# Wellposedness of Boussinesq system

<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the viscous, incompressible, nonlinear Boussinesq
system in two and three spatial dimension. We study the existence and
regularity of solutions to the Boussinesq system with nonhomogeneous boundary
conditions for which the normal component for the velocity is not necessary
equal to zero. We establish the existence of global weak solutions to the
three-dimensional instationary nonlinear Boussinesq system but without any
smallness assumption on the initial and boundary conditions.

###### Key words and phrases:

. Boussinesq system, Navier-Stokes equation, Nonhomogeneous boundary
conditions.

###### 1991 Mathematics Subject Classification:

Primary: 35Q30, 76B03, 76D03, 76D05; Secondary: 35K51.

Arnab Roy

Technische Universität Darmstadt,

Schloßgartenstraße 7, 64289 Darmstadt, Germany.

##  1\. Introduction

In this paper we are mainly interested in the following initial boundary value
problem for non-stationary nonlinear Boussinesq system that describes the flow
of a viscous incompressible fluid in $\Omega\subset\mathbb{R}^{N},\,N=2,3$
with smooth boundary $\Gamma$, subject to convective heat transfer :

(1) $\displaystyle\left\\{\begin{aligned} &\frac{\partial\mathbf{z}}{\partial
t}-\nu\Delta\mathbf{z}+(\mathbf{z}.\nabla)\mathbf{z}+\nabla
p=\bm{\beta}\theta+\mathbf{f}_{1}\mbox{ in }Q=\Omega\times(0,T),\\\ &\mbox{div
}\mathbf{z}=0\mbox{ in }Q=\Omega\times(0,T);\\\ &\mathbf{z}=\mathbf{g}\mbox{
on }\Sigma=\Gamma\times(0,T),\quad\mathbf{z}(0)=\mathbf{z}_{0}(x)\mbox{ in
}\Omega,\\\ &\frac{\partial\theta}{\partial
t}-\mu\Delta\theta+(\mathbf{z}.\nabla)\theta=f_{2}\mbox{ in }\mbox{ Q },\\\
&\frac{\partial\theta}{\partial n}=h\mbox{ on
}\Sigma,\quad\theta(0)=\theta_{0}\mbox{ in }\Omega.\end{aligned}\right.$

Here $\mathbf{z}=(z_{1},z_{2},....,z_{n})$ is the fluid velocity, $p$ denotes
the pressure and $\theta$ is the temperature. Here $i$-th component of
$(\mathbf{z}.\nabla)\mathbf{z}$ is $\sum_{j=1}^{n}z_{j}\frac{\partial
z_{i}}{\partial x_{j}}$ and $\frac{\partial\theta}{\partial n}$ denotes the
outer normal derivative of $\theta$ at $x$ on $\Gamma$. In equation (1),
$\bm{\beta}$ is gravitational vector function, $\nu>0$ is kinematic viscosity
and $\mu>0$ is the thermal diffusivity, $\mathbf{f}_{1}$ is an external force
and $f_{2}$ is a heat source applied to the fluid. Also, the boundary
conditions $\mathbf{g}(x,t)$ and $h(x,t)$ are functions defined on
$\Gamma\times(0,T)$. In this Boussinesq approximation, the fluid is treated as
incompressible when formulating the Navier-Stokes mass and momentum
conservation equations and here the effect of temperature change is taken into
account in the buoyancy term $\bm{\beta}\theta$ which drives convection.

In this paper, our aim is to prove the existence of the weak solution for
system (1) and analyse its regularity. In [13], the author also considers the
same non-homogeneous problem (1) in two and three dimension but with very
smooth boundary data. Motivated by boundary control problems, here our main
interest is in investigating the case when boundary data are not so regular.
In that case, we cannot apply the extension method as in [13]. We are also
interested in finding a sufficient condition on $(\mathbf{g},h)$ so that a
weak solution to equation (1) exists.

We denote the outward unit normal to the boundary $\Gamma$ by $\mathbf{n}$.
When the normal component of $\mathbf{g}$ is zero, i.e, in the case of
$\mathbf{g}.\mathbf{n}=0$, the authors in [7] and [8] studied existence and
regularity results of Navier-Stokes equation by pseudo-differential
techniques. We can adopt similar analysis in the case of equation (1) if
$\mathbf{g}.\mathbf{n}=0$. But for engineering and other practical
applications (e.g [9]), the interesting case to study is when
$\mathbf{g}.\mathbf{n}\neq 0$. To overcome this difficulty we will follow a
similar approach as described in [16]. In [16] the author writes Stokes and
Oseen equations in the form of a system of two operator equations and obtains
the optimal regularity results. The first one is an evolution equation
satisfied by the projection of the solution on the Stokes space - the space of
divergence free vector fields with the normal trace equal to zero. The second
one is a quasi-stationary equation satisfied by the projection of the solution
on the orthogonal complement of the Stokes space. In our case, we have to deal
with coupled system (1).

We explore here the existence and regularity of solutions to linearized and
nonlinear Boussinesq system with nonhomogeneous boundary conditions. The
novelty of our work is that we consider boundary conditions with low
regularity and the Dirichlet boundary condition for velocity for which the
normal component is not equal to zero and Neumann boundary condition for
temperature, that makes the problem more interesting and challenging. We also
establish the existence of weak solution to the three dimensional nonlinear
Boussinesq system (1) without any smallness assumption to initial condition.

Our main result concerning the existence and regularity of solution to system
(1) is the following:

######  Main Theorem.

Let $(P\mathbf{z}(0),\theta(0))\in\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$ and $\mathbf{g}\in\mathbf{V}^{s,s}(\Sigma)$,
$h\in L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap
H^{s}(0,T;(H^{1}(\Gamma))^{\prime})$ with $\frac{1}{2}<s<1$,
$\mathbf{f}_{1}\in L^{2}(0,T;\mathbf{V}^{-1}(\Omega))$,
$f_{2}\in L^{2}(0,T;(H^{1}(\Omega))^{\prime})$.Then equation (1) admits at
least one weak solution $(\mathbf{z},\theta)$ in
$C_{w}([0,T];\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega))\cap
L^{2}(0,T;\mathbf{V}^{1}(\Omega))\times L^{2}(0,T;H^{1}(\Omega))$ and
$p\in\mathcal{D}^{\prime}(0,T;L^{2}(\Omega))$, where
$C_{w}([0,T];\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega))$ is the subspace of
$L^{\infty}(0,T;\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega))$ which are
continuous from $[0,T]$ into $\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)$
equipped with its weak topology.

Here $P$ is the usual Léray projector and the notations
$\mathbf{V}^{0}(\Omega),\,\mathbf{V}^{1}(\Omega),\,\mathbf{V}^{s,s}(\Sigma)$
are as in section 2. The proof of this existence and regularity result rely on
combination of techniques described in [13] and [16]. Actually, we split the
system into two parts-homogeneous and non-homogeneous boundary data. We split
the system such that the part with homogeneous boundary data has nonlinear
terms and the part with non homogeneous boundary condition has only linear
terms. Then we study the part with homogeneous boundary data as described in
[13]. To analyse the non homogeneous part, we will follow a similar technique
as in [16], by applying projection to the first equation and then writing the
coupled system in the form of an evolution equation and a quasi-stationary
equation.

The works in the literature which are most relevant to our present paper are
[16],[3],[14],[13]. In [3], authors showed the existence of a unique local in
time weak solution in $\mathbb{R}^{2}\times(0,T)$. They also prove a global
existence theorem of weak solution for small initial data, if the exterior
force field depends on the temperature (in suitable spaces). Recently, in the
case of $\Omega=\mathbb{R}^{2}$, the result of global existence of smooth
solutions to nonlinear Boussinesq system is generalized to the cases of
’partial viscosity’ (i.e., either $\nu>$ 0 and $\mu=$ 0, or $\nu=$ 0 and
$\mu>$ 0) by Hou-Li [10] and Chae [4] independently for the case of smooth
initial data but without smallness assumption.

Ôeda studied the moving boundary case with Dirichlet boundary condition in
[14]. The author considered the time dependent domain
$\hat{\Omega}=\cup_{0\leq t\leq T}\Omega(t)\times\\{t\\}$, where $\Omega(t)$
is a bounded set in $\mathbb{R}^{N}(N=2\mbox{ or }3)$. Under certain
assumptions the existence of a weak solution on any interval $[0,T]$ and of a
unique strong solution on a small time interval $[0,\tau_{0}]$ is proved.

In [13], the author treats the initial value problem (1) on a bounded domain
in $\mathbb{R}^{N}\,(N=2\mbox{ or }3)$, with no-slip boundary conditions for
the velocity and allows the temperature to be prescribed (as a function of
space and time) on one part of the boundary, while the temperature flux is
prescribed on the rest of the boundary. He defines an appropriate notion of
weak solution, and proves existence and uniqueness (for $N=2$) results for
such solutions. But for the definition of weak solution, the author uses an
extension of boundary value $\mathbf{g}$ to $\overline{\Omega}\times(0,T)$ and
solve the system of equations corresponding to it. Assuming $\mathbf{g}\in
C^{1}(\overline{\Gamma}\times(0,T))$, i.e, sufficiently regular boundary data,
the proof of existence is based on the construction of approximate solution by
Galerkin method and passage to the limit using apriori estimates.

The rest of the paper is organized as follows. Section 2 is dedicated to
notations and general functional framework. We study the steady linearized
Boussinesq system in Section 3. We define a weak solution to this system and
establish existence and regularity results. We also introduce transposition
solution in the case of not so regular boundary data. Section 4 is devoted to
unsteady linearized Boussinesq system. At first we consider linearization
around stationary state and then we study a linearization around an
instationary state which is needed in section 5 to analyze the nonlinear
Boussinesq system with nonhomogeneous boundary condition. In section 5, we
prove the existence of weak solution to nonlinear Boussinesq system in 3D
case. To the best of our knowledge, this existence and regularity results for
three dimensional nonlinear Boussinesq system seems to be new.

##  2\. Notation and general functional framework

We will use boldface letters to denote functions with values in
$\mathbb{R}^{n}$ and spaces of functions with values in $\mathbb{R}^{n}$. Here
we denote by $\mathbf{L}^{2}(\Omega)$ the space
$(L^{2}(\Omega))^{n}=L^{2}(\Omega;\mathbb{R}^{n})$. Similarly the spaces
$(H_{0}^{1}(\Omega))^{n}$ and $(H^{-1}(\Omega))^{n}$ are
$\mathbf{H_{0}^{1}}(\Omega)$ and $\mathbf{H^{-1}}(\Omega)$. We also introduce
different spaces of divergence free functions and some corresponding trace
spaces :

$\displaystyle\mathcal{V}(\Omega)$
$\displaystyle=\\{\mathbf{u}\in\mathbf{C}_{c}^{\infty}(\Omega)|\mbox{
div}\,\mathbf{u}=0\mbox{ in }\Omega\\},$
$\displaystyle\textbf{V}_{0}^{1}(\Omega)$
$\displaystyle=\\{\mathbf{u}\in\mathbf{H}_{0}^{1}(\Omega)|\mbox{
div}\,\mathbf{u}=0\mbox{ in }\Omega,\,\mathbf{u}=0\mbox{ on }\Gamma\\},$
$\displaystyle\textbf{V}^{2}(\Omega)$
$\displaystyle=\\{\mathbf{u}\in\mathbf{H}^{2}(\Omega)|\mbox{
div}\,\mathbf{u}=0\mbox{ in }\Omega\\},$
$\displaystyle\textbf{V}_{n}^{0}(\Omega)$
$\displaystyle=\\{\mathbf{u}\in\mathbf{L}^{2}(\Omega)|\mbox{
div}\,\mathbf{u}=0\mbox{ in }\Omega,\,\mathbf{u}.n=0\mbox{ on }\Gamma\\},$
$\displaystyle\mathbf{V}^{1}(\Gamma)$
$\displaystyle=\\{\mathbf{u}\in\mathbf{H}^{1}(\Omega)|\int_{\Gamma}\mathbf{u.n}=0\mbox{
on }\Gamma\\}.$

Similarly we can define
$\mathbf{V}^{s}(\Omega),\mathbf{V}_{n}^{s}(\Omega),\mathbf{V}_{0}^{s}(\Omega),\mathbf{V}^{s}(\Gamma)$
for $s\geq 0$. For $s<0,\mathbf{V}^{s}(\Gamma)$ is the dual space of
$\mathbf{V}^{-s}(\Gamma)$, with $\mathbf{V}^{0}(\Gamma)$ as pivot space. For
notational convenience, we define the space

$\displaystyle\mathcal{H}^{s}(\Omega)=\begin{cases}H^{s}(\Omega)/\mathbb{R}\quad&\mbox{if
}s>0,\\\ (H^{-s}(\Omega)/\mathbb{R})^{\prime}\quad&\mbox{if }s<0.\end{cases}$

We introduce here some spaces for time dependent functions : For $s\geq 0$ and
$\sigma\geq 0$, define

$\displaystyle\mathbf{V}^{s,\sigma}(Q)$
$\displaystyle=H^{\sigma}(0,T;\mathbf{V}^{0}(\Omega))\cap
L^{2}(0,T;\mathbf{V}^{s}(\Omega)),$
$\displaystyle\mathbf{V}^{s,\sigma}(\Sigma)$
$\displaystyle=H^{\sigma}(0,T;\mathbf{V}^{0}(\Gamma))\cap
L^{2}(0,T;\mathbf{V}^{s}(\Gamma)).$

We also introduce

$\displaystyle W(0,T;X,Y)=\\{\mathbf{u}\in
L^{2}(0,T;X)|\,\mathbf{u}^{\prime}\in L^{2}(0,T;Y)\\},$

where $X,Y$ are two Banach spaces.

Let us denote by P, the orthogonal projection operator from
$\mathbf{L}^{2}(\Omega)$ onto $\mathbf{V}_{n}^{0}(\Omega)$( see [6, Chap III,
Theorem 1.1] for details). We recall that for all $s\in[0,2]$
$P\in\mathcal{L}(\mathbf{H}^{s}(\Omega),\mathbf{V}^{s}_{n}(\Omega))$ (see [6,
Chap III, Lemma 1.2]). The operator P can be extended to a bounded operator
from $\mathbf{H}^{-1}(\Omega)$ to $\mathbf{V}^{-1}(\Omega)$, that we still
denote by P.

Throughout this paper, the letter C denotes a positive constant that may
change from line to line. When particular positive constants are required, we
use $C_{0},C_{1}$, etc. If H is a Hilbert space, we denote by $||.||_{H}$ its
corresponding norm, by $H^{{}^{\prime}}$ its dual space and by
$\langle.,.\rangle_{H^{{}^{\prime}},H}$ the $H^{{}^{\prime}}-H$ duality
pairing. For any two Hilbert spaces $H_{1}$ and $H_{2}$ we use the notation
$H_{1}\hookrightarrow H_{2}$ to indicate that $H_{1}$ is continuously embedded
into $H_{2}$. We denote by $\mathcal{L}(H_{1},H_{2})$ the space of all bounded
linear operators from $H_{1}$ to $H_{2}$ and we use the notation
$\mathcal{L}(H)$ for $\mathcal{L}(H,H)$.

##  3\.  Steady linearized Boussinesq system

We first recall the existence result for stationary Boussinesq system with
homogeneous boundary conditions :

$\displaystyle-\nu\Delta\mathbf{z}_{s}+(\mathbf{z}_{s}.\nabla)\mathbf{z}_{s}+\nabla
q_{s}$ $\displaystyle=\mathbf{\beta}\theta_{s}+\mathbf{f}_{1}\quad\mbox{in
}\Omega,$ $\displaystyle\mbox{div}\,\mathbf{z}_{s}=0\quad\mbox{in }\Omega,$
$\displaystyle-\mu\Delta\theta_{s}+(\mathbf{z}_{s}.\nabla)\theta_{s}=f_{2}\quad\mbox{in
}\Omega,$ $\displaystyle\mathbf{z}_{s}=0\quad\mbox{on
}\Gamma;\quad\frac{\partial\theta_{s}}{\partial\mathbf{n}}=0\mbox{ on
}\Gamma,$

where $\bm{\beta}\in\mathbf{L}^{\infty}(\Omega)$ is a vector. Then from [11,
Proposition 2.3], we know that for $\mathbf{f}_{1}\in\mathbf{L}^{2}(\Omega)$
and $f_{2}\in L^{2}(\Omega)$, there exists
$(\mathbf{z}_{s},\theta_{s},q_{s})\in\mathbf{V}_{0}^{1}(\Omega)\cap\mathbf{H}^{2}(\Omega)\times
H^{2}(\Omega)\times L^{2}(\Omega)/\mathbb{R}\cap H^{1}(\Omega).$

For $\lambda_{0}>0$, let us consider steady linearised Boussinesq System
(linearized around $(\mathbf{z}_{s},\theta_{s})$) with homogeneous boundary
condition :

(2) $\left.\begin{aligned}
&\lambda_{0}\mathbf{u}-\Delta\mathbf{u}+(\mathbf{u}.\nabla)\mathbf{z}_{s}+(\mathbf{z}_{s}.\nabla)\mathbf{u}+\nabla
p=\bm{\beta}\phi+\mathbf{f}_{1}\mbox{ in }\Omega,\\\ &\mbox{div
}\mathbf{u}=0\mbox{ in }\Omega;\quad\mathbf{u}=0\mbox{ on }\Gamma,\\\
&\lambda_{0}\phi-\Delta\phi+\mathbf{z}_{s}.\nabla\phi+\mathbf{u}.\nabla\theta_{s}=f_{2}\mbox{
in }\Omega,\\\ &\frac{\partial\phi}{\partial n}=0\mbox{ on
}\Gamma.\end{aligned}\right\\}$

In this section, for notational convenience, we will use
$\mathbf{z}_{s}=\mathbf{z}$ and $\theta_{s}=\theta$. Without loss of
generality we will assume that $\nu=\mu=1$.

