
\section{Introduction}

In this paper, we study the dynamics of a tubular structure, a flexible pipe, which conveys a moving fluid. We 
assume 
that the pipe diameter is negligible compared to its length and we model the system by a one dimensional problem. 
In this 
context, the motion of the structure is driven by the following Euler-Bernoulli beam conveying 
fluid equation
\begin{equation}\label{P}\begin{split}\begin{cases}
\left(m_p+2m_{f}\right)w_{tt}(t)+EIw_{xxxx}(t)-\left(T-2m_{f}
V(t)^2\right)w_{xx}(t) \\ 
+cw_t(t)+2m_{f}V_t(t)w_{x}(t)+4m_{f}V(t)w_{xt}(t)=0,\ \ \ \ \ \ \ \ \ \ \ \ &0<x<L,\ t>0,\\
\end{cases}\end{split}
\end{equation}
endowed with the boundary conditions
\begin{equation}\label{P_boundaries}\begin{split}\begin{cases}
w_{xx}(0,t)=w_{xx}(L,t)=w(0,t)=0, &\ \ \ \ \ \ t>0,\\
 EIw_{xxx}(L,t)-(T-2m_{f}V(t)^2)w_{x}(L,t)+2m_{f}V(t)w_t(L,t)=0, &\ \ \ \ \ \ t>0.\\
\end{cases}\end{split}
\end{equation}
The parameters $L$ and $m_p$ are the length and the mass per unit length of the flexible pipe, while 
$EI$ and $T$ are its bending 
stiffness and its tension. The terms $m_{f}$ and $V(t)$ are the mass per unit 
length and the velocity of the internal fluid. We assume that the velocity $V(t)$ is strictly positive or 
strictly negative (we refer to Section \ref{conclusion} for further details on the general situation). The 
solution of the system \eqref{P} with boundary conditions \eqref{P_boundaries} represents 
the displacement of the flexible pipe at the position $x$ and time
$t$ (see Figure \ref{pipe}). We study the well-posedness of \eqref{P}-\eqref{P_boundaries} and the exponential 
stability of its solutions.




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\caption{The figure represents the displacement of a flexible pipe which conveys a moving fluid. }\label{pipe}
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The equation \eqref{P}-\eqref{P_boundaries} has been 
deduced when the velocity $V(t)$ is constant by Liu et al in \cite{Liu1} in 
presence of the environmental disturbances and a boundary control. The 
authors consider the problem of a flexible marine
riser and they compute the equation via a suitable energy functional. Finally, they prove the exponential decay of 
the 
solutions by Lyapunov method in presence of an additional boundary condition $w_x(0,t)=0$. 



In our work, we assume that the 
tension $T$ is larger than a specific value $T^*$ depending on the parameters of the problems. By exploiting 
such hypotheses, we firstly ensure the well-posedness of the \eqref{P}-\eqref{P_boundaries}. Secondly, we 
show the exponential stability of the energy of the solutions of \eqref{P}-\eqref{P_boundaries} without 
considering additional boundary conditions.



Our stability result is obtained by exploiting a different
energy functional from the one considered in \cite{Liu1} which validity is due to the assumption imposed on $T$. 
Such condition is not surprising from a practical 
point of 
view. It is reasonable to assume that the tension of the pipe has to be sufficiently strong, compared to 
the density and the velocity of the conveyed fluid, in order to have the stability. Otherwise, the fluid inside 
the pipe may dynamically interacts
with its motion, possibly causing the flexible pipe to vibrate.





Euler-Bernoulli beam conveying 
fluid equations are found in many practical applications. They are used to model for instance risers of 
offshore platforms, pipes carrying chemical
fluids, exhaust pipes in the
engines, flue-gases stacks, air-conditioning ducts, tubes in heat
exchangers and power plants, etc. A similar dynamics to \eqref{P} is studied in \cite{Khemmoudj2} by Khemmoudj 
where the internal damping $cw_t(t,x)$ is replaced by a viscoelastic term. There, the author considers suitable 
boundary conditions and he proves the exponential stability by Lyapunov 
method.
In \cite{Conrad}, Conrad et al. consider the equation $w_{tt}+w_{xxxx}=0$ and they prove exponential 
stability in presence of specific dynamical boundary conditions. For other similar results, we refer 
to \cite{Canbolat,Seghour}.
 
 
 \vspace{3mm}
 
 
The paper is organized as follows. In Section \ref{wellposedness}, we present the well-posedness of the equation 
\eqref{P}-\eqref{P_boundaries} in Theorem \ref{well}. In Section \ref{stability}, we ensure our stability result 
in 
Theorem 
\ref{TH1} 
by 
introducing a suitable Lyapunov functional.







\section{Well-posedness of the problem}\label{wellposedness}
The aim of this section is to prove the existence and uniqueness of solutions for \eqref{P}-\eqref{P_boundaries}. 
To this purpose, we 
define the space $$\mathcal{V}=\Big\{w\in H^2\big((0,L),\mathbb{R}\big)\ \big|\ w(0)=0\Big\}$$ and the Hilbert space
$$\mathcal{H}=\Big\{(w,v)\ \big|\ w\in \mathcal{V},\ v\in L^2\big((0,L),\mathbb{R}\big)\Big\}$$
equipped with the norm $\|\cdot\|_{\mathcal{H}}$ induced by the scalar product
$$\big\langle f_1,f_2\big\rangle_\mathcal{H}=\int_0^L \Big(\partial_{x}^2w_1\partial_x^2w_2+\partial_{x}w_1\partial_xw_2+v_1v_2\Big)dx,\ \ \ 
f_1=\big(w_1,v_1\big),\ 
f_2=\big(w_2,v_2\big)\in \mathcal{H}.$$

