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% Asymptotic Dynamical Difference between the Nonlocal and Local
% Swift-Hohenberg Models
% By Lin, Gao , Duan and Ervin
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% June 11, 1998
% Submitted to : J. Math. Phys.
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\begin{document}
\newcommand{\e}{\epsilon}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\newcommand{\om}{\omega}
\newcommand{\La}{\Lambda}
\newcommand{\la}{\lambda}
\newcommand{\p}{\partial}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\id}{\int_D}
\newcommand{\na}{\nabla}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\def\sqg{G(\sqrt{(x-\xi)^2+(y-\eta)^2})}
\def\kernel{G(\sqrt{x^2+y^2})}
\def\be{\begin{eqnarray}}
\def\ee{\end{eqnarray}}
\def\qed{\hbox{\vrule width 6pt height 6pt depth 0pt}}
\title{Asymptotic Dynamical Difference between the Nonlocal and Local
Swift-Hohenberg Models
\footnote{Author for correspondence: Professor Jinqiao Duan,
Department of Mathematical Sciences,
Clemson University, Clemson, South Carolina 29634, USA.
$\;$ E-mail: duan@math.clemson.edu; $\;$ Fax: (864)656-5230.} }
\author{Guoguang Lin$^1$, Hongjun Gao$^2$,
Jinqiao Duan$^3$ and Vincent J. Ervin$^3$
\\
\\
\\1. Graduate School, Chinese Academy of Engineering Physics \\
P. O. BOX 2101, Beijing 100088, and
Department of Mathematics \\
Yunnan University, Kunming 650091, China.
\\
\\2. Laboratory of Computational Physics\\
Institute of Applied Physics and Computational Mathematics\\
Beijing, 100088, China.
\\
\\3. Department of Mathematical Sciences\\
Clemson University, Clemson, South Carolina 29634, USA. }
\date{June 11, 1998 }
\maketitle
\begin{abstract}
In this paper the difference in the asymptotic dynamics between
the nonlocal and local two-dimensional Swift-Hohenberg models
is investigated. It is shown that the bounds for the dimensions
of the global attractors for the nonlocal
and local Swift-Hohenberg models differ by an absolute constant,
which depends only on the Rayleigh number,
and upper and lower bounds of the kernel of the nonlocal
nonlinearity. Even when this kernel of the nonlocal operator
is a constant function, the dimension bounds of the global attractors
still differ by an absolute constant depending on the Rayleigh number.
\bigskip
\bigskip
{\em Running Title:} Nonlocal Swift-Hohenberg Model
\bigskip
{\em Key Words:} asymptotic behavior, nonlocal nonlinearity, global attractor,
dimension estimates
\bigskip
{\em PACS Numbers:} 02.30, 03.40, 47.20
\end{abstract}
\newpage
\section{Introduction}
Fluid convection due to density gradients arises in
geophysical fluid flows in the atmosphere, oceans
and the earth's mantle.
The Rayleigh-Benard convection is a prototypical model
for fluid convection, aiming at predicting spatio-temporal
convection patterns.
The mathematical model for the
Rayleigh-Benard convection involves nonlinear Navier-Stokes
partial differential equations coupled with the temperature
equation. When the Rayleigh number is near the onset of the
convection, the Rayleigh-Benard convection model may be approximately
reduced to an amplitude or order parameter equation, as derived
by Swift and Hohenberg (\cite{Swift}).
In the current literature, most work on the
Swift-Hohenberg model deals with the following one-dimensional equation
for $w(x,t)$, which is a
localized, one-dimensionalized version of the model
originally derived by Swift and Hohenberg (\cite{Swift}),
\begin{eqnarray}
w_t & = & \mu w -(1+\p_{xx})^2 w - w^3.
\label{sh2}
\end{eqnarray}
The cubic term $w^3$ is used as an approximation of
a nonlocal integral term.
For the (local) one-dimensional Swift-Hohenberg
equation (\ref{sh2}), there has been some recent research on
propagating or steady patterns (e.g., \cite{Eckmann_Wayne}, \cite{Hilali},
\cite{Lerman}).
Mielke and Schneider(\cite{Mielke}) proved the existence of
the global attractor
in a weighted Sobolev space on the whole real line.