######  Definition 3.1.

$(\mathbf{u},\phi)\in\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)$ is a weak
solution to equation (2) if for every
$(\mathbf{\Phi},\psi)\in\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)$ and
$(\mathbf{f}_{1},f_{2})\in\mathbf{L}^{2}(\Omega)\times L^{2}(\Omega)$ :

$\displaystyle\lambda_{0}\int_{\Omega}\mathbf{u}.\mathbf{\Phi}+\int_{\Omega}\nabla\mathbf{u}.\nabla\mathbf{\Phi}+\int_{\Omega}((\mathbf{u}.\nabla)\mathbf{z}).\mathbf{\Phi}+\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{u}).\mathbf{\Phi}-\int_{\Omega}\bm{\beta}\phi.\mathbf{\Phi}$
$\displaystyle+\lambda_{0}\int_{\Omega}\phi\psi+\int_{\Omega}\nabla\phi.\nabla\psi+\int_{\Omega}(\mathbf{z}.\nabla\phi)\psi+\int_{\Omega}(\mathbf{u}.\nabla\theta)\psi=\int_{\Omega}\mathbf{f}_{1}.\mathbf{\Phi}+\int_{\Omega}f_{2}\psi.$

We have the following existence theorem for weak solution :

######  Theorem 3.2.

Let $(\mathbf{z},\theta)\in\mathbf{V}^{1}(\Omega)\times H^{1}(\Omega)$. Then
there exists large $\lambda_{0}>0$ for which there exists unique weak solution
$(\mathbf{u},p,\phi)\in\mathbf{V}_{0}^{1}(\Omega)\times
L^{2}(\Omega)/\mathbb{R}\times H^{1}(\Omega)$ to equation (2) for all
$(\mathbf{f}_{1},f_{2})\in\mathbf{L}^{2}(\Omega)\times L^{2}(\Omega)$.

###### Proof.

Define the bilinear form on $\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)$ :

$\displaystyle a((\mathbf{u},\phi),(\mathbf{\Phi},\psi))=$
$\displaystyle\lambda_{0}\int_{\Omega}\mathbf{u.\Phi}+\int_{\Omega}\nabla\mathbf{u}.\nabla\mathbf{\Phi}+\int_{\Omega}((\mathbf{u}.\nabla)\mathbf{z}).\mathbf{\Phi}+\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{u}).\mathbf{\Phi}-\int_{\Omega}\bm{\beta}\phi.\mathbf{\Phi}$
$\displaystyle+\lambda_{0}\int_{\Omega}\phi\psi+\int_{\Omega}\nabla\phi.\nabla\psi+\int_{\Omega}(\mathbf{z}.\nabla\phi)\psi+\int_{\Omega}(\mathbf{u}.\nabla\theta)\psi$

This bilinear form is continuous on $\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)$. We want to show that it is coercive if $\lambda_{0}>0$ is
large. For that we have to consider

$\displaystyle a((\mathbf{u},\phi),(\mathbf{u},\phi))=$
$\displaystyle\lambda_{0}\int_{\Omega}|\mathbf{u}|^{2}+\int_{\Omega}|\nabla\mathbf{u}|^{2}+\int_{\Omega}((\mathbf{u}.\nabla)\mathbf{z}).\mathbf{u}+\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{u}).\mathbf{u}-\int_{\Omega}\bm{\beta}\phi.\mathbf{u}$
$\displaystyle+\lambda_{0}\int_{\Omega}\phi^{2}+\int_{\Omega}|\nabla\phi|^{2}+\int_{\Omega}(\mathbf{z}.\nabla\phi)\phi+\int_{\Omega}(\mathbf{u}.\nabla\theta)\phi.$

Now we will estimate term by term. Let us start with the third term and by
H$\ddot{o}$lder’s inequality :

$\displaystyle\int_{\Omega}|((\mathbf{u}.\nabla)\mathbf{z}).\mathbf{u}|\leq||\mathbf{u}||^{2}_{\mathbf{L}^{4}(\Omega)}||\mathbf{z}||_{H^{1}(\Omega)}.$

Now observe that :

$\displaystyle\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{u}).\mathbf{\Phi}=-\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{\Phi}).\mathbf{u}\,\,.$

Hence, we get :
$\displaystyle\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{u}).\mathbf{u}=0$. Due
to the similar reasoning, we also have :

$\displaystyle\int_{\Omega}(\mathbf{z}.\nabla\phi)\phi=0.$

Also by H$\ddot{o}$lder’s inequality we have :

$\displaystyle\int_{\Omega}\mathbf{\beta}\phi.\mathbf{u}$
$\displaystyle\leq||\bm{\beta}||_{\infty}(\int_{\Omega}\phi^{2})^{1/2}(\int_{\Omega}|\mathbf{u}|^{2})^{1/2}$
$\displaystyle\leq\frac{||\bm{\beta}||_{\infty}}{2}(\int_{\Omega}\phi^{2}+\int_{\Omega}|\mathbf{u}|^{2}).$

By using the fact that $H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$, the other
coupling term gives :

$\displaystyle\displaystyle\int_{\Omega}|(\mathbf{u}.\nabla\theta)\phi|$
$\displaystyle\leq||\theta||_{H^{1}(\Omega)}||\phi||_{L^{4}(\Omega)}||\mathbf{u}||_{\mathbf{L}^{4}(\Omega)}$
$\displaystyle\leq||\theta||_{H^{1}(\Omega)}||\phi||_{H^{1}(\Omega)}||\mathbf{u}||_{\mathbf{L}^{4}(\Omega)}.$

So, we obtain :

$\displaystyle a((\mathbf{u},\phi),(\mathbf{u},\phi))\geq\,$
$\displaystyle\lambda_{0}\int_{\Omega}|\mathbf{u}|^{2}+\int_{\Omega}|\nabla\mathbf{u}|^{2}-||\mathbf{u}||^{2}_{\mathbf{L}^{4}(\Omega)}||\mathbf{z}||_{\mathbf{H}^{1}(\Omega)}-\frac{||\bm{\beta}||_{\infty}}{2}(\int_{\Omega}\phi^{2}+\int_{\Omega}|\mathbf{u}|^{2})$
$\displaystyle+\lambda_{0}\int_{\Omega}\phi^{2}+\int_{\Omega}|\nabla\phi|^{2}-||\theta||_{H^{1}(\Omega)}||\phi||_{H^{1}(\Omega)}||\mathbf{u}||_{\mathbf{L}^{4}(\Omega)}.$

Now by using
$\mathbf{V}^{1}_{0}(\Omega)\hookrightarrow\mathbf{L}^{4}(\Omega)$, we will
have :

$\displaystyle||\mathbf{u}||^{2}_{\mathbf{L}^{4}(\Omega)}||\mathbf{z}||_{\mathbf{H}^{1}(\Omega)}$
$\displaystyle\leq
C||\mathbf{u}||^{2}_{\mathbf{V}^{1}_{0}(\Omega)}||\mathbf{z}||_{\mathbf{H}^{1}(\Omega)}$
$\displaystyle=C||\mathbf{u}||^{7/4}_{\mathbf{V}^{1}_{0}(\Omega)}||\mathbf{u}||^{1/4}_{\mathbf{V}^{1}_{0}(\Omega)}||\mathbf{z}||_{\mathbf{H}^{1}(\Omega)}.$

Thus we will get :

$\displaystyle-||\mathbf{u}||^{2}_{\mathbf{L}^{4}(\Omega)}||\mathbf{z}||_{\mathbf{H}^{1}(\Omega)}$
$\displaystyle\geq-C||\mathbf{u}||^{7/4}_{\mathbf{V}^{1}_{0}(\Omega)}\big{(}||\mathbf{z}||_{\mathbf{H}^{1}(\Omega)}||\mathbf{u}||^{1/4}_{\mathbf{V}_{n}^{0}(\Omega)}\big{)}$
$\displaystyle\geq-\frac{1}{4}||\mathbf{u}||^{2}_{\mathbf{V}^{1}_{0}(\Omega)}-\frac{7^{7}\times
C^{8}}{8\times
2^{7}}||\mathbf{z}||^{8}_{\mathbf{H}^{1}(\Omega)}||\mathbf{u}||^{2}_{\mathbf{V}^{0}_{n}(\Omega)},$

where the last inequality will come from Young’s inequality. Similarly, we can
estimate :

$\displaystyle||\theta||_{H^{1}(\Omega)}||\mathbf{u}||_{\mathbf{L}^{4}(\Omega)}||\phi||_{H^{1}(\Omega)}$
$\displaystyle\leq\frac{1}{2}\Big{(}||\theta||^{2}_{H^{1}(\Omega)}||\mathbf{u}||^{2}_{L^{4}(\Omega)}+||\phi||^{2}_{H^{1}(\Omega)}\Big{)}$
$\displaystyle=\frac{1}{2}||\theta||^{2}_{H^{1}(\Omega)}||\mathbf{u}||^{1/4}_{L^{4}(\Omega)}||\mathbf{u}||^{7/4}_{L^{4}(\Omega)}+\frac{1}{2}||\phi||^{2}_{H^{1}(\Omega)}$

Hence,

$\displaystyle-||\theta||_{H^{1}(\Omega)}||\phi||_{H^{1}(\Omega)}||\mathbf{u}||_{\mathbf{L}^{4}(\Omega)}$
$\displaystyle\geq-\frac{1}{2}||\phi||^{2}_{H^{1}(\Omega)}-C_{1}||\mathbf{u}||^{7/4}_{\mathbf{V}^{1}_{0}(\Omega)}||\theta||^{2}_{H^{1}(\Omega)}||\mathbf{u}||^{1/4}_{\mathbf{V}_{n}^{0}(\Omega)}$
$\displaystyle\geq-\frac{1}{2}||\phi||^{2}_{H^{1}(\Omega)}-\frac{1}{4}||\mathbf{u}||^{2}_{\mathbf{V}^{1}_{0}(\Omega)}-\frac{7^{7}\times
C_{1}^{8}}{8\times
2^{7}}||\theta||^{16}_{H^{1}(\Omega)}||\mathbf{u}||^{2}_{\mathbf{V}^{0}_{n}(\Omega)}$

Thus we will get :

$\displaystyle a((\mathbf{u},\phi),(\mathbf{u},\phi))\geq$
$\displaystyle\lambda_{0}\int_{\Omega}|\mathbf{u}|^{2}+\int_{\Omega}|\nabla\mathbf{u}|^{2}-\frac{1}{4}||\mathbf{u}||^{2}_{\mathbf{V}^{1}_{0}(\Omega)}-\frac{7^{7}\times
C^{8}}{8\times
2^{7}}||\mathbf{z}||^{8}_{\mathbf{H}^{1}(\Omega)}||\mathbf{u}||^{2}_{\mathbf{V}^{0}_{n}(\Omega)}$
$\displaystyle-\frac{||\bm{\beta}||_{\infty}}{2}(\int_{\Omega}\phi^{2}+\int_{\Omega}|\mathbf{u}|^{2})+\lambda_{0}\int_{\Omega}\phi^{2}+\int_{\Omega}|\nabla\phi|^{2}-\frac{1}{2}||\phi||^{2}_{H^{1}(\Omega)}$
$\displaystyle-\frac{1}{4}||\mathbf{u}||^{2}_{\mathbf{V}^{1}_{0}(\Omega)}-\frac{7^{7}\times
C_{1}^{8}}{8\times
2^{7}}||\theta||^{16}_{H^{1}(\Omega)}||\mathbf{u}||^{2}_{\mathbf{V}^{0}_{n}(\Omega)}.$

Now it is sufficient to choose $\lambda_{0}\geq\max\\{1+\frac{7^{7}\times
C^{8}}{8\times
2^{7}}||\mathbf{z}||^{8}_{\mathbf{H}^{1}(\Omega)}+\frac{||\bm{\beta}||_{\infty}}{2},\,1+\frac{7^{7}\times
C_{1}^{8}}{8\times
2^{7}}||\theta||^{16}_{H^{1}(\Omega)}+\frac{||\bm{\beta}||_{\infty}}{2}\\}$,
such that following holds:

$\displaystyle
a((\mathbf{u},\phi),(\mathbf{u},\phi))\geq\frac{1}{2}\Big{(}||\mathbf{u}||_{\mathbf{V}^{1}_{0}(\Omega)}^{2}+||\phi||_{H^{1}(\Omega)}^{2}\Big{)}.$

So by Lax-Milgram theorem we have existence of the weak solution
$(\mathbf{u},\phi)\in\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)$ to
equation (2) for all $(\mathbf{f}_{1},f_{2})\in\mathbf{L}^{2}(\Omega)\times
L^{2}(\Omega)$ . Then by applying de Rham theorem, we can recover pressure
$p\in L^{2}(\Omega)/\mathbb{R}$ as in the steady Stokes system in [6]. ∎

######  Theorem 3.3.

For all $(\mathbf{f}_{1},f_{2})\in\mathbf{L}^{2}(\Omega)\times L^{2}(\Omega)$,
the weak solution $(\mathbf{u},p,\phi)$ for system (2) given by theorem (3.2)
belongs to $\mathbf{V}^{2}(\Omega)\times\mathcal{H}^{1}(\Omega)\times
H^{2}(\Omega)$.

###### Proof.

We already know the existence of the solution
$(\mathbf{u},p,\phi)\in\mathbf{V}^{1}_{0}(\Omega)\times
L^{2}(\Omega)/\mathbb{R}\times H^{1}(\Omega)$ from theorem 3.2. So, consider

$\displaystyle\lambda_{0}\mathbf{u}-\Delta\mathbf{u}+(\mathbf{u}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\mathbf{u}+\nabla
p=\bm{\beta}\phi+\mathbf{f}_{1}\mbox{ in }\Omega$

Here as $(\bm{\beta}\phi+\mathbf{f}_{1})\in\mathbf{L}^{2}(\Omega)$, using the
regularity result [16, Appendix B, lemma B.1] we can conclude that
$(\mathbf{u},p)\in\mathbf{V}^{2}(\Omega)\times(\mathcal{H}^{1}(\Omega)/\mathbb{R})$.
Then consider the equation

$\displaystyle\lambda_{0}\phi-\Delta\phi+\mathbf{z}.\nabla\phi=f_{2}-\mathbf{u}.\nabla\theta\quad\mbox{
in }\Omega.$

Here
$\mathbf{u}\in\mathbf{H}^{2}(\Omega)\hookrightarrow\mathbf{L}^{\infty}(\Omega)$
for two and three dimensions. So, $\mathbf{u}.\nabla\theta\in L^{2}(\Omega)$.
By elliptic regularity we can conclude that $\phi\in H^{2}(\Omega)$. ∎

Now onwards we fix $\lambda_{0}$ as in theorem 3.2. We consider linearised
Boussinesq system with non homogeneous boundary data :

(3) $\left.\begin{aligned}
&\lambda_{0}\mathbf{u}-\Delta\mathbf{u}+(\mathbf{u}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\mathbf{u}+\nabla
p=\bm{\beta}\phi+\mathbf{f}_{1}\mbox{ in }\Omega,\\\ &\mbox{div
}\mathbf{u}=0\mbox{ in }\Omega;\quad\mathbf{u}=\mathbf{g}\mbox{ on }\Gamma,\\\
&\lambda_{0}\phi-\Delta\phi+\mathbf{z}.\nabla\phi+\mathbf{u}.\nabla\theta=f_{2}\mbox{
in }\Omega,\\\ &\frac{\partial\phi}{\partial n}=h\mbox{ on
}\Gamma.\end{aligned}\right\\}$

We first prove the existence of a unique solution when
$(\mathbf{g},h)\in\mathbf{V}^{3/2}(\Gamma)\times H^{1/2}(\Gamma)$. If
$\mathbf{g}\in\mathbf{V}^{3/2}(\Gamma)$, we can define the Dirichlet operator
as $D_{z}\mathbf{g}=(\mathbf{w},\pi)$ using the solution to equation:

$\displaystyle\lambda_{0}\mathbf{w}-\Delta\mathbf{w}+(\mathbf{w}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\mathbf{w}+\nabla{\pi}=0\mbox{
in }\Omega,$ (4) $\displaystyle\mbox{ div}\,\mathbf{w}=0\mbox{ in
}\Omega,\quad\mathbf{w}=\mathbf{g}\mbox{ on }\Gamma.$

Then $D_{z}$ is linear and continuous from $\mathbf{V}^{3/2}(\Gamma)$ to
$\mathbf{V}^{2}(\Omega)\times\mathcal{H}^{1}(\Omega)/\mathbb{R}$. For details
see [16, Appendix B, Corollary B.1].
Then consider $\widetilde{\mathbf{w}}=\mathbf{u-w},\widetilde{p}=p-\pi$. Here
$(\widetilde{\mathbf{w}},\widetilde{p})$ satisfies :

$\displaystyle\lambda_{0}\widetilde{\mathbf{w}}-\Delta\widetilde{\mathbf{w}}+(\widetilde{\mathbf{w}}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\widetilde{\mathbf{w}}+\nabla\widetilde{p}=\bm{\beta}\phi+\mathbf{f}_{1}\mbox{
in }\Omega,$ $\displaystyle\mbox{div }\widetilde{\mathbf{w}}=0\mbox{ in
}\Omega;\quad\widetilde{\mathbf{w}}=0\mbox{ on }\Gamma.$

Now system (3) can be reduced to the following system:

(5) $\left.\begin{aligned}
&\lambda_{0}\widetilde{\mathbf{w}}-\Delta\widetilde{\mathbf{w}}+(\widetilde{\mathbf{w}}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\widetilde{\mathbf{w}}+\nabla\widetilde{p}=\bm{\beta}\phi+\mathbf{f}_{1}\mbox{
in }\Omega,\\\ &\mbox{div }\widetilde{\mathbf{w}}=0\mbox{ in
}\Omega;\quad\widetilde{\mathbf{w}}=0\mbox{ on }\Gamma,\\\
&\lambda_{0}\phi-\Delta\phi+\mathbf{z}.\nabla\phi+\widetilde{\mathbf{w}}.\nabla\theta=f_{2}-\mathbf{w}.\nabla\theta=\widetilde{f_{2}}\mbox{
in }\Omega,\\\ &\frac{\partial\phi}{\partial n}=h\mbox{ on
}\Gamma.\end{aligned}\right\\}$

For this system with inhomogeneous boundary condition, we define a weak
solution in the same lines as in definition 3.1.

######  Definition 3.4.

Let $h\in H^{1/2}(\Gamma)$,
$(\mathbf{z},\theta)\in\mathbf{V}^{1}(\Omega)\times H^{1}(\Omega)$ and
$(\mathbf{f}_{1},f_{2})\in\mathbf{L}^{2}(\Omega)\times L^{2}(\Omega)$. Then
$(\widetilde{\mathbf{w}},\tau)\in\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)$ is a weak solution to equation (5) if for every
$(\mathbf{\Phi},\psi)\in\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)$, the
following holds:

$\displaystyle\lambda_{0}\int_{\Omega}\widetilde{\mathbf{w}}.\mathbf{\Phi}+\int_{\Omega}\nabla\widetilde{\mathbf{w}}.\nabla\mathbf{\Phi}+\int_{\Omega}((\widetilde{\mathbf{w}}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\widetilde{\mathbf{w}}).\mathbf{\Phi}-\int_{\Omega}\mathbf{\beta}\tau.\mathbf{\Phi}$
$\displaystyle+\lambda_{0}\int_{\Omega}\phi\psi+\int_{\Omega}\nabla\phi.\nabla\psi+\int_{\Omega}(\mathbf{z}.\nabla\phi)\psi+\int_{\Omega}(\widetilde{\mathbf{w}}.\nabla\theta)\psi=\int_{\Omega}\mathbf{f}_{1}.\mathbf{\Phi}+\int_{\Omega}\widetilde{f_{2}}\psi+\int_{\Gamma}h\psi$

######  Theorem 3.5.

Let $(\mathbf{z},\theta)\in\mathbf{V}^{1}(\Omega)\times H^{1}(\Omega)$. For
all $(\mathbf{f}_{1},f_{2})\in\mathbf{L}^{2}(\Omega)\times L^{2}(\Omega)$ and
$(\mathbf{g},h)\in\mathbf{V}^{3/2}(\Gamma)\times H^{1/2}(\Gamma)$, there
exists a weak solution $(\mathbf{u},p,\phi)\in\mathbf{V}^{1}(\Omega)\times
L^{2}(\Omega)\times H^{1}(\Omega)$ to equation (3). Moreover, we have
$(\mathbf{u},p,\phi)\in\mathbf{V}^{2}(\Omega)\times H^{1}(\Omega)\times
H^{2}(\Omega)$.

