\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 109, pp. 1--25.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2003 Texas State University-San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/109\hfil A phase field model for solidification]
{Existence and regularity of solutions of a phase field model
for solidification with convection of pure materials in  two dimensions}

\author[Jos\'e Luiz Boldrini \& Cristina L\'{u}cia Dias Vaz\hfil EJDE--2003/109\hfilneg]
{Jos\'e Luiz Boldrini \& Cristina L\'{u}cia Dias Vaz} % in alphabetical order

\address{Jos\'e Luiz Boldrini \hfill\break
Department of Mathematics,
UNICAMP-IMECC, Brazil}
\email{boldrini@ime.unicamp.br}

 \address{Cristina L\'{u}cia Dias Vaz \hfill\break
Department of Mathematics,
Universidade Federal do Par\'{a}, Brazil}
\email{cvaz@ufpa.br}

\date{}
\thanks{Submitted September 14, 2001. Published November 3, 2003.}
\thanks{J. L. B. was partially supported by grant 300513/87-9 from CNPq, Brazil}
\subjclass[2000]{76E06, 80A22, 82B26, 76D05}
\keywords{Phase-field, phase transition, solidification, convection,\hfill\break\indent
        Navier-Stokes equations}

\begin{abstract}
  We study the existence and regularity of weak solutions of a
  phase field type model for pure material solidification in
  presence of natural convection. We assume that the non-stationary
  solidification process occurs in a two dimensional bounded domain.
  The governing equations of the model are the phase field equation
  coupled with a nonlinear heat equation and a modified Navier-Stokes
  equation. These equations include buoyancy forces modelled by
  Boussinesq approximation and a Carman-Koseny term to model the
  flow in mushy regions. Since these modified Navier-Stokes equations
  only hold in the non-solid regions, which are not known a priori,
  we have a  free boundary-value problem.
\end{abstract}
\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

One of the first papers to consider phase field models applied to change of
phases was written by  Fix \cite{Fix}. This article fostered many other studies
in this subject; see  for instance the sequence of papers
\cite{Caginalp1986}, \cite{Caginalp1994}, \cite{CaginalpJones},
\cite{Caginalp1989}.
Caginalp and others took over the task of understanding the phase field
approach, both in its mathematical aspects and in its relationship to the
classical approach of using sharp interfaces to separate the phases, which
gives rise to what is known by Stefan type problems. We remark that, for the
derivation of the kinetic equation for the phase field, Caginalp and others
used the free energy functional as a basis of the argument (see also Hoffman and
Jiang \cite{Hoffman}). An alternative derivation, suggested by
Peronse and Fife \cite{Penrose1993,Penrose1990}, uses an entropy
functional which gives a kinetic equation for the phase field ensuring
monotonic increase of the entropy in time. Peronse and Fife exhibit a
specific choice of entropy density which essentially recovers the phase field
model employed by Caginalp \cite{Caginalp1986} by linearizing the heat
flux. Thus, phase field models have a sound physical basis and provide simple
and elegant descriptions of phase transition processes. Moreover, it is more
versatile than the enthalpy method because it allows effects such as supercooling
to be included.
An important example of the utility of the phase field approach
is its use for the numerical study of dendritic growth (see Caginalp
\cite{Caginalp1994} and Kobayshi \cite{Kobayshi}).


One point of importance is that many papers interested in the
mathematical analysis of these models, whatever the approach used to model phase
change, have neglected the possibility of flow occurring in non solidified
portions of the material. In many practical situations, however,
this assumption is not satisfactory because the existence of such motions may
affect in important ways the outcome of the process of phase change. In fact,
melt convection adds new length and time scales to the problem and results in
morphologies that are potentially much different from those generated by
purely diffusive heat and solute transport. Moreover, not only does
convection influence the solidification pattern, but the evolving
microstructure can also trigger unexpected and complicated phenomena. For
instance, Coriell {\it et all} \cite{Coriell} and Davis \cite{Davis} studied
in detail the coupled convective and morphological instabilities at a growth
front. This suggests that models that do not consider melt convection may
have some limitations.

One must realize, however, that the inclusion of this possibility
brings another very difficult aspect to an already difficult
problem. For this, it is enough to observe that such a flow must
occur only in an a priori unknown non-solid region, and thus one
may be left with a rather difficult free boundary-value problem.

In recent years, some authors have considered convective effects; for
instance: Cannon \textit{et al} \cite{Cannon1983,Cannon1980},
 DiBenedetto and Friedman \cite{DiBenedettoFriedman}, DiBenedetto and
O'Leary\cite{DiBenedettoOLeary} and O'Leary \cite{OLeary}, who addressed  such
questions by using weak formulations of the Stefan type approach. Blanc
\textit{et al.} \cite{Blanc}, Pericleouns \textit{et al.} \cite{Pericleous}
and Voller \textit{et al.} \cite{VollerPrakash,VollerCrossMarkatos}
considered convective effects in phase change problems  by using the enthalpy
approach to describe change of phases, together with modified Navier-Stokes
equations to model the flow. In these works, the phases may be distinguished
by the values of a variable corresponding to the solid fraction that is
associated to the enthalpy; this same variable is used in a term that is
added to the Navier-Stokes equations to cope with the influence of the mushy
zones in the flow. Particular expressions for this term may be obtained by
modelling such mushy zones as porous media. Beckermann {\it et al.}
\cite{Beckermann} and Diepers {\it et al.} \cite{Diepers}  used the phase
field methodology to obtain models including phase change and melt
convection. By using numerical simulations they studied the influence of the
convection in the melt on phenomena like dendritic growth and coarsening. An
interesting discussion of the application of diffusive-interface methods
(phase field being one of them) to fluid mechanics can be found in Anderson
and McFadden \cite{Anderson}.

In this paper we are interested in the mathematical analysis of a
model problem having some of the main aspects that a reasonable
model for a solidification process with convection should have. We
will consider a rather simple situation of this sort in the hope
to obtain a better understanding of the mathematical difficulties
brought by the coupling of terms describing phase change and the
terms describing convection. We also restrict the subject to the
analysis of solidification of pure materials (the corresponding
mathematical analysis for alloys will be considered elsewhere.)


As in \cite{Beckermann} and \cite{Diepers}, we employ a phase
field methodology to model phase change; we also assume the solid
phase to be rigid and stationary. However, differently from their
models, convective effects will be included by using the ideas
suggested by Blanc \textit{et al} \cite{Blanc} and Voller
\textit{et al} \cite{VollerPrakash}. Since the indicator of phase
in these last papers is the solid fraction, we relate the two
approaches by postulating a functional relationship between the
solid fraction and the phase field. The governing equations of the
model are the following: the phase field equation is as in Hoffman
and Jiang \cite{Hoffman}; it is coupled with equations for the
temperature and velocity that are based on usual conservation
principles. These last equations become respectively a nonlinear
heat equation and modified a  Navier-Stokes equations which include
buoyancy forces modelled by Boussinesq approximation and a
Carman-Koseny type term to model the flow in mushy regions. Since
these modified Navier-Stokes equations only hold in a priori
unknown non-solid regions, we actually have a free boundary value
problem.

We remark that the phase field model that we consider here is rather simple
and does not take care of several important transition events such as
nucleation or spinodal decomposition. However, more complete and complex
phase field models for phase change could be similarly considered. Our choice
in this paper was just guided by mathematical simplicity. We do not present a
detailed derivation of the model equations directly from physical background
because this would be basically a repetition of the arguments of the
references (they use just the usual balance of internal energy and linear
momentum arguments arguments suitably adapted to the situation) and they
would be lengthy in an already large paper. We give the idea of such
adaptations for the balance of internal energy in Section \ref{equations};
for the balance of linear momentum the argument is exactly as in Voller
\textit{et al} \cite{VollerCrossMarkatos} (see also \cite{VollerPrakash}.)
The details of the model problem can be found in Section \ref{equations},
equations (\ref{2Dpuromodel}); the corresponding weak formulation can be
found in Definition \ref{defpuro}.

Our objective is to present a result on the existence and regularity
of solutions of these model equations corresponding to a
nonstationary phase change process in a bounded domain, which for
technical reasons in this paper is assumed to be two dimensional.

Existence will be obtained by using a regularization technique
similar to the one used by  Blanc \textit{et al}
in \cite{Blanc}: an auxiliary positive parameter will be introduced
in the equations in such way that the original free boundary value
problem will be transformed in a more standard (penalized) one. We
say that this is the regularized problem. By solving this, one
hopes to recover the solution of the original problem as the
parameter approaches zero. To accomplish such program, we will
solve the regularized problem by using Leray-Schauder
degree theory (see Section 8.3, p. 56 in Deimling
\cite{Deimling}); and use results for
certain modified Navier-Stokes equations presented in
Vaz \cite{Vaz}. Then, by taking a sequence of values of the
parameter approaching zero, we will correspondingly have a
sequence of approximate solutions. By obtaining suitable uniform
estimates for this sequence, we will then be able to take the
limit along a subsequence and, by compactness arguments, to show
that we have in fact a solution of the original problem. The
stated regularity of this solution will be obtained by applying
the $L_{p}$-theory of the parabolic linear equations together with
bootstrapping arguments.


We should stress that the ideas presented in this mathematical analysis, in
particular the penalization used for obtaining the approximate solutions,
suggest a convenient discretization scheme for numerical simulations of phase
change problems with melt convection. Such scheme would be similar to, but
different from, the ones in \cite{Beckermann} and \cite{Diepers}. Moreover,
such methods would not rely on specifying a variable viscosity across the
diffusive interface regions that tend to a large value in the rigid solid, as
several other methods propose. This would be realistic only for certain
classes of materials, and certainly difficult to specify for rigid solids. We
also remark that the mathematical analysis corresponding to the models
presented in \cite{Beckermann} and \cite{Diepers} are presently under
investigation.

This paper is organized as follows. In Section 2, we describe the
mathematical model and its variables. In Section 3, we fix the
notation and describe the the basic functional spaces to be used;
we recall certain results and present auxiliary problems; we also
state assumptions holding throughout the paper and  define the
concept of generalized solution. In Section 4, we consider the
question of existence, uniqueness and regularity of solutions of
the regularized problem. Section 5 is dedicated to the proof
of the existence of a solution of the original free boundary value
problem.

Finally, as it is usual in papers of this sort,  $C$ will denote a
generic constant depending only on a priori known quantities.

\section{Model Equations}\label{equations}

The model problem presented here has aspects of the models studied
in the works of Blanc \cite{Blanc}, Caginalp \cite{Caginalp1986}
and Voller \textit{et al}
\cite{VollerPrakash,VollerCrossMarkatos}. As we said in the
Introduction, the phase of the material will be described by using
the phase field methodology, which in its simplest approach
assumes that there is a scalar field $\varphi(x,t)$, the phase
field, depending on the spatial variable $x$ and time $t$ and real
values $\varphi_s < \varphi_\ell$ such that if  $\varphi (x,t)
\leq \varphi_s$ then the material at point $x$ at time $t$ is in
solid state; if $\varphi_\ell \leq \varphi(x,t)$ then the material
at point $x$ at time $t$ is in liquid state; if $\varphi_s <
\varphi (x,t) < \varphi_\ell$ then, at time $t$ the point $x$ is
in the mushy region. We follow Caginalp \cite{Caginalp1986} and
Hoffman and Jiang\cite{Hoffman} and take the phase field equation
as
\[
 \frac{\partial \varphi}{\partial t} - \alpha \Delta \varphi
=a\varphi + b \varphi^{2} - \varphi^3 + \theta,
\]
 where $\theta$ is the temperature; $\alpha$ is a (small)
fixed positive constant, and $a$ and $b$ are known functions which
regularity will be described later on.

We observe that the function $g(s) = a s + b s^2 - s^3$ used at
the right hand side of the above equation is the classical
possibility coming from the classical double-well potential (see
Hoffman and Jiang \cite{Hoffman}). Other possibilities for the
double-well potential can be found for instance in Caginalp
\cite{Caginalp1986} and Penrose \cite{Penrose1990}.

To obtain a equation for the temperature, we observe that when
there is phase change, the thermal energy has the following
expression:
\[
e = \theta + \frac{\ell}{2}(1-f_{s}),
\]
 where $\theta$ and $\ell /2$ represent respectively the sensible
heat (for simplicity of notation, we took the specific heat coefficient to be
one) and latent heat. $f_s$ is the solid fraction ($1-f_s$ is the non-solid
fraction), which for simplicity we assume to be a known function only  of the
phase field (obviously dependent on the material being considered.)

Then, the energy balance in pure material solidification process
may be written (see Vaz \cite{Vaz}) as follows:
\[
 \frac{\partial\theta}{\partial t} -\Delta\theta +
v.\nabla\theta = \frac{\ell}{2} \frac{\partial
f_{s}}{\partial\varphi}(\varphi)\frac{\partial\varphi}{\partial t}
\]
 where $v$ represent the velocity of the material.

We will assume that only non solid portions of the material can
move, and this is done as an incompressible flow. Consequently, in
non-solid regions Navier- Stokes type equations are required.
According to Voller \textit{et al} \cite{VollerPrakash} and Blanc
\textit{et al} \cite{Blanc} these equations can be taken as
\begin{gather*}
 \frac{\partial v}{\partial t}-\nu \Delta v + (v.\nabla)v +
\nabla p = G(f_s,v)+ F(\theta) \\
\mathop{\rm div} v = 0
\end{gather*}
where $v$ is velocity, $p$ is pressure, $\nu$ is
viscosity and $G(f_{s},v)$ and $F(\theta)$ are source terms which
will be defined below.

Assuming the Boussinesq treatment to be valid, natural convection
effects can be accounted for by defining the buoyancy source term
to be
\[
F(\theta) = C\rho\textbf{g}(\theta - \theta _r)
\]
 where $\rho $ is the mean value of the density,
\textbf{g }is the gravity, $C$ is a constant and $\theta_r$ is a
reference temperature. In order to simplify the calculations let
us consider $F(\theta) =\vec{\sigma}\theta$.