\smallskip


\noindent
We define the family $\{\mathcal{A}_0(t)\}_{t\geq 0}$ of operators in $\mathcal{H}$ such that
\begin{equation*}\begin{split}
\mathcal{A}_0(t)f&=\mathcal{A}_0(t)\left(
   \begin{array}{cc}
     w\\
v\\
    \end{array}
 \right)
\\
&=\left(
   \begin{array}{cc}
    v\\
     -\frac{EI}{m_p+2m_f} w_{xxxx}+\frac{T-2m_fV^2(t)}{m_p+2m_f}w_{xx}-\frac{4m_fV(t)}{{m_p+2m_f}} v_{x}\\
       \end{array}
 \right), 
\end{split}\end{equation*}
 for every $f=(w,v)$ in the domain $D(\mathcal{A}_0(t))$ defined by
\begin{equation*}\begin{split}D(\mathcal{A}_0(t))=\Big\{&(w,v)\in \mathcal{H}\ |\ w\in \mathcal{V}\cap H^4,\ v\in 
\mathcal{V},\ w_{xx}(0)=w_{xx}(L)=0,\\
& {2m_fV(t)} v(L)=-{EI}w_{xxx}(L)+{(T-2m_fV^2(t))}w_{x}(L)\Big\}.\end{split}\end{equation*}


\smallskip

\noindent
Let $\{B(t)\}_{t\geq 0}$ be the family of bounded operators in $\mathcal{H}$ such that
\begin{equation*}
B(t)f=\left(
   \begin{array}{cc}
   0\\
      \frac{c}{{m_p+2m_f}}v+\frac{2m_fV_t(t) }{m_p+2m_f}w_x \\
   \end{array}
 \right),\ \ \ \ \ \forall f\in \mathcal{H}, 
\end{equation*}

\smallskip

\noindent
By imposing $v=w_t$, the Cauchy problem \eqref{P}-\eqref{P_boundaries} becomes the following one in $\mathcal{H}$


\begin{equation}\label{P_1}
\begin{array}{ll}
\left(
   \begin{array}{cc}
     w_t\\
v_t\\
    \end{array}
 \right)=
\mathcal{A}(t)\left(
   \begin{array}{cc}
     w\\
v\\
    \end{array}
 \right),\ \ \ \ \ \ \ \ \ \  & \mathcal{A}(t):=\big(\mathcal{A}_0(t)+B(t)\big), \\\end{array}
 \end{equation}
endowed with the boundary conditions \eqref{P_boundaries}. The main result 
of this section is the 
well-posedness of the dynamics of \eqref{P_1}-\eqref{P_boundaries} in $\mathcal{H}$ which is
presented by the following theorem.

\begin{definition}
A two parameter family of bounded linear operators $U( t, s)$ with $0\leq  s\leq t$ on $\mathcal{H}$ is called an 
evolution system if the following conditions are satisfied:
\begin{itemize}
 \item $U(s, s) = I$ for every $s\geq 0$; 
 \item $U(t, r)U(r, s) = U(t, s)$ for every $0\leq s\leq r\leq t$;
\item for every $f\in\mathcal{H}$, the function $(t, s) \rightarrow U(t, s)f$ in $\mathcal{H}$ is continuous with 
respect to the norm of $\mathcal{H}$ for $0\leq s\leq t$.
\end{itemize}
\end{definition}


\begin{theorem}\label{well}
Let $V\in C^2\big([0,+\infty),\mathbb{R}\big)$ be a strictly positive or strictly negative function. Let 
$T>0$ 
be such that $T> 
2m_f\sup_{t\geq 0}V^2(t)$. There exists a unique evolution system $U(t,s)$ 
with $0\leq  s\leq t$ in $\mathcal{H}$ satisfying
\begin{itemize}
 \item $U(t, s)D(\mathcal{A}_0(s))= D(\mathcal{A}_0(t))$ for every $0\leq  s\leq t$; 
 \item for every $s\geq 0 $ and $f\in D(\mathcal{A}_0(s))$, the function $t\rightarrow U(t, s)f$ in $\mathcal{H}$ 
is continuously derivable with respect to the norm of $\mathcal{H}$  for $ s\leq t$ and
$$\frac{d}{d t}U(t,s) f= \mathcal{A}(t)U(t,s)f.$$
\end{itemize}
\end{theorem}
Theorem \ref{well} ensures the well-posedness of \eqref{P_1} endowed with the boundary 
conditions \eqref{P_boundaries}, 
and then of \eqref{P}-\eqref{P_boundaries}. The result is guaranteed when the tension $T$ is sufficiently large 
with respect to the velocity 
$V(t)$ and to the mass 
$m_f$. 



The following proposition is the core of the proof of Theorem \ref{well}. It ensures the existence of a unique 
evolution system on $\mathcal{H}$ generated by the 
family 
of operators $\left(\mathcal{A}_0(t), D(\mathcal{A}_0(t))\right)$. Such result leads to Theorem \ref{well} thanks 
to the properties of the family of operators $B(t)$ . 