Hsieh et al. (\cite{Hsieh}, \cite{Hsieh2})
remarked that the elemental instability mechanism is
the negative diffusion term $-w_{xx}$.
Roberts (\cite{Roberts}, \cite{Roberts2})
recently re-examined the rationale for using the
Swift-Hohenberg model as a reliable model of the spatial
pattern evolution in specific physical systems.
He argued that,
although the localization
approximation used in (\ref{sh2})
makes some sense in the one-dimensional case,
this approximation is deficient
in the two-dimensional convection problem and
one should use the nonlocal Swift-Hohenberg model
(\cite{Swift}, \cite{Roberts}, \cite{Roberts2}):
\begin{eqnarray}
u_t = \mu u -(1+\De)^2 u
- u \int_D G(\sqrt{(x-\xi)^2+(y-\eta)^2}) u^2(\xi, \eta, t)d\xi d\eta ,
\label{sh}
\end{eqnarray}
where $u=u(x,y,t)$ is the unknown amplitude function,
$\mu$ measures the difference of the Rayleigh number
from its critical onset value,
$\De = \p_{xx} + \p_{yy}$ is the Laplace operator, and
$G(r)$ is a given radially symmetric function ($r=\sqrt{x^2+y^2}$).
The equation is defined for
$t>0$ and $(x,y) \in D$, where $D$ is a bounded planar domain with
smooth boundary $\p D$.
The two-dimensional version of the local Swift-Hohenberg equation
for $u(x, y, t)$ is
\begin{eqnarray}
u_t & = & \mu u -(1+\De)^2 u - u^3.
\label{sh3}
\end{eqnarray}
Here $u^3$ is used to approximate the nonlocal term in (\ref{sh}).
Roberts (\cite{Roberts}, \cite{Roberts2}) noted that the range of Fourier
harmonics generated by the nonlinearities is fundamentally
different in two-dimensions than in one-dimension. This difference
requires a more sophisticated treatment of two-dimensional
convection problem, which leads to nonlocal nonlinearity
in the Swift-Hohenberg model. He also argued
that nonlocal operators naturally appear in systematic derivation of
simplified models for pattern evolution, and nonlocal operators
also permit symmetries which are consisitent with
physical considerations.
In this paper, we discuss the difference between
nonlocal and local two-dimensional
Swift-Hohenberg models (\ref{sh}), (\ref{sh3}),
from a viewpoint of asymptotic dynamics.
We show that the bounds for the dimensions
of the global attractors for the nonlocal
and local Swift-Hohenberg models differ by an absolute
constant, which depends only on
the the Rayleigh number, and upper and
lower bounds of the kernel of the nonlocal
nonlinearity. Even when this kernel is a constant function,
the dimension bounds of the global attractors
still differ by a constant depending on the Rayleigh number.
In \S 2 and \S 3, we will consider the
nonlocal and local Swift-Hohenberg models, respectively.
Finally in \S 4, we summarize the results.
\section{Nonlocal Swift-Hohenberg Model}
In this section, we discuss the global attractor
and its dimension estimate for the nonlocal
Swift-Hohenberg model (\ref{sh}).
In the following we use the abbreviations $L^2=L^2(D)$,
$L^{\infty}=L^{\infty}(D)$, $H^k = H^k(D)$ and $H^k_0 =H^k_0(D)$ ($k$ is
a non-negative integer)
for the standard Sobolev spaces.
Let $(\cdot, \cdot)$,
$\| \cdot \| \equiv \| \cdot \|_2$ denote the standard
inner product and norm in $L^2$,
respectively. The norm for $H^k_0$ is
$\| \cdot \|_{H^k_0}$. Due to the Poincar\'e inequality,
$\|D^k u\|$ is an equivalent norm in $ H^k_0$.
We rewrite the two-dimensional nonlocal Swift-Hohenberg equation (\ref{sh})
as
\begin{eqnarray}
u_t + \a u +2\De u + \De^2 u
+ u \int_D G(\sqrt{(x-\xi)^2+(y-\eta)^2}) u^2(\xi, \eta, t)d\xi d\eta=0,
\label{sh4}
\end{eqnarray}
where $\a = 1 - \mu$.