###### Proof.

Let $(\widetilde{\mathbf{w}},\widetilde{p},\phi)$ satisfy equation (5). As in
the previous existence theorem, we can define a continuous, coercive bilinear
form $a((\widetilde{\mathbf{w}},\phi),(\mathbf{\Phi},\psi))$ and continuous
linear form $L((\mathbf{\Phi},\psi))$ on $\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)$. By Lax-Milgram, we can conclude that
$(\widetilde{\mathbf{w}},\phi)\in\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)$ is a weak solution of (5). Hence by applying de Rham’s theorem,
we have
$(\widetilde{\mathbf{w}},\widetilde{p},\phi)\in\mathbf{V}^{1}_{0}(\Omega)\times
L^{2}(\Omega)\times H^{1}(\Omega)$.

Now we know that
$(\mathbf{u},p)=(\widetilde{\mathbf{w}}+\mathbf{w},\widetilde{p}+\pi)$, where
$(\mathbf{w},\pi)$ is the solution of (3) that belongs to
$\mathbf{V}^{2}(\Omega)\times H^{1}(\Omega)/\mathbb{R}$. So we can conclude
that $(\mathbf{u},p,\phi)\in\mathbf{V}^{1}(\Omega)\times L^{2}(\Omega)\times
H^{1}(\Omega)$.

Now for proving regularity, we consider
$(\mathbf{u},p)=(\mathbf{u}_{1}+\mathbf{u}_{2},p_{1}+p_{2})$ such that :

(6) $\left.\begin{aligned}
&\lambda_{0}\mathbf{u}_{1}-\Delta\mathbf{u}_{1}+\nabla p_{1}=0\mbox{ in
}\Omega,\,\mbox{ div}\,\mathbf{u}_{1}=0,\mbox{ in
}\Omega,\,\quad\mathbf{u}_{1}=\mathbf{g}\mbox{ on }\Gamma\\\
&\lambda_{0}\mathbf{u}_{2}-\Delta\mathbf{u}_{2}+(\mathbf{u}_{2}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\mathbf{u}_{2}+\nabla
p_{2}=-(\mathbf{u}_{1}.\nabla)\mathbf{z}-(\mathbf{z}.\nabla)\mathbf{u}_{1}+\mathbf{\beta}\phi+\mathbf{f}_{1}\mbox{
in }\Omega,\\\ &\mbox{div }\mathbf{u}_{2}=0\mbox{ in
}\Omega;\quad\mathbf{u}_{2}=0\mbox{ on }\Gamma.\end{aligned}\right\\}$

And $\phi=(\phi_{1}+\phi_{2})$ satisfies :

(7) $\left.\begin{aligned}
&\lambda_{0}\phi_{1}-\Delta\phi_{1}+(\mathbf{z}.\nabla)\phi_{1}=0\mbox{ in
}\Omega,\,\quad\frac{\partial\phi_{1}}{\partial n}=h\mbox{ on }\Gamma,\\\
&\lambda_{0}\phi_{2}-\Delta\phi_{2}+\mathbf{z}.\nabla\phi_{2}+\mathbf{u}_{2}.\nabla\theta=-\mathbf{u}_{1}.\nabla\theta+f_{2}\mbox{
in }\Omega,\\\ &\frac{\partial\phi_{2}}{\partial n}=0\mbox{ on
}\Gamma.\end{aligned}\right\\}$

Now the first equation of (6) is the Stokes Problem and we know that
$(\mathbf{u}_{1},p_{1})\in\mathbf{V}^{2}(\Omega)\times H^{1}(\Omega)$ if
$\mathbf{g}\in\mathbf{V}^{3/2}(\Gamma)$. Now the R.H.S of second equation in
(6) is in $\mathbf{L}^{2}(\Omega)$. Thus by [16, Appendix B, lemma B.1], we
can conclude $(\mathbf{u}_{2},p_{2})\in\mathbf{V}^{2}(\Omega)\times
H^{1}(\Omega)$. Hence, we can conclude that
$(\mathbf{u},p)\in\mathbf{V}^{2}(\Omega)\times H^{1}(\Omega)$.

Also from first equation of (7), we have $\phi_{1}\in H^{2}(\Omega)$ when
$h\in H^{1/2}(\Gamma)$. Then consider the equation :

$\displaystyle\lambda_{0}\phi_{2}-\Delta\phi_{2}+\mathbf{z}.\nabla\phi_{2}=-\mathbf{u}_{2}.\nabla\theta-\mathbf{u}_{1}.\nabla\theta+f_{2}\mbox{
in }\Omega$ $\displaystyle\frac{\partial\phi_{2}}{\partial n}=0\mbox{ on
}\Gamma$

Now the R.H.S of the above equation is in $L^{2}(\Omega)$. Thus we have
$\phi_{2}\in H^{2}(\Omega)$ . ∎

Now we consider the boundary data
$(\mathbf{g},h)\in\mathbf{V}^{-1/2}(\Gamma)\times H^{-3/2}(\Gamma)$. In this
case we can use transposition method to define a solution to equation (3). For
simplicity let us assume that $(\mathbf{f}_{1},f_{2})=(0,0)$.

######  Definition 3.6.

Assume that $(\mathbf{g},h)\in\mathbf{V}^{-1/2}(\Gamma)\times
H^{-3/2}(\Gamma).$ A function
$(\mathbf{u},p,\phi)\in\mathbf{V}^{0}(\Omega)\times\mathcal{H}^{-1}(\Omega)\times
L^{2}(\Omega)$ is a transposition solution to system (3) if

(8)
$\displaystyle\langle(\mathbf{u},\phi),(\mathbf{f}_{3},f_{4})\rangle_{\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega)}=\langle-\frac{\partial\mathbf{r}}{\partial\mathbf{n}}+\widetilde{\pi}\mathbf{n}-c(\widetilde{\pi})\mathbf{n},\mathbf{g}\rangle_{\mathbf{V}^{1/2}(\Gamma),\mathbf{V}^{-1/2}(\Gamma)}+\langle
s,h\rangle_{H^{3/2}(\Gamma),H^{-3/2}(\Gamma)}$

for every $(\mathbf{f}_{3},f_{4})\in\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega)$, where
$c(\widetilde{\pi})=\frac{1}{|\Gamma|}\int_{\Gamma}\widetilde{\pi}$ and
$(\mathbf{r},s)$ is the solution to :

(9) $\left.\begin{aligned}
&\lambda_{0}\mathbf{r}-\Delta\mathbf{r}+(\nabla\mathbf{z})^{T}\mathbf{r}-(\mathbf{z}.\nabla)\mathbf{r}+\nabla\tilde{\pi}+(\nabla\theta)^{T}s=\mathbf{f}_{3}\mbox{
in }\Omega,\\\ &\mbox{div }\mathbf{r}=0\mbox{ in
}\Omega;\quad\mathbf{r}=0\mbox{ on }\Gamma,\\\ &\lambda_{0}s-\Delta
s-\mathbf{z}.\nabla s-\bm{\beta.r}=f_{4}\mbox{ in }\Omega,\\\ &\frac{\partial
s}{\partial n}=0\mbox{ on }\Gamma,\end{aligned}\right\\}$

and

$\displaystyle\langle
p,k\rangle_{\mathcal{H}^{-1}(\Omega),\mathcal{H}^{1}(\Omega)}=\langle\frac{\partial\mathbf{r}_{1}}{\partial\mathbf{n}}-\pi_{1}\mathbf{n}+c(\pi_{1})\mathbf{n},\mathbf{g}\rangle_{\mathbf{V}^{1/2}(\Gamma),\mathbf{V}^{-1/2}(\Gamma)}-\langle
s_{1},h\rangle_{H^{3/2}(\Gamma),H^{-3/2}(\Gamma)}$

for every $k\in\mathcal{H}^{1}(\Omega)$ with $\int_{\Omega}k=0$ and
$(\mathbf{r}_{1},s_{1})$ is the solution to

(10) $\left.\begin{aligned}
&\lambda_{0}\mathbf{r}_{1}-\Delta\mathbf{r}_{1}+(\nabla\mathbf{z})^{T}\mathbf{r}_{1}-(\mathbf{z}.\nabla)\mathbf{r}_{1}+\nabla\pi_{1}+(\nabla\theta)^{T}s_{1}=0\mbox{
in }\Omega\\\ &\mbox{div }\mathbf{r}_{1}=k\mbox{ in
}\Omega;\quad\mathbf{r}_{1}=0\mbox{ on }\Gamma\\\ &\lambda_{0}s_{1}-\Delta
s_{1}-\mathbf{z}.\nabla s_{1}-\bm{\beta}.\mathbf{r}_{1}=0\mbox{ in }\Omega\\\
&\frac{\partial s_{1}}{\partial n}=0\mbox{ on }\Gamma\end{aligned}\right\\}$

######  Theorem 3.7.

For all $(\mathbf{g},h)\in\mathbf{V}^{-1/2}(\Gamma)\times H^{-3/2}(\Gamma)$,
system (3) admits a unique transposition solution
$(\mathbf{u},p,\phi)\in\mathbf{V}^{0}(\Omega)\times\mathcal{H}^{-1}(\Omega)\times
L^{2}(\Omega)$ in the sense of above definition.

###### Proof.

As $(\mathbf{f}_{3},f_{4})\in\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)$, by
theorem 3.3 we have
$(\mathbf{r},\tilde{\pi},s)\in\mathbf{V}^{2}(\Omega)\times\mathcal{H}^{1}(\Omega)\times
H^{2}(\Omega)$. Let us define the operator $\Lambda$ from
$\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)$ into
$\mathbf{V}^{1/2}(\Gamma)\times H^{3/2}(\Gamma)$ by :

$\displaystyle\Lambda(\mathbf{f}_{3},f_{4})=\Big{(}-\frac{\partial\mathbf{r}}{\partial\mathbf{n}}+\widetilde{\pi}\mathbf{n}-c(\widetilde{\pi})\mathbf{n},s\Big{)},$

where $(\mathbf{r},s)$ is solution to (9). Now we have :

$\displaystyle||\Lambda(\mathbf{f}_{3},f_{4})||_{\mathbf{V}^{1/2}(\Gamma)\times
H^{3/2}(\Gamma)}$
$\displaystyle=||\Big{(}-\frac{\partial\mathbf{r}}{\partial\mathbf{n}}+\widetilde{\pi}\mathbf{n}-c(\widetilde{\pi})\mathbf{n},s\Big{)}||_{\mathbf{V}^{1/2}(\Gamma)\times
H^{3/2}(\Gamma)}$ $\displaystyle\leq
C(||\mathbf{r}||_{\mathbf{V}^{2}(\Omega)}+||\tilde{\pi}||_{H^{1}(\Omega)}+||s||_{H^{2}(\Omega)})$
$\displaystyle\leq
C(||\mathbf{f}_{3}||_{\mathbf{V}^{0}(\Omega)}+||f_{4}||_{L^{2}(\Omega)})$
$\displaystyle=C||(\mathbf{f}_{3},f_{4})||_{\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega)}.$

Then it follows that $\Lambda$ is a bounded, linear operator. Then equation
(8) can be rewritten as :

$\displaystyle\langle(\mathbf{u},\phi),(\mathbf{f}_{3},f_{4})\rangle_{\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega)}$
$\displaystyle=\langle\Lambda(\mathbf{f}_{3},f_{4}),(\mathbf{g},h)\rangle_{\mathbf{V}^{1/2}(\Gamma)\times
H^{3/2}(\Gamma),\mathbf{V}^{-1/2}(\Gamma)\times H^{-3/2}(\Gamma)}$
$\displaystyle=\langle(\mathbf{f}_{3},f_{4}),\Lambda^{*}(\mathbf{g},h)\rangle_{\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega)}$

where $\Lambda^{*}$ is the adjoint of $\Lambda$ and
$\Lambda^{*}\in\mathcal{L}(\mathbf{V}^{-1/2}(\Gamma)\times
H^{-3/2}(\Gamma),\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega))$. The function
$(\mathbf{u},\phi)=\Lambda^{*}(\mathbf{g},h)$ is clearly a solution to system
(3) in the sense of definition 3.6.

Now we want to prove the existence of pressure $p$. For
$k\in\mathcal{H}^{1}(\Omega)$ with $\int_{\Omega}k=0$, the solution
$(\mathbf{r}_{1},s_{1})$ to equation (10) belongs to
$\mathbf{V}^{2}(\Omega)\times H^{2}(\Omega)$. The operator $\Lambda_{1}$ from
$\mathcal{H}^{1}(\Omega)$ to $\mathbf{V}^{\frac{1}{2}}(\Gamma))\times
H^{\frac{3}{2}}(\Gamma)$ can be defined as :

$\displaystyle\Lambda_{1}(k)=\Big{(}-\frac{\partial\mathbf{r}_{1}}{\partial\mathbf{n}}+\pi_{1}\mathbf{n}-c(\pi_{1})\mathbf{n},\,s_{1}\Big{)},$

where $(\mathbf{r}_{1},s_{1})$ is the solution to equation (10). It can be
proved similarly that the function

$\displaystyle p=\Lambda_{1}^{*}\begin{pmatrix}\mathbf{g}\\\
h\end{pmatrix}\in\mathcal{H}^{-1}(\Omega)$

is a solution to equation (3) in the sense of definition 3.6.

To prove uniqueness, let $(\mathbf{u},\phi)$ be a solution corresponding to
$(\mathbf{g},h)=(0,0).$ Thus from equation (8), for all
$(\mathbf{f}_{3},f_{4})\in\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)$, we have
:

$\displaystyle\langle(\mathbf{u},\phi),(\mathbf{f}_{3},f_{4})\rangle_{\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega)}=0.$

Hence, we will get $(\mathbf{u},\phi)=(0,0)$. ∎

Our next aim is to define lifting operator $L_{\mathbf{z}}$ for the coupled
system (3). We denote by $L_{\mathbf{z}}(\mathbf{g},h)=(\mathbf{w},p,\xi)$ and
it satisfies the equation :

$\displaystyle\lambda_{0}\mathbf{w}-\nu\Delta\mathbf{w}+(\mathbf{w}.\nabla)\mathbf{z}+(\mathbf{z}.\nabla)\mathbf{w}+\nabla
p=\bm{\beta}\xi\mbox{ in }\Omega,$ $\displaystyle\mbox{div }\mathbf{w}=0\mbox{
in }\Omega;\quad\mathbf{w}=\mathbf{g}\mbox{ on }\Gamma,$
$\displaystyle\lambda_{0}\xi-\mu\Delta\xi+\mathbf{z}.\nabla\xi+\mathbf{w}.\nabla\theta=0\mbox{
in }\Omega,$ $\displaystyle\frac{\partial\xi}{\partial n}=h\mbox{ on }\Gamma.$

######  Theorem 3.8.

The operator $L_{\mathbf{z}}$ is linear and continuous from
$\mathbf{V}^{s+1}(\Gamma)\times H^{s}(\Gamma)$ into
$\mathbf{V}^{s+3/2}(\Omega)\times\mathcal{H}^{s+1/2}(\Omega)/\mathbb{R}\times
H^{s+3/2}(\Omega)$ for all $-3/2\leq s\leq 1/2$.

###### Proof.

Due to theorem 3.5, $L_{\mathbf{z}}$ is continuous from
$\mathbf{V}^{3/2}(\Gamma)\times H^{1/2}(\Gamma)$ to
$\mathbf{V}^{2}(\Omega)\times H^{2}(\Omega).$ From theorem 3.7,
$L_{\mathbf{z}}$ is also a bounded operator from
$\mathbf{V}^{-1/2}(\Gamma)\times H^{-3/2}(\Gamma)$ into
$\mathbf{V}^{0}(\Omega)\times(H^{1}(\Omega)/\mathbb{R})^{\prime}\times
L^{2}(\Omega)$. Then our required result follows from interpolation theorem
[12, Chapter 1, theorem 9.6]. ∎

##  4\. Unsteady linearized Boussinesq system

###  4.1. Linearization around stationary state

In this section we study first the linearized Boussinesq system around zero
solution and later around nonzero solution with non homogeneous boundary
conditions.

####  4.1.1. Linearization around zero solution

Here we are mainly interested in the following equation :

(11) $\left.\begin{aligned} &\frac{\partial\mathbf{u}}{\partial
t}-\Delta\mathbf{u}+\nabla p=\bm{\beta}\phi\mbox{ in }Q,\\\ &\mbox{div
}\mathbf{u}=0\mbox{ in }Q;\quad\mathbf{u}=\mathbf{g}\mbox{ on }\Sigma,\\\
&\mathbf{u}(x,0)=\mathbf{u}_{0}(x)\mbox{ in }\Omega,\\\
&\frac{\partial\phi}{\partial t}-\Delta\phi=0\mbox{ in }Q,\\\
&\frac{\partial\phi}{\partial n}=h\mbox{ on
}\Sigma,\quad\phi(0)=\phi_{0}\mbox{ in }\Omega.\end{aligned}\right\\}$

If the boundary data is regular then there are some known existence and
regularity results for system (1). For instance, in [13], the author proved
that if $\mathbf{g}\in C^{1}(\Sigma)$ and $h\in L^{2}(\Sigma)$, then
$(\mathbf{z},p,\theta)\in L^{\infty}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))\times L^{\infty}(0,T;L^{2}(\Omega))$. So, our
interest lies in the case when the boundary data are not in such regular
spaces.

We want to define a solution to equation (11) when the boundary data
$(\mathbf{g},h)$ belongs to $L^{2}(0,T;\mathbf{V}^{-1/2}(\Gamma))\times
L^{2}(0,T;H^{-3/2}(\Gamma))$. In this case, we are going to define the
solution via transposition method.

