilde{\theta}^{k}_{\epsilon}(s)\|^{2}+\nu\,\|\nabla\textit{{v}}^{k}_{\epsilon}(s)\|^{2})\,ds\leq
C(T),$ (80)

and consequently inequality (69).

###### Proposition 5.4

Let
$(\tilde{c}_{\epsilon}^{k},\tilde{\theta}_{\epsilon}^{k},\textit{{v}}_{\epsilon}^{k})$
be a solution of (64)-(68). Then it is satisfied the estimate

$\mbox{\large e}\,^{\beta
T}\,\|Z^{k}(T)\|^{2}\leq\|Z^{k}(0)\|^{2}+C(\beta,T),$ (81)

where $\beta=C_{\Omega}^{-1}\min\\{\eta,\kappa,\nu\\}$.

Proof: Using the Poincare’s inequality ($\|Z^{k}(t)\|^{2}\leq
C_{\Omega}\|\nabla Z^{k}(t)\|^{2}$) and the definition of $\beta$, from (5)
and (80), we have

$\frac{d}{dt}\|Z^{k}(t)\|^{2}+\beta\|Z^{k}(t)\|^{2}\leq
C\,(T)+C\,\|\theta_{\delta}(t)\|^{2}_{H^{1}}+C\,\|\frac{\partial}{\partial
t}\theta_{\delta}(t)\|^{2}.$

Consequently, for all $t\in[0,T]$

$\displaystyle\frac{d}{dt}(\,\mbox{\large e}\,^{\beta t}\,\|Z^{k}(t)\|^{2})$
$\displaystyle\leq$ $\displaystyle\mbox{\large e}\,^{\beta
T}\,(\,C(T)+C\,\|\theta_{\delta}(t)\|^{2}_{H^{1}}+C\,\|\frac{\partial}{\partial
t}\theta_{\delta}(t)\|^{2}\,).$ (82)

Now, taking into account (S3) ($\theta_{\delta}\in H^{1}(H^{1})$) and
integrating (82) from $0$ to $T$, we obtain inequality (81).

In the following Lemma we will prove the existence of reproductive solution of
the approximated problem (64)-(68).

###### Lemma 5.5

Under the hypotheses of Theorem 5.1, the problem (64)-(68) admits at least one
reproductive solution
$(\tilde{c}_{\epsilon}^{k},\tilde{\theta}_{\epsilon}^{k},$
$\textit{{v}}_{\epsilon}^{k}),\ \forall\,k\geq 1$. Furthermore, there exists
$(\tilde{c}_{\epsilon},\tilde{\theta}_{\epsilon},\textit{{v}}_{\epsilon})$
such that as $k\rightarrow\infty$,

$\displaystyle\tilde{c}^{k}_{\epsilon}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{c}_{\epsilon}\quad\mbox{weakly in
}L^{2}(H^{1}),\quad\mbox{weakly -* in }L^{\infty}(L^{2}),$ (83)
$\displaystyle\tilde{\theta}^{k}_{\epsilon}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\theta}_{\epsilon}\quad\mbox{ weakly in
}L^{2}(H_{\theta}),\quad\mbox{weakly -* in }L^{\infty}(L^{2}),$ (84)
$\displaystyle\textit{{v}}^{k}_{\epsilon}$ $\displaystyle\rightarrow$
$\displaystyle\textit{{v}}_{\epsilon}\quad\mbox{ weakly in
}L^{2}(\textit{{H}}_{v}),\quad\mbox{weakly -* in }L^{\infty}(\textit{{H}}).$
(85)

Proof: Let be the operator $\mathbf{Y}:[0,T]\mapsto\mathbb{R}$ defined as

$\mathbf{Y}(t)=(\,c_{k1}^{\epsilon}(t),\,\cdots,\,c_{kk}^{\epsilon}(t),\,d_{k1}^{\epsilon}(t),\,\cdots,\,d_{kk}^{\epsilon}(t),\,e_{k1}^{\epsilon}(t),\,\cdots,\,e_{kk}^{\epsilon}(t)\,)$

where the time dependent functions
$\\{c_{ki}^{\epsilon}(t),\,d_{ki}^{\epsilon}(t),\,e_{ki}^{\epsilon}(t)\\}_{i=1}^{k}$
are the coefficients of the expansion $Z^{k}\,(x,t)$ given in (63). Thus,
taking into account Remark 5.2, we have

$\|{\bf Y}(t)\|_{\mathbb{R}^{3k}}=\|Z^{k}(t)\|,\quad\forall\,t\in[0,T].$ (86)

We define the operator

$\displaystyle\ \Phi\,:\ \mathbb{R}^{3k}$ $\displaystyle\longmapsto$
$\displaystyle\mathbb{R}^{3k}$ $\displaystyle\mathbf{r}\ $
$\displaystyle\longmapsto$
$\displaystyle\Phi(\mathbf{r})=\mathbf{Y}(T)\qquad\qquad\qquad$

where for each $k\geq 1$ and $\mathbf{r}=(r_{1},r_{2},\dots,r_{3k})$,
$\mathbf{Y}(T)$ is the vector of coefficients of the solution of the problem
(64)-(68) at time $T$, with initial condition given by

$\tilde{c}^{k}_{0}(x)=\sum_{i=1}^{k}r_{i}\,\varphi^{i}(x),\
\quad\tilde{\theta}^{k}_{0}(x)=\sum_{i=1}^{k}r_{k+i}\,\psi^{i}(x),\
\quad\textit{{v}}^{k}_{0}(x)=\sum_{i=1}^{k}r_{2k+i}\,\textit{{w}}^{i}(x).$

We will prove that $\Phi$ has at least one fixed point, as consequence of
Leray-Schauder’s homotopy Theorem. For this purpose, it is sufficient to show
that for any $\lambda\in[0,1]$, a solution of the equation

$\lambda\,\Phi(\,\mathbf{r}(\lambda)\,)={\bf r}(\lambda),$ (87)

is bounded independently of $\lambda$. Since ${\bf r}(0)={\bf 0}$, the proof
will be restrict to $\lambda\in(0,1]$.

Since $\lambda\leq 1$, by the definition of $\Phi$ together equalities (86)
and (87), we have

$\|\mathbf{r}(\lambda)\|^{2}_{\mathbb{R}^{3k}}\leq\|\Phi(\mathbf{r}(\lambda))\|^{2}_{\mathbb{R}^{3k}}\leq\|Z^{k}(T)\|^{2}.$
(88)

Then, from (88) and (81), we obtain

$\mbox{\large e}\,^{\beta
T}\,\|\mathbf{r}(\lambda)\|^{2}_{\mathbb{R}^{3k}}\leq\mbox{\large e}\,^{\beta
T}\,\|Z^{k}(T)\|^{2}\leq\|\,\mathbf{r}(\lambda)\|^{2}_{\mathbb{R}^{3k}}+C(\beta,T)$

thus $\ \displaystyle(\mbox{\large e}\,^{\beta T}-1)\,\|\,{\bf
r}(\lambda)\,\|^{2}_{\mathbb{R}^{3k}}\leq C(\beta,T)$, and consequently

$\|\,{\bf r}(\lambda)\,\|^{2}_{\mathbb{R}^{3k}}\leq\frac{1}{\mbox{\large
e}\,^{\beta T}-1}\,C(\beta,T),\ \forall\,\lambda\in[0,1].$ (89)

Since the upper bound in (89) is independent on $\lambda\in[0,1]$, we have
established that operator $\Phi$ has at least one fixed point, and from (87)
the fixed point of $\Phi$ is denoted by ${\bf r}(1)$ and satisfies (89).
Therefore, we have proved the result of the Lemma 5.5. Moreover, (83)-(85) are
a consequence of Proposition 5.3.