The source term $G(f_{s},v)$ is used to modify the Navier-Stokes
equations in the mushy regions, and according to
\cite{VollerPrakash,VollerCrossMarkatos}, can be taken of
form $G(f_{s},v) = - k(f_{s})v$. Usually the function $k(f_{s})$
is taken as the Carman-Koseny expression (see again
\cite{VollerPrakash,VollerCrossMarkatos}), which is
\[
k(f_s) = \frac{f_s^2}{(1-f_s )^3}.
\]

As in Blanc \emph{et al} \cite{Blanc}, we will consider a more
general situation including the previous one. We will assume that
assuming that $k$ is a nonnegative function in $C^{0}(-\infty,1)$,
$k = 0$ in $\mathbb{R}^{-}$ and
${ \lim_{\mbox{y}\rightarrow 1} k(\mbox{y})}= + \infty$,
and in this case, we will refer to $G$ as the Carman-Kosen type term.

To complete the description of the model problem, we must define
the regions where the above equations are valid. By using the
solid fraction, the following subsets of $Q$, denoted by  $Q_l$,
$Q_m$ and $Q_s$ and corresponding respectively to the liquid,
mushy and solid regions, are defined as:
\begin{gather*}
Q_l = \{(x,t) \in Q :  f_s (\varphi(x,t)) =0\}\\
Q_s = \{(x,t) \in Q :  f_s (\varphi(x,t)) =1\}\\
Q_m = \{(x,t) \in Q :  0 < f_s (\varphi(x,t)) <1\}
\end{gather*}
In the following, $Q_{ml}= Q\setminus \bar{Q}_s$ will
denote the non-solid part of $Q$. Moreover, for each time
$t \in [0,T]$, we define
$\Omega_{s}(t) = \{x \in \Omega :  f_{s}(\varphi(x,t))=1 \}$,
$\Omega_{ml}(t) = \Omega\setminus \bar{\Omega}_s(t)$ and
$S_{ml}=\{(x,t) \in\,\bar{Q} : x \in
\partial \Omega_{ml}(t)\}$.

We must emphasize that this model is the free boundary problem
since that $Q_l$, $Q_m$ and $Q_s$ are  a priori unknown.
Now, we can now summarize the formulation of the problem to be
analyzed as:
\begin{equation}\label{2Dpuromodel}
\begin{gathered}
 \frac{\partial\varphi}{\partial t} - \alpha \Delta\varphi =
a\varphi+b\varphi^{2}-\varphi^{3}+\theta \quad
\mbox{in } Q, \\
 \frac{\partial\theta}{\partial t}-\Delta\theta + v.\nabla\theta
=\frac{\ell}{2} \frac{\partial f_s}{\partial\varphi}(\varphi)
 \frac{\partial\varphi}{\partial t} \quad \mbox{in } Q,\\
\frac{\partial v}{\partial t} - \nu \Delta
v+(v.\nabla)v+\nabla p+ k(f_{s}(\varphi)) v =
\vec{\sigma} \theta \quad  \mbox{in }Q_{ml},\\
\mathop{\rm div} v = 0 \quad \mbox{in } Q_{ml},\\
v = 0 \quad \mbox{in } \overset{\circ}Q_{s},
\end{gathered}
\end{equation}
 subject to the boundary conditions
\begin{equation}\label{2Dpurofront}
\begin{gathered}
\frac{\partial \varphi}{\partial n}  =  0 \quad\mbox{on } S,\\
\theta = 0 \quad\mbox{on } S,\\
v = 0  \quad \mbox{on }S_{ml}.
\end{gathered}
\end{equation}
 and to the initial conditions
\begin{equation}\label{2Dpurodata}
\begin{gathered}
\varphi(x,0)=\varphi_0(x)\quad \mbox{in }\Omega,\\
\theta(x,0)=\theta_0(x)\quad \mbox{in }\Omega,\\
v(x,0)=v_0(x) \quad \mbox{in }\Omega _{ml}(0),
\end{gathered}
\end{equation}
 where $\varphi_0$, $\theta_0$ and $v_0$ are suitably
given functions such that for compatibility $v_0$ is identically
zero outside $\Omega_{ml}(0)$.

\section{Preliminaries and Main Result}\label{preliminary}

\subsection{Notation, functional  spaces and auxiliary results}\label{spaces}

Let $\Omega\subset \mathbb{R}^{2}$ be an open and bounded domain with a
sufficiently smooth boundary $\partial\Omega $ and Q = $\Omega
\times [0,T] $ the space-time cylinder with lateral surface
$S=\partial\Omega\times [0,T]$. For $t \in [0,T]$, we denote $Q_t
= \Omega \times [0,t]$.


We denote by $W_{q}^{p}(\Omega)$ the usual Sobolev space and
$W_q^{2,1}(Q)$ the Banach space consisting of functions $u(x,t)$
in $L^q(Q)$ whose generalized derivatives $D_x u$, $D_{x}^2 u$,
$u_t$ are $L^q$-integrable ($q\geq 1$). The norm in $W_q^{2,1}(Q)$
is defined by
\begin{equation}\label{normsobolev}
\left\|u\right\|_{q,Q}^{(2)} = \left\|u\right\|_{q,Q} + \left\|D_x
u\right\|_{q,Q} + \left\|D_x^{2} u\right\|_{q,Q} +
\left\|u_t\right\|_{q,Q}
\end{equation}
 where $D_x^{s}$ denotes any partial derivatives with
respect to variables $x_1, x_2,..., x_n$ of order s=1,2 and
$\|\cdot\|_q$ the usual norm in the space $L^q(Q)$.\medskip

Moreover, $W^{1,0}_{2}(Q)$ is a Hilbert space for the scalar
product
\[
(u,v)_{W_{2}^{1,0}(Q)}=\int_Q  u v + \nabla u.\nabla v \,dx\,dt
\]
and $\overset{0}{W}{}^{1,1}_{2}(Q)$ is a Hilbert space
for the scalar product
\[
(u,v)_{W_{2}^{1,1}(Q)}=\int_Q  u v + \nabla u.\nabla v + u_t v_t
\,dx\,dt
\]
 whose functions vanish on $S$ in the sense of
traces.

We also denote by $V_{2}(Q)$ the Banach space consisting of
function $u(x,t)$ in $W^{1,0}_{2}(Q)$ having the following finite
norm
\begin{equation}
\label{normabaixa}
|u|_{V_2(Q)}= \mathop{\rm ess\,sup}_{0 \leq t \leq T} \left\|u(x,t)\right\|_{2,\Omega}
+ \left\|\nabla u(x,t)\right\|_{2,Q}.
\end{equation}

Let $\overset{0}{V}_{2}(Q)$ be the Banach space consisting of those
elements of $V_{2}(Q)$ that vanish on $S$ in the sense of
traces.

We now define spaces consisting of functions that are continuous
in the sense of h\"older. We say that a function
$u(x,t)$ defined in $\overline{Q}$ is h\"older
continuous in $x$ and $t$, respectively with exponents
$\alpha$ and $\beta \in (0,1)$, if following
quantities, called h\"older constants, are finite:
\begin{gather*}
\langle u \rangle _{x}^{(\alpha)}
= \sup_{(x_1,t),(x_2,t)\,\in  \overline{Q},\; x_1 \ne x_2}
\frac{|u(x_1,t)-u(x_2,t)|}{|x_1-x_2|^{\alpha}} \\
\langle u \rangle _{t}^{(\beta)}
= \sup_{(x,t_1),(x,t_2)\,\in  \overline{Q},\; t_1 \ne t_2}
\frac{|u(x,t_1)-u(x,t_2)|}{|t_1-t_2|^{\beta}}
\end{gather*}

Then, we define the h\"older space $H^{\tau ,\tau/2}(Q)$, with $0 \le \tau < 1$,
(see Ladyzenskaja \emph{et al} \cite{Ladyzenskaja}),
as the Banach space of functions u(x,t) that
are continuous in $\overline{Q}$, having finite norm given by:
\begin{equation}\label{normHolder}
|u|_{Q}^{(\tau)}= \max_{\overline{Q}}|u| +\langle D_{x}u \rangle
_{x}^{(\tau)} +\langle u \rangle _{t}^{(\tau/2)}.
\end{equation}

For the functional spaces associated to the velocity field, we
denote $\mathcal{D} = \{u \in \mathcal{C}^{\infty}(\Omega)^2 :
\mathop{\rm supp} u \subset \Omega\}$ and $\mathcal{V} = \{ u \in
\mathcal{D} : \mathop{\rm div} u = 0\}$. The closure of $\mathcal{V}$ in
$L^2(\Omega)^2$ is denoted by H and the closure of $\mathcal{V}$
in $\overset{0}{W}{}_2^{1}(\Omega)^{2}$ is denoted by V.
These functional spaces appear in the mathematical theory of the
Navier-Stokes equations; their properties can be found for
instance in Temam \cite{Temam}.

The following two lemmas are particular case of Lemma 3.3 in
Ladyzenskaja \emph{et al} (\cite{Ladyzenskaja}). They are
stated here for ease of reference.
The first lemma is immediate consequence of Lemma 3.3 in
\cite[p.\ 80]{Ladyzenskaja}, by taking there $l=1$, $n = 2$ and
$r=s=0$.

\begin{lemma}\label{imbeddingLp}
Let $\Omega$ and $Q$ as in the beginning of this section. Then for
any function $u \in W_{q}^{2,1}(Q)$ we also have $u \in L^{p}(Q)$,
and it is valid the following inequality
\begin{equation}
\left\|u\right\|_{p,Q} \le C \left\|u\right\|_{q,Q}^{(2)},
\end{equation}
provided that
\[
p =\begin{cases}
\infty & \mbox{if } \frac{1}{q} - \frac{1}{2}< 0\\
p \geq 1 & \mbox{if }  \frac{1}{q} - \frac{1}{2}=0\\
\big(\frac{1}{q} - \frac{1}{2}\big)^{-1} &\mbox{if }
\frac{1}{q} - \frac{1}{2}>0\,.
\end{cases}
\]
The constant $C>0$ depends only on T, $\Omega$, $p$ and $q$.
\end{lemma}

The second lemma is immediately obtained from Lemma 3.3 in
\cite[p.\ 80]{Ladyzenskaja},  by taking there $l =1$, $n = 2$,
$r=s=0$ and $q = 3$.

\begin{lemma}\label{imbeddingHolder}
Let $\Omega$ and $Q$ be as in the beginning of this section. Then for any
function $u \in W_{3}^{2,1}(Q)$ we also have $u \in H^{2/3,1/3}(Q)$
satisfying the estimate
\begin{equation}\label{imersaoHolder}
|u|_{Q}^{(2/3)} \le C \left\| u\right\| _{3,Q}^{(2)}\,.
\end{equation}
The constant $C>0$ depends only on T and $\Omega$.
\end{lemma}

In the following we will consider two auxiliary problems,
respectively related to the phase field  and the velocity
equations.

The first problem is
\begin{equation}\label{auxphase}
\begin{gathered}
 \frac{\partial \varphi}{\partial t}- \alpha \Delta
\varphi  = a \varphi + b \varphi^2 - \varphi^3 + g(x,t)
\quad \mbox{in } Q, \\
\frac{\partial \varphi}{\partial \eta } = 0 \quad \mbox{on }S,\\
\varphi(x,0) =  \varphi_0(x)  \quad \mbox{in }\Omega.
\end{gathered}
\end{equation}
 where $\alpha$ in a positive constant.
This problem  was treated by Hoffman and Jaing
\cite{Hoffman} when the initial date satisfies $\varphi_0  \in
W_{\infty}^2(\Omega)$. Since we will need an existence result for
$\varphi_0 \in W_{q}^{2-2/q}(\Omega) \cap
W_{2}^{3/2-\delta}(\Omega)$, with $\delta \in (0,1)$, we restate
the result of \cite{Hoffman}. We remark that exactly the same
proof presented in \cite{Hoffman} holds in this situation (see
also Vaz \cite{Vaz} for details, where some other specific results
concerning (\ref{auxphase}) are proved.)

\begin{proposition}\label{auxphasefield}
Let $\Omega$ and $Q$ be as in the beginning of this section.
Assume that $a(x,t)$ and $b(x,t)$ in $L^{\infty}(Q)$, $g\in
L^{q}(Q)$, $\varphi_0 \in W_{q}^{2-2/q}(\Omega)\cap
W_{2}^{3/2-\delta}(\Omega)$, where $q \geq 2$,  $\delta \in (0,1)$
and $\frac{\partial \varphi_0}{\partial\eta} = 0$ in
$\partial\Omega$. Then there exists an unique solution $\varphi
\in W_q^{2,1}(Q)$ of problem \mbox{(\ref{auxphase})}, which
satisfies the estimate
\begin{equation}\label{estimatephase}
\left\| \varphi \right\| _{q,Q}^{(2)} \leq C
\left(\left\|\varphi_0\right\|_{W_{q}^{2-2/q}(\Omega)} + \left\| g
\right\|_{q,Q}\right),
\end{equation}
 where C depends only on T, $\alpha$, $\Omega$,
$\left\|a(x,t)\right\|_{\infty,Q}$ and on
$\left\|b(x,t)\right\|_{\infty,Q}$.
\end{proposition}


The second auxiliary problem is
\begin{equation}\label{velocaux}
\begin{gathered}
 \frac{\partial v}{\partial t} - \nu \Delta
v+(v.\nabla)v+\nabla p+ k(x,t) v = f(x,t) \quad \mbox{in } Q, \\
\mathop{\rm div}v = 0 \quad \mbox{in } Q,\\
v = 0 \quad \mbox{on } S,\\
v(x,0) = v_0(x)\quad \mbox{in } \Omega.
\end{gathered}
\end{equation}

\begin{proposition}\label{auxvelocity}
Let $\Omega$ and $Q$ be as in the beginning of this section.
Assume that \,$k(x,t) \in C^0(Q)$, $k(x,t) \ge 0$, $f(x,t) \in
L^2(Q)^2$ and $v_0(x) \in H$. Then there exists an unique solution
$v(x,t) \in L^2(0,T; \mbox{V}) \cap L^\infty (0,T; \mbox{H})$
of problem \eqref{velocaux} which satisfies the estimate
\begin{equation}\label{estimateveloc}
\left\| v\right\| _{L^{\infty}(0,T,H)} + \left\| v\right\|
_{L^2(0,T,V)} \le C \big(\left\|v_0\right\|_{H} +
\left\|f\right\|_{2,Q}\big)
\end{equation}
Moreover, by interpolation results $v \in L^4(Q)^2$ and
\begin{equation}\label{velocL4}
\left\| v\right\| _{4,Q} \le C \big(\left\|v_0\right\|_{H} +
\left\|f\right\|_{2,Q}\big),
\end{equation}
 where $C$ depends only on $T$ and on \,$\Omega$.
\end{proposition}

The proof of Proposition \ref{auxvelocity} is done by using the same
arguments used in the classical theory of weak solutions of the Navier-Stokes
equations. As in this classical situation, the fact that the domain is two
dimensional is important to obtain uniqueness of solutions (see Temam
\cite{Temam}, p.282, for instance.)