######  Definition 4.1.

Assume that $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{-1/2}(\Gamma))\times
L^{2}(0,T;H^{-3/2}(\Gamma))$ and
$(\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{{}^{\prime}}$.
A function $(\mathbf{u},p,\phi)\in L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;\mathcal{H}^{-1}(\Omega))\times\\\ L^{2}(0,T;L^{2}(\Omega))$ is a
solution to equation (11) in the sense of transposition iff

$\displaystyle\int_{Q}\mathbf{u.f}_{3}+\int_{Q}\phi f_{4}=$
$\displaystyle\int_{0}^{T}\langle-\frac{\partial\mathbf{r}}{\partial\mathbf{n}}(t)+\widetilde{\pi}(t)\mathbf{n}-c(\widetilde{\pi})\mathbf{n},\mathbf{g}(t)\rangle_{\mathbf{V}^{1/2}(\Gamma),\mathbf{V}^{-1/2}(\Gamma)}+$
$\displaystyle\langle
s(t),h(t)\rangle_{H^{3/2}(\Gamma),H^{-3/2}(\Gamma)}+\langle\mathbf{u}_{0},\mathbf{r}(0)\rangle_{\mathbf{H}^{-1}(\Omega),\mathbf{H}_{0}^{1}(\Omega)}+$
(12)
$\displaystyle\langle\phi_{0},s(0)\rangle_{(H^{1}(\Omega))^{\prime},H^{1}(\Omega)},$

where $c(\tilde{\pi})=\frac{1}{|\Gamma|}\int_{\Gamma}\widetilde{\pi}$,
$(\mathbf{f}_{3},f_{4})\in L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega)),$ and $(\mathbf{r},s)$ is the solution to the
equation

(13) $\left.\begin{aligned} &-\frac{\partial\mathbf{r}}{\partial
t}-\Delta\mathbf{r}+\nabla\tilde{\pi}=\mathbf{f}_{3}\mbox{ in }Q\\\ &\mbox{div
}\mathbf{r}=0\mbox{ in }Q;\quad\mathbf{r}=0\mbox{ on }\Sigma\\\
&\mathbf{r}(T)=0\mbox{ in }\Omega\\\ &-\frac{\partial s}{\partial t}-\Delta
s-\mathbf{\beta.r}=f_{4}\mbox{ in }Q\\\ &\frac{\partial s}{\partial n}=0\mbox{
on }\Sigma;\quad s(T)=0\mbox{ in }\Omega\end{aligned}\right\\}$

and

$\displaystyle\int_{0}^{T}\langle
k,p\rangle_{H^{1}(\Omega),\mathcal{H}^{1}(\Omega)}=\int_{0}^{T}\langle\frac{\partial\mathbf{r}_{1}}{\partial\mathbf{n}}(t)-\pi_{1}(t)\mathbf{n}+c(\pi_{1})\mathbf{n},\mathbf{g}(t)\rangle_{\mathbf{V}^{1/2}(\Gamma),\mathbf{V}^{-1/2}(\Gamma)}$
(14) $\displaystyle-\int_{0}^{T}\langle
s_{1}(t),h(t)\rangle_{H^{3/2}(\Gamma),H^{-3/2}(\Gamma)}-\langle\mathbf{u}_{0},\mathbf{r}_{1}(0)\rangle_{\mathbf{H}^{-1}(\Omega),\mathbf{H}_{0}^{1}(\Omega)}-\langle\phi_{0},s(0)\rangle_{(H^{1}(\Omega))^{\prime},H^{1}(\Omega)},$

for every $k\in L^{2}(0,T;H^{1}(\Omega))\cap
H^{\frac{3}{4}}(0,T;H^{-\frac{1}{2}}(\Omega))$ with $\int_{Q}k=0$ and
$(\mathbf{r}_{1},s_{1})$ is the solution to

(15) $\left\\{\begin{aligned} &-\frac{\partial\mathbf{r}_{1}}{\partial
t}-\Delta\mathbf{r}_{1}+\nabla\pi_{1}=0\mbox{ in }Q\\\ &\mbox{div
}\mathbf{r}_{1}=k\mbox{ in }Q;\quad\mathbf{r}_{1}=0\mbox{ on
}\Sigma,\quad\mathbf{r}_{1}(T)=0\mbox{ in }\Omega\\\ &-\frac{\partial
s_{1}}{\partial t}-\Delta s_{1}-\mathbf{\beta}.\mathbf{r}_{1}=0\mbox{ in }Q\\\
&\frac{\partial s_{1}}{\partial n}=0\mbox{ on }\Sigma,\quad s_{1}(T)=0\mbox{
in }\Omega\end{aligned}\right.$

######  Theorem 4.2.

For all $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{-1/2}(\Gamma))\times
L^{2}(0,T;H^{-3/2}(\Gamma))$ and
$(\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{\prime}$,
equation (11) admits a unique transposition solution
$(\mathbf{u},p,\phi)\in L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;\mathcal{H}^{-1}(\Omega))\times L^{2}(0,T;L^{2}(\Omega))$ in the
sense of
above definition.

###### Proof.

Let $(\mathbf{f}_{3},f_{4})\in L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))$, the solution $(\mathbf{r},s)$ to equation (13)
belongs to $\mathbf{V}^{2,1}(Q)\times L^{2}(0,T;H^{2}(\Omega))$. Let us define
the operator $\Lambda$ from $L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))$ into the space
$L^{2}(0,T;\mathbf{V}^{\frac{1}{2}}(\Gamma))\times
L^{2}(0,T;H^{\frac{3}{2}}(\Gamma))\times\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)$ by :

$\displaystyle\Lambda(\mathbf{f}_{3},f_{4})=\Big{(}-\frac{\partial\mathbf{r}}{\partial\mathbf{n}}+\widetilde{\pi}\mathbf{n}-c(\widetilde{\pi})\mathbf{n},\,s,\,\mathbf{r}(0),\,s(0)\Big{)},$

where $(\mathbf{r},s)$ is the solution to equation (13). Now, we can rewrite
equation (4.1) in the form :

(16) $\displaystyle\Big{\langle}\begin{pmatrix}\mathbf{u}\\\
\phi\end{pmatrix},\begin{pmatrix}\mathbf{f}_{3}\\\
f_{4}\end{pmatrix}\Big{\rangle}_{L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))}=\Big{\langle}\Lambda\begin{pmatrix}\mathbf{f}_{3}\\\
f_{4}\end{pmatrix},\begin{pmatrix}\mathbf{g}\\\ h\\\ \mathbf{u}_{0}\\\
\phi_{0}\end{pmatrix}\Big{\rangle}_{1}\,\,,$

where, $\Big{\langle}.,.\Big{\rangle}_{1}$ is the duality product between
${L^{2}(0,T;\mathbf{V}^{\frac{1}{2}}(\Gamma))\times
L^{2}(0,T;H^{\frac{3}{2}}(\Gamma))\times\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)}$ and its dual. We can check that $\Lambda$ is a linear, bounded
operator. Also $\Lambda^{*}$, the adjoint of $\Lambda$ is bounded and it is
from $L^{2}(0,T;\mathbf{V}^{-\frac{1}{2}}(\Gamma))\times
L^{2}(0,T;H^{-\frac{3}{2}}(\Gamma))\times\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{\prime}$
into $L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times L^{2}(0,T;L^{2}(\Omega))$. Thus
we have :

(17) $\displaystyle\Big{\langle}\Lambda\begin{pmatrix}\mathbf{f}_{3}\\\
f_{4}\end{pmatrix},\begin{pmatrix}\mathbf{g}\\\ h\\\ \mathbf{u}_{0}\\\
\phi_{0}\end{pmatrix}\Big{\rangle}_{1}=\Big{\langle}\begin{pmatrix}\mathbf{f}_{3}\\\
f_{4}\end{pmatrix},\Lambda^{*}\begin{pmatrix}\mathbf{g}\\\ h\\\
\mathbf{u}_{0}\\\
\phi_{0}\end{pmatrix}\Big{\rangle}_{L^{2}(0,T;\mathbf{V}^{0}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))}.$

From (16) and (17), it is clear that the function

$\displaystyle\begin{pmatrix}\mathbf{u}\\\
\phi\end{pmatrix}=\Lambda^{*}\begin{pmatrix}\mathbf{g}\\\ h\\\
\mathbf{u}_{0}\\\ \phi_{0}\end{pmatrix}$

is a solution to equation (11) in the sense of definition 4.1.

Existence of pressure $p$ can also be proved in the same way as above. For
$k\in L^{2}(0,T;\mathcal{H}^{1}(\Omega))$ with $\int_{Q}k=0$, the solution
$(\mathbf{r}_{1},s_{1})$ to equation (15) belongs to
$\mathbf{V}^{2,1}(Q)\times L^{2}(0,T;H^{2}(\Omega))$. The operator
$\Lambda_{1}:$

$\displaystyle L^{2}(0,T;\mathcal{H}^{1}(\Omega))\mapsto
L^{2}(0,T;\mathbf{V}^{\frac{1}{2}}(\Gamma))\times
L^{2}(0,T;H^{\frac{3}{2}}(\Gamma))\times\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)$

can be defined as :

$\displaystyle\Lambda_{1}(k)=\Big{(}-\frac{\partial\mathbf{r}_{1}}{\partial\mathbf{n}}+\pi_{1}\mathbf{n}-c(\pi_{1})\mathbf{n},\,s_{1},\,\mathbf{r}_{1}(0),\,s_{1}(0)\Big{)},$

where $(\mathbf{r}_{1},s_{1})$ is the solution to equation (15). It can be
proved similarly that the function

$\displaystyle p=\Lambda_{1}^{*}\begin{pmatrix}\mathbf{g}\\\ h\\\
\mathbf{u}_{0}\\\ \phi_{0}\end{pmatrix}\in
L^{2}(0,T;\mathcal{H}^{-1}(\Omega))$

is a solution to equation (11) in the sense of definition 4.1.

To prove uniqueness, let $(\mathbf{u},\phi)$ be a solution corresponds to
$(\mathbf{g},h,\mathbf{u}_{0},\phi_{0})=(0,0,0,0)$. Then for all
$(\mathbf{f}_{3},f_{4})\in L^{2}(0,T;\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega))$, by using equation (4.1) we will get :

$\displaystyle\langle(\mathbf{u},\phi),(\mathbf{f}_{3},f_{4})\rangle_{L^{2}(0,T;\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega))}=0.$

Hence we have $(\mathbf{u},\phi)=(0,0).$ ∎

Now we want to write equation (11) in the operator form when boundary data are
regular.

######  Lemma 4.3.

Let $(\mathbf{g},h)\in C^{1}(0,T;\mathbf{V}^{3/2}(\Gamma))\times
C^{1}(0,T;H^{1/2}(\Gamma))$. Then $(\mathbf{u},\phi)$, the solution of (11)
satisfies the following equation:

(18) $\displaystyle\widetilde{P}\mathbf{u_{\phi}}(t)$
$\displaystyle=e^{t\mathcal{A}}\widetilde{P}\mathbf{u_{\phi}}(0)+\int_{0}^{t}(-\mathcal{A})e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)ds$

where $\widetilde{P}=\begin{pmatrix}P&0\\\
0&I\end{pmatrix},\,\,\mathbf{u_{\phi}}(t)=\begin{pmatrix}\mathbf{u}(t)\\\
\phi(t)\end{pmatrix},\,\,e^{t\mathcal{A}}\mbox{ is the semigroup generated by
}\mathcal{A}=\begin{pmatrix}P\Delta&A_{2}\\\ 0&\Delta\end{pmatrix}$ with
$A_{2}(\tau)=P(\beta\tau)$ and $(\mathbf{w},\pi,\psi)=L_{0}(\mathbf{g},h)$
satisfies the following system:

(19) $\left.\begin{aligned}
&-\Delta\mathbf{w}(t)+\nabla\pi(t)=\bm{\beta}\psi(t)\mbox{ in }\Omega\\\
&\mbox{div }\mathbf{w}(t)=0\mbox{ in
}\Omega;\quad\mathbf{w}(t)=\mathbf{g}(t)\mbox{ on }\Gamma,\\\
&-\Delta\psi(t)=0\mbox{ in }\Omega\mbox{ and }\,\,\frac{\partial\psi}{\partial
n}(t)=h(t)\mbox{ on }\Gamma.\end{aligned}\right\\}$

###### Proof.

Let $(\mathbf{w},\pi,\psi)=L_{0}(\mathbf{g},h)$ satisfy system (19). Thus we
will have $(\mathbf{w},\pi,\psi)\in C^{1}(0,T;\mathbf{V}^{2}(\Omega))\times
C^{1}(0,T;H^{1}(\Omega)/\mathbb{R})\times C^{1}(0,T;H^{2}(\Omega))$. Let us
define $\mathbf{y}=\mathbf{u-w},q=p-\pi$ and $\tau=\phi-\psi$ and we will get
:

(20) $\left.\begin{aligned} &\frac{\partial\mathbf{y}}{\partial
t}-\Delta\mathbf{y}+\nabla q=\bm{\beta}\tau-\frac{\partial\mathbf{w}}{\partial
t}\mbox{ in }Q,\\\ &\mbox{div }\mathbf{y}=0\mbox{ in
}Q;\quad\mathbf{y}=0\mbox{ on }\Sigma,\\\
&\mathbf{y}(0)=\mathbf{u}_{0}-\mathbf{w}(0)\mbox{ in }\Omega,\\\
&\frac{\partial\tau}{\partial t}-\Delta\tau=-\frac{\partial\psi}{\partial
t}\mbox{ in }Q,\\\ &\frac{\partial\tau}{\partial n}=0\mbox{ on
}\Sigma,\quad\tau(0)=\phi_{0}-\psi(0)\mbox{ in }\Omega.\end{aligned}\right\\}$

Now observe that $(\mathbf{y},q,\tau)\in
W(0,T;\mathbf{V}^{1}_{0}(\Omega),\mathbf{V}^{-1}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))\times L^{2}(0,T;H^{2}(\Omega))$. Apply Leray
projector $P$ in the first equation of (20) and by using
$P\mathbf{y}(t)=\mathbf{y}(t)$, we will get :

$\displaystyle\mathbf{y}^{\prime}(t)$
$\displaystyle=A\mathbf{y}(t)+A_{2}\tau(t)-P\mathbf{w}^{\prime}(t),$
$\displaystyle\mathbf{y}(0)$ $\displaystyle=P(\mathbf{u}_{0}-\mathbf{w}(0)),$

where
$A:D(A)=\mathbf{V}^{2}(\Omega)\cap\mathbf{V}^{1}_{0}(\Omega)\mapsto\mathbf{V}^{0}_{n}(\Omega)$
is defined by : $A\mathbf{y}=P\Delta\mathbf{y}$ and
$A_{2}:L^{2}(\Omega)\mapsto\mathbf{V}^{0}_{n}(\Omega)$ is given by
$A_{2}\tau=P(\bm{\beta}\tau)$. So we can rewrite system (20) as :

$\displaystyle\widehat{\mathbf{z}}^{\prime}(t)$
$\displaystyle=\mathcal{A}\widehat{\mathbf{z}}(t)-\mathbf{b}^{\prime}(t),$
$\displaystyle\widehat{\mathbf{z}}(0)$
$\displaystyle=\widehat{\mathbf{z}}_{0},$

where

$\widehat{\mathbf{z}}(t)=\begin{pmatrix}\mathbf{y}(t)\\\
\tau(t)\end{pmatrix},\mathbf{b}(t)=\begin{pmatrix}P\mathbf{w}(t)\\\
\psi(t)\end{pmatrix}$

Now we can write $\begin{pmatrix}P\Delta&P(\bm{\beta}\tau)\\\
0&\Delta\end{pmatrix}=\begin{pmatrix}P\Delta&0\\\
0&\Delta\end{pmatrix}+\begin{pmatrix}0&A_{2}\\\
0&\Delta\end{pmatrix}=\mathcal{A}_{1}+\mathcal{A}_{2}$. Clearly,
$\mathcal{A}_{1}$ will generate analytic semigroup in
$\mathbf{V}^{0}_{n}(\Omega)\times L^{2}(\Omega)$. Observe that $\mathcal{A}$
is a bounded perturbation of $\mathcal{A}_{1}$. Hence it will also generate an
analytic semigroup. Denoting the semigroup generated by $\mathcal{A}$ as
$e^{t\mathcal{A}}$, the solution can be written as :

$\displaystyle\widehat{\mathbf{z}}(t)$
$\displaystyle=e^{t\mathcal{A}}\widehat{\mathbf{z}}_{0}-\int_{0}^{t}e^{(t-s)\mathcal{A}}\mathbf{b}^{\prime}(s)ds$
$\displaystyle=e^{t\mathcal{A}}(\widehat{\mathbf{z}}_{0}+\mathbf{b}(0))-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\mathbf{b}(s)ds-\mathbf{b}(t).$

So we have :

$\displaystyle\widehat{\mathbf{z}}(t)+\mathbf{b}(t)=e^{t\mathbf{A}}(\widehat{\mathbf{z}}_{0}+\mathbf{b}(0))-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\mathbf{b}(s)ds.$

Thus we obtain

$\begin{pmatrix}P\mathbf{u}(t)\\\
\phi(t)\end{pmatrix}=e^{t\mathcal{A}}\begin{pmatrix}P\mathbf{u}(0)\\\
\phi(0)\end{pmatrix}-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\begin{pmatrix}P\mathbf{w}(s)\\\
\psi(s)\end{pmatrix}ds.$

So,

$\displaystyle\widetilde{P}\mathbf{u}_{\phi}(t)=e^{t\mathcal{A}}\widetilde{P}\mathbf{u}_{\phi}(0)+\int_{0}^{t}(-\mathcal{A})e^{(t-s)\mathcal{A}}\widetilde{P}L_{D,N}(\mathbf{g},h)(s)ds.$

∎

######  Remark 1.

If we just differentiate equation (18) formally, then we can get

$\displaystyle\widetilde{P}\mathbf{u_{\phi}}^{\prime}=\mathcal{A}\widetilde{P}\mathbf{u_{\phi}}+(-\mathcal{A})\widetilde{P}L_{0}(\mathbf{g},h).$

But to make sense of the term $\mathcal{A}\widetilde{P}\mathbf{u_{\phi}}$, we
want to extend the operator $\mathcal{A}$ to $\widetilde{\mathcal{A}}$ such
that $\widetilde{P}\mathbf{u_{\phi}}$ belongs to the domain of
$\widetilde{\mathcal{A}}$. We can extend the operator $\mathcal{A}$ to an
unbounded operator $\widetilde{\mathcal{A}}$ with domain
$D(\widetilde{A})=\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$ in
$(D(\mathcal{A^{*}}))^{\prime}$. [ See [2] for details ]

If the boundary data $\mathbf{g}.\mathbf{n}\neq 0$, then it is not possible to
write equation (11) in the following operator form :

$\displaystyle\mathbf{u_{\phi}}^{\prime}$
$\displaystyle=\widetilde{\mathcal{A}}\mathbf{u_{\phi}}+(-\widetilde{\mathcal{A}})L_{0}(\mathbf{g},h)\mbox{
with }\mathbf{u_{\phi}}(0)=\begin{pmatrix}\mathbf{u}_{0}\\\
\phi_{0}\end{pmatrix},$

as we look for solution $\mathbf{u_{\phi}}$ in the space
$\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)$ but the operator
$\widetilde{\mathcal{A}}$ is defined on $\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$. To overcome this difficulty we split
$\mathbf{u}_{\phi}=\widetilde{P}\mathbf{u}_{\phi}+(I-\widetilde{P})\mathbf{u}_{\phi}$,
where an evolution equation is satisfied by $\widetilde{P}\mathbf{u}$ and a
quasi-stationary equation is satisfied by
$(I-\widetilde{P})\mathbf{u}_{\phi}$.