Nevertheless, the convergences given in Lemma 5.5 are insufficient to prove
that the limit functions become a reproductive weak solution of the problem
(39)-(48). Therefore, we need to prove strong convergence.

###### Lemma 5.6

Under the hypotheses of Lemma 5.5, as $k\rightarrow\infty,$ the sequences
$\\{\tilde{c}_{\epsilon}^{k}\\},\\{\tilde{\theta}_{\epsilon}^{k}\\}$ and
$\\{\textit{{v}}_{\epsilon}^{k}\\}$ converge weakly in $L^{2}(\Omega)$,
uniformly for $t\in[0,T]$.

Proof: For fixed $j$ and $k\geq j$, we will prove that
$\,\xi_{kj}(t)=(\tilde{c}_{\epsilon}^{k}(t),\varphi^{j}),\
\chi_{kj}(t)=(\tilde{\theta}_{\epsilon}^{k}(t),\psi^{j})$ and
$\zeta_{kj}(t)=(\textit{{v}}_{\epsilon}^{k}(t),\textit{{w}}^{j})$ are
uniformly bounded and equicontinuous families of functions on $[0,T]$, and
then applying the Theorem of Ascoli-Arzelà, we obtain the uniform convergence
on $[0,T]$.

The uniform boundedness of the family
$\\{\xi_{kj}(t),\,\chi_{kj}(t),\,\zeta_{kj}(t)\\}$ follows from Proposition
5.3, therefore, we are going to prove the equicontinuity.

The equalities (64)-(66), can be rewritten in the form

$\displaystyle\xi_{kj}^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\eta\,(\nabla\tilde{c}^{k}_{\epsilon},\nabla\varphi^{j})+(\textit{{v}}^{k}_{\epsilon}\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}^{k}),\varphi^{j}),$ (90)
$\displaystyle\chi_{kj}^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\kappa\,(\nabla\tilde{\theta}^{k}_{\epsilon},\nabla\psi^{j})-\rho\,C_{p}\,(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\tilde{\theta}^{k}_{\epsilon},\psi^{j})-\rho\,C_{p}\,(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\theta_{\delta},\psi^{j})$
(91)
$\displaystyle-\kappa\,(\nabla\theta_{\delta},\nabla\psi^{j})-(\frac{\partial}{\partial
t}\theta_{\delta},\psi^{j}),$ $\displaystyle\zeta_{kj}^{\prime}(t)$
$\displaystyle=$
$\displaystyle-\nu\,(\nabla\textit{{v}}^{k}_{\epsilon},\nabla\textit{{w}}^{j})-\rho\,(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\textit{{v}}^{k}_{\epsilon},\textit{{w}}^{j})-(F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\,\textit{{v}}^{k}_{\epsilon},\textit{{w}}^{j})$
(92) $\displaystyle+({\bf
F}_{e}(\textit{{z}}_{\epsilon}^{k}),\textit{{w}}^{j}),$

for $j=1,\dots,k$.

By integrating (90)-(92) with respect to $t$ from $t$ to $t+\Delta t$ and
estimating the right-hand side by using Schwarz’s inequality, we obtain

$\displaystyle|\xi_{kj}(t+\Delta t)-\xi_{kj}(t)|\leq\eta\,\int_{t}^{t+\Delta
t}|\nabla\tilde{c}^{k}_{\epsilon}\cdot\nabla\varphi^{j}|\,dt+\int_{t}^{t+\Delta
t}|\textit{{v}}^{k}_{\epsilon}c_{l}(\textit{{z}}_{\epsilon}^{k})\cdot\nabla\varphi^{j}|\,dt$
$\displaystyle\hskip 14.22636pt\leq\eta\int_{t}^{t+\Delta
t}\|\nabla\tilde{c}_{\epsilon}^{k}(t)\|\,\|\nabla\varphi^{j}\|dt+\int_{t}^{t+\Delta
t}\|\textit{{v}}^{k}_{\epsilon}(t)\|\,\|\nabla\varphi^{j}\|\,\|c_{l}(\textit{{z}}_{\epsilon}^{k})\|_{L^{\infty}}dt$
$\displaystyle\hskip 14.22636pt\leq C\,\int_{t}^{t+\Delta
t}\|\nabla\,\tilde{c}_{\epsilon}^{k}(t)\|\,dt+C\,\int_{t}^{t+\Delta
t}\|\textit{{v}}^{k}_{\epsilon}(t)\|\,dt$ $\displaystyle\hskip 14.22636pt\leq
C\sqrt{\Delta t}\,\\{(\int_{t}^{t+\Delta
t}\|\nabla\tilde{c}_{\epsilon}^{k}(t)\|^{2}dt)^{1/2}+(\int_{t}^{t+\Delta
t}\|\textit{{v}}^{k}_{\epsilon}(t)\|^{2}dt)^{1/2}\\}.$ (93)

The terms
$(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\tilde{\theta}^{k}_{\epsilon},\psi^{j})$
and $(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\theta_{\delta},\psi^{j})$ of (91)
can be bounded, after integrating by parts, as
$(\textit{{v}}^{k}_{\epsilon}\cdot\nabla
c_{l}(\textit{{z}}_{\epsilon}^{k}),\varphi^{j})$ in (5) or without integrating
as follows:

$\displaystyle|\chi_{kj}(t+\Delta t)-\chi_{kj}(t)|$ $\displaystyle\hskip
14.22636pt\leq C\,\int_{t}^{t+\Delta
t}\|\nabla\,\tilde{\theta}_{\epsilon}^{k}(t)\|\,\|\nabla\psi^{j}\|\,dt+C\,\int_{t}^{t+\Delta
t}\|\textit{{v}}^{k}_{\epsilon}(t)\|_{L^{3}}\,\|\nabla\tilde{\theta}_{\epsilon}^{k}\|\,\|\psi^{j}\|_{L^{6}}\,dt$
$\displaystyle\hskip 28.45274pt+C\,\int_{t}^{t+\Delta
t}\|\textit{{v}}^{k}_{\epsilon}(t)\|_{L^{3}}\,\|\nabla\theta_{\delta}(t)\|\,\|\psi^{j}\|_{L^{6}}\,dt$
$\displaystyle\hskip 28.45274pt+C\,\int_{t}^{t+\Delta
t}(\,\|\delta\theta_{\delta}(t)\|\,\|\nabla\psi^{j}\|+\|\frac{\partial}{\partial
t}\theta_{\delta}(t)\|\,\|\psi^{j}\|\,)\,dt$ $\displaystyle\hskip
14.22636pt\leq C\,\sqrt{\Delta t}(\int_{t}^{t+\Delta
t}\|\nabla\tilde{\theta}_{\epsilon}^{k}(t)\|^{2}dt)^{1/2}$
$\displaystyle\hskip 28.45274pt+C\,\sqrt[4]{\Delta t}(\int_{t}^{t+\Delta
t}\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}dt)^{1/4}(\int_{t}^{t+\Delta
t}\|\nabla\tilde{\theta}_{\epsilon}^{k}(t)\|^{2}dt)^{1/2}$
$\displaystyle\hskip 28.45274pt+C\,\sqrt[4]{\Delta t}(\int_{t}^{t+\Delta
t}\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{2}dt)^{1/4}(\int_{t}^{t+\Delta
t}\|\tilde{\theta}_{\delta}(t)\|^{2}_{H^{1}}dt)^{1/2}$ $\displaystyle\hskip
28.45274pt+C\sqrt{\Delta t}\,\Big{(}(\int_{t}^{t+\Delta
t}\|\theta_{\delta}(t)\|^{2}_{H^{1}}dt)^{1/2}+(\int_{t}^{t+\Delta
t}\|\frac{\partial}{\partial t}\theta_{\delta}(t)\|^{2}dt)^{1/2}\Big{)}.$ (94)