This equation is supplemented with the initial condition
\be
u(x,y,0)=u_0(x,y),
\label{sh5}
\ee
and the boundary conditions
\be
u|_{\p D} = 0, \;\; \frac{\p u}{\p n} |_{\p D} = 0,
\label{sh6}
\ee
where $n$ denotes the unit outward normal vector of the boundary $\p D$.
In this paper, we assume the following conditions
for every $t \geq 0 $ and $(x,y) \in D$,
\be
0 < b \leq \kernel \leq a, \; \mbox{and} \;
G, \; \na G, \; \De G \in L^{\infty}(D),
\label{sh7}
\ee
where $a, b > 0$ are some positive constants
and $\na = (\p_x, \p_y)$ is the gradient operator.
Denote $K_1= \|\na G\|_{\infty}$ and $K_2=\|\De G\|_{\infty}$.
To study the global attractor, we need to derive
some a priori estimates about solutions.
\begin{lemma}
Suppose $u$ is a solution of (\ref{sh4})-(\ref{sh6}).
Then $u$ is uniformly (in time)
bounded, and the following estimates hold for $t>0$
\be
\|u(x,y,t)\|^2 \leq \|u_0(x,y)\|^2 exp(-2\mu t)+ \frac{\mu}{b}, \;
\label{sh8}
\ee
and thus
\be
{\lim \sup}_{t\rightarrow +\infty} \|u(x,y,t)\| \leq \sqrt{\frac{\mu}{b}} \equiv R ,
\label{sh9}
\ee
where $R = \sqrt{\frac{\mu}{b}}$.
\label{bound}
\end{lemma}
{\bf Proof.} Taking the inner product of (\ref{sh4}) with $u$, we have
\be
\frac{1}{2}\frac{d}{dt}\|u\|^2 + \|\De u\|^2 + 2(\De u,u) + \a \|u\|^2
\nonumber \\
+ (u^2,\id \sqg u^2 (\xi,\eta)d\xi d\eta) = 0.
\label{sh10}
\ee
Note that
$$ 2|(\De u,u)|\leq 2\|\De u\|\|u\| \leq \|\De u\|^2 + \|u\|^2,$$
$$ (u^2,\id \sqg u^2(\xi,\eta)d\xi d\eta) $$
$$ = \id u^2(\id \sqg u^2(\xi,\eta)d\xi d\eta)dxdy$$
$$ \geq b\id u^2(x,y)dxdy \id u^2(\xi,\eta)d\xi d\eta = b\|u\|^4.
$$
Then from (\ref{sh10}) we get
\be
\frac{d}{dt}\|u\|^2 + 2(\a-1)\|u\|^2 + 2b\|u\|^4 \leq 0.
\label{L2norm}
\ee
It is easy to see that if $\a \geq 1$, i.e., $\mu \leq 0$,
then all solutions
approach zero in $L^2$. We will not consider this simple dynamical
case. In the rest of this paper we assume that $\mu > 0$, i.e.,
$\a < 1$.
Thus we have, for any constant $\e >0$,
\be
\frac{d}{dt}\|u\|^2 + 2\e \|u\|^2 +2(\a-1-\e)\|u\|^2+ 2b\|u\|^4 \leq 0,
\ee
or
\be
\frac{d}{dt}\|u\|^2 + 2\e \|u\|^2
+ [ \frac{(\a-1-\e)}{\sqrt{2b}} + \sqrt{2b}\|u\|^2 ]^2
\leq \frac{(\a-1-\e)^2}{2b}.
\ee
So
\be
\frac{d}{dt}\|u\|^2 + 2\e \|u\|^2 \leq \frac{(\a-1-\e)^2}{2b}.
\ee
By the usual Gronwall inequality (\cite{Temam}) we obtain
\be
\|u\|^2 \leq \|u_0\|^2 exp(-2\e t)+ \frac{(\a-1-\e)^2}{4b\e}.
\ee
When $\e = 1-\a = \mu$, we get the optimal or tight estimate
\be
\|u\|^2 \leq \|u_0\|^2 exp(-2\mu t) + \frac{\mu}{b}.
\ee
This completes the proof of
Lemma \ref{bound}. $\hfill \qed$
Moreover, higher order derivatives of $u$ are also
uniformly bounded.
\begin{lemma}
Suppose $u$ is a solution of (\ref{sh4})-(\ref{sh6}). Then $\|\na u\|$
and $\|\De u\|$ are uniformly (in time) bounded.