Now we are in a position to state a new definition of weak solution involving
$\widetilde{P}\mathbf{u}_{\phi}$ and $(I-\widetilde{P})\mathbf{u}_{\phi}$ :

######  Definition 4.4.

Assume that $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$ and
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$.
A function $(\mathbf{u},\phi)\in L^{2}(0,T;V^{0}(\Omega)\times L^{2}(\Omega))$
is a weak solution to equation (11) if $\widetilde{P}\mathbf{u_{\phi}}$ is a
weak solution of the following evolution system :

(21) $\displaystyle\widetilde{P}\mathbf{u_{\phi}}^{\prime}$
$\displaystyle=\widetilde{\mathcal{A}}\widetilde{P}\mathbf{u_{\phi}}+(-\widetilde{\mathcal{A}})\widetilde{P}L_{0}(\mathbf{g},h),\mbox{
with }\widetilde{P}\mathbf{u_{\phi}}(0)=\widetilde{P}\mathbf{u_{\phi}}^{0},$
(22) $\displaystyle\mbox{ and }(I-\widetilde{P})\mathbf{u_{\phi}}(.)$
$\displaystyle=(I-\widetilde{P})L_{0}(\mathbf{g}(.),h(.)).$

where $\widetilde{\mathcal{A}}$ is extension of the operator $\mathcal{A}$ as
discussed in remark 1.

Now by definition of weak solution of an evolution equation, a function
$\widetilde{P}\mathbf{u_{\phi}}\in L^{2}(0,T;V_{n}^{0}(\Omega)\times
L^{2}(\Omega))$ is a weak solution to (21) iff for all
$(\mathbf{\Phi},\psi)\in D(\mathcal{A}^{*})$, the mapping
$t\mapsto\int_{\Omega}(P\mathbf{u}(t)\mathbf{\Phi}+\phi(t)\psi)$ belongs to
$H^{1}(0,T)$ and satisfies

(23)
$\displaystyle\frac{d}{dt}\int_{\Omega}(P\mathbf{u}(t)\mathbf{\Phi}+\phi(t)\psi)=\langle(P\mathbf{u}(t)+\phi(t)\psi),\mathcal{A}^{*}(\mathbf{\Phi},\psi)\rangle_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$
$\displaystyle+\langle(-\widetilde{\mathcal{A})}\widetilde{P}L_{0}(\mathbf{g}(t),h(t)),(\mathbf{\Phi},\psi)\rangle_{(D(A^{*}))^{\prime},D(A^{*})}.$

######  Theorem 4.5.

(i) For all $(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$ and $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$, equation (21)-(22) admits a unique
solution $(P\mathbf{u},\phi)$ belongs to
$\mathbf{V}^{1/2-\epsilon,1/4-\epsilon/2}(Q)\times
H^{1/2-\epsilon,1/4-\epsilon/2}(Q)$ for any $\epsilon>0$. This solution
$(P\mathbf{u},\phi)=\widetilde{P}\mathbf{u}_{\phi}$ and
$(I-\widetilde{P})\mathbf{u}_{\phi}$ satisfies :

$\displaystyle||\widetilde{P}\mathbf{u}_{\phi}||_{L^{2}(0,T;\mathbf{V}_{n}^{1/2-\epsilon}(\Omega)\times
H^{1/2-\epsilon}(\Omega))}+$
$\displaystyle||\widetilde{P}\mathbf{u}_{\phi}||_{H^{1/4-\epsilon/2}(0,T;\mathbf{V}^{0}(\Omega)\times
L^{2}(\Omega))}+||(\widetilde{I}-\widetilde{P})\mathbf{u}_{\phi}||_{L^{2}(0,T;\mathbf{V}^{1/2}(\Omega)\times
H^{1/2}(\Omega))}$ (24) $\displaystyle\leq
C\left(||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}+||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})}\right),\quad\,\forall\,\epsilon>0.$

(ii) For all $(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)$ and $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{2}(\Gamma))\cap
H^{1}(0,T;\mathbf{V}^{0}(\Gamma))\times L^{2}(0,T;H^{1}(\Gamma))\cap
H^{1}(0,T;(H^{1}(\Gamma))^{\prime})$, equation (21)-(22) admits a unique weak
solution $(P\mathbf{u},\phi)\in\mathbf{V}^{5/2-\epsilon,5/4-\epsilon/2}\times
H^{5/2-\epsilon,5/4-\epsilon/2}$ under the compatibility condition :

(25)
$\displaystyle\left.\left(\widetilde{P}\left[(\mathbf{u}_{0},\phi_{0})-L_{0}(\mathbf{g}(0),h(0))\right]\right)\right|_{\Gamma}=0.$

(iii) Let $0\leq s<1,$ for all
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{0\vee(s-1/2)}(\Omega)\times
H^{0\vee(s-1/2)}(\Omega)$ and
$(\mathbf{g},h)\in\mathbf{V}^{s,s/2}(\Sigma)\times(L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap
H^{s/2}(0,T;((H^{1}(\Gamma))^{\prime})))$, equation (21)-(22) admits a unique
weak solution
$(P\mathbf{u},\phi)\in\mathbf{V}^{s+1/2-\epsilon,s/2+1/4-\epsilon/2}(Q)\times
H^{s+1/2-\epsilon,s/2+1/4-\epsilon/2}(Q)$.

(iv) Let $1<s\leq 2,$ for all
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{(s-1/2)}(\Omega)\times
H^{(s-1/2)}(\Omega)$ and
$(\mathbf{g},h)\in\mathbf{V}^{s,s/2}(\Sigma)\times(L^{2}(0,T;H^{s-1}(\Gamma))\cap
H^{s/2}(0,T;((H^{1}(\Gamma))^{\prime})))$, equation (21)-(22) admits a unique
weak solution
$(P\mathbf{u},\phi)\in\mathbf{V}^{s+1/2-\epsilon,s/2+1/4-\epsilon/2}(Q)\times
H^{s+1/2-\epsilon,s/2+1/4-\epsilon/2}(Q)$ under the compatibility condition
(25).

###### Proof.

$(i)$ We will follow the technique of proof used in [16] for the Stokes
equation. We have :

(26)
$\widetilde{P}\mathbf{u}_{\phi}(t)=e^{t\mathcal{A}}\widetilde{P}\mathbf{u}_{\phi}(0)-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)(s)ds.$

Our aim is to prove
$\widetilde{P}\mathbf{u}_{\phi}(t)\in\mathbf{V}_{n}^{1/2-\epsilon}(\Omega)\times
H^{1/2-\epsilon}(\Omega)$. Let us consider $0<\epsilon\leq\frac{1}{2}$. Now
from [2, Chapter II.1.6, proposition 6.1], we know that
$D((-\mathcal{A})^{1/4-\epsilon/4})\simeq\mathbf{V}^{1/2-\epsilon/2}(\Omega)\times
H^{1/2-\epsilon}(\Omega)$. So, it is enough to show
$\widetilde{P}\mathbf{u}_{\phi}(t)\in D((-\mathcal{A})^{1/4-\epsilon/4})$.

$\widetilde{P}\circ L_{0}$ is linear and continuous from
$\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime}$ to
$\mathbf{V}_{n}^{1/2}(\Omega)\times H^{1/2}(\Omega)$. Thus if
$(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$, then
$(-\mathcal{A})^{1/4-\epsilon/4}\widetilde{P}L_{0}(\mathbf{g},h)\in
L^{2}(0,T;\mathbf{V}_{n}^{0}(\Omega))\times L^{2}(0,T;L^{2}(\Omega))$. We also
have from (26) :

$\displaystyle||(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(t)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$
$\displaystyle\leq||e^{t\mathcal{A}}(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}+$
$\displaystyle\int_{0}^{t}||(-\mathcal{A})^{1-\epsilon/4}e^{(t-s)\mathcal{A}}||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}||(-\mathcal{A})^{1/4-\epsilon/4}\widetilde{P}L_{0}(\mathbf{g}(s),h(s))||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}ds$ $\displaystyle\leq
Ct^{-1/4+\epsilon/2}||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}+$
$\displaystyle\int_{0}^{t}(t-s)^{-1+\epsilon/4}||(-\mathcal{A})^{1/4-\epsilon/4}\widetilde{P}L_{0}(\mathbf{g}(s),h(s))||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}ds$ $\displaystyle(\mbox{ By using
\cite[cite]{[\@@bibref{}{PAZY}{}{}, Chapter 2.6, theorem 6.13]}})$
$\displaystyle\leq
Ct^{-1/4+\epsilon/2}||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Gamma)}+\int_{0}^{t}(t-s)^{-1+\epsilon/4}C||(\mathbf{g}(s),h(s))||_{\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime}}$
$\displaystyle(\mbox{By using \cite[cite]{[\@@bibref{}{PAZY}{}{}, Chapter 2,6,
theorem 6.10]} })$ $\displaystyle\leq
Ct^{-1/4+\epsilon/2}||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}+$ $\displaystyle
C\left(\int_{0}^{t}(t-s)^{-2+\epsilon/2}ds\right)\left(\int_{0}^{t}||(\mathbf{g}(s),h(s))||_{\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime}}^{2}ds\right)$
$\displaystyle\leq
Ct^{-1/4+\epsilon/2}||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}+Ct^{-1+\epsilon/2}||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})}.$

Hence : $\widetilde{P}\mathbf{u}_{\phi}(t)\in
D((-\mathcal{A})^{1/4-\epsilon/4})$. Also,

$\displaystyle||\widetilde{P}\mathbf{u}_{\phi}||_{L^{2}(0,T;\mathbf{V}^{1/2-\epsilon}(\Omega))\times
H^{1/2-\epsilon}(\Omega)}^{2}=\int_{0}^{T}||(-\mathcal{A})^{-(1/4-\epsilon/2)}(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(t)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}^{2}dt$ $\displaystyle\leq
C\int_{0}^{T}||(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(t)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}^{2}\quad(\mbox{By \cite[cite]{[\@@bibref{}{PAZY}{}{}, Chapter
2.6, lemma 6.3]}})$ $\displaystyle\leq
C\int_{0}^{T}\left(t^{-1/2+\epsilon}||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}^{2}+t^{-2+\epsilon}||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma))}^{2}dt\right)\,$
$\displaystyle(\mbox{As }(a+b)^{2}\leq 2(a^{2}+b^{2}))$

Hence we deduce that:

(27)
$||\widetilde{P}\mathbf{u}_{\phi}||_{L^{2}(0,T;V^{1/2-\epsilon}(\Omega)\times
H^{1/2-\epsilon}(\Omega))}\leq
C\left(||\widetilde{P}\mathbf{u}_{\phi}(0)||_{V^{0}_{n}(\Omega)\times
L^{2}(\Omega)}+||(\mathbf{g},h)||_{L^{2}(0,T;V^{0}(\Gamma))\times(H^{1}(\Gamma))^{\prime}}\right).$

Our next aim is to show that : $\widetilde{P}\mathbf{u}_{\phi}\in
H^{1/4-\epsilon/2}(0,T;\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega))$. Moreover
from equation (26) after differentiating w.r.t t:

$\displaystyle\frac{dP\mathbf{u}_{\phi}}{dt}$
$\displaystyle=\mathcal{A}e^{t\widetilde{\mathcal{A}}}\widetilde{P}\mathbf{u}_{\phi}(0)-\mathcal{A}\widetilde{P}L_{0}(\mathbf{g},h)(t)-\mathcal{A}\int_{0}^{t}\mathcal{A}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)(s)ds$
$\displaystyle=-(-\mathcal{A})^{3/4+\epsilon/2}e^{t\mathcal{A}}(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(0)+(-\mathcal{A})^{3/4+\epsilon/2}(-\widetilde{\mathcal{A}})^{1/4-\epsilon/2}\widetilde{P}L_{0}(\mathbf{g},h)(t)$
$\displaystyle+(-\mathcal{A})^{3/4+\epsilon/2}(-\mathcal{A})^{1/4-\epsilon/2}\int_{0}^{t}\mathcal{A}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)(s)ds$
$\displaystyle=(-\mathcal{A})^{3/4+\epsilon/2}[-(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(t)+(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}L_{0}(\mathbf{g},h)(t)]$

That is we have:

$\displaystyle(-\mathcal{A})^{-3/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}^{\prime}=[-(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(t)+(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}L_{0}(\mathbf{g},h)(t)].$

That is we have:

$\displaystyle(-\mathcal{A})^{-3/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}^{\prime}=[-(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}(t)+(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}L_{0}(\mathbf{g},h)(t)].$

We deduce that:

$\displaystyle||\widetilde{P}\mathbf{u}_{\phi}^{\prime}||_{L^{2}(0,T;[D((-\mathcal{A})^{3/4+\epsilon/2})]^{\prime})}$
$\displaystyle\leq
C\left(||(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}\mathbf{u}_{\phi}||_{L^{2}(0,T;\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega))}+||(-\mathcal{A})^{1/4-\epsilon/2}\widetilde{P}L_{0}(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}_{n}^{0}(\Omega)\times(H^{1}(\Gamma))^{\prime})}\right)$
$\displaystyle\leq
C\left(||\widetilde{P}\mathbf{u}_{\phi}||_{L^{2}(0,T;V^{1/2-\epsilon}(\Omega)\times
H^{1/2-\epsilon}(\Omega))}+||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})}\right)$

Hence we have:

(28)
$||\widetilde{P}\mathbf{u}_{\phi}^{\prime}||_{L^{2}(0,T;[D((-\mathcal{A})^{3/4+\epsilon/2})]^{\prime})}\leq
C\left(||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}+||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})}\right).$

By using interpolation theorem [12, Chapter 1, theorem 9.7], we obtain:

(29)
$\displaystyle||\widetilde{P}\mathbf{u}_{\phi}||_{H^{1/4-\epsilon/2}(0,T;\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega))}\leq
C\left(||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}^{0}_{n}(\Omega)\times
L^{2}(\Omega)}+||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})}\right).$

Now let $(\mathbf{g},h)\in
L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})$ and we know
that the operator $L_{0}$ is linear and continuous from
$\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime}$ to
$\mathbf{V}^{1/2}(\Omega)\times H^{1/2}(\Gamma)$. Observe that :

$\displaystyle||(I-\widetilde{P})\mathbf{u}_{\phi}||_{L^{2}(0,T;\mathbf{V}^{1/2}(\Omega)\times
H^{1/2}(\Omega))}^{2}$
$\displaystyle=\int_{0}^{T}||(I-\widetilde{P})\mathbf{u}_{\phi}(t)||_{\mathbf{V}^{1/2}(\Omega)\times
H^{1/2}(\Omega)}^{2}$
$\displaystyle=\int_{0}^{T}||(I-\widetilde{P})L_{0}(\mathbf{g},h)(t)||_{\mathbf{V}^{1/2}(\Omega)\times
H^{1/2}(\Omega)}^{2}$ (30) $\displaystyle\leq
C||(\mathbf{g},h)||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime}}^{2}.$

Thus by the relations (27), (29) and (4.1.1), we can establish the estimate
(4.5).

$(ii)$ Let us consider the case when $(\mathbf{g},h)\in V^{2,1}(\Sigma)\times
L^{2}(0,T;H^{1}(\Gamma))\cap H^{1}(0,T;(H^{1}(\Gamma))^{\prime})$. Also it is
given that $(P\mathbf{u}_{0},\phi(0))\in V^{3/2}(\Omega)\times
H^{3/2}(\Omega)$ and
$\Big{(}\widetilde{P}\Big{[}(\mathbf{u}_{0},\phi_{0})-L_{0}(\mathbf{g}(0),h(0))\Big{]}\Big{)}|_{\Gamma}=0$.
Our aim is to prove that
$\widetilde{P}\mathbf{u}_{\phi}\in\mathbf{V}^{5/2-\epsilon,5/4-\epsilon/2}(Q)\times
H^{5/2-\epsilon,5/4-\epsilon/2}(Q)$ for all $\epsilon>0$. We have:

$\displaystyle\widetilde{P}\mathbf{u}_{\phi}(t)$
$\displaystyle=e^{t\mathcal{A}}P\mathbf{u}_{\phi}(0)-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)(s)ds$
$\displaystyle=e^{t\mathcal{A}}\widetilde{P}\mathbf{u}_{\phi}(0)+\int_{0}^{t}\frac{d}{ds}(e^{(t-s)\mathcal{A}})\widetilde{P}L_{0}(\mathbf{g},h)(s)ds$
$\displaystyle=e^{t\mathcal{A}}\widetilde{P}\mathbf{u}_{\phi}(0)-\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime}(s),h^{\prime}(s))ds+$
$\displaystyle\quad\widetilde{P}L_{0}(\mathbf{g},h)(t)-e^{t\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)(0).$

Thus we obtain :

$\displaystyle\widetilde{P}\mathbf{u}_{\phi}(t)=e^{t\mathcal{A}}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))+\widetilde{P}L_{0}(\mathbf{g},h)(t)-$
(31)
$\displaystyle\quad\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds.$

At first we want to show that $\widetilde{P}\mathbf{u}_{\phi}\in
L^{2}(0,T;\mathbf{V}^{5/2-\epsilon}(\Omega)\times H^{5/2-\epsilon}(\Omega))$.
Now by [2, Chapter II.1.6, proposition 6.1], we have
$D((-\mathcal{A})^{5/4-\epsilon/2})\simeq\mathbf{V}^{5/2-\epsilon}(\Omega)\times
H^{5/2-\epsilon}(\Omega)$. So it is enough to show
$\widetilde{P}\mathbf{u}(t)\in D((-\mathcal{A})^{5/4-\epsilon/2})$.