\begin{proposition}\label{lemma_well}
Let $V\in C^1\big([0,+\infty),\mathbb{R}\big)$ be a strictly positive or strictly negative function. Let 
$T>0$ 
be such that $T> 
2m_f\sup_{t\geq 0}V^2(t)$. There exists a unique evolution system $U(t,s)$ 
with $0\leq  s\leq t$ in $\mathcal{H}$ satisfying
\begin{itemize}
 \item $U(t, s)D(\mathcal{A}_0(s))= D(\mathcal{A}_0(t))$ for every $0\leq  s\leq t$; 
 \item for every $s\geq 0$ and $f\in D(\mathcal{A}_0(s))$, the function $t\rightarrow U(t, s)f$ in $\mathcal{H}$ 
is 
continuously derivable with respect to the norm of $\mathcal{H}$  for $t\geq s$ and
$$\frac{d }{d t}U(t,s) f= \mathcal{A}_0(t)U(t,s)f.$$
\end{itemize}
\end{proposition}
\begin{proof}
Thanks to the assumptions on the velocity $V(t)$, we can define the norm $\|\cdot\|_{t}$ for every $t\geq 0$ 
of 
$\mathcal{H}$ induced by the scalar product
$$
\langle f_1,f_2\rangle_{t}=\int_0^L \big(\alpha(t)\partial_{x}^2w_1\partial_x^2w_2+\beta(t)\partial_{x}w_1\partial_xw_2+\gamma(t) 
v_1v_2\big)dx,
$$ for every $f_1=(w_1,v_1)$ and $f_2=(w_2,v_2)\in \mathcal{H}$ with 
$$\alpha:=\frac{EI}{2m_f},\ \ \ \ \ 
\beta(t)=\frac{T-2m_fV^2(t)}{2m_f},\ \ \ \ \ 
\gamma=\frac{m_p+2m_f}{2m_f}.$$
The assumptions on the potential $V$ and on the tension $T$ yield the existence of $C>1$ such that
\begin{align}\label{normequi}C^{-1}\|\cdot\|_{\mathcal{H}}\leq \|\cdot\|_t\leq C\|\cdot\|_{\mathcal{H}},\ \ \ \ \ \ \  \ \ \forall 
 t\geq 0.\end{align}
The domain of $\mathcal{A}_0(t)$ can be rewritten in terms of the parameters $\alpha$ and $\beta(t)$ as follows
\begin{equation*}\begin{split}D(\mathcal{A}_0(t))=\Big\{&(w,v)\in \mathcal{H}\ |\ w\in \mathcal{V}\cap H^4,\ v\in 
\mathcal{V},\ w_{xx}(0)=w_{xx}(L)=0,\\
&V(t)v(L)=-\alpha w_{xxx}(L)+\beta(t) w_{x}(L)\Big\}.\end{split}\end{equation*}

\smallskip
\noindent
{\bf 1) Dissipative property.} First, we prove that $\mathcal{A}_0(t)$ is dissipative for every $t\geq 0$ in $\mathcal{H}$ 
with 
respect to the scalar product $\langle\cdot,\cdot\rangle_{t}$. Let us denote 
$$a_1:=-\frac{EI}{m_p+2m_f},\ \ \ \  \ \ \ a_2(t):=\frac{T-2m_fV^2(t)}{m_p+2m_f},\ \ \ \ \  \ 
a_3(t):=-\frac{4m_fV(t)}{{m_p+2m_f}},$$
For every $f\in D(\mathcal{A}_0(t))$, thanks to the relations $$\gamma a_1=-\alpha,\ \ \ \ \ \ 
\ \  \ \gamma a_2(t)=\beta(t), \ \ \ \ \ \ \ \gamma(t)a_3(t)=-2V(t),$$ there holds 
\begin{align*}\langle f,\mathcal{A}_0(t) f\rangle_{t}=&\int_0^L\Big(\alpha w_{xx}v_{xx}+ \beta(t) w_{x}v_{x}+\gamma\big(a_1 
w_{xxxx}+a_2(t)w_{xx}+a_3(t)v_{x}\big)v\Big)dx\\
&=-\alpha w_{xxx}(L)v(L)+\beta(t) w_{x}(L)v(L)\\
&+\int_0^L\Big(\big(\alpha+\gamma a_1\big) w_{xxxx}+ 
\big(\beta(t)-\gamma a_2(t)\big) w_{xx}+\gamma a_3(t)v_{x}\big)v\Big)dx \\
&=\big(-\alpha w_{xxx}(L)+\beta(t) w_{x}(L)\big)v(L)-\int_0^L 2V(t)v_{x}v dx \\
&=V(t)v(L)^2-V(t)v(L)^2=
0.\end{align*}


