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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 27, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/27\hfil Regularity of vibrations]
{Regularity result for the problem of vibrations of a
nonlinear beam}
\author[M. F. M'Bengue, M. Shillor\hfil EJDE-2008/27\hfilneg]
{M'Bagne F. M'Bengue, Meir Shillor}
\address{M'Bagne F. M'Bengue \newline
Department of Mathematics and Statistics \\
Oakland University, Rochester, MI 48309, USA}
\email{mfmbengu@oakland.edu}
\address{Meir Shillor \newline
Department of Mathematics and Statistics \\
Oakland University, Rochester, MI 48309, USA}
\email{shillor@oakland.edu}
\thanks{Submitted January 25, 2008. Published February 28, 2008.}
\subjclass[2000]{74H30, 35L75, 74K10, 74D10, 74H45}
\keywords{Nonlinear beam; existence and uniqueness; energy balance;
\hfill\break\indent pseudomonotone operators; regularity}
\begin{abstract}
A model for the dynamics of the Gao nonlinear beam, which allows for
buckling, is studied. Existence and uniqueness of the local weak solution
was established in Andrews {\it et al.} (2008). In this work the further
regularity in time of the weak solution is shown using recent results for
evolution problems. Moreover, the weak solution is shown to be global,
existing on each finite time interval.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction} \label{sec:int}
A model for the vibrations of a nonlinear beam that takes into
account the beam's thickness which, however, is one-dimensional, was
derived by Gao in \cite{Gao96,Gao00}. The existence of the unique
local weak solution for the problem was established recently in
\cite{ADuMBS08}. In this work we show that the weak solution has
additional regularity in time, when the problem data is smoother.
This allows us to establish that the weak solution is also a
global solution existing on each finite time interval.
The motivation for the introduction of various models for beams
was to capture more fully the nonlinearities that beams exhibit,
in particular buckling, which is associated with a double-well in
the energy function of the beam. Among the models derived in
\cite{Gao00}, the one-dimensional model studied in
\cite{ADuMBS08} and here is the simplest. However, it is
nonlinear, it allows for buckling, and has interest in and of
itself. The literature on the beam includes also \cite{RW02} and
the references therein.
This work is the continuation of \cite{ADuMBS08} where, in
addition to the analysis, a finite difference scheme for the beam
was introduced based on the Newmark time discretization and
Hermite cubic finite elements, and the results of numerical
simulations depicted. Here, we prove that if the problem data is
more regular, then the solution has additional time regularity.
The proof is based on the problem for the time derivative of the
solution. We first study the truncated problem, and use results
for variational problems for pseudomonotone operators of
\cite{Ken86, KSpsm}. Then, we use a continuity argument to show
that there exists a unique weak solution such that for some time
the truncation is inactive. Using the energy balance and a priori
estimates derived from it, we also show that for a sufficiently
large truncation ceiling, the truncation is not active on each
preassigned time interval, and so the solution is global.
The rest of the paper is organized as follows. In Section 2 we
present shortly the classical formulation of the model, following
\cite{Gao96, Gao00, ADuMBS08}. In Section \ref{sec:Problem} the
weak formulation and the statement of the existence and uniqueness
result in \cite{ADuMBS08} is given, the weak or abstract
formulation of the problem for the time derivative presented and
the statements of the existence and uniqueness results for the
truncated problem and the full problem stated. Our main
regularity result is states in Theorem \ref{thm34}. The proof is
provided in Section \ref{sec:proof}, and is based on the results
for the truncated problem in Section \ref{sec:proofR}. In Section
\ref{sec:E} we present the energy balance equation for the
original problem, derive a priori estimates on
$\|w_x\|_{L^\infty(0, T; L^\infty(0,1))}$, and based on this
estimate we conclude that the solution of the problem is global in
time.
\section{The Model}\label{sec:Problem}
The derivation of the model was done in Gao in \cite{Gao96, Gao00}, and a
more detailed description can be found in \cite{ADuMBS08}. Here,
we just present it with very few comments. The beam's centerline
is, in dimensionless variables, $[0, 1]$ and its thickness $2h$,
and we denote by $w(x,t)$ the transverse displacement of its
central axis, for $0\leq x\leq 1$ and $0\leq t \leq T$, for
$00$, assumed to be small.
The classical formulation of the {\it dynamic model for a beam
with finite deformations} with viscosity, is as follows.