Here, we have used that
$\|\textit{{v}}^{k}_{\epsilon}(t)\|_{L^{3}}\leq\|\textit{{v}}^{k}_{\epsilon}(t)\|^{1/2}\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{1/2}$
and $\|\textit{{v}}^{k}_{\epsilon}(t)\|\leq C$.

$\displaystyle|\zeta_{kj}(t+\Delta t)-\zeta_{kj}(t)|$ $\displaystyle\hskip
14.22636pt\leq C\,\int_{t}^{t+\Delta
t}\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|\,dt+C\,\int_{t}^{t+\Delta
t}\,\|\nabla\textit{{v}}^{k}_{\epsilon}(t)\|^{3/2}\,dt$ $\displaystyle\hskip
28.45274pt+C\,\int_{t}^{t+\Delta
t}\|F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\|_{L^{\infty}}\,\|\textit{{v}}^{k}_{\epsilon}(t)\|\,dt+\,\int_{t}^{t+\Delta
t}\|\textit{{F}}_{e}(\textit{{z}}_{\epsilon}^{k})\|\,dt$ $\displaystyle\hskip
14.22636pt\leq C\,\sqrt{\Delta t}\,(\int_{t}^{t+\Delta
t}\|\nabla\,\textit{{v}}^{k}_{\epsilon}(t)\|^{2}dt)^{1/2}+C\,\sqrt[4]{\Delta
t}\,(\int_{t}^{t+\Delta
t}\|\nabla\,\textit{{v}}^{k}_{\epsilon}(t)\|^{2})^{3/4}dt$
$\displaystyle\hskip
28.45274pt+C\,(\max_{t}\|\textit{{v}}^{k}_{\epsilon}(t)\|)\,\sqrt{\Delta
t}\,(\int_{t}^{t+\Delta
t}\|F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\|_{L^{\infty}}^{2}dt)^{1/2}+C\,\Delta
t.$ (95)

Then by Lemma 5.5, we have that the right-hand side of inequalities (93), (94)
and (5) converges to zero uniformly in $k$ as $\Delta t\rightarrow 0$ and it
is follow that $\,\xi_{kj}(t),\,\chi_{kj}(t)$ and $\zeta_{kj}(t)$ are
equicontinuous families on $[0,T]$. Thus, from Lemma 5.5 and the Theorem of
Arzelà-Ascoli the proof of Lemma 5.6 is complete.

In the following lemma, we will prove strong convergence of the reproductive
solutions of the problem (64)–(68).

###### Lemma 5.7

Under the hypotheses of Lemma 5.6, the sequence
$\\{(\tilde{c}^{k}_{\epsilon},\tilde{\theta}^{k}_{\epsilon},\textit{{v}}^{k}_{\epsilon})\\}$
converges strongly in $L^{2}(L^{2})\times L^{2}(L^{2})\times
L^{2}(\textit{{H}})$ toward
$(\tilde{c}_{\epsilon},\,\tilde{\theta}_{\epsilon},\,\textit{{v}}_{\epsilon})$.

Proof: By setting

$\textit{{u}}(t)=(c_{\epsilon}^{m,n}(t),\theta_{\epsilon}^{m,n}(t),\textit{{v}}_{\epsilon}^{m,n}(t))$

in the inequality of Friedrich, Lemma 3.3, being

$\displaystyle c_{\epsilon}^{m,n}(t)$ $\displaystyle=$
$\displaystyle\tilde{c}^{k_{m}}_{\epsilon}(t)-\tilde{c}^{k_{n}}_{\epsilon}(t),\quad\theta_{\epsilon}^{m,n}(t)\
=\ \tilde{\theta}^{k_{m}}_{\epsilon}(t)-\tilde{\theta}^{k_{n}}_{\epsilon}(t),$
$\displaystyle\textit{{v}}_{\epsilon}^{m,n}(t)$ $\displaystyle=$
$\displaystyle\textit{{v}}^{k_{m}}_{\epsilon}(t)-\textit{{v}}^{k_{n}}_{\epsilon}(t),$

and integrating it with respect to $t$ from $0$ to $T$, we obtain

$\displaystyle\int_{0}^{T}(\|c_{\epsilon}^{m,n}(t)\|^{2}+\|\theta_{\epsilon}^{m,n}(t)\|^{2}+\|\textit{{v}}_{\epsilon}^{m,n}(t)\|^{2})dt$
$\displaystyle\hskip
14.22636pt\leq\sum_{i=1}^{N_{\omega}}\int_{0}^{T}[(c_{\epsilon}^{m,n}(t),\varphi^{i})^{2}+(\theta_{\epsilon}^{m,n}(t),\psi^{i})^{2}+(\textit{{v}}_{\epsilon}^{m,n}(t),\textit{{w}}^{i})^{2}]dt$
$\displaystyle\hskip 28.45274pt+\omega\,\int_{0}^{T}(\|\nabla
c_{\epsilon}^{m,n}(t)\|^{2}+\|\nabla\theta_{\epsilon}^{m,n}(t)\|^{2}+\|\nabla(\textit{{v}}_{\epsilon}^{m,n}(t)\|^{2})dt,$
(96)

for any given $\omega>0$.

From Proposition 5.3, the last integral on the right-hand side of (5) does not
exceed a fixed constant for any $k_{m}$ and $k_{n}$, and from Lemma 5.6, the
first integral can be made arbitrarily small for sufficiently large $k_{m}$
and $k_{n}$ because of the uniform convergence in $t$. Therefore, for
sufficiently large $k_{m}$ and $k_{n}$

$\int_{0}^{T}(\|\tilde{c}^{k_{m}}_{\epsilon}(t)-\tilde{c}^{k_{n}}_{\epsilon}(t)\|^{2}+\|\tilde{\theta}^{k_{m}}_{\epsilon}(t)-\tilde{\theta}^{k_{n}}_{\epsilon}(t)\|^{2}+\|\textit{{v}}^{k_{m}}_{\epsilon}(t)-\textit{{v}}^{k_{n}}_{\epsilon}(t)\|^{2})dt\rightarrow
0,$

and this implies the Lemma 5.7. Moreover, from Lemma 5.5, we conclude

$\tilde{c}_{\epsilon}^{k}\rightarrow\tilde{c}_{\epsilon},\quad\tilde{\theta}_{\epsilon}^{k}\rightarrow\tilde{\theta}_{\epsilon}\mbox{
strongly in }L^{2}(L^{2}),\mbox{ and
}\textit{{v}}_{\epsilon}^{k}\rightarrow\textit{{v}}_{\epsilon}\mbox{ strongly
in }L^{2}(\textit{{H}})$. The results of the Lemma 5.5 and Lemma 5.7 enable us
to prove the Theorem 5.1.