\smallskip
\noindent
{\bf 2) Surjectivity conditions.} Second, fixed $t\geq 0$, we ensure the surjectivity of the map $$(\lambda 
I-\mathcal{A}_0(t)):D(\mathcal{A}_0(t))\subset \mathcal{H}\longrightarrow \mathcal{H}$$ for every $\lambda>0.$ The property is equivalent to prove 
that, for every $f^*=(w^*,v^*)\in \mathcal{H}$, there exists a unique solution $f=(w,v)\in D(\mathcal{A}_0(t))$ of the equation 
$(\lambda I-\mathcal{A}_0(t))f=f^*$. In other words, we need to study the existence of a unique solution of the following 
system of equations
\begin{equation}\label{initial}\begin{split}\begin{cases}
\lambda w(x) - v(x)= w^*(x),\\
\lambda v(x) - a_1 w_{xxxx}(x)-a_2(t)w_{xx}(x)-a_3(t) v_{x}(x) = v^*(x),& \ \ \ \ \ x\in [0,L],\\
V(t)v(L)=-\alpha w_{xxx}(L)+\beta(t)w_{x}(L),\\
w_{xx}(0)=w_{xx}(L)=w(0)=0.\\
\end{cases}\end{split}\end{equation}
 Solving the previous system is equivalent to study
\begin{equation*}\begin{split}\begin{cases}
\lambda^2 w -a_1 w_{xxxx}-a_2(t)w_{xx}-a_3(t) \lambda w_x = v^*-a_3(t) w^*_x+ \lambda w^*,\\
\lambda V(t)w(L) - V(t)w^*(L)=-\alpha w_{xxx}(L)+\beta(t)w_{x}(L),\\
w_{xx}(0)=w_{xx}(L)=w(0)=0.\\
\end{cases}\end{split}\end{equation*}
Let $g^*=v^*-a_3(t) w^*_x+ \lambda w^*$. The weak formulation of the previous system is given by 
\begin{equation}\label{weak}\begin{split}
&\int_0^L\Big(\lambda^2 w\phi -a_1 w_{xx}\phi_{xx}+a_2(t)w_{x}\phi_x-a_3(t)\lambda w_x(x)\phi\Big)dx\\
&-a_1 
w_{xxx}(L) \phi(L)-a_2(t)w_{x}(L)\phi(L)= \int_0^L g^*\phi dx\\
\end{split}\end{equation}
with $ \phi\in 
\mathcal{V}$. The identity \eqref{weak} can be rewritten as
\begin{equation*}\begin{split}
&\int_0^L\Big(\lambda^2 w\phi -a_1 w_{xx}\phi_{xx}+a_2(t)w_{x}\phi_x-a_3(t)\lambda w_x(x)\phi\Big)dx\\
&+\gamma^{-1}\big(\alpha w_{xxx}(L) -\beta(t)w_{x}(L)\big)\phi(L)= \int_0^L g^*\phi dx,\\
\end{split}\end{equation*}
Thanks to the validity of the boundary conditions in $L$, we have
\begin{equation*}\begin{split}
&\int_0^L\Big(\lambda^2 w\phi -a_1 w_{xx}\phi_{xx}+a_2(t)w_{x}\phi_x-a_3(t)\lambda 
w_x\phi\Big)dx-\gamma^{-1}\lambda V(t) w(L)\phi(L)\\
&= -\gamma^{-1}V(t) w^*(L)\phi(L)+\int_0^L g^*\phi dx.\\
\end{split}\end{equation*}
We consider the bilinear form $\langle \cdot,\cdot\rangle_{\mathcal{A}_0(t) }$ in $\mathcal{V}$ such that, for every 
$w,\phi\in\mathcal{V}$,
\begin{equation*}\begin{split}
\langle w,\phi\rangle_{\mathcal{A}_0(t) }=&\int_0^L\Big(\lambda^2 w\phi -a_1 w_{xx}\phi_{xx}+a_2(t)w_{x}\phi_x-a_3(t)\lambda 
w_x\phi\Big)dx\\
&-\gamma^{-1}\lambda V(t) w(L)\phi(L).\\
\end{split}\end{equation*}
We notice that, for every $w\in \mathcal{V}$, 
$$\langle w,w\rangle_{\mathcal{A}_0(t) }=\int_0^L\big(\lambda^2 
w^2 -a_1 w_{xx}^2+a_2(t)w_{x}^2\big)dx -\Big(\frac{a_3(t)}{2}+\gamma^{-1}V(t)\Big)\lambda w(L)^2.$$ Thanks to 
the identity 
$$\frac{a_3(t)}{2}+\gamma^{-1}V(t)=0,$$
$\langle \cdot,\cdot\rangle_{\mathcal{A}_0(t) }$ is a coercive bilinear form as $-a_1>0$ and $a_2(t)>0$. From the 
Lax-Milgram theorem, the weak formulation admits an unique solution. Thanks to the regularity of $w \in H^4 ((0, 
L),\mathbb{R})$ and by using particular $\phi$, it is possible to recover the boundary conditions in $w$ and $v$ is 
defined 
by \eqref{initial} which is unique. Now, $f=(w, v)\in  D(\mathcal{A}_0(t))$. This shows the surjectivity, for every 
$\lambda>0$, of the map $$(\lambda 
I-\mathcal{A}_0(t)):D(\mathcal{A}_0(t))\subset \mathcal{H}\longrightarrow \mathcal{H}.$$ 

\smallskip
\noindent
{\bf 3) Conclusion.} Thanks to the point {\bf 1)}, the operator $\mathcal{A}(t)$ is dissipative for $t\geq 0$ with 
respect to the scalar product $\langle\cdot,\cdot\rangle_{t}$
and then, for every $\lambda>0$, $$\|(\lambda 
I-\mathcal{A}_0(t))\psi\|_t\geq \lambda \|\psi\|_t,\ \ \ \ \ \psi\in D(\mathcal{A}_0(t)).$$ Now, we define a 
new 
norm of $\mathcal{H}$ $$\|\cdot\|_t':=m(t)\|\cdot\|_t,\ \ \ \ \ \forall t\geq 0$$ by choosing a suitable strictly 
positive
function $m:\mathbb{R}^+\longrightarrow \mathbb{R}^+$ 
such that there exists $C>1$ such that
$$C^{-1}\|\cdot\|_{\mathcal{H}}\leq \|\cdot\|_t'\leq \|\cdot\|_s'\leq C\|\cdot\|_{\mathcal{H}},\ \ \ \ \ \ \  \ \ \forall 0\leq 
s\leq t$$
(by keeping in mind the validity of the inequality $\eqref{normequi}$). Thus, we have 
$$\|(\lambda I-\mathcal{A}_0(t))\psi\|'_t\geq \lambda \|\psi\|'_t,\ \ \ \ \ \  \ \ \forall \lambda>0,\ \psi\in 
D(\mathcal{A}_0(t)).$$
Let us denote, for every $t\geq 0$ and $x\in [0,L]$, $$f(t,x)= \Bigg(1+\frac{x}{L}\Big(\frac{|V(t)|}{\alpha} 
-1\Big)\Bigg),$$
$$g(t,x)=sign(V(t))\frac{\beta(t)}{\alpha}\frac{x}{L}.$$ 


\medskip

\noindent
We introduce the family of operators $\Gamma_t$ with 
$t\geq 0$ such that $$\Gamma_t:(w,v)\mapsto \big(w,f(t,\cdot) 
v-g(t,\cdot)w_x\big).$$ 
We 
notice that 


\begin{equation*}\begin{split}\Gamma_t(D(\mathcal{A}_0(t)))=\Big\{&(w,v)\in 
\mathcal{H}\ |\ w\in 
\mathcal{V}\cap H^4,\\
& v\in \mathcal{V},\ w_{xx}(0)=w_{xx}(L)=0,\,\ \ 
 v(L)=-sign{(V(t))}w_{xxx}(L)\Big\}\end{split}\end{equation*} which does not depends 
on time since $V(t)$ has constant sign. Each 
$\Gamma_t:\mathcal{H}\rightarrow\mathcal{H}$ is invertible and $\Gamma_t^{-1}$ is a family of linear bounded and invertible 
operators satisfying \cite[hypotheses (4.3) and (4.4); p. 309]{Kisynski}.
In conclusion, the validity of \cite[Theorem 4.2]{Kisynski} yields the result.
\qedhere
\end{proof}




\begin{proof}[Proof of Theorem \ref{well}]
The statement follows from Proposition \ref{lemma_well} and classical arguments of existence of evolution 
systems (see 
for instance \cite[Chapter\ 6.1; Theorem\ 1.2\ \&\ Theorem\ 
1.5]{Paz83}). The property is due to the the smoothness of the family of 
bounded operators $B(t)$ which is due to hypotheses imposed on the velocity $V(t)$.\qedhere 
\end{proof}