\subsection{Technical Hypotheses and Generalized Solution}
\label{generalized solution}

For the rest of this article we will be using the following technical
hypotheses:
\begin{itemize}
\item[(H1)] $\Omega \subset \mathbb{R}^{2}$ is an open and bounded domain
with sufficiently smooth boundary $\partial \Omega$; T is a finite
positive number; $Q= \Omega \times (0,T)$.

\item[(H2)] $a(x,t), b(x,t)$ are given functions in $L^{\infty}(Q)$;
$f_{s} \in C_b^{1,1}(\mathbb{R})$, $0 \leq f_s(z) \leq 1$
for all $z \in \mathbb{R}$;
$k(y) \in C^{0}(-\infty,1)$, $k(0)= 0$, $k(y) = 0$ in $\mathbb{R}^{-}$,
$k(y)$ is nonnegative and
$\lim_{y \rightarrow 1}k(y)=+\infty$.

\item[(H3)] $v_0$ $\in \mbox{H}$;
$\theta_0 \in W_{2}^1(\Omega)$, $\theta_0 = 0$ on $\partial
\Omega$;
$\varphi_0\in W_{3}^{4/3}(\Omega)\cap W_{2}^{3/2+\delta}(\Omega)$,
for some $\delta \in (0,1)$, $ \frac{\partial \varphi_0}{\partial \eta }= 0 $ on
$\partial \Omega$.
\end{itemize}

Now  we  explain in what sense we will understand
a solution of (\ref{2Dpuromodel}), (\ref{2Dpurofront}),
(\ref{2Dpurodata}).

\begin{definition}\label{defpuro} \rm
By a {\bf generalized solution} of the problem
\mbox{(\ref{2Dpuromodel}), (\ref{2Dpurofront}),
(\ref{2Dpurodata})}, we mean a triple of functions $(\varphi,
\theta, v)$ such that $\varphi \in V_{2}(Q)$, $\theta \in
\overset{0}{V}{}_{2}(Q)$ and $v \in L^{2}(0,T; V)\cap
L^{\infty}(0,T; H)$. Moreover, being
\begin{gather*}
Q_s = \{(x,t) \in Q :  f_s (\varphi(x,t)) =1\},\\
\Omega_{s}(t) = \{x \in \Omega : f_{s}(\varphi(x,t))=1 \},\\
Q_{ml}= Q\setminus\bar{Q}_s,\\
\Omega_{ml}(0)=\Omega\setminus\bar{\Omega}_s(0)
\end{gather*}
 we have $v = 0$ a.e in $\overset{\circ}Q_{s}$, and
$\varphi$, $\theta$ and $v$ satisfy the integral relations
\begin{gather}\label{generalizedphase}
\begin{aligned}
 &-\int_{Q}\varphi\,\beta_{t}\,dx\,dt +
\alpha\int_{Q}\nabla\varphi \nabla\beta\,dx\, dt\\
&= \int_{Q}\left(a + b\varphi - \varphi^{2} \right)
\varphi\, \beta dx\,dt  +\int_{Q}\theta\beta\, dx\, dt +
\int_{\Omega}\varphi_0(x)\beta(x,0) dx\,,
\end{aligned}\\
\label{generalizedtemp}
\begin{aligned}
 &-\int_{Q}\theta \,\xi_{t} dx\,dt + \int_{Q} \nabla
\theta \nabla \xi dx\,dt + \int_{Q}v.\nabla\theta\,\xi\, dx\, dt \\
& = \int_{Q} \frac{\ell}{2} \frac{\partial f_{s}}
{\partial\varphi}(\varphi) \, \varphi_t \, \xi\,  dx\, dt +
\int_{\Omega}\theta_0(x)\xi(x,0) dx\,,
\end{aligned} \\
\label{generalizedveloc}
\begin{aligned}
& -\int_{Q_{ml}} v \,\phi_{t} \, dx\, dt
+\nu \int_{Q_{ml}} \nabla v \nabla \phi \, dx\, dt\\
&+\int_{Q_{ml}}(v.\nabla)v\,\phi \,dx\,dt
+\int_{Q_{ml}}k(f_{s}(\varphi)) v \, \phi \, dx\, dt\\
&=\int_{Q_{ml}}\vec{\sigma}\theta \phi \, dx\, dt
+\int_{\Omega_{ml}(0)} v_0(x) \phi(x,0) dx\,,
\end{aligned}
\end{gather}
for all $\beta$ in $W_{2}^{1,1}(Q)$ such that
$\beta(x,T)=0$; for all $\xi$ in
$\overset{0}{W}{}_{2}^{1,1}(Q)$ such that $\xi(x,T)=0$, and
for all $\phi \in C([0,T]; W_{2}^{1}(\Omega_{ml}(t)))$ such that
$\phi(. ,T) = 0$,  $\mathop{\rm div} \phi(. ,t) = 0$ for all
$t \in [0,T]$ and $\mathop{\rm supp}\phi(x,t) \subset Q_{ml} \cup
\Omega_{ml}(0)$.
\end{definition}

Note that due to our technical hypotheses and choice of functional
spaces,  all of the integrals in Definition \ref{defpuro} are well
defined.


            % Main Result
            %%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Existence of Generalized Solutions}
The purpose of this paper is to prove the following result

\begin{theorem}\label{2Dpuro}
Under the hypotheses (H1), (H2), (H3), there is a
generalized solution of the problem \mbox{(\ref{2Dpuromodel}),
(\ref{2Dpurofront}), (\ref{2Dpurodata})} in the sense of the
Definition \mbox{\ref{defpuro}}. Moreover, when $\varphi_0\in
W_{q}^{2-2/q}(\Omega) \cap W_{2}^{3/2+\delta}(\Omega)$ for some
$\delta\in$ (0,1) and $ q \ge 3$, and $\theta_0\in
W_{p}^{2-2/p}(\Omega)$ with $3 \leq p < 4$, then such solution
satisfies $ \varphi\in W_{q}^{2,1}(Q)\cap L^{\infty}(Q)$, $\theta
\in W_{p}^{2,1}(Q)\cap L^{\infty}(Q)$, $v\in L^{2}(0,T;V)\cap
L^{\infty}(0,T;H)$.
\end{theorem}

The proof of the previous result is long and will be done in the
following sections. Here we want just to sketch it: existence of a
solution of problem (\ref{2Dpuromodel}), (\ref{2Dpurofront}),
(\ref{2Dpurodata}) will proved by using a regularization technique
already used by Blanc \textit{et al} in \cite{Blanc}. The purpose
this regularization is to deal with the Navier-Stokes equations in
whole domain instead of unknown regions. Thus, the problem will be
adequately regularized with the help of a positive parameter, and
the existence of solutions for this regularized problem will
obtained by using the  Leray-Schauder degree theory (see Theorem
\ref{2Dpuroteo}). Then, as this parameter approaches zero, a
sequence of regularized solutions is obtained. With the help of
suitable estimates and compactness arguments, a limit of a
subsequence is then proved to exist and to be a solution of
problem (\ref{2Dpuromodel}), (\ref{2Dpurofront}),
(\ref{2Dpurodata}).

We also remark that the phase field equation admits classic solution when
$\varphi_0$ is sufficiently smooth. In fact, its right hand side term
satisfies $a\varphi+b\varphi^{2}-\varphi^{3} + \theta \in L^{\infty}(Q)$,
and, in particular when $\varphi_0\in W_{q}^{2-2/q}(\Omega)\cap
W_{2}^{3/2+\delta}(\Omega)$, with $q \geq 2$, we obtain a strong solution
with the equation  satisfied in the  a.e-sense. The boundary and initial
conditions are also satisfied in the pontual sense because $\varphi\in
C^{1}(Q)$. When  $\theta_0\in W_{p}^{2-2/p}(\Omega)$, with $3 \leq p < 4$,
the same sort of arguments applies and the solution is strong with $\theta
\in C^{0}(Q)$; the temperature equation and the boundary and initial
conditions are valid in point wise sense. Unfortunately, we are not able to
improve the regularity of the corresponding solution even if the initial
velocity is very regular. Thus, we only generalized solutions are obtained
for the velocity equation.

\section{Regularized Problem}\label{2Dregpuro}

In this section we regularize problem (\ref{2Dpuromodel}),
(\ref{2Dpurofront}), (\ref{2Dpurodata}) by changing the term
$k(f_s(\varphi))v$ in the velocity equation. We will obtain a
result of the existence, uniqueness and regularity for this
associated regularized problem.

\begin{theorem}\label{2Dpuroteo}
Fix $\varepsilon \in (0,1]$. Under the hypotheses $(H_1)$,
$(H_2)$, $(H_3)$, there exists an unique solution
$(\varphi_\varepsilon, v_\varepsilon, \theta _\varepsilon) \in
W_{3}^{2,1}(Q) \times (L^2(0,T;V)\cap L^\infty(0,T;H)) \times
W_{2}^{2,1}(Q) \subset L^6(Q) \times L^2(0,T;H) \times L^3(Q)$ of
the following problem:
\begin{equation}\label{2Dpuroreg}
\begin{gathered}
\frac{\partial \varphi_\varepsilon}{\partial t}
- \alpha \Delta\varphi_\varepsilon
= a \varphi_\varepsilon + b\varphi_\varepsilon^2 - \varphi_\varepsilon^3
+ \theta_\varepsilon, \\
\frac{\partial v_\varepsilon}{\partial t} - \nu \Delta v_\varepsilon
 + (v_\varepsilon .\nabla ) v_\varepsilon + \nabla p_\varepsilon +
k(f_s(\varphi_\varepsilon) - \varepsilon) v_{\varepsilon} =
\vec{\sigma}\theta_\varepsilon, \\
\mathop{\rm div} v_\varepsilon = 0, \\
\frac{\partial \theta_\varepsilon}{\partial t}
- \Delta\theta_\varepsilon + v_\varepsilon . \nabla \theta_\varepsilon
= \frac{\ell}{2} \frac{\partial f_{s}}{\partial \varphi}(\varphi_{\varepsilon})
\frac{\partial\varphi_{\varepsilon}}{\partial t}
\end{gathered}\end{equation}
in $Q$;
\begin{equation}\label{2Dpuroregfront}
\frac{\partial\varphi _{\varepsilon}}{\partial n}= 0, \quad
\theta_{\varepsilon }=0 ,\quad
v_\varepsilon = 0
\end{equation}
on $S$; and
\begin{equation}\label{2Dpuroregdata}
\varphi_{\varepsilon}(x,0)  =  \varphi_0(x),\quad
\theta _\varepsilon(x,0) =  \theta _0(x), \quad
v_\varepsilon(x,0)  =  v_0(x)
 \end{equation}
in $\Omega$.
Moreover, as $\varepsilon$ varies in $[0,1]$, such solutions
$(\varphi_\varepsilon, v_\varepsilon, \theta _\varepsilon)$ are
uniformly bounded with respect to $\varepsilon$ in $W_{3}^{2,1}(Q)
\times (L^2(0,T;V)\cap L^\infty(0,T;H)) \times W_{2}^{2,1}(Q)$.
\end{theorem}

This theorem will be proven the end of this
section, after some preparation and auxiliary lemmas. The
solvability of problem (\ref{2Dpuroreg}), (\ref{2Dpuroregfront}),
(\ref{2Dpuroregdata}) will be proved by applying the
Leray-Schauder degree theory (see Deimling \cite{Deimling}) as in
Morosanu and Motreanu \cite{Morosanu}. For this, we will
reformulate the  problem as $T(1,\varphi, v, \theta) = (\varphi,
v, \theta)$, where $T(\lambda, \cdot)$ is a compact homotopy
depending on a parameter $\lambda \in [0,1]$ to be described
shortly.

Basic tools in our argument are $L_p-$theory of parabolic
equations and Theorems \ref{auxphasefield} and \ref{auxvelocity}
in Section \ref{preliminary}. Moreover, we emphasize that the
regularity of solution of Navier-Stokes and phase field equations
plays an essential role in this proof. Such connection is strictly
related with a selection of the order of the equations in
quasilinear problem, mainly in deriving \textit{a priori
estimates} for possible solutions. Moreover, since that the phase
field has smooth solution (classical solution), the regularity of
Navier-Stokes equations becomes very important but this regularity
is governed by the additional Carman-Koseny type term
$k(f_s(\varphi))v$ that one not permits one to obtain uniform
estimate in some different as $L^{2}(0,T;\mbox{V})\cap L^{\infty}(0,T;H)$.