Proof of the Theorem 5.1

We going to prove that
$(\,\tilde{c}_{\epsilon}\,(x,t),\,\tilde{\theta}_{\epsilon}\,(x,t),\,\textit{{v}}_{\epsilon}\,(x,t)\,)$,
obtained in Lemma 5.7, is a reproductive weak solution of problem (39)-(48)
taking limit as $k\rightarrow\infty$ in (64)-(67) after integrating in
$[0,T]$.

In fact, considering $\phi\in C^{\infty}([0,T])$ such that $\phi(T)=0$, and
using by parts integration, we have

$\displaystyle\int_{0}^{T}(\frac{\partial}{\partial
t}(\tilde{c}^{k}_{\epsilon}-\tilde{c}_{\epsilon})(t),\varphi^{j}\phi(t))dt$
$\displaystyle=$
$\displaystyle-\int_{0}^{T}(\tilde{c}^{k}_{\epsilon}(t)-\tilde{c}_{\epsilon}(t),\varphi^{j}\,\phi^{{}^{\prime}}(t))dt$
$\displaystyle-(\tilde{c}^{k}_{\epsilon}(x,0)-\tilde{c}_{\epsilon}(x,0),\varphi^{j}\phi(0)),$

then, from (41), (68) and Lemma 5.7, we obtain that as $\ k\rightarrow\infty$

$\int_{0}^{T}(\frac{\partial}{\partial
t}(\tilde{c}^{k}_{\epsilon}-\tilde{c}_{\epsilon})(t),\varphi^{j}\phi(t))dt\longrightarrow
0.$ (97)

Similarly, if $\ k\rightarrow\infty$ we have

$\displaystyle\int_{0}^{T}(\frac{\partial}{\partial
t}(\tilde{\theta}^{k}_{\epsilon}-\tilde{\theta}_{\epsilon})(t),\,\psi^{j}\phi(t))dt\longrightarrow
0,$ (98) $\displaystyle\int_{0}^{T}(\frac{\partial}{\partial
t}(\textit{{v}}^{k}_{\epsilon}-\textit{{v}}_{\epsilon})(t),\,\textit{{w}}^{j}\phi(t))dt\longrightarrow
0.$ (99)

Also, by Lemma 5.5 when $\ k\rightarrow\infty$, we conclude

$\displaystyle\int_{0}^{T}(\nabla\tilde{c}^{k}_{\epsilon}(t)-\nabla\tilde{c}_{\epsilon}(t),\nabla\varphi^{j}\phi(t))dt\longrightarrow
0,$ (100)
$\displaystyle\int_{0}^{T}(\nabla\tilde{\theta}^{k}_{\epsilon}(t)-\nabla\tilde{\theta}_{\epsilon}(t),\nabla\psi^{j}\phi(t))dt\longrightarrow
0,$ (101)
$\displaystyle\int_{0}^{T}(\nabla\textit{{v}}^{k}_{\epsilon}(t)-\nabla\,\textit{{v}}_{\epsilon}(t),\nabla\textit{{w}}^{j}\phi(t))dt\longrightarrow
0.$ (102)

To bound the nonlinear terms, we consider $b_{\epsilon}$ that will represent
$c_{l}(\textit{{z}}_{\epsilon}^{k})$, $\tilde{\theta}^{k}_{\epsilon}$ or
$\textit{{v}}_{\epsilon}^{k}$ and $\xi^{j}$ that will be $\phi^{j}$,
$\psi^{j}$ or $\textit{{w}}^{j}$ in the following expression:

$\displaystyle|(\textit{{v}}^{k}_{\epsilon}\cdot\nabla
b^{k}_{\epsilon}-\textit{{v}}_{\epsilon}\cdot\nabla
b_{\epsilon},\xi^{j}\,\phi(t))|$ $\displaystyle\hskip
14.22636pt\leq|((\textit{{v}}_{\epsilon}-\textit{{v}}^{k}_{\epsilon})\cdot\nabla\xi^{j},b^{k}_{\epsilon})-(\textit{{v}}_{\epsilon}\cdot\nabla\xi^{j},(b^{k}_{\epsilon}-b_{\epsilon}))|\,\sup_{t\in[0,T]}|\phi(t)|$
$\displaystyle\hskip 14.22636pt\leq
C\,(\|\textit{{v}}^{k}_{\epsilon}-\textit{{v}}_{\epsilon}\|\,\|b^{k}_{\epsilon}\|_{L^{6}}+\|\textit{{v}}_{\epsilon}\|_{L^{6}}\,\|b^{k}_{\epsilon}-b_{\epsilon}\|)\,\|\nabla\xi^{j}\|_{L^{3}}$
$\displaystyle\hskip 14.22636pt\leq
C\,\|\textit{{v}}^{k}_{\epsilon}-\textit{{v}}_{\epsilon}\|\,\|\nabla
b^{k}_{\epsilon}\|+C\,\|\nabla\textit{{v}}_{\epsilon}\|\,\|b^{k}_{\epsilon}-b_{\epsilon}\|.$

Then, by integrating with respect $t$ from $0$ to $T$ and using the Schwarz’s
inequality, we obtain

$\displaystyle\int_{0}^{T}(\textit{{v}}^{k}_{\epsilon}\cdot\nabla\,b^{k}_{\epsilon}-\textit{{v}}_{\epsilon}\cdot\nabla
b_{\epsilon},\,\xi^{j})\phi(t)\,dt$ $\displaystyle\qquad\leq
C\,(\int_{0}^{T}\|\textit{{v}}^{k}_{\epsilon}\,(t)-\textit{{v}}_{\epsilon}\,(t)\|^{2}dt)^{1/2}\,(\int_{0}^{T}\|\nabla
b^{k}_{\epsilon}\,(t)\|^{2}dt)^{1/2}$
$\displaystyle\qquad\quad+C\,(\int_{0}^{T}\|\nabla\textit{{v}}_{\epsilon}\,(t)\|^{2}dt)^{1/2}\,(\int_{0}^{T}\|b^{k}_{\epsilon}\,(t)-b_{\epsilon}\,(t)\|^{2}dt)^{1/2}.$
(103)

Thus, from Proposition 5.3, the continuity of $c_{l}(\cdot,\cdot)$, Lemma 5.7
and inequality (5), as $\ k\rightarrow\infty$ we have

$\displaystyle\int_{0}^{T}(\textit{{v}}^{k}_{\epsilon}\,(t)\cdot\nabla
c_{l\epsilon}^{k}\,(t)-\textit{{v}}_{\epsilon}\,(t)\cdot\nabla
c_{l\epsilon}\,(t),\varphi^{j})\phi(t)dt\longrightarrow 0,$ (104)
$\displaystyle\int_{0}^{T}(\textit{{v}}^{k}_{\epsilon}\,(t)\cdot\nabla\tilde{\theta}^{k}_{\epsilon}\,(t)-{\bf
v}_{\epsilon}\,(t)\cdot\nabla\,\tilde{\theta}_{\epsilon}\,(t),\psi^{j})\phi(t)dt\longrightarrow
0,$ (105)
$\displaystyle\int_{0}^{T}(\textit{{v}}^{k}_{\epsilon}\,(t)\cdot\nabla\textit{{v}}^{k}_{\epsilon}\,(t)-\textit{{v}}_{\epsilon}\,(t)\cdot\nabla\textit{{v}}_{\epsilon}\,(t),\textit{{w}}^{j})\phi(t)dt\longrightarrow
0.$ (106)