\label{highbound}
\end{lemma}
In order to prove this lemma, we recall a few useful inequalities.
{\em Uniform Gronwall inequality} (\cite{Temam}).
Let $g,h,y$ be three positive locally
integrable functions on $[t_0,+\infty)$ satisfying the inequalities
$$
\frac{dy}{dt}\leq gy + h ,
$$
with $\int_t^{t+1}gds \leq a_1,$ $\int_t^{t+1}hds\leq a_2$
and $\int_t^{t+1}yds \leq a_3$ for $t\geq t_0,$
where the $a_i$(i=1,2,3) are positive constants. Then
$$
y(t+1) \leq (a_2+a_3)exp(a_1), \mbox{for} \; t \geq t_0 .
$$
{\em Gagliardo-Nirenberg inequality} (\cite{Pazy}).
Let $w \in L^q\cap W^{m,r}(D)$, where $1 \leq q,r \leq \infty$.
For any integer $j$, $0\leq j \leq m$, $\frac{j}{m}\leq \la \leq 1.$
$$
\|D^j w\|_p \leq C_0 \|w\|_q^{1-\la}\|D^m w\|_r^{\la}
\label{GN}
$$
provided
$$\frac{1}{p} = \frac{j}{n} + \la (\frac{1}{r} - \frac{m}{n})
+ \frac{1-\la}{q},$$
and $m-j-\frac{n}{r}$ is not a nonnegative integer
If $m-j-\frac{n}{r}$ is a nonnegative integer, then the inequality
(\ref{GN}) holds for
$\la = \frac{j}{m}$.
{\em Poincar\'e inequality} (\cite{Friedman}).
For $w \in H_0^1(D)$,
$$ \la_1 \| w \|^2 \leq \|\na w \|^2 , $$
where $\la_1$ is the first eigenvalue of $-\De$ on the domain $D$, with zero
Dirichlet boundary condition
on $\p D$.
{\bf Proof of Lemma \ref{highbound}}.
Due to the boundary condition (\ref{sh6}) on $ \na u$
and the Poincar\'e inequality, we get
$ \| \na u \|^2 \leq \la_1^{-1} \|\De u \|^2$. Hence it is
sufficient to prove that $ \|\De u\|$ is
bounded. We first show that $\int_t^{t+1}\|\De u\|^2ds$
is bounded. In fact, using
$$ 2|(\De u,u)| \leq 2\|\De u\|\|u\| \leq \frac12 \|\De u\|^2 + 2\|u\|^2,$$
in (\ref{sh10}), we get
\be
\frac{d}{dt} \|u\|^2 + \|\De u\|^2 + 2(\a-2)\|u\|^2 + 2b\|u\|^4 \leq 0.
\ee
Since
$$2b\|u\|^4 + 2(\a-2)\|u\|^2 = b\|u\|^2 +
2\b(\|u\|^4 + \frac{2\a-4-\b}{2\b}\|u\|^2)$$
$$=b\|u\|^2 + 2\b(
\|u\|^2 + \frac{2\a-4-\b}{4\b})^2 -
\frac{(2\a -4 -\b)^2}{8\b}$$
$$\geq b\|u\|^2 - \frac{(2\a -4 -\b)^2}{8\b},$$
we conclude
\be
\frac{d}{dt} \|u\|^2 + \|\De u\|^2 + b\|u\|^2 \leq
\frac{(2\a -4 -\b)^2}{8\b} =\frac{(2 + 2\mu + \b)^2}{8\b}.
\label{sh11.5}
\ee
Integrating (\ref{sh11.5}) with respect to $t$ from $t$ to $t+1$ and noting
Lemma \ref{bound}, we see that $\int_t^{t+1}\|\De u\|^2ds$ is bounded.
Now, multiplying (\ref{sh4}) by $\De^2 u$ and integrating over $D$,
it follows that
$$
\frac{1}{2}\frac{d}{dt}\|\De u\|^2 + \|\De^2 u\|^2 +
2\id \De u\De^2udxdy + \a \|\De u\|^2$$
\be
+\id (u \id \sqg u^2(\xi,\eta)d\xi d\eta)\De^2 udxdy = 0.