Since $(\mathbf{g},h)\in L^{2}(0,T;V^{2}(\Gamma))\cap
H^{1}(0,T;V^{0}(\Gamma))\times L^{2}(0,T;H^{1}(\Gamma))\cap
H^{1}(0,T;(H^{1}(\Gamma))^{\prime})$, we have:

$\displaystyle\widetilde{P}L_{0}(\mathbf{g},h)\in
L^{2}(0,T;\mathbf{V}^{5/2}(\Omega)\times H^{5/2}(\Omega))\cap
H^{1}(0,T;\mathbf{V}^{1/2}(\Omega)\times H^{1/2}(\Omega)).$

Now,

$\displaystyle||(-\mathcal{A})^{5/4-\epsilon/2}\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$
$\displaystyle=||\int_{0}^{t}(-\mathcal{A})^{1-\epsilon/4}e^{(t-s)\mathcal{A}}(-\mathcal{A})^{1/4-\epsilon/4}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-1+\epsilon/4}||(-\mathcal{A})^{1/4-\epsilon/4}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}ds$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-1+\epsilon/4}||(\mathbf{g}^{\prime},h^{\prime})(s)||_{\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime}}ds$
$\displaystyle=Ct^{\epsilon/4}||(\mathbf{g}^{\prime},h^{\prime})||_{L^{2}(0,T;\mathbf{V}^{0}(\Gamma)\times(H^{1}(\Gamma))^{\prime})}.$

Also we obtain :

$\displaystyle||(-\mathcal{A})^{5/4-\epsilon/2}e^{t\mathcal{A}}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$
$\displaystyle=||(-\mathcal{A})^{1/2-\epsilon/4}e^{t\mathcal{A}}(-\mathcal{A})^{3/4-\epsilon/4}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$ $\displaystyle\leq
Ct^{-1/2+\epsilon/4}||(-\mathcal{A})^{3/4-\epsilon/4}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))||_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}$ $\displaystyle\leq
Ct^{-1/2+\epsilon/4}||(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))||_{\mathbf{V}^{3/2-\epsilon/2}(\Omega)\times
H^{3/2-\epsilon/2}}$ $\displaystyle\leq
Ct^{-1/2+\epsilon/4}(||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)}+||\widetilde{P}L_{0}(\mathbf{g},h)(0)||_{\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)}).$

Thus we have deduced that:

$\displaystyle||\widetilde{P}\mathbf{u}_{\phi}||_{L^{2}(0,T;\mathbf{V}^{5/2-\epsilon}(\Omega)\times
H^{5/2-\epsilon})}$ (32) $\displaystyle\leq
C(||\widetilde{P}\mathbf{u}_{\phi}(0)||_{\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)}+||(\mathbf{g},h)||_{\mathbf{V}^{2,1}(\Sigma)\times
L^{2}(0,T;H^{1}(\Gamma))\cap H^{1}(0,T;(H^{1}(\Gamma))^{\prime})}).$

Our next aim is to prove that $\widetilde{P}\mathbf{u}_{\phi}\in
H^{1}(0,T;\mathbf{V}^{5/4-\epsilon/2}(\Omega))\times
H^{1}(0,T;H^{5/4-\epsilon/2}(\Omega)).$

Now by differentiating the expression (4.1.1) of
$\widetilde{P}\mathbf{u}_{\phi}$, we shall get:

(33)
$\displaystyle\frac{d\widetilde{P}\mathbf{u}_{\phi}}{dt}=\mathcal{A}e^{t\mathcal{A}}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds.$

Now $(\mathbf{g}^{\prime},h^{\prime})\in L^{2}(0,T;V^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$ and by applying (i) of theorem (4.5) we
already have $\widetilde{P}\mathbf{u}_{\phi}\in
H^{1/4-\epsilon/2}(0,T;V^{0}(\Omega)\times L^{2}(\Omega))$. Now recall that

$\displaystyle\widetilde{P}\mathbf{u}_{\phi}(t)=e^{t\mathcal{A}}\widetilde{P}\mathbf{u}_{\phi}(0)-\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g},h)(s)ds.$

As $\widetilde{P}\mathbf{u}_{\phi}(0)\in\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)$, we have
$e^{t\mathcal{A}}\widetilde{P}\mathbf{u}_{\phi}(0)\in
H^{1/4}(0,T;\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega))$. So, we deduce that :

(34)
$\displaystyle\mathcal{A}\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds\in
H^{1/4-\epsilon/2}(0,T;V^{0}(\Omega)\times L^{2}(\Omega)).$

Moreover it is given that
$\widetilde{P}(\mathbf{u}_{\phi}(0)-L_{0}(\mathbf{g},h)(0))\in\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)$. Now by an isomorphism theorem [2, Chapter II.3.2, theorem
2.1] with
$D(\mathcal{A})=\mathbf{V}^{2}(\Omega)\cap\mathbf{V}_{0}^{1}(\Omega)\times
H^{2}(\Omega)$ and $H=\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega),$ we will
get for initial condition in $\mathbf{V}^{1}(\Omega)\times H^{1}(\Omega)$,
solution is in $L^{2}(0,T;\mathbf{V}^{2}(\Omega)\times H^{2}(\Omega))\cap
H^{1}(0,T;L^{2}(\Omega)\times L^{2}(\Omega))$.

We can apply same result by replacing $D(\mathcal{A})$ by $D(\mathcal{A}^{2})$
and $H$ by $D(\mathcal{A})$ and we can conclude that for initial condition in
$\mathbf{V}^{3}(\Omega)$, solution belongs to
$L^{2}(0,T;\mathbf{V}^{4}(\Omega)\times H^{4}(\Omega))\cap
H^{1}(0,T;\mathbf{V}^{2}(\Omega)\times H^{2}(\Omega))$ .

Then by interpolation theorem [12, Chapter 1, theorem 9.7], we can conclude
that for
$(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))\in\mathbf{V}^{3/2}(\Omega)\times
H^{3/2}(\Omega)$, we have
$e^{t\mathcal{A}}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))\in
L^{2}(0,T;\mathbf{V}^{5/2}(\Omega)\times H^{5/2}(\Omega))\cap
H^{1}(0,T;\mathbf{V}^{1/2}(\Omega)\times H^{1/2}(\Omega))\hookrightarrow
H^{1/4}(0,T;\mathbf{V}^{2}(\Omega)\times H^{2}(\Omega)).$Thus

(35)
$\displaystyle\mathcal{A}e^{t\mathcal{A}}(\widetilde{P}\mathbf{u}_{\phi}(0)-\widetilde{P}L_{0}(\mathbf{g},h)(0))\in
H^{1/4}(0,T;\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)).$

Therefore, by using the relations (33), (34) and (35), we can conclude that :

(36) $\displaystyle\frac{d\widetilde{P}\mathbf{u}_{\phi}}{dt}\in
H^{1/4-\epsilon/2}(0,T;\mathbf{V}^{0}(\Omega)\times L^{2}(\Omega)).$

It is clear from (4.1.1) and (36) that :
$(P\mathbf{u},\phi)\in\mathbf{V}^{5/2-\epsilon,5/4-\epsilon/2}\times
H^{5/2-\epsilon,5/4-\epsilon/2}$ .

$(iii)$ As we have already proved the results for $s=0$ and $s=2$, we can get
our required result by interpolation. ∎

We would like to answer the question if we can take $\epsilon=0$ in the
results of theorem 4.5. We give a complete answer to this question in the next
theorem, following the argument of [8] as in the case of Navier-Stokes
equation.

######  Theorem 4.6.

Assume that
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{0\vee(s-1/2)}(\Omega)\times
H^{0\vee(s-1/2)}(\Omega)$, $(\mathbf{g},h)\in\mathbf{V}^{s,s/2}(\Sigma)\times
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap
H^{s/2}(0,T;(H^{1}(\Gamma))^{\prime})$, with $s\in[0,1)$ and
$(\mathbf{g},h)\in\mathbf{V}^{s,s/2}(\Sigma)\times
L^{2}(0,T;H^{s-1}(\Gamma))\cap H^{s/2}(0,T;(H^{1}(\Gamma))^{\prime})$ with
$s\in(1,2]$. If $(\mathbf{u}_{0},\phi_{0})$ and $(\mathbf{g}(0),h(0))$ satisfy
the compatibility condition (25), then :

$\displaystyle||(P\mathbf{u},\phi)||_{\mathbf{V}^{s+1/2,s/2+1/4}(Q)\times
H^{s+1/2,s/2+1/4}(Q)}\leq
C(||P\mathbf{u}_{0}||_{\mathbf{V}^{0\vee(s-1/2)}(\Omega)}+$ (37)
$\displaystyle||\phi_{0}||_{H^{0\vee(s-1/2)}(\Omega)}+||\mathbf{g}||_{\mathbf{V}^{s,s/2}(\Sigma)}+||h||_{L^{2}(0,T;H^{s-1}(\Omega))\cap
H^{s/2}(0,T;(H^{1}(\Omega))^{\prime})}).$

###### Proof.

We can write the first equation of (11) as :

(38) $\displaystyle\frac{\partial P\mathbf{u}}{\partial t}-\Delta
P\mathbf{u}+\nabla
p=\mathrm{\beta}\phi+\Delta((I-P)\mathbf{u})-\frac{\partial(I-P)\mathbf{u}}{\partial
t}.$

Let us consider $\mathbf{g}\in\mathbf{V}^{2,1}(\Sigma)$. Due to theorem 4.5,
we have $P\mathbf{u}\in\mathbf{V}^{5/2-\epsilon,5/4-\epsilon/2}(Q)$ and
$(I-P)\mathbf{u}\in L^{2}(0,T;\mathbf{V}^{5/2}(\Omega))\cap
H^{1}(0,T;\mathbf{V}^{1/2}(\Omega))$, $\phi\in
H^{5/2-\epsilon,5/4-\epsilon/2}(Q)$. Thus from equation (38), we know the
pressure $p\in L^{2}(0,T;H^{1}(\Omega))$. Also, from characterisation of
$(I-P)$ we know that $(I-P)\mathbf{u}=\nabla q$, where $q\in
L^{2}(0,T;H^{2}(\Omega)/\mathbb{R})$ satisfies :

$\displaystyle\Delta q(t)=0\quad\mbox{in }\Omega,$
$\displaystyle\frac{\partial
q}{\partial\mathbf{n}}=\mathbf{g}(t).\mathbf{n}\quad\mbox{on }\Gamma$

Thus we can write equation (38) as :

(39) $\displaystyle\frac{\partial P\mathbf{u}}{\partial t}-\Delta
P\mathbf{u}+\nabla\pi=\mathrm{\beta}\phi,$

where $\pi=p-\Delta q+\frac{\partial q}{\partial t}=p+\frac{\partial
q}{\partial t}L_{z}$. Also observe that :

$\displaystyle P\mathbf{u}|_{\Sigma}.\mathbf{n}$
$\displaystyle=\mathbf{u}|_{\Sigma}.\mathbf{n}-(I-P)\mathbf{u}|_{\Sigma}.\mathbf{n}$
$\displaystyle=\mathbf{g}.\mathbf{n}-\nabla q|_{\Sigma}.\mathbf{n}$
$\displaystyle=0.$

Hence $P\mathbf{u}$ satisfies Stokes equation (39) with the condition
$P\mathbf{u}|_{\Sigma}.\mathbf{n}=0$. Now we are in a position to use the
regularity results for instationary Stokes equation with nonhomogeneous
boundary condition as in [8] to conclude that
$P\mathbf{u}\in\mathbf{V}^{5/2,5/4}(Q)$. Also in (11), $\phi$ satisfies heat
equation. So we can use regularity result of heat equation to conclude that
$\phi\in H^{5/2,5/4}(Q)$.
Now we can follow same steps as in [16] to get our desired result for
$s\in[0,1)\cup(1,2]$. ∎

We want to find an appropriate boundary condition for which we can get
continuous in time solution for equation (11). The next theorem will give us
such a kind of boundary condition :

######  Corollary 1.

If $(\mathbf{g},h)\in\mathbf{V}^{3/4,3/4}(\Sigma)\times
L^{2}(0,T;(H^{1/4}(\Gamma))^{\prime})\cap
H^{3/4}(0,T;(H^{1}(\Gamma))^{\prime})$,
$(P\mathbf{u}_{0},\phi_{0})\in(\mathbf{V}^{3/4}_{n}(\Omega)\times
H^{3/4}(\Omega))$ and if the following condition

$\displaystyle\widetilde{P}\Big{[}(\mathbf{u}_{0},\phi_{0})-L_{0}(\mathbf{g}(0),h(0))\Big{]}=0,$

holds, then :
$(\mathbf{u},\phi)\in C([0,T];\mathbf{V}^{3/4}(\Omega))\cap
L^{2}(0,T;\mathbf{V}^{5/4}(\Omega))\times C([0,T];H^{3/4}(\Omega))\cap
L^{2}(0,T;H^{5/4}(\Omega))$.

###### Proof.

From definition 4.4, we know that :
$(I-\widetilde{P})\mathbf{u_{\phi}}=(I-\widetilde{P})L_{0}(\mathbf{g},h)$ . As
$(\mathbf{g},h)\in\mathbf{V}^{3/4,3/4}(\Sigma)\times
L^{2}(0,T;(H^{1/4}(\Gamma))^{\prime})\cap
H^{3/4}(0,T;(H^{1}(\Gamma))^{\prime})$, we will obtain

(40) $\displaystyle(I-\widetilde{P})\mathbf{u_{\phi}}\in$ $\displaystyle
L^{2}(0,T;\mathbf{V}^{5/4}(\Omega)\times H^{5/4}(\Omega))\cap
H^{3/4}(0,T;\mathbf{V}^{1/2}\times H^{1/2}(\Omega)).$
$\displaystyle\hookrightarrow C([0,T];\mathbf{V}^{3/4}(\Omega))\times
C([0,T];H^{3/4}(\Omega)).$

Our aim is to show : $\widetilde{P}\mathbf{u}_{\phi}\in
C([0,T];\mathbf{V}^{3/4}(\Omega))\times C([0,T];H^{3/4}(\Omega))$.

When $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$, from $(i)$ of theorem (4.5), we know
that

(41) $\displaystyle\widetilde{P}\mathbf{u}_{\phi}\in\mathbf{V}^{1/2,1/4}\times
H^{1/2,1/4}.$

If $(\mathbf{g},h)\in\mathbf{V}^{1.1}(\Sigma)\times
L^{2}(0,T;L^{2}(\Gamma))\cap H^{1}(0,T;(H^{1}(\Gamma))^{\prime})$ and
$\widetilde{P}\Big{[}(\mathbf{u}_{0},\phi_{0})-L_{0}(\mathbf{g}(0),h(0))\Big{]}=0$,
then (4.1.1) gives :

(42)
$\displaystyle\widetilde{P}\mathbf{u}_{\phi}(t)=\widetilde{P}L_{0}(\mathbf{g},h)(t)-\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds.$

Here

$\displaystyle\int_{0}^{t}e^{(t-s)\mathcal{A}}\widetilde{P}L_{0}(\mathbf{g}^{\prime},h^{\prime})(s)ds$
$\displaystyle\in L^{2}(0,T;\mathbf{V}^{2}(\Omega)\times H^{2}(\Omega))\cap
H^{1}(0,T;\mathbf{V}^{1/4}(\Omega)\times H^{1/4}(\Omega))$ (43)
$\displaystyle\hookrightarrow H^{2/3}(0,T;\mathbf{V}^{5/6}(\Omega)\times
H^{5/6}(\Omega)).$

Also we have :

$\displaystyle\widetilde{P}L_{0}(\mathbf{g},h)$ $\displaystyle\in
L^{2}(0,T;\mathbf{V}^{3/2}(\Omega)\times H^{3/2}(\Omega))\cap
H^{1}(0,T;\mathbf{V}^{1/2}(\Omega)\times H^{1/2}(\Omega))$ (44)
$\displaystyle\hookrightarrow H^{2/3}(0,T;\mathbf{V}^{5/6}(\Omega)\times
H^{5/6}(\Omega)).$

Thus by using (4.1.1) and (4.1.1), the relation (42) gives : when
$(\mathbf{g},h)\in\mathbf{V}^{1.1}(\Sigma)\times L^{2}(0,T;L^{2}(\Gamma))\cap
H^{1}(0,T;(H^{1}(\Gamma))^{\prime})$,

(45) $\displaystyle\widetilde{P}\mathbf{u}_{\phi}\in
H^{2/3}(0,T;\mathbf{V}^{5/6}(\Omega)\times H^{5/6}(\Omega)).$

By interpolation between (41) and (45), for
$(\mathbf{g},h)\in\mathbf{V}^{3/4,3/4}(\Sigma)\times
L^{2}(0,T;(H^{1/4}(\Gamma))^{\prime})\cap
H^{3/4}(0,T;(H^{1}(\Gamma))^{\prime})$, we get :

$\displaystyle\widetilde{P}\mathbf{u}_{\phi}$ $\displaystyle\in
C([0,T];\mathbf{V}^{3/4}(\Omega))\times C([0,T];H^{3/4}(\Omega)),$ (46)
$\displaystyle\mbox{ and }\widetilde{P}\mathbf{u}_{\phi}$ $\displaystyle\in
L^{2}(0,T;\mathbf{V}^{5/4}(\Omega)\times H^{5/4}(\Omega)).$

∎

######  Remark 2.

In the proof of the above corollary, we need the condition
$\widetilde{P}\Big{[}(\mathbf{u}_{0},\phi_{0})-L_{0}(\mathbf{g}(0),h(0))\Big{]}=0$.
Here,
$(\mathbf{g},h)\in\mathbf{V}^{3/4,3/4}(\Sigma)\times
L^{2}(0,T;(H^{1/4}(\Gamma))^{\prime})\cap
H^{3/4}(0,T;(H^{1}(\Gamma))^{\prime})$. So,

$\displaystyle\widetilde{P}L_{0}(\mathbf{g},h)$ $\displaystyle\in
L^{2}(0,T;\mathbf{V}_{n}^{5/4}(\Omega)\times H^{5/4}(\Omega))\cap
H^{3/4}(0,T;\mathbf{V}_{n}^{1/2}(\Omega)\times H^{1/2}(\Omega)),$
$\displaystyle\hookrightarrow C([0,T];\mathbf{V}_{n}^{3/4}(\Omega)\times
H^{3/4}(\Omega)).$

i.e,
$\widetilde{P}L_{0}(\mathbf{g}(0),h(0))\in\mathbf{V}_{n}^{3/4}(\Omega)\times
H^{3/4}(\Omega).$ That’s why we need the condition
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}_{n}^{3/4}(\Omega)\times
H^{3/4}(\Omega)$ in corollary 1.