\section{Exponential stability of the problem}\label{stability}
\section*{\small Main result}
In this section, we ensure the stability result for the
solution of the problem \eqref{P}-\eqref{P_boundaries}. To the purpose, we introduce the 
following assumptions,

\begin{assumption1}
Let $V\in C^2\big([0,+\infty),\mathbb{R}\big)$ be a strictly positive or strictly negative function. 
Let $T>0$ be such that $$
T> 2m_f\sup_{t\geq 0}V(t)^2+\max\{T_1,T_2\},
$$
where we denote
$$
T_1=\frac{L^2}{4}(m_p+2m_f)+2\sqrt{2} Lm_f \sup_{t\geq 0}|V(t)|$$
and
$$T_2=\frac{c^2L^2}{8(c-m_p-2m_f)}+2m_f\sup_{t\geq 0}|V_t(t)V(t)|.
$$
\end{assumption1}

{\it Assumptions I} ensure that the tension $T$ of 
the beam is sufficiently strong with respect to the velocity $V$ and to the parameters of the problem. In this 
framework, the quantity $\frac{T}{2}-m_f V^2(t)>0$ for every $t\geq 0$ and it is valid the well-posedness 
result provided in the previous section by Theorem \ref{well}. In addition, the assumption on $T$ allows us to 
consider the 
following energy 
functional associated to the solutions of \eqref{P}-\eqref{P_boundaries}
\begin{equation}\label{en1}\begin{split}
E(t)=&\frac{1}{2}(m_p+2m_f)\int^L_0w_t^2(t)\,dx+\frac{EI}{2}\int^L_0w_{xx}^2(t)\,dx\\
&+\Big(\frac{T}{2}-m_f 
V^2(t)\Big)\int^L_0w_{x}^2(t)\,dx.
\end{split}\end{equation}
We are finally ready to state the stability result of the 
problem \ref{P}-\eqref{P_boundaries}.



\begin{theorem}\label{TH1}
Let the problem \eqref{P}-\eqref{P_boundaries} satisfy {\it{Assumptions I}}. There 
exist two 
positive constants $k_0$  and $k_1$, such that for any solution $w$ of \eqref{P}-\eqref{P_boundaries}, the energy 
functional 
\eqref{en1} decays as follows
\begin{equation}\label{BB}
E(t)\leq k_0 E(s) e^{-k_1 (t-s)}\,\,\, \ \forall\  0\leq s \leq t .
\end{equation}
\end{theorem}


The proof of Theorem \ref{TH1} is established in the final part of the section by gathering different results. We 
firstly compute the time derivative of the energy functional
$\frac{d}{dt}E(t)$. After, we introduce a suitable Lyapunov functional $\mathcal{L}(t)$ for which 
there 
exist $C_1,C_2>0$ so that $C_1 E(t)\leq \mathcal{L}(t)\leq C_2 E(t) $. Finally, we show 
the existence of 
$C_3>0$ such that $\frac{d}{dt}\mathcal{L}(t)\leq - C_3 \mathcal{L}(t)$. This identity implies 
$\mathcal{L}(t)\leq \mathcal{L}(s) e^{-C_3 (
t-s)}$ and Theorem \ref{TH1} is proved by gathering the previous results.

\section*{\small Some preliminaries}




We start by recalling the following
Sobolev-Poincar\'e inequality (see \cite{adam} for further details). 





 \begin{lemma}\label{L1}(Sobolev-Poincar\'e inequality) 
For every $v\in \{w\in H^1(0,L)\ |\ w(0)=0\}$, there holds 
$$
\int_0^Lv^2\,dx \leq P \int_0^L v_x^2\,dx,\ \ \ \ \ \ \ \text{with}\ \ \ \ \ \ P:=\frac{L^2}{2}.
$$\end{lemma}
\begin{lemma}\label{L1A}
Let $w$ be a solution of the problem \eqref{P}-\eqref{P_boundaries}. The energy functional 
defined by \eqref{en1} satisfies
\begin{equation*}
\begin{split}
\frac{d}{dt}E(t)
=&-c\int_0^L w^2_t(t)\,dx -2m_{f}V_t(t)\int_0^L w_t(t)w_{x}(t)\,dx \\
&-2m_fV_t(t)V(t)\int_0^Lw_x^2(t)\,dx.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
By multiplying the first equation in (\ref{P}) with $w_t$ and by integrating over  $(0,L)$, we obtain
\begin{equation}\label{8MAA}
\begin{split}
&\int^L_0w_t(t)\Big[(m_p+2m_f)w_{tt}(t)+EIw_{xxxx}(t)\\
&-(T-2m_fV^2(t))w_{xx}(t)\Big]\,dx+4m_{f}
V(t)\int_0^Lw_t(t) w_{xt}(t)\,dx\\
&=-c\int_0^L w^2_t(t)\,dx-2m_{f}V_t(t)\int_0^L  w_t(t)w_{x}(t)\,dx.
\end{split}
\end{equation}
We study each term appearing in first integral of \eqref{8MAA}
\begin{equation}\label{A1}
(m_p+2m_f)\int^L_0w_t(t)w_{tt}(t)\,dx=\frac{1}{2}(m_p+2m_f)\frac{d}{dt}\int^L_0w_t^2(t)\,dx,
\end{equation}
\begin{equation}\label{A2}
EI\int^L_0w_t(t)w_{xxxx}(t)\,dx=EI w_t(L,t)w_{xxx}(L,t)+\frac{EI}{2}\frac{d}{dt}\int^L_0w_{xx}^2(t)\,dx
\end{equation}
and 
\begin{equation}\label{A3}
\begin{split}
-(T-2m_fV^2(t))\int^L_0w_{t}(t)w_{xx}(t)\,dx=&\,\big(2m_fV^2(t)-T\big)\,w_t(L,t)w_x(L,t)\\
&+\frac{T}{2}\frac{d}{dt}\int^L_0w_{x}
^2(t)\,dx-m_fV^2(t)\frac{d}{dt}\int^L_0w_{x}^2(t)\,dx.\\
\end{split}
\end{equation}
The right-hand side of \eqref{A3} can be rewritten as follows
\begin{equation}\label{A4}
\begin{split}
&(2m_fV^2(t)-T)w_t(L,t)w_x(L,t)+\frac{T}{2}\frac{d}{dt}
\int^L_0w_{x}
^2(t)\,dx\\
&-m_f\frac{d}{dt}\Big(V^2(t)\int_0^Lw_x^2\,dx\Big)+ 2m_fV_t(t)V(t)\int_0^Lw_x^2\,dx.
\end{split}
\end{equation}
Now, we investigate the remaining term in the left-hand side of \eqref{8MAA}
\begin{equation}\label{A5}
4m_{f}V(t)\int_0^Lw_t(t) w_{xt}(t)\,dx= 2m_fV(t)w^2_t(L,t).
\end{equation}
By using (\ref{A1})-(\ref{A5}) into (\ref{8MAA}) yields the following expression
\begin{equation*}
\begin{split}
&\frac{d}{dt}E(t)+\Big(EIw_{xxx}(L,t)-(T-2m_{f}V^2(t))w_{x}(L,t)+2m_{f}V(t)w_t(L,t)\Big)w_t(L,t)\\&
=-c\int_0^L w^2_t(t)\,dx -2m_{f}V_t(t)\int_0^L w_t(t)w_{x}(t)\,dx 
-2m_fV_t(t)V(t)\int_0^Lw_x^2(t)\,dx.
\end{split}
\end{equation*}
The boundary conditions \eqref{P_boundaries} complete the proof.\qedhere
\end{proof}