\label{sh12}
\ee
Note that
\be
2|\id\De u\De^2 udxdy | \leq \frac{1}{2}\|\De^2u\|^2 + 2\|\De u\|^2,
\ee
and
$$
| \id ( u \id \sqg u^2(\xi,\eta)d\xi d\eta) \De^2 udxdy |
$$
$$
= | \id (\De u)^2(\id \sqg u^2(\xi,\eta)d\xi d\eta)dxdy
$$
$$
+ \id u\De u (\id(\De \sqg))u^2(\xi,\eta)d\xi d\eta)dxdy
$$
$$
+ 2\id \na u\De u(\id(\na \sqg ))u^2(\xi,\eta)d\xi d\eta)dxdy|
$$
$$
\leq (a\|u\|^2+2\la_1^{-\frac12} \|\na G \|_{\infty}\|u\|^2
+ \frac12 \|\De G \|_{\infty}\|u\|^2 ) \|\De u\|^2
+ \frac12 \|\De G \|_{\infty} \|u\|^4
$$
$$
\leq (a +2\la_1^{-\frac12} K_1 + \frac12 K_2 )\|u\|^2 \| \De u \|^2
+ \frac12 K_2 \|u\|^4,
$$
where $a, K_1, K_2$ are various upper bounds of $G$ defined
in (\ref{sh7}), and $R$ is the $L^2$ bound of the
solution $u$ as in Lemma 1.
Hence by (\ref{sh12}) we get
\be
\frac{d}{dt}\|\De u\|^2 \leq 2 [(a +2\la_1^{-\frac12} K_1 + \frac12 K_2)\|u\|^2
-\a +2] \|\De u\|^2 + K_2 \|u\|^4.
\label{sh13}
\ee
Finally, applying the uniform Gronwall inequality (\ref{sh13})
and noting Lemma \ref{bound}, we conclude
that $\| \De u\|^2$ is uniformly
bounded for all $t\geq 0.$
This proves Lemma \ref{highbound}. $\hfill \qed$
We now have the following global existence and uniqueness
result.
\begin{theorem}
Let $u_0(x,y) \in L^2(D)$ and $G$ satisfies (\ref{sh7}),
then the initial-boundary value problem
$(\ref{sh}), (\ref{sh5}), (\ref{sh6})$ has a unique global
solution $u\in L^{\infty}(0,\infty; H^2_0(D))$. Moreover,
the corresponding solution semigroup $S(t)$, defined by
$$
u = S(t) u_0,
$$
has a bounded absorbing set
$$
B_0 = \{ u \in H^2_0(D): (\|u\|^2+\|\na u\|^2
+\|\De u\|^2)^{\frac{1}{2}} \leq \tilde{R} \},
$$
where $\tilde{R} $ is a postive constant which depending on the
uniform bound of $\|u\|, \|\na u\|, \|\De u\|$.
Finally, the solution semigroup $S(t)$, when restricted on
$H^2_0(D)$, is continuous from
$H^2_0(D)$ into $H^2_0(D)$ for $t > 0$.
\label{global}
\end{theorem}
{\bf Proof.}
The global existence, uniqueness and absorbing property
follow from standard arguments (e.g., \cite{Temam})
together with Lemmas \ref{bound}, \ref{highbound} above.
The absorbing property also follows from these two lemmas.
We now prove that $S(t)$ is continuous in $H^2(D)\cap H_0^1(D)$.
Suppose that $u_0,v_0\in H^2_0(D)$ with $\|\De u_0\|,
\|\De v_0\|\leq 2R_1,$
we denote by $u(t), v(t)$ the corresponding solutions, i.e.,
$u(t) = S(t)u_0, v(t) = S(t)v_0$. Let
$w(t)=u(t)-v(t).$ Then $w(t)$ satisfies
$$
w_t + \De^2 w + 2 \De w + \a w + w\id \sqg u^2(\xi,\eta)
d\xi d\eta+$$
\be
v\id \sqg (u(\xi,\eta)+v(\xi,\eta))w(\xi,\eta)d\xi d\eta = 0.