We can extend the result of last corollary with boundary conditions
$\mathbf{g}\in\mathbf{V}^{s,s}(\Sigma)$, $h\in
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap H^{s}(0,T;(H^{1}(\Gamma))^{\prime})$
in the following way :

######  Corollary 2.

Assume that $\frac{1}{2}<s<1$. If
$(\mathbf{g},h)\in\mathbf{V}^{s,s}(\Sigma)\times
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap
H^{s}(0,T;(H^{1}(\Gamma))^{\prime})$,
$(P\mathbf{u}_{0},\phi_{0})\in(\mathbf{V}^{s}_{n}(\Omega)\times
H^{s}(\Omega))$ and if the following condition

$\displaystyle\widetilde{P}\Big{[}(\mathbf{u}_{0},\phi_{0})-L_{0}(\mathbf{g}(0),h(0))\Big{]}=0,$

holds, then:
$(\mathbf{u},\phi)\in C([0,T];\mathbf{V}^{s}(\Omega))\cap
L^{2}(0,T;\mathbf{V}^{s+\frac{1}{2}}(\Omega))\times C([0,T];H^{s}(\Omega))\cap
L^{2}(0,T;H^{s+\frac{1}{2}}(\Omega))$.

###  4.2. Regularity of pressure

We have a fairly good understanding about the existence of the solution
$(\mathbf{u},\phi)$ of equation (11). Now we want to understand the space
where the pressure lies. We have the following result

######  Proposition 1.

Assume that $(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{s-1/2}_{n}(\Omega)\times
H^{s-1/2}(\Omega)$,
$(\mathbf{g},h)\in\mathbf{V}^{s,s/2}(\Sigma)\times
L^{2}(0,T;H^{s-1}(\Gamma))\cap H^{s/2}(0,T;(H^{1}(\Gamma))^{\prime})$ with
$s\in(1,2]$.
$1.$ If $(P\mathbf{u},\phi)\in\mathbf{V}^{s+1/2,s/2+1/4}(Q)\times
H^{s+1/2,s/2+1/4}(Q)$ obeys (21)-(22), then there exists a unique pressure
$p\in H^{s/2-1}(0,T;\mathcal{H}^{s-1/2}(\Omega))$ such that
$(\mathbf{u},p,\phi)$ satisfies (11).

We have analysed the space of the solution $(\mathbf{u},\phi)$ in the case of
$\mathbf{g}\in\mathbf{V}^{s,s}(\Sigma)$, $h\in
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap H^{s}(0,T;(H^{1}(\Gamma))^{\prime})$
[Corollary 1 and Corollary 2]. Now we want to see the space where pressure
lies.

######  Proposition 2.

Assume that $(\mathbf{g},h)\in\mathbf{V}^{s,s}(\Sigma)\times
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap\\\
H^{s/2}(0,T;(H^{1}(\Gamma))^{\prime})$ and
$(P\mathbf{u}_{0},\phi_{0})\in(\mathbf{V}^{s}_{n}(\Omega)\times
H^{s}(\Omega))$ with $s\in(\frac{1}{2},1)\cup(1,2]$.
If $(P\mathbf{u},\phi)\in\mathbf{V}^{s+1/2,s/2+1/4}(Q)\times
H^{s+1/2,s/2+1/4}(Q)$ obeys (21)-(22), then there exists a unique pressure
$p\in H^{-1/2}(0,T;\mathcal{H}^{s-1/2}(\Omega))$ such that
$(\mathbf{u},p,\phi)$ satisfies (11).

Now we want to see the regularity of pressure when we provide a better time
regularity to the boundary conditions.

######  Proposition 3.

Assume that $(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}^{s-1/2}_{n}(\Omega)\times
H^{s-1/2}(\Omega)$,
$(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{s}(\Gamma))\cap
H^{\frac{s}{2}+\frac{1}{4}}(0,T;V^{-\frac{1}{2}}(\Gamma))\times
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap\\\
H^{s/2}(0,T;(H^{\frac{3}{2}}(\Gamma))^{\prime})$, with $s\in(1/2,1)$,
$(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{s}(\Gamma))\cap
H^{\frac{s}{2}+\frac{1}{4}}(0,T;V^{-\frac{1}{2}}(\Gamma))\times
L^{2}(0,T;(H^{s-1}(\Gamma)))\cap\\\
H^{s/2}(0,T;(H^{\frac{3}{2}}(\Gamma))^{\prime})$, with $s\in(1,2]$ under the
compatibility condition (25).
If $(P\mathbf{u},\phi)\in\mathbf{V}^{s+1/2,s/2+1/4}(Q)\times
H^{s+1/2,s/2+1/4}(Q)$ obeys (21)-(22), then there exists a unique pressure
$p\in H^{s/2-3/4}(0,T;\mathcal{H}^{s-1/2}(\Omega))$ such that
$(\mathbf{u},p,\phi)$ satisfies (11).

###### Proof.

We have $(\mathbf{u},\phi)\in\mathbf{V}^{s+1/2,s/2+1/4}(Q)\times
H^{s+1/2,s/2+1/4}(Q)$ as in [16, Lemma 3.2]. We want to determine the correct
space for the pressure. Define:

$\displaystyle\mathbf{U}(\cdot)=\int\limits_{0}^{\cdot}u(t)\,dt\in
H^{1}(0,T;\mathbf{V}^{s+\frac{1}{2}}(\Omega)).$

Also we have

$\displaystyle\mathbf{u}\in L^{2}(0,T;\mathbf{V}^{s+\frac{1}{2}}(\Omega))\cap
H^{\frac{s}{2}+\frac{1}{4}}(0,T;\mathbf{V}^{0}(\Omega))\hookrightarrow
C([0,T];\mathbf{V}^{s-\frac{1}{2}}(\Omega)).$

We can say that, for all $\mathbf{\phi}\in\mathbf{V}^{0}(\Omega)$, we have

$\displaystyle\left\langle\mathbf{u}(t)-\mathbf{u}_{0}-\Delta\mathbf{U}(t),\mathbf{\phi}\right\rangle_{\mathbf{L}^{2}(\Omega)}=0,$

there exists $P(t)$ such that

$\displaystyle\nabla
P(t)=\mathbf{u}(t)-\mathbf{u}_{0}-\Delta\mathbf{U}(t)\in\mathbf{V}^{s-\frac{3}{2}}(\Omega).$

Also, we obtain $\nabla P\in
H^{\frac{s}{2}+\frac{1}{4}}(0,T;\mathbf{V}^{s-\frac{3}{2}}(\Omega))$. We can
define, the pressure function $p=\frac{dP}{dt}$ and we have $p\in
H^{\frac{s}{2}-\frac{3}{4}}(0,T;\mathcal{H}^{s-1/2}(\Omega))$. ∎

######  Remark 3.

In [1], the author assumes that the solution is in the space
$L^{2}(0,T;\mathbf{V}^{1}(\Omega))\cap H^{1}(0,T;\mathbf{V}^{-1}(\Omega))$ and
also the boundary data has good time regularity. So its not possible to
compare our result with [1, theorem 4.9].

####  4.2.1. Linearization around nonzero stationary state

In the previous section, we consider linearization around zero solution. Here
we are going to extend the results of previous section to the following
linerized system around a stationary state
$(\mathbf{z},\theta)\in\mathbf{V}^{1}(\Omega)\times H^{1}(\Omega)$ :

(47) $\left.\begin{aligned} &\frac{\partial\mathbf{u}}{\partial
t}-\Delta\mathbf{u}+(\mathbf{z}.\nabla)\mathbf{u}+(\mathbf{u}.\nabla)\mathbf{z}+\nabla
p=\bm{\beta}\phi\mbox{ in }Q;\\\ &\mbox{div }\mathbf{u}=0\mbox{ in }Q;\\\
&\mathbf{u}=\mathbf{g}\mbox{ on
}\Sigma,\quad\mathbf{u}(0)=\mathbf{u}_{0}(x)\mbox{ in }\Omega;\\\
&\frac{\partial\phi}{\partial
t}-\Delta\phi+(\mathbf{z}.\nabla)\phi+(\mathbf{u}.\nabla)\theta=0\mbox{ in
}\mbox{ Q };\\\ &\frac{\partial\phi}{\partial n}=h\mbox{ on
}\Sigma,\quad\phi(0)=\phi_{0}\mbox{ in }\Omega.\end{aligned}\right\\}$

To analyze equation (47), we will introduce following operators :

$\displaystyle A_{1}:D(A_{1})\rightarrow\mathbf{V}_{n}^{0}(\Omega);\quad
A_{1}\mathbf{u}=P\left(\Delta\mathbf{u}-(\mathbf{u.}\nabla)\mathbf{z}-(\mathbf{z.}\nabla)\mathbf{u}\right),$
$\displaystyle A_{2}:L^{2}(\Omega)\rightarrow\mathbf{V}_{n}^{0}(\Omega);\quad
A_{2}\phi=P(\bm{\beta}\phi),$ $\displaystyle
A_{3}:\mathbf{L}^{2}(\Omega)\rightarrow L^{2}(\Omega);\quad
A_{3}\mathbf{u}=-\mathbf{u.}\nabla\theta$ $\displaystyle
A_{4}:D(A_{4})\rightarrow L^{2}(\Omega);\quad
A_{4}\phi=\Delta\phi-\mathbf{z.}\nabla\phi$

where,

$\displaystyle
D(A_{1})=\mathbf{V}^{2}(\Omega)\cap\mathbf{V}_{0}^{1}(\Omega)\quad\mbox{and}\quad
D(A_{4})=\\{\tau\in H^{2}(\Omega)|\,\,\frac{\partial\tau}{\partial n}=0\\}.$

Let us define an operator
$\mathcal{A}_{\mathbf{z},\theta}=\begin{pmatrix}A_{1}&A_{2}\\\
A_{3}&A_{4}\end{pmatrix}:D(\mathcal{A}_{\mathbf{z},\theta})\mapsto\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$ with $D(\mathcal{A}_{\mathbf{z},\theta})=D(A_{1})\times
D(A_{4})$.

In this section, we will denote by $\lambda_{0}>0$ an element in the resolvent
set of $\mathcal{A}_{\mathbf{z},\theta}$ satisfying:

(48)
$\displaystyle\langle(\lambda_{0}I-\mathcal{A}_{\mathbf{z},\theta})(\mathbf{\Phi},\psi),(\mathbf{\Phi},\psi))\rangle_{\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)}\geq\omega_{0}|(\mathbf{\Phi},\psi))|_{\mathbf{V}_{0}^{1}(\Omega)\times
H_{0}^{1}(\Omega)},$

for some $\omega_{0}>0$.

######  Remark 4.

We can use the regularity of $(\mathbf{z},\theta)$ to establish the relation
(48). Use (48) to conclude that
$(\lambda_{0}I-\mathcal{A}_{\mathbf{z},\theta},D(\mathcal{A}_{\mathbf{z},\theta}))$
generates an analytic semigroup on $\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$.
Here also we can extend the operator $\mathcal{A}_{\mathbf{z},\theta}$ to an
unbounded operator $\widetilde{\mathcal{A}_{\mathbf{z},\theta}}$ with domain
$D(\widetilde{A}_{\mathbf{z},\theta})=\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$ in $(D(\mathcal{A}_{\mathbf{z},\theta}^{*}))^{\prime}$.

We can define weak solution as in the case of linearization around zero
solution :

######  Definition 4.7.

Assume that $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$ and
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$.
A function $(\mathbf{u},\phi)\in L^{2}(0,T;V^{0}(\Omega)\times L^{2}(\Omega))$
is a weak solution to equation (47) if $\widetilde{P}\mathbf{u_{\phi}}$ is a
weak solution of the following evolution system :

(49) $\displaystyle\widetilde{P}\mathbf{u_{\phi}}^{\prime}$
$\displaystyle=\widetilde{\mathcal{A}_{\mathbf{z},\theta}}\widetilde{P}\mathbf{u_{\phi}}+(\lambda_{0}I-\widetilde{\mathcal{A}_{\mathbf{z},\theta}})\widetilde{P}L_{\mathbf{z}}(\mathbf{g},h),\mbox{
with }\widetilde{P}\mathbf{u_{\phi}}(0)=\widetilde{P}\mathbf{u_{\phi}}^{0},$
$\displaystyle\mbox{ and }(I-\widetilde{P})\mathbf{u_{\phi}}(.)$
$\displaystyle=(I-\widetilde{P})L_{0}(\mathbf{g}(.),h(.)).$

######  Theorem 4.8.

Assume that $(\mathbf{z},\theta)\in\mathbf{V}^{1}(\Omega)\times
H^{1}(\Omega)$. For all
$(P\mathbf{u}_{0},\phi_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$
and $(\mathbf{g},h)\in L^{2}(0,T;\mathbf{V}^{0}(\Gamma))\times
L^{2}(0,T;(H^{1}(\Gamma))^{\prime})$, equation (47) admits a unique solution
$(P\mathbf{u},\phi)\in\mathbf{V}^{1/2-\epsilon,1/4-\epsilon/2}\times
H^{1/2-\epsilon,1/4-\epsilon/2}.$

###### Proof.

To prove this we will follow the same steps as in theorem 4.5 by replacing
$\mathcal{A},e^{t\mathcal{A}},L_{0}$ by
$\mathcal{A}_{\mathbf{z},\theta},e^{t\mathcal{A}_{\mathbf{z},\theta}},L_{\mathbf{z}}$.
∎

###  4.3. Linearization around an instationary state

In this section, our aim is to study the linearized Boussinesq system around
an instationary state $(\mathbf{z},\theta)$, with homogeneous boundary
conditions :

(50) $\left.\begin{aligned} &\frac{\partial\mathbf{y}}{\partial
t}-\Delta\mathbf{y}+(\mathbf{z}.\nabla)\mathbf{y}+(\mathbf{y}.\nabla)\mathbf{z}+\nabla
p=\bm{\beta}\tau+\mathbf{f}_{1}\mbox{ in }Q;\\\ &\mbox{div }\mathbf{y}=0\mbox{
in }Q;\\\ &\mathbf{y}=\mathbf{0}\mbox{ on
}\Sigma,\quad\mathbf{y}(0)=\mathbf{y}_{0}(x)\mbox{ in }\Omega\\\
&\frac{\partial\tau}{\partial
t}-\Delta\tau+(\mathbf{z}.\nabla)\tau+(\mathbf{y}.\nabla)\theta=f_{2}\mbox{ in
}\mbox{ Q }\\\ &\frac{\partial\tau}{\partial n}=0\mbox{ on
}\Sigma,\quad\tau(0)=\tau_{0}\mbox{ in }\Omega.\end{aligned}\right\\}$

Here we assume that $(\mathbf{z},\theta)\in
L^{2}(0,T;\mathbf{V}^{1}(\Omega))\cap
L^{\infty}(0,T;\mathbf{L}^{4}(\Omega))\times L^{2}(0,T;H^{1}(\Omega))\cap
L^{\infty}(0,T;L^{4}(\Omega))$ and $(\mathbf{f}_{1},f_{2})\in
L^{2}(0,T;\mathbf{H}^{-1}(\Omega))\times L^{2}(0,T;H^{-1}(\Omega))$.

We define the operator
$\mathcal{A}_{\mathbf{z},\theta}(t)\in\mathcal{L}(\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega),\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{\prime})$ by :

$\displaystyle\langle\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix},\begin{pmatrix}\mathbf{\Phi}\\\ \psi\end{pmatrix}\rangle=$
$\displaystyle-\int_{\Omega}\nabla\mathbf{y}.\nabla\mathbf{\Phi}-\int_{\Omega}((\mathbf{z}.\nabla)\mathbf{y}).\mathbf{\Phi}-\int_{\Omega}((\mathbf{y}.\nabla).\mathbf{z}).\mathbf{\Phi}+\int_{\Omega}\mathbf{\beta}\tau.\mathbf{\Phi}$
(51)
$\displaystyle-\int_{\Omega}\nabla\tau.\nabla\psi-\int_{\Omega}(\mathbf{z}.\nabla\tau)\psi-\int_{\Omega}(\mathbf{y}.\nabla\theta)\psi$

for all $(\mathbf{\Phi},\psi)\in\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)$. We can rewrite equation (50) in the form :

$\displaystyle\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}^{\prime}=\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}+\tilde{P}\begin{pmatrix}\mathbf{f}_{1}\\\
f_{2}\end{pmatrix}\mbox{ with }\begin{pmatrix}\mathbf{y}(0)\\\
\tau(0)\end{pmatrix}=\begin{pmatrix}\mathbf{y}_{0}\\\ \tau_{0}\end{pmatrix}.$

######  Lemma 4.9.

Assume that $(\mathbf{z},\theta)\in L^{2}(0,T;\mathbf{V}^{1}(\Omega))\cap
L^{\infty}(0,T;\mathbf{L}^{4}(\Omega))\times L^{2}(0,T;H^{1}(\Omega))\cap
L^{\infty}(0,T;L^{4}(\Omega))$. Then there exist $\lambda_{0}>0$ and $M>0$
such that :

(52)
$\displaystyle(i)\quad|\langle\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix},\begin{pmatrix}\mathbf{\Phi}\\\
\psi\end{pmatrix}\rangle|\leq M||\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}||_{\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)}||\begin{pmatrix}\mathbf{\Phi}\\\
\psi\end{pmatrix}||_{\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)}$

and

(53) $\displaystyle(ii)\quad\langle\lambda_{0}\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}-\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix},\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}\rangle\geq\frac{1}{2}\Big{(}||\mathbf{y}||_{\mathbf{V}^{1}_{0}(\Omega)}^{2}+||\tau||^{2}_{H^{1}(\Omega)}\Big{)}.$

where
$\langle,\rangle=\langle,\rangle_{\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{\prime},\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)}$.

###### Proof.

$(i)$ For all $(\mathbf{y},\tau)\in\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)$ and all $(\Phi,\psi)\in\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)$ and $t\in(0,T)$, we have

(54) $|\langle\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix},\begin{pmatrix}\mathbf{\Phi}\\\
\psi\end{pmatrix}\rangle|\leq||\mathbf{y}||_{\mathbf{V}_{0}^{1}(\Omega)}||\Phi||_{\mathbf{V}^{1}_{0}(\Omega)}+||\mathbf{z}||_{L^{\infty}(0,T;\mathbf{L}^{4}(\Omega))}||\mathbf{y}||_{\mathbf{V}^{1}_{0}(\Omega)}||\Phi||_{\mathbf{L}^{4}(\Omega)}\\\
+||\mathbf{y}||_{\mathbf{L}^{4}(\Omega)}||\Phi||_{\mathbf{V}_{0}^{1}(\Omega)}||\mathbf{z}||_{L^{\infty}(0,T;\mathbf{L}^{4}(\Omega))}+||\bm{\beta}||_{L^{\infty}}||\tau||_{L^{2}(\Omega)}||\Phi||_{\mathbf{L}^{2}(\Omega)}\\\
+||\tau||_{H^{1}(\Omega)}||\psi||_{H^{1}(\Omega)}+||\mathbf{z}||_{L^{\infty}(0,T;\mathbf{L}^{4}(\Omega))}||\tau||_{H^{1}(\Omega)}||\psi||_{L^{4}(\Omega)}+\\\
||\mathbf{y}||_{\mathbf{L}^{4}(\Omega)}||\psi||_{H^{1}(\Omega)}||\theta||_{L^{\infty}(0,T;L^{4}(\Omega))}\leq
M||\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}||_{\mathbf{V}_{0}^{1}(\Omega)\times
H^{1}(\Omega)}||\begin{pmatrix}\mathbf{\Phi}\\\
\psi\end{pmatrix}||_{\mathbf{V}_{0}^{1}(\Omega)\times H^{1}(\Omega)}.$

Here the last inequality can be obtained by using the fact that
$H^{1}(\Omega)\hookrightarrow L^{4}(\Omega)$ is continuous.