For simplicity of notation, we omit the subscript $\varepsilon $
in the rest of this section.

\begin{definition}\label{homotopydef} \rm
Define the homotopy $T: [0,1]\times L^6(Q)\times L^2(0,T;H)\times
L^3(Q) \rightarrow L^6(Q)\times L^2(0,T;H)\times L^3(Q)$ as
\begin{equation}\label{homotopy}
T(\lambda,\phi,u,\omega) = (\varphi, v, \theta)
\end{equation}
 where $(\varphi, v, \theta)$ is the unique solution of
the  quasilinear problem:
\begin{equation}\label{2Dpuroreglinear}
\begin{gathered}
 \frac{\partial \varphi}{\partial t} -\alpha\Delta\varphi
 = a \varphi + b \varphi^{2} - \varphi^{3} +\lambda\omega\,, \\
\frac{\partial v}{\partial t} - \nu \Delta v  + (v.\nabla ) v
+ \nabla p + k(f_s(\varphi) - \varepsilon) v
= \lambda \vec{\sigma}\omega \,,\\
\mathop{\rm div}v = 0\,,\\
\frac{\partial\theta}{\partial t} - \Delta\theta + v.\nabla \theta
= \lambda \frac{\ell}{2} \frac{\partial f_{s}}{\partial\varphi}(\varphi)
 \frac{\partial\varphi}{\partial t}
\end{gathered} \end{equation}
in $Q$;
\begin{equation}\label{2Dpuroreglinearfront}
 \frac{\partial \varphi}{\partial n}  =  0\,, \quad
\theta  =  0\,, \quad
v  =  0
\end{equation}
on $S$;  and
\begin{equation}\label{2Dpuroreglineardata}
\varphi(x,0) = \varphi_0(x)\,,\quad
\theta(x,0)=\theta_0(x)\,,\quad
v(x,0)  =  v_0(x)
\end{equation}
in $\Omega$.
\end{definition}


We observe that the homotopy $T(\lambda, \cdot)$ is well defined.
In fact, for fixed $\lambda \in [0,1]$, by using Proposition
\ref{auxphasefield}  and Lemma \ref{imbeddingLp}, we conclude that
first equation of problem (\ref{2Dpuroreglinear}),
(\ref{2Dpuroreglinearfront}), (\ref{2Dpuroreglineardata}) has a
unique solution $\varphi\in W_{3}^{2,1}(Q)\cap L^{\infty}(Q)$.
Once $\varphi$ is known, Proposition \ref{auxvelocity} implies
that the modified Navier-Stokes equations has an unique solution
$v \in L^{2}(0,T;V)\cap L^{\infty}(0,T; H)$. By usual
interpolation, it results that $v\in L^4(Q)^{2}$. Now that
$\varphi$ and $v$ are known, the $L_{p}-$theory of parabolic
equations , that also is valid for Neumann boundary condition (see
Ladyzenskaja \emph{et al} \cite[p.351]{Ladyzenskaja}), Lemma
\ref{imbeddingLp} and the facts that
$ \frac{\partial\varphi}{\partial t} \in L^3(Q)$,
$v \in L^{4}(Q)^{2}$, and $f_s \in C^{1,1}_b (\mathbb{R})$ imply that there is a
unique solution $\theta\in W_{3}^{2,1}(Q)\cap L^{\infty }(Q)$ for
the third equation of (\ref{2Dpuroreglinear}).

\begin{lemma}\label{compact}
Under assumptions (H1), (H2), (H3), the mapping $T:
[0,1]\times L^6(Q)\times L^2(0,T;H)\times L^3(Q) \rightarrow
L^6(Q)\times L^2(0,T;H)\times L^3(Q)$ is a compact mapping, i.e,
it is continuous and maps bounded sets into relatively compact
sets.
\end{lemma}

\begin{proof}  Let us check the continuity of
$T(\lambda,.)$. For this, let $\lambda_n \rightarrow \lambda$ in
[0,1] and $(\phi_n, u_n,\omega_n) \rightarrow (\phi,u,\omega)$ in
$L^6(Q)\times L^2(0,T;\mbox{H})\times L^3(Q)$. Denoting
$T(\lambda_n,\phi_n,u_n,\omega_n)$ = $(\varphi_n,v_n,\theta_n)$,
from (\ref{homotopy}), we write
\begin{gather}\label{2Dpurophasecompact}
 \frac{\partial \varphi_n}{\partial t} - \alpha \Delta \varphi_n
=  a \varphi_n + b\varphi_n^{2}- \varphi_n^{3} +\lambda_n\omega_n \,,
\\
 \frac{\partial v_n}{\partial t} - \nu \Delta v_n +
(v_n.\nabla)v_n + \nabla p_n + k(f_{s}(\varphi_n)-\varepsilon) v_n
=\lambda_n\vec{\sigma}\omega_n \,,
\label{2Dpuroveloccompact}\\
\mathop{\rm div}\,v_n = 0\,,  \label{div} \\
\frac{\partial \theta_n}{\partial t} - \Delta
\theta_n + v_n.\nabla \theta_n
=  \lambda_n \frac{\ell}{2}\frac{\partial f_s}{\partial\varphi} (\varphi_n)
\frac{\partial \varphi_n}{\partial t} \label{2Dpuroheatcompact}
\end{gather}
in $Q$;
\begin{equation} \label{2Dpurofrontcompact}
 \frac{\partial\varphi_n}{\partial \eta}=0\,,\quad
v_n = 0 \,,\quad
\theta_n  = 0\,,
\end{equation}
on $S$; and
\begin{equation} \label{2Dpuroinitialcompact}
\varphi_n(x,0)=\varphi_0(x)\,, \quad
v_n(x,0)  =  v_0(x)\,, \quad
\theta_n(x,0)=\theta_0(x)
 \end{equation}
in $\Omega$.
By applying Proposition \ref{auxphasefield} with $\omega_n \in
L^2(Q)$, we obtain the following estimate for the phase-field
equation (\ref{2Dpurophasecompact})
\begin{equation}\label{2Dpurophasecomp}
\| \varphi_n \|_{2,Q}^{(2)} \leq C \big( |\lambda_n|
\left\| \omega_n \right\|_{2,Q} + \left\| \varphi_0
\right\|_{W_{2}^{1}(\Omega)} \big)
\end{equation}
Now, by applying Proposition \ref{auxvelocity}, we obtain the
following estimates for the velocity equation
(\ref{2Dpuroveloccompact})
\begin{equation}\label{2Dpuroveloccomp}
\| v_n\| _{L^{\infty}(0,T,H)} + \| v_n\|_{L^2(0,T,V)}
\le C \big(\|v_0\|_{H} +|\lambda_n|\|\omega_n\|_{2,Q}\big),
\end{equation}
which by usual interpolation implies
\begin{equation}\label{2DpuroveloccompL4}
\| v_n\| _{4,Q} \le C \big(\|v_0\|_{H} + |\lambda_n|\|\omega_n\|_{2,Q}\big)\,.
\end{equation}
For (\ref{2Dpuroheatcompact}), the $L_p$-theory of the parabolic
equation (see Ladyzenskaja \cite[p.\ 351]{Ladyzenskaja}) with the
facts that
$ \frac{\partial\varphi_n}{\partial t} \in L^2(Q)$,
$ \frac{\partial f_s}{\partial\varphi}(\varphi_n) \in L^{\infty}(Q)$,
$v_n \in L^4(Q)^2$ and
$\theta_0 \in W_2^1(\Omega)$ provides the estimate
\begin{equation}
\label{2Dpuroheatcomp}
\|\theta_n \|_{2,Q}^{(2)} \leq C \Big(\|v_n\|_{4,Q}\|\theta_0\|_{W_2^1(\Omega)}
+ |\lambda_n| \big\| \frac{\partial\varphi_n}{\partial t} \big\|_{2,Q}
+ \| \theta_0 \|_{W_{2}^{1}(\Omega)} \Big)\,.
\end{equation}
Since the sequences $(\omega_n)$ and $(\lambda_n)$ are
respectively bounded in $L^2(Q)$ and [0,1], from
(\ref{2Dpurophasecomp}) and (\ref{2Dpuroveloccomp}) we obtain for
all $n$ that
\begin{equation}
\label{2Dpurophaseveloccomp}
\left\| \varphi_n \right\|_{2,Q}^{(2)} \leq C\quad
\mbox{and}\quad  \left\| v_n\right\| _{L^{\infty}(0,T,H)} +
\left\| v_n\right\| _{L^2(0,T,V)} \le C\,.
\end{equation}
Consequently, from (\ref{2Dpuroheatcomp}) we have for all $n$
\begin{equation}
\label{2Dpurotempcomp}
\left\| \theta_n \right\|_{2,Q}^{(2)} \leq C\,.
\end{equation}
 From (\ref{2Dpurophaseveloccomp}) and (\ref{2Dpurotempcomp}),
it follows
$\{T(\lambda_n, \phi_n, u_n, \omega_n)\}= \{(\varphi_n, v_n, \theta_n)\}$
is uniformly bounded sequence with respect to $n$ in the functional space
$W_2^{2,1}(Q)\times( L^2(0,T;V)\cap L^{\infty}(0,T;H)) \times
W_2^{2,1}(Q)$.
Moreover, we observe that for fixed $\varepsilon \in (0,1]$, from
the properties of $k(y)$ (see the conditions stated in $(H_2)$),
there is a finite positive constant $C$ depending only on
$\varepsilon$ such that ${\rm sup} \{ k(y - \varepsilon)\} \leq
C$. By using this and our previous estimates as in Lions
\cite[p.\ 71]{Lions}, we conclude that for all $n$,
\begin{equation}\label{regular_v_nt}
\|(v_n)_t\|_{L^2(0,T; V')} \leq C(\varepsilon).
\end{equation}

Thus, the previous estimates, and the Aubin-Lions Lemma
(see Temam \cite{Temam} or Lions \cite{Lions}), allow us to select
a subsequence, which we denote $\{T(\lambda_k, \phi_k, u_k,
\omega_k)\}= \{(\varphi_k, v_k, \theta_k)\}$ such that
\begin{gather}
\varphi_k  \rightharpoonup \varphi   \mbox{in }   W_{2}^{2,1}(Q)
         \label{2Dpurophasefracocomp} \\
v_k \rightharpoonup v \quad \mbox{in }  L^2(0,T; V)
         \label{2Dpurovelocfracocomp}\\
v_k\stackrel{\ast}{\rightharpoonup} v  \quad \mbox{in }
L^{\infty}(0,T;H)\\ \theta_k \rightharpoonup \theta \quad \mbox{in } W_{2}^{2,1}(Q)
         \label{2Dpuroheatfracocomp} \\
(v_k)_t \rightharpoonup v_t \quad \mbox{in }  L^2(0,T; V')
         \label{2Dpurovelocderivfracocomp}\\
\varphi_k \rightarrow \varphi \quad \mbox{in }  L^{6}(Q)
         \label{2Dpurophasefortecomp}\\
v_k \rightarrow v  \quad \mbox{in }  L^2(0,T;H)
         \label{2Dpurovelocfortecomp}\\
\theta_k \rightarrow \theta \quad \mbox{in }  L^{3}(Q)
         \label{2Dpuroheatfortecomp}
\end{gather}

Now, let us verify that $T(\lambda,\phi,u,\omega) =
(\varphi,v,\theta)$, in other words, that $(\varphi,v,\theta)$ is
solution of (\ref{2Dpuroreglinear}),
(\ref{2Dpuroreglinearfront}),(\ref{2Dpuroreglineardata}). For
this, we are going to pass to the limit with respect to the above
subsequence in equations
(\ref{2Dpurophasecompact})-(\ref{2Dpuroheatcompact}) together with
the conditions
(\ref{2Dpurofrontcompact})-(\ref{2Dpuroinitialcompact}).

Let us prove that the equations are satisfied in the sense
distribution. For this, fix in the sequel  $g \in
C^{\infty}_c(Q)$, and let us describe the process of taking the
limit only for those terms of the equations that are neither
trivial nor standard.
We observe that by using (\ref{2Dpurophasefortecomp}) and
$\lambda_k \rightarrow \lambda$, we obtain
\begin{equation}\label{limitphasecomp}
\int_Q \lambda_k (a\varphi_k +b\varphi_k^2-\varphi_k^3)g\,dx\,dt
\rightarrow \int_Q \lambda (a\varphi +b\varphi^2-\varphi^3)g\,dx\,dt
\hspace{.5in}\forall\,g
\end{equation}

Thus, passing to the limit in phase field equation
(\ref{2Dpurophasecompact}), using the convergence
(\ref{2Dpurophasefracocomp}), (\ref{2Dpurophasefortecomp}) and
(\ref{limitphasecomp}), we obtain the first equation in
(\ref{2Dpuroreglinear}).

To verify the convergence
\begin{equation}\label{limitveloccomp}
\int_Q k(f_s(\varphi_k) - \varepsilon) v_k\, g\,dx\,dt \rightarrow
\int_Q k(f_s(\varphi) - \varepsilon) v\, g \, dx\,dt
\end{equation}
we use (\ref{2Dpurovelocfortecomp}), the fact that for
fixed $\varepsilon \in (0,1]$, $k(f_s(\cdot) - \varepsilon$ is
bounded, and following argument. Consider $h_k =|k(f_s(\varphi_k)
-\varepsilon) - k(f_s(\varphi)-\varepsilon)|^2$. Since
$k(f_s(\cdot) -\varepsilon)$ is continuous and
(\ref{2Dpurophasefortecomp}) is valid, passing to a subsequence if
necessary, we know that $h_k \rightarrow 0$ almost everywhere in
Q. Moreover, $|h_k| \leq C \left\|f_s(\varphi)\right\|_{\infty}^2$
a.e and therefore $h_k \rightarrow 0$ in $L^1(Q)$ by Lebesgue
dominated convergence theorem. Thus, $k(f_s(\varphi_k) -
\varepsilon) \rightarrow k(f_s(\varphi)- \varepsilon)$ in
$L^2(Q)$, what together with (\ref{2Dpurovelocfracocomp}) implies
(\ref{limitveloccomp}).
By passing to the limit in velocity equation
(\ref{2Dpuroveloccompact}), using the convergence
(\ref{2Dpurovelocfracocomp}), (\ref{2Dpurovelocfortecomp}) and
(\ref{limitveloccomp}) we obtain the second equation in
(\ref{2Dpuroreglinear}).

Now, we use (\ref{2Dpurophasefracocomp}),
(\ref{2Dpurophasefortecomp}), $\lambda_k \rightarrow \lambda$ and
arguments similar to the ones previously with
$ \frac{\partial f_s}{\partial\varphi}(\varphi_k)$
in place of $f_s$ to obtain
\begin{equation}
\label{limitheatcomp}
\int_Q \lambda_k\, \frac{\partial f_s}{\partial\varphi}(\varphi_k)
\frac{\partial\varphi_k}{\partial t}\, g\,dx\,dt
\rightarrow
\int_Q  \lambda\, \frac{\partial f_s}{\partial\varphi}(\varphi)
\frac{\partial\varphi}{\partial t}\, g\,dx\,dt
\end{equation}
By passing to the limit in temperature equation
(\ref{2Dpuroheatcompact}), using the convergence
(\ref{2Dpuroheatfracocomp}), (\ref{2Dpuroheatfortecomp}) and
(\ref{limitheatcomp}), we obtain the third equation in
(\ref{2Dpuroreglinear})

The required boundary conditions are included in the definitions
of the functional spaces where $(\varphi,v,\theta)$ is in. Also,
with the estimates we have obtained, it is standard to prove that
$\varphi$, $v$ and $\theta$ satisfy the required initial
conditions. Hence, $(\varphi,v,\theta)$ is solution of
(\ref{2Dpurophasecompact})-(\ref{2Dpuroinitialcompact}).