\subsection*{Problem $P_{cl}$}
Find the displacement field
$w=w(x, t)$ for $x\in (0, 1)$ and $t\in (0, T)$, such that
\begin{gather}
w_{tt} + k w_{xxxx}+ \gamma w_{txxxx}
-(aw_x^2 - \nu p) w_{xx}= f, \label{p1}\\
w(0,t)=w_x(0,t) =0, \label{p3}\\
w(1,t)=w_x(1,t) =0, \label{p4}\\
w(x, 0)=w_0(x),\quad w_t(x,0)=v_0(x). \label{p5}
\end{gather}
The system is nonlinear and the existence and uniqueness of the
local weak solution to the problem has been established in
\cite{ADuMBS08}. We first present the weak formulation of the
problem and then show that additional assumptions on the problem
data lead to an improved regularity, in time, of the solution.
\section{Weak formulation and results}\label{sec:weak}
We first describe the weak formulation of Problem $P_{cl}$, the
assumptions on the problem data, and state the existence and
uniqueness result for local weak solutions. We follow
\cite{ADuMBS08}. Then, we describe the problem obtained by
differentiating Problem $P_{cl}$ with respect to time.
We denote by $(\cdot, \cdot)$ the inner product in $H=L^2(0, 1)$,
and let
\[
W=H^2_0(0,1)=\{u\in H^2(0, 1): u=u_x=0\;\mbox{at }\; x=0, 1\},
\]
be a Hilbert space endowed with the inner product $(w, u)_W =
\int_0^1 w_{xx} u_{xx}\,dx,$ and the associated norm
$\|w\|_W^2=(w,w)_W$ which, in view of the boundary conditions and
the Poincare theorem, is equivalent to the usual $H^2(0,1)$ norm
on $W$. The dual of $W$ is denoted by $W'$, and since
$W\subseteq H \subseteq W'$, by identifying $H'=H$, it follows that
$(W, H, W')$ is a Gelfand triple. Next, let $\mathcal{H}=L^2(0, T; H)$,
and $\mathcal{W}=L^{2}(0,T; W)$ with inner product
$(\cdot, \cdot)_\mathcal{W}$ and duality pairing
$\langle \cdot, \cdot \rangle _\mathcal{W}$ between $\mathcal{W}$
and its dual $\mathcal{W'}$, which we write as
$\langle \cdot, \cdot \rangle$.
Again, we have
\[
\mathcal{W} \subseteq \mathcal{H}= \mathcal{H'}\subseteq \mathcal{W'} .
\]
Next, proceeding as usual, we obtain the following
{\it variational formulation of the problem of vibrations of a nonlinear
beam}.
\subsection*{Problem $P_V$}
Find the displacement field $w : [0, T] \to W$ and the
velocity $v=w_t$, such that for a.a. $t \in [0, T]$ and $ \psi \in W$,
\begin{gather}
\begin{gathered}
\langle v_{t}(t), \psi \rangle_W + k (w_{xx}(t), \psi_{xx})
+ \gamma (v_{xx}(t), \psi_{xx})\\
+\frac13 a (w_x^3(t), \psi_x)
- \nu p(t)( w_{x}(t), \psi_{x})= ( f(t) , \psi),
\end{gathered} \label{v1}\\
w(0)=w_0,\quad w_t(0) =v_0. \label{v3}
\end{gather}
We make the following assumptions on the problem data:
\begin{gather}
w_{0}, \; v_{0} \in W, \quad \|w_{0xx}\|_{L^2(0, 1)},
\|w_{0x}\|_{L^{\infty}(0, 1)}\leq R^*, \label{w01}\\
p\in C^1([0,T]),\quad |p|, |p'| \leq p^*, \label{p}\\
f \in \mathcal{H}. \label{f0}
\end{gather}
Here, $R^*$ and $ p^*$ are two positive constants.
The main existence and uniqueness result in \cite{ADuMBS08} is the following.
\begin{theorem} \label{thm-main}
Assume that \eqref{w01}--\eqref{f0} hold. Then there exists $T^*>0$ and a
unique solution $w$ to Problem $P_{V}$ on the time interval $[0, T^*)$
such that
\begin{equation}
\label{36}
w, v \in L^{\infty}(0, T^*; W),\quad v' \in L^2(0, T^*; { W}').
\end{equation}
\end{theorem}
To establish additional regularity of $w$, we study the problem for $v=w'$,
where here and below we denote by a prime the (weak) time derivative.
We differentiate equation (\ref{v1}) with
respect to t, set $z=w''=v'$, and, for $\psi \in W$, we obtain
\begin{equation}
\begin{gathered}
\langle z'(t), \psi \rangle_W + k (v_{xx}(t), \psi _{xx})
+ \gamma (z_{xx}(t), \psi_{xx})
+ a (w_x^2(t)v_x(t), \psi_x) \\
-\nu p'(t)(w_{x}(t), \psi_{x})-\nu p(t)(v_{x}(t),
\psi_{x})= ( f'(t) , \psi).