$(ii)$ We can follow the same steps as in the proof of theorem (3.2) to get
our desired estimate. ∎

######  Theorem 4.10.

Assume that $(\mathbf{z},\theta)\in L^{2}(0,T;\mathbf{V}^{1}(\Omega))\cap
L^{\infty}(0,T;\mathbf{L}^{4}(\Omega))\times\\\ L^{2}(0,T;H^{1}(\Omega))\cap
L^{\infty}(0,T;L^{4}(\Omega))$. For all
$(\mathbf{y}_{0},\tau_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$
and $(\mathbf{f}_{1},f_{2})\in L^{2}(0,T;\mathbf{H}^{-1}(\Omega))\times
L^{2}(0,T;H^{-1}(\Omega))$, equation (50) admits a unique weak solution
$(\mathbf{y},p,\tau)\in
W(0,T;\mathbf{V}^{1}_{0}(\Omega),\mathbf{V}^{-1}(\Omega))\times
L^{2}(0,T;L^{2}(\Omega))\times W(0,T;H^{1}(\Omega),(H^{1}(\Omega))^{\prime}).$

###### Proof.

Here $(\mathbf{y}_{0},\tau_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$ and $(\mathbf{f}_{1},f_{2})\in
L^{2}(0,T;\mathbf{H}^{-1}(\Omega))\times L^{2}(0,T;H^{-1}(\Omega))$. Observe
that : $\mathbf{V}^{1}_{0}(\Omega)\times
H^{1}(\Omega)\hookrightarrow\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)\hookrightarrow\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{\prime}$.
From (52), we have $\mathcal{A}_{\mathbf{z},\theta}(t)$ is a continuous linear
operator from $\mathbf{V}^{1}_{0}(\Omega)\times H^{1}(\Omega)$ to
$\mathbf{V}^{-1}(\Omega)\times(H^{1}(\Omega))^{\prime})$. Also coercivity of
$\mathcal{A}_{\mathbf{z},\theta}(t)$ follows from (53). Then the theorem is a
direct consequence of lemma 4.9 and [5, Chapter 18, Section 3.2, Theorem 1 and
2]. ∎

##  5\. Nonlinear Boussinesq system

In this section , we want to study non homogeneous boussinesq system (1). Let
us study at first following homogeneous system:

(55) $\left.\begin{aligned} &\frac{\partial\mathbf{y}}{\partial
t}-\Delta\mathbf{y}+(\mathbf{y}.\nabla)\mathbf{y}+\nabla
q=\bm{\beta}\tau+\mathbf{f}_{1}\mbox{ in }Q\\\ &\mbox{div }\mathbf{y}=0\mbox{
in }Q;\\\ &\mathbf{y}=\mathbf{0}\mbox{ on
}\Sigma,\quad\mathbf{y}(0)=\mathbf{y}_{0}(x)\mbox{ in }\Omega\\\
&\frac{\partial\tau}{\partial
t}-\Delta\tau+(\mathbf{y}.\nabla)\tau=f_{2}\mbox{ in }\mbox{ Q }\\\
&\frac{\partial\tau}{\partial n}=0\mbox{ on }\Sigma,\tau(0)=\tau_{0}\mbox{ in
}\Omega.\end{aligned}\right\\}$

######  Remark 5.

Equation (55) can be written as :

(56) $\left.\begin{aligned} &\begin{pmatrix}\mathbf{y}^{\prime}\\\
\tau^{\prime}\end{pmatrix}=\mathcal{A}\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}-\widetilde{P}\begin{pmatrix}(\mathbf{y}.\nabla)\mathbf{y}\\\
(\mathbf{y}.\nabla)\tau\end{pmatrix}-\widetilde{P}\begin{pmatrix}\mathbf{f}_{1}\\\
f_{2}\end{pmatrix},\\\ &\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}(0)=\begin{pmatrix}\mathbf{y}_{0}\\\
\tau_{0}\end{pmatrix}.\end{aligned}\right\\}$

where, $\mathcal{A}=\begin{pmatrix}P\Delta&P(\bm{\beta}\tau)\\\
0&\Delta\end{pmatrix}$ .

######  Theorem 5.1.

For given $(\mathbf{f}_{1},f_{2})\in
L^{2}(0,T;(\mathbf{V}^{1}_{0}(\Omega))^{\prime})\times
L^{2}(0,T;(H^{1}(\Omega))^{\prime})$ and
$(\mathbf{y}_{0},\tau_{0})\in\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega),$
then system (55) has at least one weak solution $(\mathbf{y},q,\tau)$ belongs
to $L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))\times L^{2}(0,T;L^{2}(\Omega))\times
L^{2}(0,T;H^{1}(\Omega)).$ Moreover, $(\mathbf{y},\tau)$ belongs to
$L^{\infty}(0,T;\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega))$ and
$(\mathbf{y},\tau)$ is weakly continuous from $[0,T]$ into
$\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$.

###### Proof.

The proof is based on the construction of approximate solution by the Galerkin
method. Since $\mathbf{V}^{1}_{0}(\Omega)$ is separable and
$\mathcal{V}(\Omega)$ is dense in $\mathbf{V}^{1}_{0}(\Omega)$, there exists
sequence of elements of $\mathcal{V}(\Omega)$, which is basis of
$\mathbf{V}^{1}_{0}(\Omega)$. Similarly, there exists a sequence of elements
of $C_{c}^{\infty}(\Omega)$ which is a basis of $H^{1}(\Omega)$. Then a
passage to the limit using suitable a priori estimate for approximate solution
and compactness theorems help us to complete the proof. (See e.g. [17, Chapter
3, Theorem 3.1] for Navier-Stokes equation). As equation (55) is with
homogeneous boundary data, we can follow the same steps as in [13] to complete
the proof. ∎

######  Remark 6.

Our main aim is to solve nonlinear Boussinesq system (1). Now we write
$\mathbf{z}=\mathbf{u}+\mathbf{y}$ and $\theta=\phi+\tau$, where
$(\mathbf{z},p,\theta)$ solves (1) and $(\mathbf{u},\phi)=\mathbf{u}_{\phi}$
satisfies :

(57) $\left.\begin{aligned}
\widetilde{P}\mathbf{u_{\phi}}&=\widetilde{\mathcal{A}}\widetilde{P}\mathbf{u_{\phi}}+(-\widetilde{\mathcal{A}})\widetilde{P}L_{0}(\mathbf{g},h),\quad\widetilde{P}\mathbf{u}_{\phi}(0)=\widetilde{P}L_{0}(\mathbf{g}(0),h(0)),\\\
\mbox{ and
}&(I-\widetilde{P})\mathbf{u_{\phi}}(.)=(I-\widetilde{P})L_{0}(\mathbf{g}(.),h(.)).\end{aligned}\right\\}$

Here $(\mathbf{y},q,\tau)$ is the solution of :

(58) $\left.\begin{aligned} &\frac{\partial\mathbf{y}}{\partial
t}-\Delta\mathbf{y}+(\mathbf{u}.\nabla)\mathbf{y}+(\mathbf{y}.\nabla)\mathbf{u}+(\mathbf{y}.\nabla)\mathbf{y}+(\mathbf{u}.\nabla)\mathbf{u}+\nabla
q=\bm{\beta}\tau\mbox{ in }Q;\\\ &\mathrm{div}\,\mathbf{y}=0\mbox{ in }Q;\\\
&\mathbf{y}=\mathbf{0}\mbox{ on }\Sigma,\\\ &\frac{\partial\tau}{\partial
t}-\Delta\tau+(\mathbf{u}.\nabla)\tau+(\mathbf{u}.\nabla)\phi+(\mathbf{y}.\nabla)\phi+(\mathbf{y}.\nabla)\tau=0\mbox{
in }\mbox{ Q }\\\ &\frac{\partial\tau}{\partial n}=0\mbox{ on }\Sigma,\\\
&\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}(0)=\widetilde{P}\begin{pmatrix}\mathbf{z}(0)\\\
\theta(0)\end{pmatrix}-\widetilde{P}L_{0}\begin{pmatrix}\mathbf{g}(0)\\\
h(0)\end{pmatrix}\quad\mbox{ in }\quad\Omega.\end{aligned}\right\\}$

######  Lemma 5.2.

Let $\mathbf{u}_{\phi}$ be the solution of (57) with
$(\mathbf{g},h)\in\mathbf{V}^{3/4,3/4}(\Sigma)\times\\\
C([0,T])L^{2}(0,T;(H^{1/4}(\Gamma))^{\prime})\cap
H^{3/4}(0,T;(H^{1}(\Gamma))^{\prime})$. Then $(\mathbf{u}.\nabla)\mathbf{u}\in
L^{2}(0,T;\mathbf{V}^{-1}(\Omega))$ and $(\mathbf{u}.\nabla)\phi\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime})$.

###### Proof.

As $(\mathbf{g},h)\in\mathbf{V}^{3/4,3/4}(\Sigma)\times
L^{2}(0,T;(H^{1/4}(\Gamma))^{\prime})\cap
H^{3/4}(0,T;(H^{1}(\Gamma))^{\prime})$, from corollary 1 we know that :

$\displaystyle(\mathbf{u},\phi)\in C([0,T];\mathbf{V}^{3/4}(\Omega))\cap
L^{2}(0,T;\mathbf{V}^{5/4}(\Omega))\times C([0,T];H^{3/4}(\Omega))\cap
L^{2}(0,T;H^{5/4}(\Omega)).$

Consider $\mathbf{w}\in L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))$. To show
$(\mathbf{u}.\nabla)\mathbf{u}\in L^{2}(0,T;\mathbf{V}^{-1}(\Omega))$, it is
enough to establish that :

$\displaystyle\int_{0}^{T}|\langle(\mathbf{u}(t).\nabla)\mathbf{u}(t),\,\mathbf{w}(t)\rangle_{\mathbf{V}^{-1}(\Omega),\mathbf{V}^{1}_{0}(\Omega)}|^{2}<+\infty.$

Now, we have :

$\displaystyle\|(\mathbf{u}.\nabla)\mathbf{u}\|_{L^{2}(0,T;\mathbf{V}^{-1}(\Omega))}$
$\displaystyle=\sup_{\|\mathbf{w}\|_{L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))}=1}\int_{0}^{T}|\int_{\Omega}(\mathbf{u}(t).\nabla)\mathbf{u}(t).\mathbf{w}(t)|^{2}$

On the other hand,

$\displaystyle\int_{0}^{T}|\int_{\Omega}(\mathbf{u}(t).\nabla)\mathbf{u}(t).\mathbf{w}(t)|^{2}$
$\displaystyle=\int_{0}^{T}|\int_{\Omega}(\mathbf{u}(t).\nabla)\mathbf{w}(t).\mathbf{u}(t)|^{2}$
$\displaystyle\leq\int_{0}^{T}\big{(}\int_{\Omega}|(\mathbf{u}(t).\nabla)\mathbf{w}(t).\mathbf{u}(t)|\big{)}^{2}$
$\displaystyle\leq\int_{0}^{T}\big{(}||\mathbf{u}(t)||_{\mathbf{L}^{4}(\Omega)}||\mathbf{w}(t)||_{\mathbf{V}^{1}_{0}(\Omega)}||\mathbf{u}(t)||_{\mathbf{L}^{4}(\Omega)}\big{)}^{2}$
$\displaystyle\leq\int_{0}^{T}\big{(}||\mathbf{u}(t)||_{\mathbf{V}^{3/4}(\Omega)}||\mathbf{w}(t)||_{\mathbf{V}^{1}_{0}(\Omega)}||\mathbf{u}(t)||_{\mathbf{V}^{3/4}(\Omega)}\big{)}^{2}$
$\displaystyle\leq||\mathbf{u}||^{4}_{C([0,T];\mathbf{V}^{3/4}(\Omega))}||\mathbf{w}||^{2}_{L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))}<+\infty.$

Similar calculations as above also ensures that $(\mathbf{u}.\nabla)\phi\in
L^{2}(0,T;(H^{1}(\Omega))^{\prime})$. ∎

######  Theorem 5.3.

Let $(P\mathbf{z}(0),\theta(0))\in\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega)$ and $(\mathbf{g},h)\in\mathbf{V}^{s,s}(\Sigma)\times\\\
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap
H^{s}(0,T;(H^{1}(\Gamma))^{\prime})$. Then equation (58) admits at least one
weak solution $(\mathbf{y},\tau)\in
C_{w}([0,T];\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega))\cap(L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))\times
L^{2}(0,T;H^{1}(\Omega)))$ and $q\in\mathcal{D}^{\prime}(0,T;L^{2}(\Omega))$,
where $C_{w}([0,T];\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega))$ is the
subspace of
$L^{\infty}(0,T;\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega))$ which are
continuous from $[0,T]$ into $\mathbf{V}_{n}^{0}(\Omega)\times L^{2}(\Omega)$
equipped with its weak topology.

###### Proof.

We can rewrite equation (58) satisfied by $(\mathbf{y},\tau)$ as :

(59) $\left.\begin{aligned} &\begin{pmatrix}\mathbf{y}^{\prime}\\\
\tau^{\prime}\end{pmatrix}=\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}-\widetilde{P}\begin{pmatrix}(\mathbf{y}.\nabla)\mathbf{y}\\\
(\mathbf{y}.\nabla)\tau\end{pmatrix}-\widetilde{P}\begin{pmatrix}(\mathbf{u}.\nabla)\mathbf{u}\\\
(\mathbf{u}.\nabla)\phi\end{pmatrix},\\\ &\begin{pmatrix}\mathbf{y}\\\
\tau\end{pmatrix}(0)=\widetilde{P}\begin{pmatrix}\mathbf{z}(0)\\\
\theta(0)\end{pmatrix}-\widetilde{P}L_{0}\begin{pmatrix}\mathbf{g}(0)\\\
h(0)\end{pmatrix}.\end{aligned}\right\\}$

Since $(\mathbf{g},h)\in\mathbf{V}^{s,s}(\Sigma)\times
L^{2}(0,T;(H^{1-s}(\Gamma))^{\prime})\cap
H^{s}(0,T;(H^{1}(\Gamma))^{\prime})$, according to Corollary 2, we conclude
that $\widetilde{P}L_{0}\begin{pmatrix}\mathbf{g}(0)\\\
h(0)\end{pmatrix}\in\mathbf{V}^{s}(\Omega)\times H^{s}(\Omega)$ and
$(\mathbf{u},\phi)\in C([0,T];\mathbf{V}^{s}(\Omega))\cap
L^{2}(0,T;\mathbf{V}^{s+\frac{1}{2}}(\Omega))\times C([0,T];H^{s}(\Omega))\cap
L^{2}(0,T;H^{s+\frac{1}{2}}(\Omega))$. Thus it is clear that
$\widetilde{P}\begin{pmatrix}(\mathbf{u}.\nabla)\mathbf{u}\\\
(\mathbf{u}.\nabla)\phi\end{pmatrix}\in
L^{2}(0,T;\mathbf{V}^{-1}(\Omega))\times L^{2}(0,T;(H^{1}(\Omega))^{\prime})$.
A function $(\mathbf{y},\tau)\in C_{w}([0,T];\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega))\cap(L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))\times
L^{2}(0,T;H^{1}(\Omega)))$ is a weak solution to (59) iff
$\begin{pmatrix}\widehat{\mathbf{y}}(t)\\\
\widehat{\tau}(t)\end{pmatrix}=e^{-\lambda_{0}t}\begin{pmatrix}\mathbf{y}(t)\\\
\tau(t)\end{pmatrix}$ is a solution to the system

(60) $\left.\begin{aligned} &\begin{pmatrix}\widehat{\mathbf{y}}^{\prime}\\\
\widehat{\tau}^{\prime}\end{pmatrix}=\mathcal{A}_{\mathbf{z},\theta}(t)\begin{pmatrix}\widehat{\mathbf{y}}\\\
\widehat{\tau}\end{pmatrix}-\lambda_{0}\begin{pmatrix}\widehat{\mathbf{y}}\\\
\widehat{\tau}\end{pmatrix}-\widetilde{P}\begin{pmatrix}(\widehat{\mathbf{y}}.\nabla)\widehat{\mathbf{y}}\\\
(\widehat{\mathbf{y}}.\nabla)\widehat{\tau}\end{pmatrix}-\lambda_{0}\begin{pmatrix}\widehat{\mathbf{y}}\\\
\widehat{\tau}\end{pmatrix}-\widetilde{P}\begin{pmatrix}(\mathbf{u}.\nabla)\mathbf{u}\\\
(\mathbf{u}.\nabla)\phi\end{pmatrix},\\\
&\begin{pmatrix}\widehat{\mathbf{y}}\\\
\widehat{\tau}\end{pmatrix}(0)=\widetilde{P}\begin{pmatrix}\mathbf{z}(0)\\\
\theta(0)\end{pmatrix}-\widetilde{P}L_{0}\begin{pmatrix}\mathbf{g}(0)\\\
h(0)\end{pmatrix}.\end{aligned}\right\\}$

Thanks to Lemma 4.9, the existence in
$C_{w}([0,T];\mathbf{V}_{n}^{0}(\Omega)\times
L^{2}(\Omega))\cap(L^{2}(0,T;\mathbf{V}^{1}_{0}(\Omega))\times
L^{2}(0,T;H^{1}(\Omega)))$ of $(\widehat{\mathbf{y}},\widehat{\tau})$
satisfying the weak formulation of equation (60) can be proved as in the case
of Navier-Stokes equations (see [16, Theorem 5.1]). ∎

###### Proof of Theorem Main Theorem.

As we split solution $(\mathbf{z},p,\theta)$ in the form
$(\mathbf{z},p,\theta)=(\mathbf{u},p_{1},\phi)+(\mathbf{y},q,\tau)$, theorem
Main Theorem follows from corollary 2, remark 6 and theorem 5.3. ∎

## References

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Received xxxx 20xx; revised xxxx 20xx.