Note that for any given subsequence of
$\{T(\lambda_n,\varphi_n, v_n,\theta_n)\}$,  the above
arguments can be applied to conclude that this subsequence admits another
subsequence converging to a solution of
(\ref{2Dpurophasecompact})-(\ref{2Dpuroinitialcompact}). Since $(\phi, u,
\omega)$ is also fixed and the solution of this last problem is unique, we
conclude that $\{T(\lambda_n,\varphi_n, v_n,\theta_n)\}$ is a sequence  with
the property that any one of its subsequences has by its turn a subsequence
converging to a limit that is independent of the chosen subsequence. Hence,
$\{T(\lambda_n,\varphi_n, v_n,\theta_n)\}$ converges to this limit, and the
continuity of $T$ is proved.\medskip

The same arguments prove that mapping $T$ is a compact
mapping. In fact, if  $\{(\phi_n, u_n,\omega_n)\}$ is any bounded
sequence in $L^6(Q)\times ( L^2(0,T;V)\cap L^{\infty}(0,T;H)
)\times L^3(Q)$, the above arguments can be applied to obtain
exactly the same sort of estimates for
$T(\lambda_n,\phi_n,u_n,\omega_n)$. These imply that
$\{(\varphi_n,v_n,\theta_n)\}$ is relatively compact in
$L^6(Q)\times L^2(0,T;H)\times L^3(Q)$, and thus there exists a
subsequence of $T(\lambda_n,\phi_n,u_n,\omega_n)$ converging in
$L^6(Q)\times L^2(0,T;H)\times L^3(Q)$. Therefore, the compactness
is proved.
\end{proof}

The next lemma give us an uniform estimate for any
possible fix point of $T(\lambda,\cdot)$.

\begin{lemma}\label{fixedpoint}
Under assumptions (H1), (H2), (H3), there exists a
positive number $\rho$, depending only on the given data of the
problem and in particular independent of $\lambda \in [0,1]$, with
the property any fix point of $T(\lambda,.)$ is in the interior of
the ball of radius $\rho$ in $L^6(Q)\times L^2(0,T;H)\times L^3(Q)$.
That is,
\begin{equation}\label{normfixed}
T(\lambda, \varphi, v, \theta) = (\varphi, v, \theta) \Rightarrow
\left\|(\varphi, v, \theta) \right\| < \rho ,
\end{equation}
 where $\|\cdot\|$ denotes the norm in
$L^6(Q)\times L^2(0,T;H)\times L^3(Q)$.
\end{lemma}

\begin{proof} Using (\ref{homotopy}), the condition
$T(\lambda, \varphi, v, \theta) = (\varphi, v, \theta)$ is
equivalent to
\begin{gather}
\frac{\partial \varphi}{\partial t} - \alpha \Delta \varphi
=a \varphi + b\varphi^{2}- \varphi^{3} + \lambda \theta,\label{2Dpurophasefixed}
\\
\frac{\partial v}{\partial t} - \nu \Delta v + (v.\nabla)v
+ \nabla p + k(f_{s}(\varphi)-\varepsilon) v
= \lambda\vec{\sigma}\theta, \label{2Dpurovelocfixed}
\\
\mathop{\rm div} v=0, \label{2Dpurodivfixed} \\
\frac{\partial\theta}{\partial t} - \Delta\theta + v.\nabla \theta
= \lambda \frac{\ell}{2} \frac{\partial f_s}{\partial\varphi}(\varphi)
\frac{\partial \varphi}{\partial t}  \label{2Dpuroheatfixed}
\end{gather}
in  $Q$;
\begin{equation}\label{2Dpurofrontfixed}
\frac{\partial\varphi}{\partial\eta}=0,\quad
\theta =0, \quad
v = 0
\end{equation}
on $S$;
\begin{equation}\label{2Dpuroinitialfixed}
\varphi(x,0)=\varphi_0(x), \quad
\theta(x,0)=\theta_0(x), \quad
v(x,0)=v_0(x)
\end{equation}
in $\Omega$.
To obtain estimates for $(\varphi, v, \theta)$, we start by
multiplying the first equation (\ref{2Dpurophasefixed}) by
$\varphi$. After integrating of the result over $Q_t$
($t \in(0,T]$), using Fubini's theorem, Green's formula and Young's
inequality, we get
\begin{equation}\label{estimatephase1}
\begin{aligned}
&\int_{\Omega }\varphi ^{2} dx+ \int_0^{t}\int_{\Omega } |\nabla\varphi| ^{2}
dx\,dt +\frac{\lambda }{2} \int_0^{t}\int_{\Omega}\varphi^{4} dx\,dt\\
&\leq C\Big(\left\|\varphi_0 \right\|_{2,\Omega}^{2}
+ \int_0^{T} \int_{\Omega} |\theta|^{2} dx\,dt
+ \int_0^{T} \int_{\Omega}|\varphi| ^{2} dx\,dt\Big),
\end{aligned}
\end{equation}
 where $C$ depends on $\alpha$ and
${\max_{(x,t)\in Q} \Big(a(x,t)+ b(x,t)s -\frac{1}{2}s^{2}\Big)}$.
Applying Gronwall's inequality in (\ref{estimatephase1}), we get
\begin{equation}\label{estimatephase2}
\int_0^{T}\int_{\Omega}|\varphi|^{2} dx\,dt
\le C\Big(\|\varphi _0\|_{2,\Omega}^{2}+\int_0^{T}
\int_{\Omega }|\theta|^{2} dx\,dt\Big).
\end{equation}
Thus, combining (\ref{estimatephase1}) and (\ref{estimatephase2}), we conclude
\begin{gather}
\label{estimatephase3}
\int_0^{T}\int_{\Omega}|\nabla\varphi|^{2} dx\,dt
\leq C \Big(\left\|\varphi _0\right\|_{2,\Omega}^{2}+\int_0^{T}
\int_{\Omega }|\theta \|^{2} dx\,dt\Big) \\
\label{estimatephase4}
\frac{\lambda}{2}\int_0^{T}\int_{\Omega } \varphi^4 dx\,dt
\leq C \Big(\left\|\varphi _0\right\|_{2,\Omega}^{2}
+ \int_0^{T} \int_{\Omega }|\theta|^{2} dx\,dt\Big)
\end{gather}
Now, we multiply equation (\ref{2Dpuroheatfixed}) by
$\theta$ and integrate over $Q_t$. Then we use  the fact that
$ \frac{\partial f_s}{\partial \varphi} \in L^{\infty}(\mathbb{R})$,
(\ref{2Dpurodivfixed}), Green's formula and also Poincar\'e's
and Young's inequalities to obtain
\begin{equation}
\label{estimateheat1}
\int_{\Omega} \theta ^{2} dx
+ \int_0^{t} \int_{\Omega}\left| \nabla \theta \right|^{2} dx\,dt
\leq
C\Big(\left\|\theta _0\right\| _{2,\Omega}^{2}
+ \int_0^{T}\int_{\Omega}|\varphi_t|^2\,dx\,dt\Big),
\end{equation}
 where $C$ depends on $\Omega$, $\ell$ and
$\| \frac{\partial f_s}{\partial\varphi}\|_{\infty}$.

Multiplying the first equation (\ref{2Dpurophasefixed}) by
$ \frac{\partial\varphi}{\partial t}$, integrating over $Q_t$,
using Green's formula and Young's inequality, we obtain
\begin{equation}
\label{estimatephase5}
\begin{aligned}
&\int_0^{t}\int_{\Omega}\big(\frac{\partial\varphi}{\partial t}\big)^{2} dx\,dt
+ \int_0^{t}\int_{\Omega}| \nabla\varphi|^{2} dx\,dt\\
&\leq
C\Big(\left\|\nabla \varphi _0\right\|_{2,\Omega}^{2}
+ \int_0^{T}\int_{\Omega}|\varphi|^2 dx\,dt
+ \int_0^{T}\int_{\Omega}|\theta|^2 dx\,dt \Big),
\end{aligned}
\end{equation}
where $C$ depends on $\alpha$ and
$ {\max_{(x,t)\in Q} \left(a(x,t)+ b(x,t)s - s^{2}\right)}$.
Using (\ref{estimatephase2}) in (\ref{estimatephase5}) and
applying the resulting  estimate in (\ref{estimateheat1}), we get
\begin{equation}\label{estimateheat2}
\int_{\Omega} \theta ^{2} dx+\int_0^{t} \int_{\Omega}\left| \nabla \theta \right|^{2} dx\,dt
\leq
C \Big(\left\|\theta_0\right\| _{2,\Omega}^{2}
+ \left\|\varphi _0\right\|_{W_2^1(\Omega)}^{2}
+ \int_0^{T}\int_{\Omega}|\theta|^2\,dx\,dt\Big)
\end{equation}
Applying Gronwall inequality in (\ref{estimateheat2}), we
obtain
\begin{equation}
\label{estimateheat3}
\|\theta\|_{2,Q}
\leq C \Big( \left\| \theta_0 \right\|_{2,\Omega}
+\left\| \varphi_0 \right\|_{W_{2}^{1}(\Omega)} \Big),
\end{equation}
and, consequently,
$\|\nabla\theta\|_{2,Q} \leq C \big( \left\| \theta_0 \right\|_{2,\Omega}
+ \left\| \varphi_0 \right\|_{W_{2}^{1}(\Omega)} \big)$.
Moreover, by interpolation results (see Ladyzenskaja
\cite[p.\ 74]{Ladyzenskaja}), we have
\begin{equation}
\label{estimateheat4}
\|\theta\|_{4,Q} \leq  M \left( \left\| \theta_0
\right\|_{2,\Omega} +\left\| \varphi_0
\right\|_{W_{2}^{1}(\Omega)} \right).
\end{equation}
Using (\ref{estimateheat3}) in (\ref{estimatephase2}),
(\ref{estimatephase3}) and (\ref{estimatephase4}), we conclude
that
\begin{gather}
\left\| \varphi \right\|_{2,Q} \leq  C \left( \left\| \theta_0
\right\|_{2,\Omega} +\left\| \varphi_0
\right\|_{W_{2}^{1}(\Omega)} \right), \label{estimatephase7}\\
\left\|\nabla\varphi\right\|_{2,Q}
\leq C \left( \left\| \theta_0 \right\|_{2,\Omega}
+ \left\| \varphi_0 \right\|_{W_{2}^{1}(\Omega)} \right), \label{estimatephase8}
\\
\lambda\left\| \varphi \right\|_{4,Q}
\leq C \left( \left\| \theta_0 \right\|_{2,\Omega}
+ \left\| \varphi_0 \right\|_{W_{2}^{1}(\Omega)} \right). \label{estimatephase9}
\end{gather}
Using (\ref{estimateheat3}) and (\ref{estimatephase7}) in
(\ref{estimatephase5}), we have
\begin{equation}
\label{estimatephase10}
\| \frac{\partial\varphi}{\partial t}\|_{2,Q}
\leq
C \left( \left\| \theta_0 \right\|_{2,\Omega}
+\left\| \varphi_0 \right\|_{W_{2}^{1}(\Omega)}\right).
\end{equation}
Multiplying the first equation (\ref{2Dpurophasefixed})
equation by $-\Delta \varphi $ integrating over $Q_t$, using Green
formula and Young inequalities, we obtain
\begin{equation}\label{estimatephase11}
\begin{aligned}
& \int_{\Omega }|\nabla \varphi| ^{2} dx+
\int_0^{t}\int_{\Omega }|\Delta \varphi|^{2} dx\,dt +
3\lambda\int_0^{t}\int_{\Omega }\varphi^{2}|\nabla \varphi|^{2} dx\,dt
\\
&\leq C \Big(\left\|\nabla \varphi _0\right\|_{2,\Omega }^{2}
+ \int_0^{T}\int_{\Omega}|\theta|^{2} dx\,dt+\int_0^{T} \int_{\Omega}|\varphi|^{2} dx\,dt
+ \lambda\int_0^{T}\int_{\Omega}| \varphi|^{4} dx\,dt \Big),
\end{aligned}
\end{equation}
where $C$ depends on $\Omega$, $\alpha$,
$\left\|a\right\|_{\infty,Q}$, $\left\|b\right\|_{\infty,Q}$ and
$ \| \frac{\partial f_s}{\partial \varphi}\|_{\infty}$.
Using (\ref{estimateheat3}), (\ref{estimatephase7}) and
(\ref{estimatephase9}) in (\ref{estimatephase11}), we obtain
\begin{equation}\label{estimatephase12}
\left\|\Delta\varphi \right\|_{2,Q} \le  C \left( \left\|
\theta_0 \right\|_{2,\Omega} +\left\| \varphi_0
\right\|_{W_{2}^{1}(\Omega)} \right)
\end{equation}
Combining estimates (\ref{estimatephase7}),
(\ref{estimatephase8}), (\ref{estimatephase10}) and
(\ref{estimatephase12}), using the imbedding in Lemma
\ref{imbeddingLp}, we have
\begin{equation}
\label{estimatephase13}
 \left\| \varphi \right\|_{p,Q} \le C \left\|
\varphi \right\|_{2,Q}^{(2)} \le C \left( \left\| \theta_0
\right\|_{2,\Omega} + \left\| \varphi_0
\right\|_{W_{2}^{1}(\Omega)} \right)\quad (p \geq 6).
\end{equation}

Now, by multiplying the second equation (\ref{2Dpurovelocfixed})
by $v$, integrating over $Q_t$, using Green's formula, and Poncar\'e's
and Young's inequalities, we get
\begin{equation}
\label{estimateveloc1}
\begin{aligned}
&\frac{1}{2}\int_{\Omega }v^{2} dx
+ \frac{\nu }{2}\int_0^{t}\int_{\Omega }\left| \nabla v\right| ^{2} dx\,dt
+ \int_0^{t}\int_{\Omega }k\left( f_{s}\left( \varphi \right) -\varepsilon \right)v^{2} dx\,dt
\\
&\leq C \Big( \left\|v_0\right\|_{\mathrm{H}}
+ \int_0^{T}\int_{\Omega} \left|\theta \right|^{2} dx\,dt \Big).
\end{aligned}
\end{equation}
Combining (\ref{estimateheat3}) and (\ref{estimateveloc1}), using
that $k(f_s(\varphi)- \epsilon) \geq 0$, we conclude that
\[
\left\|v\right\|_{L^{\infty}(0,T;\mathrm{H})}+
\left\|v\right\|_{L^{2}(0,T;\mathrm{V})} \le C
\left(\left\|v_0\right\|_{\mathrm{H}}+\left\|\theta
_0\right\|_{2,\Omega}+
\left\|\varphi_0\right\|_{W_{1}^{2}(\Omega)}\right).
\]
Finally, by the interpolation result given in Theorem
(\ref{auxvelocity}),
\begin{equation}\label{estimateveloc2}
\left\|v\right\|_{4,Q} \le C
\left(\left\|v_0\right\|_{\mathrm{H}}+\left\|\theta_0\right\|_{2,\Omega}+
\left\|\varphi_0\right\|_{W_{1}^{2}(\Omega)}\right).
\end{equation}
\end{proof}