\end{gathered}\label{reg1}
\end{equation}
To obtain the initial condition for $z$, we formally set $t=0$ in
(\ref{p1}) and obtain condition (\ref{reg3}) below.
We have the following problem for the triple $\{w, v, z \}$.
\subsection*{Problem $P_{Vz}$}
Find the displacement field $w: [0,T]\to W$, the velocity field
$v:[ 0,T]\to W$, and the acceleration $z:[0,T] \to W$ such that
for a.a. $t\in[0,T]$
and every $\psi \in W$ the variational equation (\ref{reg1}) holds,
together with
\begin{gather}
w(t)=w_0+\int_{0}^{t} v(\tau)\, d\tau, \quad v(t)
=v_0+\int_{0}^{t} z(\tau)\,d\tau,
\label{reg2} \\
z(0)= - k w_{0xxxx}- \gamma v_{0xxxx}
+aw^2_{0x} w_{0xx} - \nu p(0) w_{0xx}+ f(0). \label{reg3}
\end{gather}
Problem \eqref{reg1}--\eqref{reg3} makes sense only if we assume,
in addition to (\ref{w01})--(\ref{f0}), that
\begin{equation}
f, f' \in {\mathcal{W}'}, \quad f(0)\in H, \label{ff}
\end{equation}
and to ensure that $z(0)\in L^{2}( 0,1)$ we assume that
\begin{equation}
w_{0}, \; v_{0} \in H^4(0,1), \quad
w_{0} \in {H_0^2(0, 1)}, \quad
\|w_{0x}\|_{L^{\infty}(0, 1)}\leq R^*. \label{reginit}
\end{equation}
We note that the term $a w^2_{0x} w_{0xx}$ is well defined.
To deal with the term with $w_x^2v_x$ we introduce the truncation
\begin{equation}
\label{etast}
\Psi_R(r)=\begin{cases}
R & \text{for } R \leq r,\\
r & \text{for } |r| \leq R, \\
-R & \text{for } r \leq - R,
\end{cases}
\end{equation}
where $R$ is a large positive number, and we replace $w_x^2$ with
$\Psi^2_R(w_x)$. Eventually, we show that when $R$ is sufficiently
large, the truncation is inactive.
To proceed with the abstract formulation of the truncated problem,
we define the operators $ B, K: \mathcal{W}\to {\mathcal{W'}}$, and
$ K_{RN} : \mathcal{W}\times \mathcal{W} \to {\mathcal{W'}}$ by
\begin{gather}
\langle {B}(w), \psi \rangle
=\int^{T}_{0}\int^{1}_{0} w_x \psi_{x}\, dx\,dt, \label{B1}\\
\langle K( w), \psi \rangle =\int^{T}_{0}\int^{1}_{0}
w_{xx}\psi_{xx}\, dx\,dt, \label{K1} \\
\langle K_{NR}(w, v),\psi\rangle =\int_{0}^{T}\int_{0}^{1}
\Psi_R^2(w_{x})v_{x}\psi_{x}\,dx\,dt. \label{K3}
\end{gather}
We introduce the function space
\[
\mathcal{Y} =\mathcal{W}\times \mathcal{W}\times\mathcal{W},
\]
and denote its dual by $\mathcal{Y}'$.
The abstract formulation of the truncated version of Problem $P_{Vz}$ is:
\subsection*{Problem $P_{Vz R}$}
Find $(w,v,z)\in \mathcal{Y} $, with $z' \in \mathcal{W'}$ such that:
\begin{gather}
v=w', \quad z = v', \label{318}\\
z' + k K(v)+ \gamma K(z ) + aK_{NR}(w, v)
-\nu p' B(w)-\nu p B(v)=f', \label{319}
\end{gather}
in $\mathcal{W'}$, together with (\ref{reg2}) and (\ref{reginit}).
The abstract formulation of Problem $P_{Vz}$ is obtained by
reinstating $w_x^2$ in place of $\Psi^2_R(w_x)$ in $K_{NR}$.
Next, we rewrite (\ref{318}) in an equivalent form,
\[
K(v) =K(w'), \quad K(z) = K(v').
\]
The equivalence follows from the boundary conditions. This allows us to
show the coercivity of the operator $A_{\rm reg}$.
The operator $A: \mathcal{Y}\to\mathcal{Y}' $, for
$y=(w, v, \varphi )\in \mathcal{Y} $, is defined by
\[
A(w,v, \varphi) =k K(v) +\gamma K(\varphi)
+a K_{NR}(w, v)-\nu p' B(w) -\nu p B(v).