For the external force, taking into account that

${\bf F}_{e}(\textit{{z}}_{\epsilon}^{k})-{\bf
F}_{e}(\textit{{z}}_{\epsilon})=\rho\,{\bf
g}[\alpha\,(\tilde{\theta}_{\epsilon}^{k}-\tilde{\theta}_{\epsilon})+\beta\,(\tilde{c}_{\epsilon}^{k}-\tilde{c}_{\epsilon})],$

we have

$\displaystyle|({\bf F}_{e}(\textit{{z}}_{\epsilon}^{k})-{\bf
F}_{e}(\textit{{z}}_{\epsilon}),\textit{{w}}^{j}\,\phi(t))|$
$\displaystyle\hskip 14.22636pt\leq C\,\|{\bf
g}\|_{\infty}(\,\|\tilde{\theta}_{\epsilon}^{k}-\tilde{\theta}_{\epsilon}\|+\|\tilde{c}_{\epsilon}^{k}-\tilde{c}_{\epsilon}\|)\|\textit{{w}}^{j}\|\,\sup_{t\in[0,T]}|\phi(t)|$
$\displaystyle\hskip 14.22636pt\leq
C\,(\|\tilde{\theta}_{\epsilon}^{k}-\tilde{\theta}_{\epsilon}\|+\|\tilde{c}_{\epsilon}^{k}-\tilde{c}_{\epsilon}\|),$

and, by integrating and using the Schwarz’s inequality, we get

$\displaystyle\int_{0}^{T}({\bf F}_{e}(\textit{{z}}_{\epsilon}^{k})-{\bf
F}_{e}(\textit{{z}}_{\epsilon}),\textit{{w}}^{j})\,\phi(t)\,dt$
$\displaystyle\hskip 28.45274pt\leq
C\,(\int_{0}^{T}\|\tilde{\theta}_{\epsilon}^{k}-\tilde{\theta}_{\epsilon}\|^{2}dt)^{1/2}+C\,(\int_{0}^{T}\|\tilde{c}_{\epsilon}^{k}-\tilde{c}_{\epsilon}\|^{2}dt)^{1/2}.$
(107)

Moreover,

$\displaystyle
F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\,\textit{{v}}_{\epsilon}^{k}-F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\,\textit{{v}}_{\epsilon}=(F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})-F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}))\,\textit{{v}}_{\epsilon}^{k}+F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\,(\textit{{v}}_{\epsilon}^{k}-\textit{{v}}_{\epsilon}),$

by Proposition 5.3, the regularity of $F_{i}^{\epsilon}(\cdot,\cdot)$, and the
Schwarz’s inequality,

$\displaystyle\int_{0}^{T}(F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})\,\textit{{v}}_{\epsilon}^{k}-F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\,\textit{{v}}_{\epsilon},\textit{{w}}^{j})\,\phi(t)\,dt$
$\displaystyle\hskip 14.22636pt\leq
C\int_{0}^{T}\|F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})-F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\|_{\infty}\|\textit{{v}}_{\epsilon}^{k}\,(t)\|\,dt$
$\displaystyle\hskip
28.45274pt+C\int_{0}^{T}\|F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\|_{\infty}\|\textit{{v}}_{\epsilon}^{k}\,(t)-\textit{{v}}_{\epsilon}\,(t)\|\,dt,$
$\displaystyle\hskip 14.22636pt\leq
C\int_{0}^{T}\|F_{i}^{\epsilon}(\textit{{z}}_{\epsilon}^{k})-F_{i}^{\epsilon}(\textit{{z}}_{\epsilon})\|_{\infty}dt$
$\displaystyle\hskip
28.45274pt+C(T,\epsilon)\,(\int_{0}^{T}\|\textit{{v}}_{\epsilon}^{k}\,(t)-\textit{{v}}_{\epsilon}\,(t)\|^{2}\,dt)^{1/2}.$
(108)

Therefore, from (97)-(102), (104)-(5) and Lemma 5.7, we conclude that the
system (64)-(67) converges to the system (59)-(62). Thus, the proof of the
Theorem 5.1 is complete.

##  6 Reproductive weak solution of the solidification problem

In this section we will prove our main result.

###### Theorem 6.1

Under the hypotheses (S1), (S2) and (S3), the problem (8)-(17) admits at least
one weak reproductive solution.

Proof: From Theorem 5.1 and Proposition 5.3, for $\epsilon\in(0,1]$, any
reproductive weak solution
$(c_{\epsilon},\theta_{\epsilon},\textit{{v}}_{\epsilon})$ of the problem
(39)-(48) is uniformly bounded with respect to $\epsilon$. Then, similarly to
Lemma 5.5 and Lemma 5.7, we can conclude that there exists
$(c,\theta,\textit{{v}})$ and a sequence, still indexed by $\epsilon$, such
that

$\displaystyle c_{\epsilon}\ \rightarrow c,\quad\theta_{\epsilon}\
\rightarrow\theta\quad\mbox{ weakly in }L^{2}(H^{1}),\ \mbox{ strongly in
}L^{2}(L^{2}),$ (109) $\displaystyle\textit{{v}}_{\epsilon}\
\rightarrow\textit{{v}}\quad\mbox{ weakly in }L^{2}(\textit{{H}}_{v}),\mbox{
strongly in }L^{2}(\textit{{H}}).$ (110)

We check now that $(c,\theta,\textit{{v}})$ is a weak reproductive solution to
problem (8)-(17).

Let $\textit{{w}}\in\textit{{C}}^{\infty}_{0}(\Omega)$ with compact support
$K\subset\Omega_{ml}$. Then, for an certain $\delta>0$ and for every $x\in K$,

$f_{s}(c(x,t),\theta(x,t))<1-\delta\quad\mbox{ a.e. in }\ (0,T),$ (111)

and this implies that

$\|F_{i}(c(t),\theta(t))\|_{L^{\infty}(K)}<C(\delta)\quad\mbox{ a.e. in }\
(0,T).$ (112)

Due to the uniform convergence of $f_{s}(c_{\epsilon},\theta_{\epsilon})$
toward $f_{s}(c,\theta)$ for any compact subset of $\Omega_{ml}$, given
$\frac{\delta}{2}$ there exist an $\epsilon_{\delta}$ such that
$\forall\,\epsilon\in(0,\epsilon_{\delta})$ and $\forall\,x\in K$,

$|f_{s}(c_{\epsilon}(x,t),\theta_{\epsilon}(x,t))-f_{s}(c(x,t),\theta(x,t)|<\frac{\delta}{2},$
(113)

thus, from (113) and (111), $\forall\,x\in K$ and a.e. in $(0,T)$ we have

$f_{s}(c_{\epsilon}(x,t),\theta_{\epsilon}(x,t))<\frac{\delta}{2}+f_{s}(c(x,t),\theta(x,t)<1-\frac{\delta}{2}.$
(114)