The next lemma tell us that there is an unique fix point in the
special case $\lambda=0$.

\begin{lemma}\label{lambdazero}
Under assumptions (H1), (H2), (H3), there exists an
unique solution of the problem $T(0,\varphi, v, \theta) =
(\varphi, v, \theta)$ ($T$ defined in \eqref{homotopy}).
\end{lemma}

\begin{proof} The equation $T(0,\varphi, v, \theta) = (\varphi, v,\theta)$
is equivalent to the nonlinear system
\begin{gather*}
\frac{\partial \varphi}{\partial t}-\alpha \Delta\varphi
= a \varphi + b \varphi^{2} - \varphi^{3},\\
\frac{\partial v}{\partial t} - \nu \Delta v
 + (v.\nabla ) v + \nabla p + k(f_s(\varphi) - \varepsilon) v = 0, \\
\mathop{\rm div}v = 0, \\
\frac{\partial\theta}{\partial t} - \Delta\theta + v.\nabla \theta = 0
\end{gather*}
in $Q$;
\[
\frac{\partial \varphi}{\partial \eta}= 0, \quad  \theta =0, \quad v= 0
\]
on $S$;
\[
\varphi(x,0) = \varphi_0(x),\quad  \theta(x,0)=\theta_0(x),\quad \\
v(x,0)= v_0(x)
\]
in $\Omega$.
For these equations, Proposition  \ref{auxphasefield} ensures the
existence and uniqueness of $\varphi$; then Proposition
\ref{auxvelocity} gives the existence and uniqueness $v$. The
$L_p-$theory of the linear parabolic equations ensures then the
existence and uniqueness of $\theta$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{2Dpuroteo}]
According to Lemma \ref{fixedpoint}, there exists a number
$\rho$ satisfying (\ref{normfixed}). Let us consider the
open ball
\[
B_{\rho} = \left\{ (\varphi, v, \theta) \in L^6(Q)\times
L^2(0,T;H)\times L^3(Q) : \left\|(\varphi, v,
\theta)\right\| < \rho\right\},
\]
where $\|\cdot\|$ is the norm in the space
$L^6(Q)\times L^2(0,T;H)\times L^3(Q)$.
Lemma \ref{compact} ensures that the mapping $T : [0,1]\times
L^6(Q)\times L^2(0,T;\mbox{H})\times L^3(Q) \rightarrow
L^6(Q)\times L^2(0,T;\mbox{H})\times L^3(Q)$ is a homotopy of
compact transformations on the closed ball $\overline{B}_{\rho}$
and Lemma \ref{fixedpoint} implies that
\[
T(\lambda, \varphi, v, \theta) \ne (\varphi, v, \theta)\quad
\forall (\varphi,v,\theta) \in \partial B_{\rho},
\; \forall\lambda \in [0,1]
\]
The foregoing properties allow us consider the Leray-Schauder
degree $D(Id - T(\lambda, ^.), B_{\rho}, 0),\;\forall\lambda \in
[0,1]$ (see Deimling \cite{Deimling}). The homotopy invariance of
Leray-Schauder degree shows that the equality below holds
\begin{equation}\label{degree}
D(Id - T(0, ^.), B_{\rho}, 0) = D(Id - T(1, ^.), B_{\rho}, 0)
\end{equation}
Moreover, the Lemma \ref{lambdazero} ensures that the problem
$T(0,\varphi, v, \theta) = (\varphi, v, \theta)$ has a unique
solution in $L^6(Q)\times L^2(0,T;\mbox{H})\times L^3(Q)$. Hence
we can choose a sufficiently large $\rho > 0$ such that the ball $
B_{\rho}$ contains this solution, it turns out that $D(Id - T(0,
^.), B_{\rho}, 0) = 1$. Then relation (\ref{degree}) ensures that
the equation $T(1, \varphi, v, \theta) - (\varphi, v, \theta) = 0$
has a solution $(\varphi, v, \theta)\in B_{\rho}\subset
L^6(Q)\times L^2(0,T;\mbox{H})\times L^3(Q)$. By
(\ref{homotopy}) with $\lambda = 1$, this is just a solution of
the problem
(\ref{2Dpuroreg})-(\ref{2Dpuroregfront})-(\ref{2Dpuroregdata}).

The uniqueness and regularity of problem (\ref{2Dpuroreg}),
(\ref{2Dpuroregfront}), (\ref{2Dpuroregdata}) are consequence of
the application of the Propositions \ref{auxphasefield} and
\ref{auxvelocity} and $L_p$-regularity theory for  linear parabolic
equations. To prove uniqueness let $\varphi_i$, $v_i$
and $\theta_i$ with {\small i = 1, 2} be two solutions of problem
(\ref{2Dpuroreg}), (\ref{2Dpuroregfront}), (\ref{2Dpuroregdata}), with
corresponding pressures $p_i$
(for simplicity of exposition, we omit the subscript $\varepsilon$).
We first observe that by using the previously obtained estimates and
arguments similar to the ones used to prove that $T_\lambda$
is well defined (Definition \ref{homotopydef}), we conclude that
$\varphi_i \in W_3^{2,1}(Q) \cap L^{\infty}(Q)$,
$v_i \in  L^{2}(0,T;V) \cap L^{\infty}(0,T;H)$ and
$\theta \in W_2^{2,1}(Q) \cap L^p(Q)$ (for any finite $p \geq 1$).

Let
$\tilde{\varphi} = \varphi_1 - \varphi_2$,
$\tilde{v} = v_1 - v_2$,
$\tilde{\theta} = \theta_1 - \theta_2$ and
$\tilde{p} = p_1 -p_2$. These functions satisfy
the following conditions:
\begin{gather}
 \frac{\partial \tilde{\varphi}}{\partial t}
- \alpha \Delta \tilde{\varphi} =
[a(x,t)+ b(x,t)(\varphi_{1}+\varphi_{2})-(\varphi_{1}^{2}
+\varphi_{1}\varphi_{2}+\varphi_{2}^{2})] \tilde{\varphi}
+ \tilde{\theta}, \label{2Dpurophaseuniq}
\\
\begin{aligned}
&\frac{\partial \tilde{v}}{\partial t}
- \nu \Delta \tilde{v} + (v_1 . \nabla) \tilde{v} + \nabla \tilde{p} +
k(f_{s}(\varphi_1)-\varepsilon) \; \tilde{v}  \\
&=\vec{\sigma} \tilde{\theta} - (\tilde{v}.\nabla)v_2
 + \{ k(f_s(\varphi_1)-\varepsilon)-k(f_s(\varphi_2)-\varepsilon)\} \tilde{v},
\end{aligned} \label{2Dpurovelocuniq}
\\
\mathop{\rm div} \tilde{v}=0,
\\
\frac{\partial \tilde{\theta}}{\partial t}
- \Delta \tilde{\theta}
+ v_1.\nabla \tilde{\theta}
=  \frac{\ell}{2} \frac{\partial f_{s}}{\partial \varphi}(\varphi_{1})
\frac{\partial \tilde{\varphi}}{\partial t}  - (\tilde{v}.\nabla) \theta_{2}
+ \frac{\ell}{2}\left( \frac{\partial f_{s}}{\partial\varphi}(\varphi_{1})
- \frac{\partial f_{s}}{\partial\varphi}(\varphi_{2})\right)
\frac{\partial\varphi_{2}}{\partial t};
 \label{2Dpuroheatuniq}
\end{gather}
\begin{equation}
\frac{\partial \tilde{\varphi}}{\partial\eta}= 0, \quad
\tilde{\theta} =0, \quad
\tilde{v} = 0
\end{equation}
on $S$; and
\begin{equation}
\tilde{\varphi}(x,0) =\tilde{\theta}(x,0) = 0, \quad
\tilde{v}(x,0) = 0 \quad
\end{equation}
in $\Omega$.
Multiplying equation (\ref{2Dpurophaseuniq}) by $\tilde{\varphi}$ and
integrating on $\Omega$, after usual integration by parts, using the fact
that $a(\cdot)$, $b(\cdot)$, $\varphi_1$, $\varphi_2 \in L^\infty (Q)$ and
Holder's inequality, we obtain
\begin{equation}
\label{tildevarphiEstimate}
\frac{d}{d t} \|\tilde{\varphi} (t)\|_{2, \Omega}^2
+ 2 \alpha \|\nabla \tilde{\varphi} (t)\|_{2, \Omega}^2
\leq C_1 \big[ \|\tilde{\varphi} (t)\|_{2, \Omega}^2
+\|\tilde{\theta} (t)\|_{2, \Omega}^2 \big].
\end{equation}
Multiply (\ref{2Dpurophaseuniq}) by
$ \frac{\partial \tilde{\varphi}}{\partial t}$ and integrate
on $\Omega$. Proceeding  as before, we obtain
\begin{equation}
\label{TimeDerivativetildevarphiEstimate}
\| \frac{\partial \tilde{\varphi}}{\partial t} (t)\|_{2, \Omega}^2
+ \frac{\alpha}{2}  \frac{d}{d t} \| \nabla \tilde{\varphi} (t)\|_{2,\Omega}^2
\leq C_ 2 \big[ \|\tilde{\varphi} (t)\|_{2, \Omega}^2
+\|\tilde{\theta} (t)\|_{2, \Omega}^2 \big].
\end{equation}
Multiply (\ref{2Dpurovelocuniq}) by $\tilde{v}$ and proceed as usual with
the help of the facts that $\mathop{\rm div}v_1 =0$,
$k(f_s(\varphi_1) - \epsilon \geq 0$ and Holder's inequality to obtain
\begin{equation}
\label{VelocityPreparaty1Estimate}
\begin{aligned}
&\frac{1}{2}\frac{d}{d t} \|\tilde{v} (t)\|_{2, \Omega}^2
+ \nu \|\nabla \tilde{v} (t)\|_{2, \Omega}^2\\
&\leq C \big[ \|\tilde{\theta} (t)\|_{2, \Omega}^2
+\|\tilde{v} (t)\|_{2, \Omega}^2
+\int_\Omega (\tilde{v}(t). \nabla) v_2 (t) \tilde{v} (t)\\
&\quad +\int_\Omega [k(f_s(\varphi_1 (t)) -\epsilon) - k(f_s(\varphi_2 (t))
 -\epsilon)] |\tilde{v}(t)|^2 \big]
\end{aligned}
\end{equation}
The integral terms on the right hand side of the previous inequality can be
estimated as follows.
\begin{align*}
| \int_\Omega (\tilde{v}(t). \nabla) v_2 (t) \tilde{v} (t)|
&\leq C \|\nabla v_2 (t)\|_{2, \Omega} \|\tilde{v} (t)\|_{4, \Omega}^2 \\
&\leq C \|\nabla v_2 (t)\|_{2, \Omega} \|\tilde{v} (t)\|_{2, \Omega}
\|\nabla \tilde{v} (t)\|_{2, \Omega} \\
& \leq C_\nu \|\nabla v_2 (t)\|_{2, \Omega}^2 \|\tilde{v} (t)\|_{2, \Omega}^2
+ \frac{\nu}{4} \|\nabla \tilde{v} (t)\|_{2, \Omega}^2
\end{align*}
Next, by using the facts that $k(\cdot)$ is a Lipschitz function on
$(-\infty, 1-\epsilon)$ and
$f_s(\cdot)$ is a $L^\infty$-function, we obtain
\begin{align*}
&|\int_\Omega [k(f_s(\varphi_1 (t)) -\epsilon) - k(f_s(\varphi_2 (t)) -\epsilon) ]
|\tilde{v} (t)|^2 dx| \\
& \leq C_\epsilon \int_\Omega |[f_s(\varphi_1 (t)) -\epsilon)]
- [f_s(\varphi_1 (t)) -\epsilon]\|\tilde{v} (t)|^2 dx \\
&= C_\epsilon \int_\Omega |f_s(\varphi_1 (t))
- f_s(\varphi_1 (t))\|\tilde{v} (t)|^2 dx
 \leq C \|\tilde{v} (t)\|_{2, \Omega}^2.
\end{align*}
Using the last two estimates in (\ref{VelocityPreparaty1Estimate}), we obtain
\begin{equation} \label{VelocityEstimate}
\frac{d}{d t} \|\tilde{v} (t)\|_{2, \Omega}^2
+ \frac{3}{2}\nu \|\nabla \tilde{v} (t)\|_{2, \Omega}^2
\leq
C_3 \|\tilde{\theta} (t)\|_{2, \Omega}^2
+ C_4(1 + \|\nabla v_2 (t)\|_{2, \Omega}^2)\|\tilde{v} (t)\|_{2, \Omega}^2
\end{equation}
We proceed by multiplying equation (\ref{2Dpuroheatuniq}) by $\tilde{\theta}$, integranting on $\Omega$.
After integration by parts and the use of the facts that $\rm{div} \;  v_1 =0$,
$ \frac{\partial f_s}{\partial\varphi} \in L^\infty(R)$, with the help of Holder's inequality,
we obtain
\begin{equation}
\label{TemperaturePreparaty1Estimate}
\begin{aligned}
&\frac{1}{2} \frac{d}{d t} \|\tilde{\theta} (t)\|_{2, \Omega}^2
+  \|\nabla \tilde{\theta} (t)\|_{2, \Omega}^2\\
&\leq C ( \|\tilde{\theta} (t)\|_{2, \Omega}^2
+ \| \frac{\partial \tilde{\varphi}}{\partial t} (t) \|_{2, \Omega}^2 )
\\
&\quad + \int_\Omega (\tilde{v} (t). \nabla) \theta_2 \, \tilde{\theta} (t) dx
+ \int_\Omega \frac{\ell}{2}
(\frac{\partial f_s}{\partial \varphi} ( \varphi_1 (t) )
- \frac{\partial f_s}{\partial \varphi} ( \varphi_1 (t) ))
\frac{\partial \varphi_2}{\partial t} (t) \; \tilde{\theta} (t) \, dx
\end{aligned}
\end{equation}
The last two integrals terms in the above inequality can be estimated as follows:
\begin{align*}
|\int_\Omega (\tilde{v} (t). \nabla) \theta_2 \tilde{\theta} (t) dx |
&= |\int_\Omega \rm{div} \; (\tilde{v} (t) \theta_2) \; \tilde{\theta} (t) dx |\\
&= |\int_\Omega  (\tilde{v} (t) \theta_2) \; \nabla \tilde{\theta} (t) dx|\\
&\leq \|\tilde{v} (t)\|_{4, \Omega} \|\theta_2 (t)\|_{4,\Omega} \|\nabla
\tilde{\theta} (t)\|_{2, \Omega} \\
&\leq 4 \|\tilde{v} (t)\|_{4, \Omega}^2 \|\theta_2 (t)\|_{4, \Omega}^2
+ \frac{1}{4}\|\nabla \tilde{\theta} (t)\|_{2, \Omega}^2 \\
&\leq C \|\tilde{v} (t)\|_{2, \Omega} \|\nabla \tilde{v} (t)\|_{2,
\Omega} \|\theta_2 (t)\|_{4, \Omega}^2
+ \frac{1}{4}\|\nabla \tilde{\theta} (t)\|_{2, \Omega}^2 \\
&\leq C_\nu  \|\theta_2 (t)\|_{4, \Omega}^4 \|\tilde{v} (t)\|_{2, \Omega}^2
+ \frac{\nu}{2} \|\nabla \tilde{v} (t)\|_{2, \Omega}^2
+ \frac{1}{4}\|\nabla \tilde{\theta} (t)\|_{2, \Omega}^2.
\end{align*}
Using the fact that $ \frac{\partial f_s}{\partial \varphi}$ is a Lipschitz function,
we obtain
\begin{align*}
&|\int_\Omega \frac{\ell}{2}
(\frac{\partial f_s}{\partial \varphi} ( \varphi_1 (t) )
- \frac{\partial f_s}{\partial \varphi} ( \varphi_1 (t) ))
\frac{\partial \varphi_2}{\partial t} (t) \, \tilde{\theta} (t) \, dx |\\
& \leq C \int_\Omega |\tilde{\varphi} (t)|\,
|\frac{\partial \varphi_2}{\partial t} (t)| \, |\tilde{\theta} (t)| \, dx \\
& \leq C \|\tilde{\varphi} (t)\|_{2, \Omega}
\|\frac{\partial \varphi_2}{\partial t} (t)\|_{3, \Omega}
\|\tilde{\theta} (t)\|_{6, \Omega}\\
& \leq C \|\tilde{\varphi} (t)\|_{2, \Omega}
\|\frac{\partial \varphi_2}{\partial t} (t)\|_{3, \Omega}
\|\nabla \tilde{\theta} (t)\|_{2, \Omega} \\
& \leq C \|\frac{\partial \varphi_2}{\partial t} (t)\|_{3, \Omega}^2
\|\tilde{\varphi} (t)\|_{2, \Omega}^2
+ \frac{1}{4} \|\nabla \tilde{\theta} (t)\|_{2, \Omega}^2.
\end{align*}
Using the last two inequalities in (\ref{TemperaturePreparaty1Estimate}),
\begin{equation}
\label{TemperatureEstimate}
\begin{aligned}
&\frac{d}{d t} \|\tilde{\theta} (t)\|_{2, \Omega}^2
+  \|\nabla \tilde{\theta} (t)\|_{2, \Omega}^2 \\
&\leq
C_5 ( \|\tilde{\theta} (t)\|_{2, \Omega}^2
+ \| \frac{\partial \tilde{\varphi}}{\partial t} (t) \|_{2, \Omega}^2 )
+ C_6 \|\theta_2 (t)\|_{4, \Omega}^4 \|\tilde{v} (t)\|_{2, \Omega}^2
+ C_7 \|\frac{\partial \varphi_2}{\partial t} (t)\|_{3, \Omega}^2
\|\tilde{\varphi}\|_{2, \Omega}^2.
\end{aligned}
\end{equation}
Now, we multiply (\ref{TimeDerivativetildevarphiEstimate}) by  $2C_5$ and add the result to
(\ref{tildevarphiEstimate}), (\ref{VelocityEstimate}) and (\ref{TemperatureEstimate}).
After some simplifications, we obtain
\begin{align*}
&\frac{d}{d t} \|\tilde{\varphi} (t)\|_{2, \Omega}^2
+ \frac{d}{d t} \|\tilde{v} (t)\|_{2, \Omega}^2
+ \frac{d}{d t} \|\tilde{\theta} (t)\|_{2, \Omega}^2
+ \alpha C_5 \frac{d}{d t} \|\nabla \tilde{\varphi} (t)\|_{2, \Omega}^2
\\
&\leq C_8 (1+ \|\frac{\partial \varphi_2 (t)}{\partial t}\|_{3, \Omega}^2 )
\|\tilde{\varphi} (t)\|_{2, \Omega}
\\
&\quad + C_9 (1 + \|\nabla v_2 (t)\|_{2, \Omega}^2 + \|\theta_2 (t) )\|_{4, \Omega}^4)
\|\tilde{v} (t)\|_{2, \Omega}
+ C_{10} \|\tilde{\theta}\|_{2, \Omega}^2.
\end{align*}
By denoting
$z(t)= \|\tilde{\varphi} (t)\|_{2, \Omega}^2
+ \|\tilde{v} (t)\|_{2, \Omega}^2
+ \|\tilde{\theta} (t)\|_{2, \Omega}^2
+ \alpha C_5\|\nabla \tilde{\varphi} (t)\|_{2, \Omega}^2$,
the last inequality implies
\[
\frac{d}{d t} z(t)
\leq C \big[1+ \|\frac{\partial \varphi_2 }{\partial t} (t)\|_{3, \Omega}^2 )
 + \|\nabla v_2 (t)\|_{2, \Omega}^2 + \|\theta_2 (t) )\|_{4, \Omega}^4 \big] z(t).
\]
This inequality implies that for $t \in [0,T]$,
\[
0 \leq z(t) \leq z(0)
\exp\big\{ C(T) [1
+ \|\frac{\partial \varphi_2 }{\partial t} (t)\|_{3, Q}^2 )
 + \| v_2 (t)\|_{L^2(0,T;V)}^2 + \|\theta_2 (t) )\|_{4, Q}^4 ] \big\}.
\]
Since
$\| \frac{\partial \varphi_2 }{\partial t} (t)\|_{L^3(Q)}^2 )
 + \| v_2 (t)\|_{L^2(0,T;V)}^2 + \|\theta_2 (t) )\|_{L^4(Q)}^4 $ is finite, due to the known
regularity of the involved functions, and $z(0)=0$, we conclude that $z(t) \equiv 0$, and therefore
$\tilde{\varphi} \equiv 0$, $\tilde{v} \equiv 0$, $\tilde{\theta} \equiv 0$, which imply the uniqueness
of the solutions.