\label{sh14}
\ee
Applying the Gagliardo-Nirenberg inequality
$$ \|u\|_{\infty} \leq C_0\|\De u\|,$$
and the Poincar\'e inequality
$$ \|w\| \leq \frac{1}{\la_1}\|\De w\|,$$
we obtain (similar to the proof of Lemma \ref{highbound}),
$$
\frac{d}{dt}\|\De w\|^2 \leq C_1 \|\De w\|^2,
$$
which implies that $\|\De w(t)\|^2 \leq \|\De w_0\|^2 exp(C_1 t)$
for some positive constant $C_1$. This shows that $S(t)$ is continuous.
$\hfill \qed$
This theorem implies that (\ref{sh4})-(\ref{sh6})
defines an infinite dimensional nonlocal dynamical system.
In the rest of this section, we consider the global attractor
for the nonlocal dynamical system (\ref{sh4})-(\ref{sh6}).
We will establish the following result about the global attractor.
\begin{theorem}
There exists a global attractor ${\cal A}$
for the nonlocal dynamical system
$(\ref{sh}), (\ref{sh5}), (\ref{sh6})$.
The global attractor is the $\omega-$limit set
of the absorbing set $B_0$ (as in Theorem \ref{global}), and it
has the following properties:
(i) $A$ is compact and $S(t){\cal A} = {\cal A},$ for $t > 0$;
(ii) for every bounded set $B\subset H^2_0(D)$,
$\lim\limits_{t\to \infty} d(S(t)B,{\cal A})=0;$
(iii)${\cal A}$ is connected in $H^2_0(D),$
where
$d(X,Y)=\sup\limits_{x\in X}\inf\limits_{y\in Y}\|x-y\|_{H^2_0(D)}$
is the Hausdorff distance.
Moreover, the global attractor ${\cal A}$ has finite Hausdorff dimension
$d_H({\cal A}) \leq m$,
where
$$
m \sim C (1+\sqrt{ \mu + (2a-b)\frac{\mu}{b} }),
\label{attractor}
$$
where $ C >0$ is a constant depending only on the domain $D$,
and $a>0, b>0$ are the upper, lower bounds of the kernel $G$,
respectively.
\end{theorem}
{\bf Proof.}
The existence and properties of ${\cal A}$ are
quite standard now (see \cite{Temam} and references therein).
We omit this part, and only estimate the dimensions below.
As in \cite{Temam}, we may use
the so-called Constantin-Foias-Temam trace formula (which works for
the semiflow $S(t)$ here) to estimate
the sum of the global Lyapunov exponents of
${\cal A}$. The sum of these Lyapunov exponents can then
be used to estimate the upper bounds of ${\cal A}$'s
Hausdorff dimension, $d_H({\cal A})$.
To this end,
we linearize equation (\ref{sh4}) about a solution $u(t)$ in the
global attractor to obtain an equation for $v(t)$ and then
use the trace formula to estimate the sum of
the global Lyapunov exponents. Doing so, we obtain
\be
v_t + L(u(t)) v = 0,
\label{sh15}
\ee
where
$$
L(u(t))v = \De^2 v + 2\De v + \a v +
v\id \sqg u^2(\xi,\eta)d\xi d\eta
$$
$$+2u\id \sqg u(\xi,\eta) v(\xi,\eta)d\xi d\eta.
$$
This equation is supplemented with
$v(x,y , 0) = \xi(x,y) \in H^2_0(D)$.
Denote by $\xi_1(x,y), \ldots , \xi_m(x,y)$,
$m$ linearly independent functions
in $H^2_0(D)$, and $v_{i}(x,y , t)$ the solution of (\ref{sh15})
satisfying $v_{i}(x,y , 0) = \xi_{i}(x,y)$, $i = 1, \ldots, m$.
Let $Q_m (t)$ represent the orthogonal projection of $H^2_0(D) $
onto the subspace spanned by $\{v_1(x,y,t), \ldots , v_m(x,y, t) \}$.
We need to estimate the lower bound of
$Tr(L(u(t)Q_m(t)))$, which gives bounds on the sum of
global Lyapunov exponents.
Note that in \cite{Temam},
the linearized equation like (\ref{sh15}) is written
as $v_t = L(u(t))v$ and in that case one needs to
estimate the upper bound of $Tr(L(u(t)Q_m(t)))$.