Consequently, considering (5), (38), (111) and (114), when
$\epsilon\rightarrow 0$,

$F_{i}^{\epsilon}(c_{\epsilon}(x,t),\theta_{\epsilon}(x,t))\longrightarrow
F_{i}(c(x,t),\theta(x,t))\ \mbox{in }C^{0}(K)\ \mbox{ a.e. in }\ (0,T).$ (115)

Then, taking into account (112) and similarly as (5), we have

$\displaystyle\int_{0}^{T}(F_{i}^{\epsilon}(c_{\epsilon},\theta_{\epsilon})\,\textit{{v}}_{\epsilon}\,(t)-F_{i}(c,\theta)\,\textit{{v}}\,(t),\,\textit{{w}}^{i})\,\phi(t)\,dt$
$\displaystyle\hskip 14.22636pt\leq
C\int_{0}^{T}\|F_{i}^{\epsilon}(c_{\epsilon},\theta_{\epsilon})-F_{i}(c,\theta)\|_{L^{\infty}(K)}dt$
$\displaystyle\hskip
28.45274pt+C(\delta)(\int_{0}^{T}\|\textit{{v}}_{\epsilon}\,(t)-\textit{{v}}\,(t)\|^{2}\,dt)^{1/2}.$
(116)

Therefore, using (115), (6), (109), (110), and passing to the limit in the
equations (54)-(56), we obtain (30)-(32), where by density the equation (32)
holds for any $\textit{{w}}\in\textit{{H}}_{v}$ with supp
$\textit{{w}}\subset\Omega_{ml}$.

To check that $\textit{{v}}=0$ in $\Omega_{s}\times(0,T)$ take a compact set
$K\subset\Omega_{s}$. Since the solid domain is open, there exists an
$\epsilon_{K}>0$ such that
$\,f_{s}(c_{\epsilon}(x,t),\theta_{\epsilon}(x,t))=1$ if $x\in K$ a.e. in
$(0,T)$, whenever $\epsilon\in(0,\epsilon_{K})$, and this implies that

$F_{i}^{\epsilon}(c_{\epsilon}(x,t),\theta_{\epsilon}(x,t))=\frac{C_{0}}{\epsilon^{3}}\quad\mbox{in
}\,x\in K,\ \mbox{ a.e. in }\ (0,T).$ (117)

Choosing $\textit{{w}}=\textit{{v}}_{\epsilon}$ in (56), observing (78) and
(117), we have

$\displaystyle\frac{1}{2}\frac{d}{dt}\|\textit{{v}}_{\epsilon}(t)\|^{2}+\frac{3\,\nu}{4}\|\nabla\textit{{v}}_{\epsilon}(t)\|^{2}+\frac{C_{0}}{\epsilon^{3}}\,\|\textit{{v}}_{\epsilon}(t)\|^{2}_{L^{2}(K)}\leq
C.$ (118)

By integrating (118) respect to $t$ from $0$ to $T$, and since
$\textit{{v}}_{\epsilon}(0)=\textit{{v}}_{\epsilon}(T)$, we get

$\frac{C_{0}}{\epsilon^{3}}\,\int_{0}^{T}\|\textit{{v}}_{\epsilon}(t)\|^{2}_{L^{2}(K)}\,dt\leq
C\,T,$ (119)

where $C$ is independent of $\,\epsilon$. Thus, as $\,\epsilon$ vanishes
$\displaystyle\,\frac{C_{0}}{\epsilon^{3}}\rightarrow\infty$, then (119)
implies that
$\displaystyle\,\int_{0}^{T}\|\textit{{v}}_{\epsilon}(t)\|^{2}_{L^{2}(K)}\,dt$
to converge to $0$ and consequently
$\|\textit{{v}}_{\epsilon}(t)\|_{L^{2}(K)}$ converge to $0$. Therefore,
$\textit{{v}}=0$ in $K$ a.e. in $(0,T)$, and the arbitrary choice of $K$
implies $\textit{{v}}=0$ in $\Omega_{s}\times(0,T)$. Thus, we complete the
proof of the Theorem 6.1.

## References

  * [1] Ahmad, N.: Numerical simulation of transport processes in multicomponent systems related to solidification problems, Ph.D.Thesis DMA-EPFL N${}^{\underline{\circ}}$ 1349 (1995).
  * [2] Badillo, A., Durán, M., Ortega-Torres, E.: Cálculo de inestabilidades de un proceso de solidificación en dominios de simetría cilíndrica, Parte II: Resultados Numericos. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, Vol. 21, 4, 307-325 (2005).
  * [3] Blanc, Ph., Gasser, L., Rappaz, J.: Existence for a Stationary Model of Binary Alloy Solidification. Mathematical Modelling and Numerical Analysis, Vol. 29, 6, 687-699 (1995).
  * [4] Combeau, H., Lesoult, G.: Simulation of freckles formation and related segregation during directional solidification of metallic alloys. Modelling of casting, welding and advanced solidification processes 6, Voller V., Piwonka T.S. & Katgerman L. eds., 201-208 (1993).
  * [5] Durán, M., Ortega-Torres, E., Rappaz, J.: Weak Solution of a Stationary Convection-Diffusion Model Describing Binary Alloy Solidification Processes. Mathematical and Computer Modelling, Vol. 42, 11/12, 1269-1286 (2005).
  * [6] Durán, M., Ortega-Torres, E., Stein, R.: Cálculo de inestabilidades de un proceso de solidificación en dominios de simetría cilíndrica, Parte I: Formulación del modelo. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, Vol. 17, 2, 127-148 (2001).
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  * [8] Gaillard, F.: Modélisation et analyse numérique d’instabilités dévelopées lors de la solidification d’alliages binaires, Ph.D. Tesis DMA-EPFL, N${}^{\underline{\circ}}$ 1868 (1998).
  * [9] Gaillard, F., Rappaz, J.: Analysis and numerical simulation for models of binary alloy solidification, Periaux Contributed Book, John Wiley & Sons Ltd (1997).
  * [10] Gasser, L.: Existence analysis and numerical schemes for models of binary alloy solidification, Ph.D.Thesis DMA-EPFL N${}^{\underline{\circ}}$ 1421 (1995).
  * [11] Hills, R.N., Loper, D.E., Roberts, P.H.: A thermodynamically consistent model of a mushy zone. Quarterly J. Mech. Appl. Math., 36, 505-539 (1983).
  * [12] Ladyszhenskaya, O.A.: The mathematical theory of viscous incompressible flow, Second edition, Gordon and Breach, New York (1969).
  * [13] Milne-Thomson, L.M.: Theoretical hydrodinamics. Dover Publications, Inc., New York 1996.
  * [14] Rappaz, M., Voller, V.R.: Modelling of micro-macrosegregation in solidification processes. Metall. Trans. A-Physical Metallurgy and Materials, Vol. 21, 3, 749-753 (1990).
  * [15] Shapiro, A.H.: The dynamics and thermodynamics of compressible flow. Ronald Press, New York 1953.
  * [16] Voller, V.R., Prakash, C.: A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems. Int. J. Heat Mass Transfer, 30, 1709-1719 (1987).