\section*{\small A suitable Lyapunov functional}



Let us introduce the functionals
\begin{equation*}\begin{split}
&G_1(t)=(m_p+2m_f)\int_0^L w(t)w_t(t)\,dx,\\
&G_2(t)=2m_fV(t)\int_0^L w(t)w_x(t)\,dx,
\end{split}\end{equation*}
and
\begin{eqnarray}
G(t)&=&G_1(t)+G_2(t).\label{gg}
\end{eqnarray}

\smallskip

\noindent
We define the Lyapunov functional $\mathcal{L}$ such that
\begin{equation}\label{L}
\mathcal{L}(t)=E(t)+G(t).
\end{equation}
\begin{lemma} 
Let the problem \eqref{P}-\eqref{P_boundaries} satisfy {\it{Assumptions I}}. There 
exist two positive 
constants $ \xi_1>0$ and $ \xi_2>0$ depending on
$m_p$,\,\, $m_f$,\,\,\, $T$ and $V(t)$ such that, for all $t\geq 0$,
\begin{equation}\label{B}
 \xi_1  E(t)\leq \mathcal{L}(t) \leq \xi_2 E(t).
\end{equation}
\end{lemma}
\begin{proof}
Let $P$ be the Poincar\'e's constant from Lemma \ref{L1} and $\alpha_1>0$. By using the Young's and the 
Poincar\'e inequalities, we obtain
\begin{equation}\label{8AM}
\begin{split}
|G_1(t)| \leq&  \frac{\alpha_1}{2}\big( m_p +2m_{f}\big) \int_0^L w^2_t(t)\,dx\\
&+\frac{P}{2\alpha_1}\big( m_p 
+2m_{f}\big) \int_0^Lw^2_x(t)  \,dx.
\end{split}
\end{equation}
By setting $\alpha_2=\sqrt{\frac{1}{P}}$, we have
\begin{equation}\label{1K00}
\begin{split}
|G_2(t)| \leq& 2m_f|V(t)| \Big(\frac{\alpha_2P}{2}+\frac{1}{2\alpha_2}\Big)\int_0^L w_x^2(t)\,dx\\
=&
2m_f|V(t)|\sqrt{P}\int_0^L w_x^2(t)\,dx
\end{split}
\end{equation}
In the last relation, we impose $\alpha_2=\sqrt{\frac{1}{P}}$ in order to minimize the function 
$\alpha_2 \mapsto 
\{\frac{P}{2}\alpha_2 +\frac{1}{2\alpha_2}\}$ which attains its minimum in $\sqrt{P}$ exactly when 
$\alpha_2=\sqrt{\frac{1}{P}}$. By combining (\ref{en1}), (\ref{8AM}) and (\ref{1K00}), we have
\begin{equation}\label{8AME}
\begin{split}
\mathcal{L}(t)&\leq \Bigg(\frac{T}{2}-m_fV^2(t)+\frac{P}{2\alpha_1}\big( m_p +2m_{f}\big) +2m_f|V(t)| 
\sqrt{P}\Bigg)\int_0^L w_x^2(t)\,dx\\&
+(m_p+2m_f)\Big(1+\frac{\alpha_1}{2}\Big)  \int_0^L w^2_t(t)\,dx+EI\int_0^L w_{xx}^2(t)\,dx.
\end{split}
\end{equation}
For the lower bound, we can see that
\begin{equation}\label{8AMK}
\begin{split}
\mathcal{L}(t)&\geq  \Bigg(\frac{T}{2}-m_fV^2(t)-\frac{P}{2\alpha_1}\big( m_p +2m_{f}\big) 
-2m_f|V(t)|\sqrt{P}\Bigg)\int_0^L w_x^2(t)\,dx\\&
+(m_p+2m_f)\Big(1-\frac{\alpha_1}{2}\Big)  \int_0^L w^2_t(t)\,dx+EI\int_0^L w_{xx}^2(t)\,dx.
\end{split}
\end{equation}
We recall that $P=\frac{L^2}{2}$ and, thanks to {\it Assumptions I},
\begin{align}\label{assump}T> 2m_f\sup_{t\geq 0}V(t)^2+\frac{L^2}{4}(m_p+2m_f)+2\sqrt{2} Lm_f\sup_{t\geq 0}|V(t)|.
\end{align}
Now, we choose $\alpha_1\in [1,2)$ in \eqref{8AME} and \eqref{8AMK} as such number 
such that
$$\frac{T}{2}-m_f\sup_{t\geq 0}V(t)^2-\frac{P}{2\alpha_1}\big( m_p +2m_{f}\big) 
-2m_f\sup_{t\geq 0}|V(t)|\sqrt{P}>0,$$
which existence is guaranteed by \eqref{assump}. In addition, $(m_p+2m_f)(1-\frac{\alpha_1}{2}) >0$ and 