Suppose that $\phi_1(t),...,\phi_m(t)$ is
an orthonormal basis ($\|\phi_j\| =1$)
of the subspace $Q_m(t) H_0^2(D)$
for any $t>0$.
Now we estimate the lower bound of
$Tr(L(u(t)Q_m(t)))$. It is easy to see that
$$Tr(L(u(t)Q_m))$$
$$
=\sum\limits^m_{j=1}(\De^2\phi_j+2\De\phi_j+\a \phi_j+\phi_j\id
\sqg u^2(\xi,\eta)d\xi d\eta,\phi_j)
$$
$$
+ \sum\limits^m_{j=1}(2u\id \sqg u(\xi,\eta) \phi_j d\xi d\eta,\phi_j).
$$
Since
$(2\De\phi_j, \phi_j) \geq -(\frac1{\e}\|\De \phi_j\|^2 +\e \|\phi_j\|^2)$
for any constant $\e > 1$, we get
$$
Tr(L(u(t)Q_m)) \geq \sum\limits^m_{j=1} [(1-\frac1{\e})\|\De \phi_j\|^2+
b\|\phi_j\|^2\|u\|^2 + (\a - \e) \|\phi_j\|^2 ]
$$
$$
+ \sum\limits^m_{j=1} 2\id u\phi_jdxdy (\id \sqg u(\xi,\eta) \phi_j d\xi d\eta)
$$
$$
\geq \sum\limits^m_{j=1} (1-\frac1{\e}) \|\De \phi_j\|^2
+ \sum\limits^m_{j=1} (b\|u\|^2+ \a-\e -2 a\|u\|^2)
$$
\be
= \sum\limits^m_{j=1} (1-\frac1{\e}) \|\De \phi_j\|^2
+ [1-\mu-\e + (b-2 a) \|u\|^2 ] m.
\label{sh18}
\ee
We introduce notation $f(x,y) = \sum\limits_{j=1}^m | \phi_j|^2$.
Note that $m = \id f(x,y)dxdy$.
By the generalized Sobolev-Lieb-Thirring
inequality (\cite{Temam}, page 462),
$$\id f^3(x,y)dxdy \leq K_0 \sum\limits^m_{j=1} \|\De \phi_j\|^2,$$
where $K_0>0$ depending only on the domain $D$.
Moreover, due to the fact that $L^3 (D) \hookrightarrow L^1(D)$,
$$m^3 = (\id f(x,y)dxdy)^3 \leq C_2 \id f^3(x,y)dxdy $$
$$\leq K_0 C_2 \sum\limits^m_{j=1} \|\De \phi_j\|^2 $$
$$ = C\sum\limits^m_{j=1} \|\De \phi_j\|^2 $$
for some constants $C_2>0, C>0$ depending only on the domain $D$.
Thus
\be
(1-\frac1{\e}) \sum\limits^m_{j=1} \|\De \phi_j\|^2
\geq (1-\frac1{\e}) \frac1{C} m^3.
\label{sh19}
\ee
Therefore, by (\ref{sh18})-(\ref{sh19}) we have
\be
Tr (L(u(t)Q_m))
& \geq & \frac{1-\frac1{\e}}{C} m^3 - (\mu-1 +\e + (2a-b) \|u\|^2) m
\nonumber \\
& \geq & \frac{1-\frac1{\e}}{C} m^3 - (\mu-1 +\e + (2a-b)\frac{\mu}{b}) m
\nonumber \\
& > & 0
\ee
whenever
\be
m >\sqrt{[ \mu-1 +\e + (2a-b) \frac{\mu}{b} ]
\frac{C}{1-\frac1{\e}}}.
\label{mm}
\ee
The right hand side of (\ref{mm}) has the minimal value of
\be
m \sim C (1+\sqrt{\mu + (2a-b)\frac{\mu}{b}})
\label{dimension}
\ee
when
$ \e = 1+\sqrt{ \mu + (2a-b)\frac{\mu}{b} }$.
As in \cite{Temam}, we conclude that the Hausdorff dimension of
${\cal A}$ is estimated as in (\ref{dimension}).