1181 Research

# Expanding Targets in Virtual Reality Environments: A Fitts’ Law Study

Rongkai Shi e-mail<EMAIL_ADDRESS>0000-0001-8845-6034
Yushi Wei e-mail<EMAIL_ADDRESS>0000-0002-6003-0557 Yue Li
e-mail<EMAIL_ADDRESS>0000-0003-3728-218X Lingyun Yu e-mail:
<EMAIL_ADDRESS>0000-0002-3152-2587 Hai-Ning Liang Corresponding
author; e-mail<EMAIL_ADDRESS>0000-0003-3600-8955 Department of
Computing, School of Advanced Technology
Xi’an Jiaotong-Liverpool University

###### Abstract

Target pointing selection is a fundamental task. According to Fitts’ law,
users need more time to select targets with smaller sizes. Expanding the
target to a larger size is a practical approach that can facilitate pointing
selection. It has been well-examined and -deployed in 2D user interfaces.
However, limited research has investigated target expansion methods using an
immersive virtual reality (VR) head-mounted display (HMD). In this work, we
aimed to fill this gap by conducting a user study using ISO 9241-411 multi-
directional pointing task to examine the effect of target expansion on target
selection performance in VR HMD. Based on our results, we found that compared
to not expanding the target, expanding the target width by 1.5 and 2.5 times
during the movement can significantly reduce the selection time. We hope that
the design and results derived from the study can help frame future work.

Human-centered computingHuman computer interaction (HCI)Interaction
paradigmsVirtual reality; Human-centered computingHuman computer interaction
(HCI)Empirical studies in HCI

##  1 Introduction

Target pointing selection is a fundamental interaction task in all user
interfaces (UIs). Expanding the target to a larger size is a practical
approach that can facilitate pointing selection. In desktops, tablets, and
other 2D UIs, users commonly need to point to the interface widgets, such as
buttons and icons. Expanding the size of the target can improve the target
selection performance. McGuffin and Balakrishnan [11, 12] showed empirical
evidence that target expansion during users reaching the cursor to the target
could reduce the pointing time. Except for the linear expansion, researchers
also developed non-linear magnification effects, such as fisheye views [8, 9]
or Voronoi-based expansions [6, 7], to support the target selection in 2D UIs.

While in 3D UIs, users are no longer pointing to 2D interface elements but to
3D objects. However, limited research has investigated target expansion
methods in 3D UIs. Early in 2007, Vanacken [15] presented 3D bubble cursor to
support target selection in dense 3D environments. Once the 3D bubble cursor
captured a spherical target, a second, expanding sphere around the captured
target would appear to be contained by the cursor. Argelaguet and Andujar [1]
proposed two target expansion strategies for 3D selection tasks in CAVE. Both
research studies focused on expanding targets to support selection in dense
and occluded virtual environments.

With recent advancements, virtual reality (VR) head-mounted displays (HMDs)
have become powerful and popular. VR HMDs offer users an immersive experience
that is largely greater than other devices. The interaction in VR HMDs also
differs from other devices. Users normally control the ray emitted from the
6-DOF controller (i.e., ray-casting [3]) to point to the target objects in
virtual environments provided by VR HMDs. Yu et al. [18] tested Expanding
Target technique in VR HMDs and found that the selection performance was
improved by expanding the size of the pointing object. However, to the best of
our knowledge, no research has been conducted to investigate the effect of
target expansion in Fitts’ law tasks in VR HMDs. This can help understand and
predict user behavior to aid designers and developers of VR environments in
providing target expansion techniques to enable efficient and accurate target
selection in different VR application scenarios.

Fitts’ law is one of the most famous HCI predictive models for human movements
[5] and has recently been studied in immersive VR environments (e.g., [2]).
Fitts’ law states that the movement time to acquire a target is affected by
the distance to and the size of the target. One of the most widely used
mathematical variations is defined in ISO 9241-411:2012 [10]:

$MovementTime=a+b*log_{2}(\frac{A}{W}+1)=a+b*ID$ (1)

where $A$ is the effective target distance, $W$ is the effective target size,
and the $log$ term represents the effective index of difficulty
($ID$).111Effective target distance, size, and index of difficulty are usually
expressed as $A_{e}$, $W_{e}$, and $ID_{e}$. To improve readability, we
directly use $A$, $W$, and $ID$ to represent effective measures in this paper.
The coefficients $a$ and $b$ are derived from linear regression using the
empirical results. According to ISO standard [10], $ID$ and movement time can
be further used to calculate the throughput:

$Throughput=(\frac{ID}{MovementTime})$ (2)

In this work, we adapted the multi-directional pointing task described in the
ISO standard [10] to investigate the effect of target expansion on target
selection performance in VR environments. We used movement time, error rate,
and throughput as measurements to examine the selection performance. Besides,
we also recorded the selection endpoints and visualized their distributions.
We conducted a user study ($N=17$) to compare three expansion factors
(expanding the target width by 1.5, 2, or 2.5 times during the movement)
against a baseline condition (no expansion) in the Fitts’ law task. Our
results showed that expanding expansion factors of 1.5 and 2.5 can
significantly reduce the movement time. This work mainly made an empirical
contribution based on a user study with seventeen participants. We hope future
work on target expansion and Fitts’ law in VR environments can be framed based
on this work’s exploration.

Figure 1: Experimental setup. (a) $W$ is the target width and $A$ is the
movement amplitude. The blue arrows indicate the trial order. (b) $W_{expand}$
represents the expanded target width. $W$, $A$, and $W_{expand}$ are all
effective distances using angular forms. The green button in the center
represents the start button.

##  2 Preliminary Study

We aimed to investigate the effect of target expansion on target selection. To
do so, we conducted a user study with seventeen participants recruited from a
local university and used a Meta Quest 2 VR HMD to provide the testing
environment. Our primary research question was how expanding targets would
affect the speed, accuracy, and throughput of target selection.

###  2.1 Task and Stimuli

We used an ISO 9241-411:2012 [10] multi-directional tapping task with 11 grey
spherical targets placed as a circle (Fitts’ ring design). Participants were
asked to use the dominant-hand controller to point at the targets and press
the trigger on the non-dominant-hand controller to perform the selection. This
setting was made to avoid the Heisenberg effect [4]. They were informed to
select the target as fast and as precisely as possible. The task started with
the target at the top of the ring after participants triggered the start
button located at the center of the ring. Once participants selected a target,
the next target was on the diagonal, and continued clockwise on the ring. The
goal target would be in blue as an indication. Participants were sitting in
the middle of an empty virtual room where visual depth cues were provided via
this setting.

The original target width was $W$, and the movement amplitude was $A$. When
participants controlled the ray-casting to move past a specified fraction $P$,
that is, the expansion point, of the total distance $A$, the target size $W$
would instantaneously expand to $W_{expand}$. In this study, we set $P=0.9*A$
[12]. Figure 1 shows the experimental setup. Following the approach described
by Yu et al. [19], we used a spherical coordinate system and angular
representations whenever feasible. Thus, $W$, $A$, and $W_{expand}$ were all
effective distances.