Next, we show that the solutions
$(\varphi_\varepsilon, v_\varepsilon, \theta _\varepsilon)\in
L^6(Q)\times L^2(0,T;\mbox{H})\times L^3(Q)$ of the problem
(\ref{2Dpuroreg}), (\ref{2Dpuroregfront}), (\ref{2Dpuroregdata})
are uniformly bounded with respect to $\varepsilon$ in the space
$W_3^{2,1}(Q)\times L^{2}(0,T;V) \cap L^{\infty}(0,T;H)\times
W_2^{2,1}(Q)$.
For this, note first that $\theta_{\varepsilon}\in L^{3}(Q)$; the
$L_{p}$-theory of parabolic linear equation combined with Theorem
\ref{auxphasefield} and Lemma \ref{imbeddingLp} allow us to
conclude that there exists an unique $\varphi_{\varepsilon} \in
W_{3}^{2,1}(Q)\cap L^{\infty}(Q)$ such that
\begin{equation}\label{2Dpurophasereg1}
\left\| \varphi_{\varepsilon} \right\|_{\infty,Q} \leq  C \left\|
\varphi_{\varepsilon} \right\|_{3,Q}^{(2)}
\leq  C \Big(
\left\|(a+b\varphi_{\varepsilon}-\varphi_{\varepsilon}^{2})
\varphi_{\varepsilon} \right\|_{3,Q} +\left\| \theta_{\varepsilon}
\right\|_{3,Q} + \left\| \varphi_{0\varepsilon}
\right\|_{W_{3}^{4/3}(\Omega)} \Big).
\end{equation}
Since $ {\max_{(x,t)\in\, Q} \left(a(x,t)+ b(x,t)s -
s^{2}\right)}$ is finite, from (\ref{2Dpurophasereg1}), we have
\begin{equation}\label{2Dpurophasereg2}
\left\| \varphi_{\varepsilon} \right\|_{\infty,Q} \leq  C \left\|
\varphi_{\varepsilon} \right\|_{3,Q}^{(2)}
\leq  C \Big(
\left\|\varphi_{\varepsilon}\right\|_{6,Q} + \left\|
\theta_{\varepsilon} \right\|_{3,Q}+\left\|
\varphi_{0\varepsilon}\right\|_{W_{3}^{4/3} (\Omega)} \Big).
\end{equation}
Combining (\ref{estimateheat4}), (\ref{estimatephase13})
and (\ref{2Dpurophasereg2}) and using usual Sobolev imbeddings, we
conclude that
\begin{equation}\label{2Dpurophasereg3}
\left\|\varphi_{\varepsilon}\right\|_{3,Q}^{(2)} \leq  C
\left(\left\|\theta_{0\varepsilon} \right\|_{2,\Omega} +
\left\|\varphi_{0\varepsilon}\right\|_{W_{3}^{4/3}(\Omega)}
\right).
\end{equation}
Moreover, Lemma \ref{imbeddingHolder}  gives us that
$\varphi_{\varepsilon}\in H^{2/3,1/3}(Q)$ such that
\begin{equation}\label{2Dpurophasereg4}
|\varphi_{\varepsilon}|_{Q}^{(2/3)} \le C
\left\|\varphi_{\varepsilon}\right\|_{3,Q}^{(2)} \le C \left(
\left\|\theta_{0\varepsilon}\right\|_{2,\Omega} +
\left\|\varphi_{0\varepsilon} \right\|_{W_{3}^{4/3}(\Omega)}
\right).
\end{equation}

We consider then the equation for the temperature. By applying the
$L_{p}$-theory of parabolic linear equations (see Ladyzenskaja
\cite{Ladyzenskaja}) together  with the facts that
$ \frac{\partial\varphi_{\varepsilon}}{\partial t}\in L^{2}(Q)$,
$f_s \in C_b^{1,1}(\mathbb{R})$ and $v_{\varepsilon} \in L^{4}(Q)^{2}$,
we have that there exists an unique
$\theta_{\varepsilon} \in W_{2}^{2,1}(Q) \cap L^{p}(Q)\,(p \geq 2)$ such that
\begin{equation}
\label{2Dpurocalorreg1}
\left\|\theta_{\varepsilon}\right\|_{2,Q}^{(2)}
\leq C \Big(\left\|v_{\varepsilon}\right\|_{4,Q}\left\|\theta_{0\varepsilon}
\right\|_{W_{2}^1(\Omega)}
+ \big\|\frac{\partial f_{s}}{\partial\varphi}\big\|_{\infty,Q}
\big\| \frac{\varphi_{\varepsilon}}{\partial t}\big\|_{2,Q}
+ \left\|\theta_{0\varepsilon}\right\|_{W_{2}^1(\Omega)}\Big),
\end{equation}
 where the estimates
$ \|v_{\varepsilon}\|_{4,Q}$ and
$ \| \frac{\varphi_{\varepsilon}}{\partial t}\|_{2,Q}$ are
given by (\ref{estimateveloc2}) and (\ref{estimatephase10}),
respectively.

Combining (\ref{estimatephase13}), (\ref{estimateveloc2}) and
(\ref{2Dpurocalorreg1}), we obtain
\begin{equation}\label{2Dpurocalorreg2}
\left\|\theta_{\varepsilon}\right\|_{2,Q}^{(2)} \leq C
\left(\left\|v_0\right\|_{\mathrm{H}} +
\left\|\varphi_0\right\|_{W_{3}^{4/3}(\Omega)}+
\left\|\theta_0\right\|_{W_{2}^1(\Omega)}\right)
\end{equation}
Therefore, the solutions $(\varphi_\varepsilon, v_\varepsilon,
\theta _\varepsilon)$ of problem (\ref{2Dpuroreg}),
(\ref{2Dpuroregfront}), (\ref{2Dpuroregdata}) are uniformly
bounded with respect to $\varepsilon$ in the space
$W_{3}^{2,1}(Q)\times (L^2(0,T;V)\cap L^{\infty}(0,T;H)) \times
W_{2}^{2,1}(Q)$, and this completes the proof of Theorem
\ref{2Dpuroteo}.
\end{proof}


\section{Proof of Theorem \ref{2Dpuro}}

In this section we  use the results of Theorem \ref{2Dpuroteo}, the
$L_p-$theory of parabolic equations, the imbedding of Lemma
\ref{imbeddingHolder} and compactness arguments to prove a result on
existence and regularity of solution for problem (\ref{2Dpuromodel}),
(\ref{2Dpurofront}), (\ref{2Dpurodata}). This will be obtained by passing to
the limit in the regularized problem (\ref{2Dpuroreg}),
(\ref{2Dpuroregfront}), (\ref{2Dpuroregdata}) as $\varepsilon$ approaches
zero. Due to the estimates we present, the convergence of almost all the
terms in the  equations of the regularized problem will be standard ones,
except for the regularized velocity equation that will require a local
argument. The stated regularity of the solutions will be obtained by using
bootstrapping arguments. Unfortunately, due to the additional Carman-Koseny
type term in the velocity equation, we cannot improve the regularity of weak
solution of Navier-Stokes equations.

\subsection*{Passing to the Limit}
As a consequence of Theorem \ref{2Dpuroteo}, for $\varepsilon \in
(0,1]$, any solution $(\varphi_\varepsilon , v_\varepsilon,
\theta_\varepsilon) \in L^6(Q)\times L^2(0,T; \mbox{H})\times
L^3(Q)$ of problem (\ref{2Dpuroreg}), (\ref{2Dpuroregfront}),
(\ref{2Dpuroregdata}) is uniformly bounded with respect to
$\varepsilon$ in the space $W_{3}^{2,1}(Q)\times (L^2(0,T;V)\cap
L^{\infty}(0,T;H)) \times W_{2}^{2,1}(Q)$.