This proves Theorem \ref{attractor}. $\hfill \qed$
\section{Local Swift-Hohenberg Model}
Similarly, for the two-dimensional local Swift-Hohenberg
equation (\ref{sh3}), we can obtain the existence of
the global attractor $\tilde {\cal A}$. We omit this part
and will only estimate the dimension
of $\tilde {\cal A}$.
\begin{theorem}
There exists the global attractor $\tilde {\cal A}$ for the
local dynamical system (\ref{sh3}), (\ref{sh5}), (\ref{sh6}).
The Hausdorff dimension of $\tilde {\cal A}$ is finite, and
$d_H(\tilde {\cal A} ) \leq m_1 \sim C (1+\sqrt{ \mu }) $,
where $ C $ is a constant depending only on the domain $D$.
\label{attractor2}
\end{theorem}
{\bf Proof.} As in the proof of Ttheorem \ref{attractor},
we consider the linearized equation of (\ref{sh3}), defined by
$$ v_t + L_1(u(t))v = 0,$$
where
$$L_1(u(t))v = \De^2v + 2\De v + \a v + 3u^2v. $$
Then we estimate
$$ Tr(L_1(u(t)Q_m)) $$
$$
=\sum\limits^m_{j=1}(\De^2\phi_j+2\De\phi_j+\a \phi_j+3u^2\phi_j ,\phi_j)
$$
$$
= \sum\limits^m_{j=1}[\|\De \phi_j\|^2 + 2(\De \phi_j, \phi_j)
+\a \|\phi_j\|^2 + 3(u^2\phi_j, \phi_j)]
$$
$$
\geq \sum\limits^m_{j=1} (1-\frac1{\e})\|\De \phi_j\|^2
+ \sum\limits^m_{j=1}(\a -\e),
$$
where we have used the fact that $3(u^2\phi_j, \phi_j) \geq 0$.
Noting again that
$m^3 \leq C\sum\limits^m_{j=1} \|\De \phi_j\|^2$ and $\a =1-\mu$,
we have
\be
Tr (L_1(u(t)Q_m))
\geq \frac{1-\frac1{\e}}{C} m^3 -(\mu -1+\e) m >0
\ee
whenever
\be
m >\sqrt{( \mu-1 +\e ) \frac{C}{1-\frac1{\e}} }.
\label{mmm}
\ee
The right hand side of (\ref{mmm}) has the minimal value of
\be
m \sim C (1+\sqrt{ \mu })
\ee
when
$ \e = 1+\sqrt{ \mu }$.
This completes the proof. $\hfill \qed$
\section{Discussions}
In this paper, we have discussed the Hausdorff dimension estimates
for the global attractors of the two-dimensional nonlocal and local
Swift-Hohenberg model for Rayleigh-Benard convection.
The Hausdorff dimension for the global attractor of the
nonlocal model is estimated as
$$
m \sim C (1+\sqrt{\mu + (2a-b)\frac{\mu}{b}}),
$$
while for the local model this estimate is
$$
m \sim C (1+\sqrt{ \mu }),
$$
where $C>0$ is an absolute constant depending only on
the fluid convection domain, and $\mu>0$ measures the
difference of the Rayleigh number from its critical
convection onset value.
Note that $a, b > 0$ are the upper and lower
bounds, respectively, of the kernel $G$ of the nonlocal nonlinearity
in (\ref{sh}).
The two dimension estimates above
differ by an absolute constant $(2a-b)\frac{\mu}{b}$,
which depends only on
the the Rayleigh number through $\mu$, and upper and
lower bounds of the kernel $G$ of the nonlocal
nonlinearity. Moreover, if the kernel $G$ is a constant function
(thus, $a=b=G$), then the dimension estimate for the
nonlocal model becomes
$$
m \sim C (1+\sqrt{ 2 \mu}),
$$
which still differs from the dimension
estimate for the local model by a constant depending on the Rayleigh number
through $\mu$.
\bigskip
{\bf Acknowledgement.}
Part of this work was done while Jinqiao Duan was visiting the
Institute for Mathematics and its Applications (IMA), Minnesota,
and the Center for Nonlinear Studies, Los Alamos National
Laboratory.
This work was supported by the Nonlinear Science Program of China,
the National Natural Science Foundation of China Grant 19701023,
the Science Foundation of Chinese
Academy of Engineering Physics Grant 970682, and the USA National Science
Foundation Grant DMS-9704345.
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