###  2.2 Experimental Design

We used three target widths $W$ (1.5°, 2.5°, and 3.5°) and two movement
amplitudes $A$ (15° and 20°), which led to six indices of difficulty (IDs).
This study followed a 6$\times$4 within-subject design with two factors: index
of difficulty $ID$ (2.402, 2.747, 2.807, 3.170, 3.459, 3.841) and expansion
factor $E$ (1, 1.5, 2, and 2.5). An expansion factor of 1 represented a
baseline condition where the target size would not change during the trial.
While for the remaining three conditions, the target size would expand—the
target width would increase to 1.5, 2, and 2.5 times the initial width. The
orders of $W$ and $E$ were counterbalanced using a Latin Square approach
separately. For a given $W$, the order of $A$ was randomized. The data from
the first trial for each $ID$ condition (i.e., each $W\times A$ combination or
each ring) was discarded. Thus, in total, we recorded 6 indices of difficulty
$\times$ 4 expansion factors $\times$ 10 repetitions $\times$ 17 participants
= 4080 trials of data.

###  2.3 Measurements

For each trial, we recorded the movement time (second, s) and endpoint
$(x,y)$. The endpoints were also represented using angular forms [19]. We
further calculated error rates (%) and throughput (°/s) for each condition per
participant. Throughput was calculated using Equation 2.

###  2.4 Procedure

Before the experiment, participants were briefly introduced to the
experimental setup, including the device, task, and procedure. After that,
they started a 3-minute training session to familiarize themselves with the VR
device and ISO pointing task. Then they began to complete the formal trials.
The whole experiment lasted about 35 minutes per participant.

###  2.5 Results and Discussion

####  2.5.1 Movement Time, Error Rate, and Throughput

We used repeated measure (RM-) ANOVA to analyze the movement time, error rate,
and throughput results. We applied Greenhouse-Geisser correction for degrees
of freedom whenever needed ($\epsilon>.75$ for our cases) and Bonferroni
correction for post hoc analyses. Results are summarized in Figure 2(a-c) and
Table 1.

We only found a significant main effect of ID ($F(2.466,39.460)=3.776$,
$p=.024$, $\eta_{p}^{2}=.191$) and a significant main effect of E on movement
time ($F(3,48)=6.734$, $p<.001$, $\eta_{p}^{2}=.296$). Post hoc analyses
revealed that an expansion factor of 1.5 ($M=.797s$, $p=.015$) and 2.5
($M=.753s$, $p=.024$) led to significantly longer movement time than an
expansion factor of 1 ($M=.872s$); that is, no expansion effect applied.

Figure 2: (a-c) Bar plots for Movement Time (a), Error Rate (b), and Throughput (c). Error bars in (a-c) represent 95% confidence intervals. (d) Fitts’ law results for the four expansion factors.  Table 1: The results of RM-ANOVAs in the study. | Factor | $DF_{n}$ | $DF_{e}$ | $F$ | $p$ | $\eta_{p}^{2}$
---|---|---|---|---|---|---
Movement | $ID$ | 2.466 | 39.460 | 3.776 | .024 | .191
Time | $E$ | 3 | 48 | 6.734 |  $<$.001 | .296
|  $ID$$\times$$E$ | 3.200 | 51.203 | .709 | .560 | .042
Error | $ID$ | 2.712 | 43.396 | 1.238 | .306 | .072
Rate | $E$ | 3 | 48 | 2.613 | .062 | .140
|  $ID$$\times$$E$ | 15 | 240 | 1.045 | .409 | .061
Through- | $ID$ | 1.151 | 18.414 | 2.358 | .139 | .128
put | $E$ | 1.186 | 18.979 | 2.105 | .162 | .116
|  $ID$$\times$$E$ | 1.261 | 20.169 | 1.206 | .299 | .070
Figure 3: Distributions of endpoints across all participants in all
conditions. $A$, $W$, and $E$ represent effective target width, effective
movement amplitude, and expansion factor, respectively. The blue circles are
the targets in initial size, the purple circles are the targets in expanded
size, and the red ellipses are 95% confidence ellipses.

####  2.5.2 Fitts’ Law Analysis and Endpoints

Using Fitts’ Law in Equation 1, we found the MT (movement time) can be modeled
as $MT=.186+.223*ID$ ($R^{2}=.972$) for $E=1$ condition, $MT=.206+.193*ID$
($R^{2}=.963$) for $E=1.5$ condition, $MT=.112+.207*ID$ ($R^{2}=.984$) for
$E=2$ condition, and $MT=.221+.173*ID$ ($R^{2}=.977$) for $E=2.5$ condition.
Figure 2(d) shows the Fitts’ law results.

We plotted all the endpoints in Figure 3 to show their distributions in each
condition.

###  2.6 Answers to the Research Question

According to the study results, expanding the target width by 1.5 and 2.5
times during the movement can significantly shorten the time for reaching the
target. As expected, expanding the target size made the selection faster.
However, no significant difference among expanding conditions was found (i.e.,
$E$ = 1.5, 2, and 2.5). One possible reason is that the expansion factors in
our study did not vary largely from each other. Yet, for a large expansion
factor, its application scenario would also be limited. Thus, we suggest VR
developers consider expanding the target (e.g., buttons or 3D objects) to a
suitable size to make the selection more efficient.

We did not identify significant improvements in the accuracy and throughput
for target selection when expanding targets. Notably, the main effect of $E$
on the error rate was close to statistically significant ($p=.062$,
$\eta_{p}^{2}=.140$). From Figure 3, we can observe that expanding the target
($E$ = 1.5, 2, and 2.5) tolerated the selection pointed outside the initial
target. While some endpoints were around the target in $E=1$ condition, which
can be potentially selected if the target can expand. Future work with more
participants, IDs, and testing trials are needed to substantiate when the
difference would appear.

##  3 Limitations and Future Work

We only applied linear magnification effects to the target. Prior work has
proposed non-linear magnification strategies [8, 9]. Future work can also test
them using Fitts’ law approach to investigate the user motor performance. In
addition, as a nature of the Fitts’ law task, the target selection tasks were
given repeatedly. It is inevitable that participants could predict the
expansion of the target, which could affect their performance. We plan to
conduct a follow-up study involving conditions in which the target could
unpredictably expand, remain unchanged, or even shrink [20] and test with more
$IDs$ varying from a wider range [14]. Besides, prior work has been conducted
to understand the endpoint distribution of pointing selection tasks [19, 16],
which can help understand users’ behavior from another perspective and can be
integrated into selection techniques to improve the selection performance
further. We intend to model endpoint distributions based on our collected
data. We envision that with the support of predictive models, expanding the
target can improve the selection performance in different VR scenarios (e.g.,
in dense scenarios where objects overlap [13] or outside of users’ view [17]).

##  4 Conclusion

In this paper, we conducted a Fitts’ law study (N = 17) to investigate how
expanding the target affects the selection performance in virtual reality (VR)
head-mounted displays (HMDs). Specifically, we tested three expansion factors
(expand the target width to 1.5, 2, and 2.5 times its initial size) and
compared them with a baseline non-expansion condition. Our results showed that
expanding the target can improve the speed of target selection compared to the
baseline condition. We then discussed the limitations of the present study,
which could represent avenues for future work. This work serves as an initial
exploration of expanding targets for pointing selection tasks in VR HMDs from
a Fitts’ law perspective and can help frame future work with relevant topics.

###### Acknowledgements.

The authors wish to thank the participants for their time and the reviewers
whose comments helped improve our paper. This work was partly supported by the
National Natural Science Foundation of China (#62207022; #62272396) and the
Natural Science Foundation of the Jiangsu Higher Education Institutions of
China (#22KJB520038).

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