\]
The operator
$A_{\rm reg}:\mathcal{Y} \to \mathcal{Y} ' $ is defined,
for $y=(w, v, \varphi )\in \mathcal{Y} $, by
\[
A_{\rm reg}(y)=\left( - K(v), - K(\varphi), A(w, v, \varphi) \right).
\]
We let $G = W\times W\times H$ with dual $G'$, we define
$\mathcal{G}=\mathcal{W}\times \mathcal{W}\times \mathcal{H}$,
the operator $D:\mathcal{G} \to \mathcal{G}' $ is defined,
for $y=(w, v, \varphi )\in \mathcal{G}$, by
\[
D(y)=\left( K(w), K(v), \varphi \right),
\]
and the functional $F:\mathcal{Y} \to\mathbb{R}^3$ as
\[
\langle F, y \rangle =(0, 0, \int_0^T \int_{0}^{1} f' \varphi\,dx\,dt).
\]
Problem $P_{Vz R}$ can now be written in the following
abstract form.
\subsection*{Problem $P_{AR}$}
Find $y=(w,v,z)\in \mathcal{Y} $ such that
\begin{gather*}
(Dy)' + A_{\rm reg}(y)= F, \quad \mbox{in } \mathcal{Y} ',\\
Dy(0)=Dy_{0}, \quad \mbox{in } G',
\end{gather*}
where $w$ and $v$ are given in (\ref{reginit}), $y_0=(w_0, v_0, z(0))$,
and $z(0)$ in (\ref{reg3}).
\begin{theorem} \label{thm32}
Assume that \eqref{w01}--\eqref{f0}, \eqref{ff} and \eqref{reginit}
hold. Then Problem $P_{AR}=P_{Vz R}$ has a unique weak solution.
\end{theorem}
The proof of the theorem is given in the next section.
The main step to the main result of this paper is the following.
\begin{theorem} \label{thm33}
Assume that \eqref{w01}--\eqref{f0}, \eqref{ff} and \eqref{reginit}
hold. Then there exists $0 6 \nu p^*/a$. Then
\[
J(t) \leq \frac a {12}\int_{\chi_+(x, t)}
\left( \frac{6 \nu p^*}{a} - w_x^2(x, t) \right)w_x^2(x, t)\,dx\leq 3 \nu^2 (p^*)^2 \equiv c_{*\nu}.
\]
Using now the Gronwall inequality yields
\begin{equation}
\label{42}
\| v(t)\|_H^2 + \|w_{xx}(t)\|_H^2 \leq \exp(C T^*)
\left((E_0 + c_{*\nu})T^* +\|f\|^2_{L^2(0, T^*, H)}
\right),
\end{equation}
for all $0\leq t 0$ such that
the solution $(w_R, v_R, z_R)$ of Problem $P_{Vz R}$ satisfies
$\Psi_R(w_x)=w_x$ a.e. on $[0, 1]\times [0, T^*)$.
\end{proposition}
\begin{proof}
It follows from Theorem \ref{thm32} that $w=w_R, v=w_R'\in L^2(0, T: W)$,
then \cite[Lemma 1.2]{Lio69} asserts that
$w \in C([0, T]; W)$.
Thus, the mappings $w_x:[0,T]\to H_0^1(0,1)$, and
$w_{xx}:[0,T]\to L^2(0,1) $
are continuous. Now, $w_x(x,t)=\int_0^x w_{xx}(r,t)\,dr$, and since
$w_x(t) \in H^1_0(0,1)$,
the H\"older inequality yields
\[
|w_x(x,t)| \leq \|w_x(t)\|_{L^{\infty}(0,1)}
\leq \int_0^1 |w_{xx}(r,t)|\,dr\leq \|w_{xx}(t)\|_{L^2(0,1)},
\]
for $0\leq t \leq T$. Let $h(t)=\|w_{xx}(t)\|_{L^2(0,1)}$
then $h:[0,T]\to \mathbb{R}$ is continuous on a compact set,
so it is bounded. Since $h(0)=\|w_{0xx}\|_{L^2(0,1)} \leq R^* < R$,
there exists $T^*\leq T$ such that $h(t) \leq R$ for all $t\in [0,T^*)$.
It follows that
\begin{equation} \label{51}
\|w_{x}(t)\|_{L^{\infty}(0,1)} \leq R,
\end{equation}
and, therefore, the truncation is inactive on the time interval $[0, T^*)$,
i.e., $\Psi_{R}^2(w_{x}) = w^2_{x}$, and so the solution
$(w_R, v_R, z_R)=(w, v, z)$ is a solution of the Problem $P_{Vz}$
on $[0, T^*)$.
\end{proof}
This completes the proof of the existence of a local solution of Problem
$P_{Vz}$.
We note in passing that it follows from the proof that if $R_1