$$\frac{T}{2}-m_f\inf_{t\geq 0}V^2(t)\geq \frac{T}{2}-m_f\sup_{t\geq 0}V^2(t) >0$$ thanks to 
\eqref{assump}. By combining 
(\ref{8AME}) and (\ref{8AMK}), we obtain  (\ref{B}). The constants $\xi_1$ and $\xi_2$ are 
explicitly given by
\begin{equation*}\begin{split}
\xi_1=\max\Bigg\{&\frac{1}{\frac{T}{2}-m_f\sup_{t\geq 0}V(t)^2}\Big(\frac{T}{2}-m_fV^2(t)+\frac{P}{
2\alpha_1 }
\Big( m_p +2m_{f}\Big) \\
&+2m_f\sup_{t\geq 0}|V(t)| \sqrt{P},\ 2\Big(1+\frac{\alpha_1}{2}\Big),\ 2\Bigg\},
\end{split}\end{equation*}
\begin{equation*}\begin{split}
\xi_2=\min\Bigg\{&\frac{1}{\frac{T}{2}-m_f\inf_{t\geq 0}V^2(t)}\Big(\frac{T}{2}-m_fV^2(t)-\frac{P}{2\alpha_1}
\big( m_p +2m_{f}\big) \\
&-2m_f\sup_{t\geq 0}|V(t)|\sqrt{P},\ 2\Big(1-\frac{\alpha_1}{2}\Big),\ 
2\Bigg\}.\qedhere\end{split}\end{equation*}
\end{proof}
\begin{lemma}\label{L2M1}
Let $w$ be the solution of \eqref{P}-\eqref{P_boundaries}. The functional $G$ defined by (\ref{gg})
satisfies
\begin{equation*}
\begin{split}
&\frac{d}{dt} G(t)=-EI\int_0^L w^2_{xx}(t)\,dx-(T-2m_fV^2(t))\int_0^L w^2_{x}(t)\,dx 
\\&  -c\int_0^L w(t)w_{t}(t)\,dx+
2m_fV(t)\int_0^L w_x(t)w_t(t)\,dx+(m_p+2m_f)\int_0^L w^2_t(t)\,dx.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
We know that $\frac{d}{dt}G(t)=\frac{d}{dt} G_1(t)+\frac{d}{dt}G_2(t)$. Now,
 \begin{equation*}
\begin{split}
\frac{d}{dt} G_1(t) &=(m_p+2m_f)\int_0^L w(t)w_{tt}(t)\,dx+(m_p+2m_f)\int_0^L w^2_t(t)\,dx\
\end{split}
\end{equation*}
We use \eqref{P} and we obtain
 \begin{equation*}
\begin{split} \frac{d}{dt} G_1(t) &=-EI 
\int_0^Lw_{xxxx}(t)w(t)+(T-2m_fV^2(t))\int_0^Lw_{xx}(t)w(t)\,dx\\
&-2m_fV_t(t) \int_0^L w_x(t)w(t)\,dx-4m_fV(t) \int_0^L w_{xt}(t)w(t)\,dx
\\& 
-c\int_0^L w(t)w_{t}(t)\,dx+(m_p+2m_f)\int_0^L w^2_t(t)\, dx
.
\end{split}
\end{equation*}
We integrate by parts and $\frac{d}{dt} G_1(t)$ becomes
 \begin{equation*}
\begin{split}
&-\Big(EI w_{xxx}(L,t)-\big(T-2m_fV^2(t)\big) w_{x}(L,t)+2m_fV(t)w_t(L,t)\Big)w(L,t)\\
&
-EI\int_0^L w^2_{xx}(t)\,dx-\big(T-2m_fV^2(t)\big)\int_0^L w^2_{x}(t)\,dx -2m_fV_t(t) \int_0^L w_x(t)w(t)\,dx \\
&-2m_fV(t) \int_0^L w_{xt}(t)w(t)\,dx-c\int_0^L w(t)w_{t}(t)\,dx+(m_p+2m_f)\int_0^L w^2_t(t)\,dx .
\end{split}
\end{equation*}
By using the boundary conditions, we obtain
 \begin{equation}\label{8Af}
\begin{split}
\frac{d}{dt} G_1(t) &=-EI\int_0^L w^2_{xx}(t)\,dx-\big(T-2m_fV^2(t)\big)\int_0^L w^2_{x}(t)\,dx \\
&-2m_fV_t(t) 
\int_0^L w_x(t)w(t)\,dx -2m_fV(t) \int_0^L w_{xt}(t)w(t)\,dx\\
&-c\int_0^L w(t)w_{t}(t)\,dx+(m_p+2m_f)\int_0^L w^2_t(t)\,dx .
\end{split}
\end{equation}
Now, we compute $\frac{d}{dt}G_2(t)$
 \begin{equation}\label{8Ag}
\begin{split}
\frac{d}{dt} G_2(t) &=2m_fV(t)\int_0^L w_{xt}(t)w(t)\,dx+2m_fV(t)\int_0^L 
w_x(t)w_t(t)\,dx\\&+2m_fV_t(t)\int_0^Lw(t)w_x(t)\,dx.
\end{split}
\end{equation}
By gathering (\ref{8Af}) and (\ref{8Ag}), we achieve the claim.\qedhere
\end{proof}