With the help of Aubin-Lions Lemma (see Temam \cite{Temam}, Lions
\cite{Lions} or Corollary 4, p. 85, in Simon \cite{Simon}), there exists
$(\varphi, v, \theta) \in L^6(Q)\times L^2(0,T; H)\times L^3(Q)$ and a
subsequence, which for simplicity of notation is still indexed by
$\varepsilon$, such that as $\varepsilon \rightarrow 0$
\begin{gather*}
\varphi_\varepsilon  \rightarrow  \varphi \quad \mbox{in } L^{q}(Q)\,(q \ge 6)\\
\nabla \varphi_\varepsilon  \rightarrow  \nabla \varphi \quad \mbox{in }
 L^{3}(Q)^2\\
\varphi_\varepsilon   \rightharpoonup  \varphi \quad \mbox{in } W_{3}^{2,1}(Q)\\
\theta_\varepsilon   \rightarrow  \theta \quad \mbox{in }
L^{p}(Q)\,(p \ge 2)\\
\nabla \theta_\varepsilon   \rightarrow  \nabla \theta \quad \mbox{in }
 L^{2}(Q)^2\\
\theta_\varepsilon   \rightharpoonup  \theta \quad \mbox{in } W_{2}^{2,1}(Q)\\
v_\varepsilon   \rightharpoonup  v  \quad \mbox{in } L^2(0,T;\mbox{V})\\
v_\varepsilon   \stackrel{\ast}{\rightharpoonup}  v  \quad \mbox{in }
 L^{\infty}(0,T;\mbox{H})
\end{gather*}
Moreover, by Lemma \ref{imbeddingHolder}, $\varphi_{\varepsilon}
\in  H^{2/3,1/3}(Q)$ and for all $\varepsilon \in [0,1]$ we have
$|\varphi_{\varepsilon}|_{Q}^{(2/3)} \leq C
\left\|\varphi_{\varepsilon}\right\| _{3}^{(2)}$. In particular,
$ \sup_{\overline{Q}}|\varphi_{\varepsilon}(x,t)|
\leq C$, and $\langle\varphi_{\varepsilon}\rangle _{t}^{(1/3)} \le
C$. Thus, $\{\varphi_{\varepsilon}\}$ is uniformly bounded and
equicontinuous family in $\overline{Q}$. By Arzela-Ascoli's
Theorem it follows that there exists a subsequence, that we
denote, for simplicity, again by $\{\varphi_{\varepsilon}\}$ such
that $\varphi _{\varepsilon }\rightarrow \varphi$ uniformly in
$\overline{Q}$.

We check now that $(\varphi, v, \theta) \in L^6(Q)\times
L^2(0,T;H)\times L^3(Q)$ is a generalized solution of problem
(\ref{2Dpuromodel}), (\ref{2Dpurofront}), (\ref{2Dpurodata}).
We start by taking $Q_s$ and $Q_{ml}$ as in Definition
\ref{defpuro} with the just obtained function $\varphi$.

Now, we have to prove that  $v = 0$ in $\overset{\circ}Q_s$. For
this, we will use an argument already  used by Blanc \emph{et al}
\cite{Blanc}: we take $K$ a compact subset in $\overset{\circ}Q_s$
and observe that $f_s \in C_b^{1,1}(\mathbb{R})$, $f_s(\varphi(x,t)) = 1$
in a neighborhood of $K$. Since $\varphi_\varepsilon \rightarrow
\varphi$ uniformly in $\overline{Q}$, we conclude that there is a
small positive $\varepsilon_K$ such that
\[
f_s( \varphi_\varepsilon (x,t)) = 1 \quad \mbox{in } \quad K
\]
whenever $\varepsilon \in (0,\varepsilon_K)$.
By multiplying the regularized velocity equation of problem
(\ref{2Dpuroreg})-(\ref{2Dpuroregfront})-(\ref{2Dpuroregdata}) by
$v_\varepsilon$, integrating over $K$, using Green's formula and
Young's inequality, we obtain
\[
k(1-\varepsilon) \left\| v_\varepsilon \right\|_{2,K} \leq C
\]
with $C$ a positive constant independent of $\varepsilon
\in (0,\varepsilon_K)$.
As $\varepsilon$ approaches zero, $k(1-\varepsilon) $ blows up and
compels $\left\| v_\varepsilon \right\| _{L ^2(K)^2}$ to converge
to 0. Therefore, $v_{\varepsilon |_K} \rightarrow 0$ in $L^2(K)$,
and consequently $v = 0$ in $K$. Since $K$ was an arbitrary
compact set of $\overset{\circ}Q_s$, we conclude that $v = 0$ in
$\overset{\circ}Q_s$.

Now we have to show that the triple of functions $(\varphi,
\theta, v)$ satisfies equations (\ref{generalizedphase}),
(\ref{generalizedtemp}) and (\ref{generalizedveloc}). We start by
proving that (\ref{generalizedveloc}) is satisfied.
For this, we  multiply the second equation in (\ref{2Dpuroreg}) by
a test function $\phi \in C([0,T];W_{2}^{1}(\Omega_{ml}(t)))$ such
that $\mathop{\rm div}\phi(. ,t)= 0$ for all $ t \in [0,T]$ ,
$\mathop{\rm supp} \phi(x,t)\subset Q_{ml} \cup \Omega_{ml}(0)$ and
$\phi(.,T) = 0$ and integrate over $Q$. After some usual integrations by
parts using (\ref{2Dpuroregfront}), (\ref{2Dpuroregdata}) and
observing the properties of $\phi$, we obtain
\begin{equation}
\label{ApproximateEquation}
\begin{aligned}
&-\int_{Q_{ml}} v_\varepsilon \,\phi_{t} \, dx\,dt
+\nu \int_{Q_{ml}} \nabla v_\varepsilon \nabla \phi \, dx\,dt
+\int_{Q_{ml}}(v_\varepsilon.\nabla)v_\varepsilon\,\phi \,dx\,dt \\
&+\int_{Q_{ml}}k(f_{s}(\varphi_\varepsilon)
-\varepsilon) v_\varepsilon \, \phi \, dx\,dt\\
&=\int_{Q_{ml}}\vec{\sigma}\theta_\varepsilon
\phi \, dx\,dt +\int_{\Omega_{ml}(0)} v_0(x) \phi(x,0) dx
\end{aligned}
\end{equation}
The stated convergence for $\varphi_\varepsilon$, $\theta_\varepsilon$ and
$v_\varepsilon$ are enough to conclude the convergence of the first and
second terms of the left hand side and also of the first term of the right
hand side of equation (\ref{ApproximateEquation}). For the convergence of the
third and fourth term of the left hand side, however, we need to be more
careful.
We first observe that
\begin{equation}
\label{kfsEstimate} k(f_s( \varphi_\varepsilon) - \varepsilon)
\rightarrow k(f_s(\varphi)) \quad \mbox{in } C^0(K_{ml})
\end{equation}
for any fixed compact $K_{ml} \subset  Q_{ml} \cup
\Omega_{ml}(0)$. In fact, in such $K_{ml}$, $k(f_s(
\varphi_\varepsilon (x,t)) -\varepsilon)$ and $k(f_s(
\varphi(x,t)))$ are bounded  continuous functions, and, since
$f_s(\varphi_\varepsilon) - \varepsilon $ converges to
$f_s(\varphi)$ in $C^0(K_{ml})$, we obtain the stated result.
In particular, this result holds for $K_{ml}$ taken as $supp \;
\phi$, and this guarantees the convergence of the last term in the
left hand side of the last equation.

For the convergence of the third term of the left hand side it is
necessary to improve the convergence of $v_\varepsilon$. For this,
we first observe that $Q_{ml}$ is an open set and can be covered
by a countable number of open cylinders $\Omega_i \times
(a_i,b_i)$, such that for each $i= 1, \ldots, \infty $, we have
$\overline{\Omega}_i \subset \Omega$ and $[a_i,b_i] \subset
(0,T)$. Thus, for each $i= 1, \ldots, \infty$, we can take the
compact set $\overline{\Omega}_i \times [a_i,b_i]$ as $K_{ml}$ in
(\ref{kfsEstimate}) and conclude that there is $\varepsilon_i \in
(0,1]$ and $C_i > 0$ independent of $\varepsilon \in (0,
\varepsilon_i]$ such that for such $\varepsilon$ we have
\[
\|k(f_s( \varphi_\varepsilon) - \varepsilon)\|
_{L^\infty(\overline{\Omega}_i \times [a_i,b_i])} \leq C_i.
\]
 This and our previous estimates allow us to work with the second
equation in (\ref{2Dpuroreg}) restricted to $\Omega_i \times (a_i,b_i)$ to
obtain that there is $C_i > 0$ independent of $\varepsilon \in (0,
\varepsilon_i]$ such that for such $\varepsilon$ we have
\[
\big\|\frac{\partial v_\varepsilon}{\partial t}\big\| _{L^2(a_i, b_i; V'(\Omega_i)}
\leq C_i,
\]
where $V'(\Omega_i)$ is the topological dual of the
Banach space
\[
V(\Omega_i)=\{ u \in\, \overset{0}{W}{}^{1}_{2}(\Omega_i)^2; \mathop{\rm div} u = 0\},
\]
considered with the norm of
$\overset{0}{W}{}^{1}_{2}(\Omega_i)^2$.

Also, our previous estimates tell us in particular that $\{v_\varepsilon \}$
for is uniformly bounded with respect to $\varepsilon \in (0, \varepsilon_i]$
in $L^2(a_i,b_i; W(\Omega_i))$, where $W(\Omega_i) =$\linebreak $\{ u \in
W^{1}_2(\Omega_i)^2; \mathop{\rm div} u = 0 \}$ is a Banach space with the
$W^1_2(\Omega_i)^2$-norm.

Consider the Banach space
\[
H(\Omega_i)= \{ u \in L^2(\Omega_i)^2; \mathop{\rm div } u = 0, \;
\mbox{and null normal trace} \}
\]
with the $L^2(\Omega_i)^2$-norm (see Temam \cite{Temam}
for properties of this and the previous Banach spaces). We observe that
$W(\Omega_i) \subset H(\Omega_i) \subset V'(\Omega_i)$, and the first
imbedding is compact, we can use Corolary 4, p. 85, in Simon \cite{Simon} to
conclude that there is a subsequence of $\{v_\varepsilon\}$ converging to $v$
in $L^2(a_i,b_i; H(\Omega_i))$. In particular, this implies that along such
subsequence $v_\varepsilon \rightarrow v \quad \mbox{in } \quad L^2(\Omega_i
\times (a_i,b_i))$.

Proceeding as above for each $i=1, \dots, \infty$, with the help
of the usual diagonal argument, we obtain a subsequence such that
\[
v_\varepsilon \rightarrow v \quad \mbox{in } \quad
L^2_{\rm loc}(Q_{ml}).
\]
Thus, along such subsequence, we can pass to the limit as
$\varepsilon \rightarrow 0$ in (\ref{ApproximateEquation}) by
proceeding exactly as in the case of the classical Navier-Stokes
equations and conclude that (\ref{generalizedveloc}) is
satisfied.\medskip

To obtain the other equations in Definition \ref{defpuro}, we
multiply the first and third equations of (\ref{2Dpuroreg})
respectively by $\beta \in W{}_2^{1,1}(Q)$ with $\beta(.\,,T)= 0$
and $\xi \in \,\overset{0}{W}{}_2^{1,1}(Q)$ with $\xi(.,T)=0$,
and proceed as before. Using arguments similar to the ones in
(\ref{limitphasecomp}) and (\ref{limitheatcomp}), we conclude that
\begin{gather*}
\int_{Q} \left(a \varphi_{\varepsilon}+b
\varphi_{\varepsilon}-\varphi_{\varepsilon}^{3} \right)\,\beta
\,dx\,dt \rightarrow \int_{Q} \left(a \varphi+b \varphi-
\varphi^{3}\right)\,\beta\,dx\,dt , \\
\int_{Q}  \frac{\partial
f_s}{\partial\varphi}(\varphi_{\varepsilon})
\frac{\partial\varphi_{\varepsilon}}{\partial t} \,\xi\,dx\,dt
\rightarrow \int_{Q}  \frac{\partial f_s}{\partial
\varphi}(\varphi) \frac{\partial \varphi}{\partial
t}\,\xi\,dx\,dt ,
\end{gather*}
as $\varepsilon \rightarrow 0$.
With these results, it is easy to to pass to the limit as
$\varepsilon \rightarrow 0$ and conclude that equations
(\ref{generalizedphase}) and (\ref{generalizedtemp}) are also
satisfied.


\subsection*{\bf Regularity of the Solution}
Now we have to examine the regularity of $(\varphi, \theta, v)$.
For this,  we remark that by interpolation (see Ladyzenskaja
\cite{Ladyzenskaja} p. 74),  $\theta\in L^{4}(Q)$. Thus,  applying
Proposition \ref{auxphasefield} with $\theta\in L^{3}(Q)$, we
conclude that $\varphi\in W_{3}^{2,1}(Q) \cap L^{\infty}(Q)$.
Also, Proposition \ref{auxvelocity} give us that $v \in
L^{4}(Q)^{2}$.

Applying the $L_{p}-$theory of parabolic equations together
with the facts that $f_{s}\in C_{b}^{1,1}(\mathbb{R})$, $v \in L^{4}(Q)^{2}$,
$ \frac{\partial\varphi}{\partial t} \in L_{2}(Q)$
and Lemma the result of \ref{imbeddingLp}, we conclude
that $\theta\in W_{2}^{2,1}(Q)\cap L^{p}(Q)$ $(p \geq 2)$.
Therefore, by using a bootstrapping argument with $\theta \in
L^{q}(Q)$ where $q \ge 3$ and smoothness of the data $\varphi_0$
we conclude that $\varphi \in W_{q}^{2,1}(Q) \cap L^{\infty}(Q)$.

Applying again the $L_{p}-$theory of parabolic equations with
$f_{s}\in C_{b}^{1,1}(\mathbb{R})$, $v\in L^{4}(Q)^{2}$,
$ \frac{\partial\varphi}{\partial t} \in L^{p}(Q)$, with
$2 \leq p < 4$, recalling the given smoothness of $\theta_0$ and
the result of Lemma \ref{imbeddingLp}, we conclude that
$\theta \in W_{p}^{2,1}(Q)\cap L^{\infty}(Q)$, with $2 \leq p < 4$.
This completes the proof of Theorem \ref{2Dpuro}.

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\end{document}