\section*{\small Proof of Theorem \ref{TH1}}


Now, we are ready to prove our main result.\\
\begin{proof}[Proof of Theorem \ref{TH1}.]
By the definition (\ref{L}), we have $\frac{d}{dt}\mathcal{L}(t)=\frac{d}{dt} E(t)+\frac{d}{dt}G(t)$. Thanks 
to Lemma \ref{L1A} and Lemma \ref{L2M1}, we obtain
\begin{equation}\label{base}
\begin{split}
\frac{d}{dt}\mathcal{L}(t)&=-c\int_0^L w^2_t(t)\,dx 
-2m_fV_t(t)V(t)\int_0^Lw_x^2(t)\,dx\\
&-EI\int_0^L w^2_{xx}(t)\,dx-(T-2m_fV^2(t))\int_0^L w^2_{x}(t)\,dx \\
&-c\int_0^L 
w(t)w_{t}(t)\,dx+(m_p+2m_f)\int_0^L w^2_t(t)\,dx.
\end{split}
\end{equation}
By using Young's inequality and Poincar\'e inequality, for every $\alpha_1>0$, we have
\begin{equation}\begin{split}\label{YP}-c\int_0^L w(t)w_{t}(t)\,dx&\leq c\frac{\alpha_1}{2} \int_0^L 
w_t(t)^2\,dx+\frac{1}{2\alpha_1}c\int_0^L w(t)^2\,dx\\
&\leq c\frac{\alpha_1}{2} \int_0^L 
w_t(t)^2\,dx+\frac{P}{2\alpha_1}c\int_0^L 
w_{x}(t)^2\,dx.\end{split}\end{equation}
Now, thanks to {\it Assumptions I}, there exists $\delta\in (0,c-m_p-2m_f)$ such that
\begin{align}\label{TT}
T>2m_f\sup_{t\geq 0}V(t)^2 +\frac{c^2 P}{4(c-m_p-2m_f-\delta)}+2m_f\sup_{t\geq 0}|V_t(t)V(t)|.
\end{align}
We set $\alpha_1=2\frac{c-m_p-2m_f-\delta}{c}$ in \eqref{YP} that we use in \eqref{base} in order to obtain
\begin{equation*}
\begin{split}
\frac{d}{dt}\mathcal{L}(t)& \leq -\gamma_0\int_0^L w^2_t(t)\,dx-\gamma_1\int_0^L w^2_x(t)\,dx-EI\int_0^L 
w^2_{xx}(t)\,dx
\end{split}
\end{equation*}
where
$$
\gamma_0=c-c\frac{\alpha_1}{2}-m_p-2m_f,$$
$$
\gamma_1=T-2m_f\sup_{t\geq 0}|V_t(t)V(t)|-2m_f\sup_{t\geq 0}V(t)^2-c\frac{P}{2\alpha_1}.
$$
We notice that $\gamma_0>0$ thanks to the choice of $\alpha_1$ and $\gamma_1>0$ thanks to \eqref{TT}. Finally, we 
see that
\begin{equation}\label{M1}
\frac{d}{dt}\mathcal{L}(t) \leq -\vartheta E(t)
\end{equation}
with 
$$\vartheta:
=\min\Bigg\{\frac{2\gamma_0}{m_p+2m_f},\frac{2\gamma_1}{T-2m_f\inf_{t\geq 0}V(t)^2},2\Bigg\}.$$
A combination of (\ref{B}) and (\ref{M1}) gives
\begin{equation}\label{M2}
\frac{d}{dt}\mathcal{L}(t) \leq -k_1\mathcal{L}(t),\,\,\,\, \forall t\geq 0,
\end{equation}
where $k_1=\frac{\vartheta}{\xi_2}$. We integrate (\ref{M2}) over $(s,t)$ and
\begin{equation}\label{M3}
\mathcal{L}(t) \leq \mathcal{L}(s) e^{-k_1(t-s)},\,\,\,\, \forall t\geq s.
\end{equation}
Finally, by combining (\ref{B}) and (\ref{M3}), we obtain (\ref{BB}) with $k_0=\frac{\xi_2}{\xi_1}$, which
completes the proof.\qedhere\end{proof}



\section{Conclusions}\label{conclusion}


In Section \ref{wellposedness} and Section \ref{stability}, we studied the well-posedness and the stability of the 
solutions of \eqref{P}-\eqref{P_boundaries}. We assumed that the velocity $V(t)$ is a sufficiently smooth function 
with constant 
sign, while the 
tension $T$ is larger than a specific value $T^*$ depending on the parameters of the problems. By exploiting 
such hypotheses, we proved the well-posedness of \eqref{P}-\eqref{P_boundaries}. Secondly, we 
ensured the exponential stability by introducing a suitable Lyapunov functional. 


The choice of considering $V$ with constant sign is due to the following reason. When $V$
vanishes for some time, the dynamical boundary condition in \eqref{P_boundaries} becomes a statical boundary 
condition. In this case, the problem lacks of a boundary condition on $w_t(t,x)$ and 
it is not clear which ``natural" boundary condition appears in such a context. Nevertheless, the 
stability result from Section \ref{stability} could still be 
valid, at least from a formal point of view.



Finally, the assumption on $T$ is not so surprising when we think to the nature of the problem modeled by 
\eqref{P}-\eqref{P_boundaries}. It is reasonable to assume that the tension of the pipe has to be 
sufficiently strong, compared to 
the density and the velocity of the conveyed internal fluid, in order to have the stability. From this 
perspective, 
it could be interesting to explore this phenomenon further, at least from a numerical point of view. One 
could seek for evidences of instability phenomena when the tension $T$ is too low. 






\smallskip
\noindent
{\bf Acknowledgments.} The authors are grateful to Ka\"is Ammari for suggesting them the problem. The third author 
also thanks the colleagues Romain Joly and Nabile Boussa\"id for the fruitful discussions on the well-posedness 
of the problem. The third author has been funded by the project ISDEEC ANR-16-CE40-0013.

