\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 152, pp. 1--9.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/152\hfil Truncated gradient flows]
{Truncated gradient flows of the van der Waals free energy}
\author[M. Grinfeld, I. Stoleriu\hfil EJDE-2006/152\hfilneg]
{Michael Grinfeld, Iulian Stoleriu} % in alphabetical order
\address{Michael Grinfeld \newline
Department of Mathematics,
University of Strathclyde, 26 Richmond Street, G1 1XH Glasgow,
Scotland, United Kingdom}
\email{m.grinfeld@strath.ac.uk}
\address{Iulian Stoleriu \newline Faculty of Mathematics, ``Al. I. Cuza"
University, Bvd. Carol I, No. 11, 700506 Ia\c{s}i,
Romania \;\; \& \newline
EML Research gGmbH, Schloss Wolfsbrunnenweg 33, 69118 Heidelberg, Germany}
\email{iulian.stoleriu@uaic.ro, iulian.stoleriu@eml-r.villa-bosch.de}
\thanks{Submitted November 1, 2006. Published December 5, 2006.}
\subjclass[2000]{47H20, 45J05, 35K55, 41A21}
\keywords{Gradient flow; van der Waals energy;
integro-differential equation; \hfill\break\indent Pad\'e approximants}
\begin{abstract}
We employ the Pad\'{e} approximation to derive a set of new partial
differential equations, which can be put forward as possible models
for phase transitions in solids. We start from a nonlocal free energy
functional, we expand in Taylor series the interface part of this energy,
and then consider gradient flows for truncations of the resulting expression.
We shall discuss here issues related to the existence and uniqueness
of solutions of the newly obtained equations, as well as the convergence
of the solutions of these equations to the solution of a nonlocal version
of the Allen-Cahn equation.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}\label{sec:int}
Solid-solid phase transitions may be well described by suitable
gradient flows of the Ginzburg-Landau free energy functional,
\begin{equation}\label{gl}
E_1(u) = \frac{\gamma}{2} \int_\Omega |\nabla u|^2 dx +
\int_\Omega F(u) dx,
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}$, $u(x, t)$ is a
suitable order parameter and $\gamma$ is a
measure of strength of intermolecular forces. In many situations, the
suitable bulk energy $F(u)$ has a double well structure. Starting
from (\ref{gl}) and considering the gradient
flow with respect to the $L^2 $-inner product, one obtains
the well-known Allen-Cahn equation,
\begin{equation} \label{ca}
u_t = \gamma\Delta u - f(u).
\end{equation}
Here $f(u)=F'(u)$ is usually a bistable function. This equation has been
used in modelling order parameter non-conserving phenomena,
such as transitions between variants of a crystalline substance
(see, for example, \cite{BC,DGS,Fife1,Fife2,Stol}).
For order parameter conserving situations, one has to consider a
constrained gradient flow.
If one uses the gradient with respect to the $H^{-1}$-inner product,
one obtains:
\begin{equation}\label{ch}
u_t = -\Delta (\gamma \Delta u -f(u)),
\end{equation}
which is the Cahn-Hilliard equation \cite{Fife1,Pen1}.
The equation (\ref{ch}) gives a qualitatively faithful description of spinodal
decomposition, of the transition from spinodal to metastable behaviour, as well
as of critical nuclei.
The problem with the Ginzburg-Landau approach (apart from the fact that
it fails to give a good quantitative fit to the course of coarsening in
a number of situations; see \cite{Hyde} for example) is that it is
totally phenomenological. Other approaches exist, all of them more or
less starting with the Ising model. Examples are the work of Penrose
\cite{Pen2} and equations derived from the free energy written down
by van der Waals \cite{vdW} and advocated by Khachaturyan \cite{Khach}
(see equation (\ref{vdw}) below).
Gradient flows of the van der Waals free energy (in the non-conserving
case) have been studied, among others, by \cite{BC,DGS,Fife2,Stol}. In
particular, the paper \cite{Fife2} sets out the general theory of these
(integro-differential) equations; \cite{BC} gives a careful derivation of
the equations directly from the Ising model and describe stationary
solutions, while \cite{DGS, Stol} deal mainly with the lack of coarsening and
non-compactness of attractors in the case of sufficiently small
$\gamma$ (this is in stark contrast to the Allen-Cahn situation).
For simplicity, we shall take below $\Omega= \mathbb{R}$. The van der
Waals free energy is then
\begin{equation}\label{vdw}
E_2(u) = \frac{\gamma}{4}\int_\mathbb{R}
\int_\mathbb{R} J(|x-y|)(u(y)-u(x))^2 dy dx
+\int_\mathbb{R} F(u(x)) dx.
\end{equation}
where $J(\cdot)$ is an $L^1(\mathbb{R})$ kernel describing
intermolecular interactions (for most of the paper we will have to
impose additional restrictions on $J(\cdot)$). The $L^2$ gradient flow
of $E_2$ is
\begin{equation}\label{nl}
u_t = \gamma \int_\mathbb{R} J(|x-y|) (u(y)-u(x)) \, dy -f(u).
\end{equation}
Note that the linear part of equation (\ref{nl}) is a bounded operator;
for small $\gamma$ it is a regular perturbation of the kinetic
equation
\begin{equation}\label{ke}
u_t = -f(u),
\end{equation}
while the Cahn-Allen equation equation
(\ref{ca}) is a singular perturbation of (\ref{ke}).
Formally, one can derive the Ginzburg-Landau functional from the van der
Waals one by performing a gradient expansion and retaining only the
leading term. Trying to retain more than the leading term is, however,
fraught with difficulties: as we show below in Section \ref{sec:ts},
retaining an even number of terms leads to an ill-posed problem, and
even in the case of an odd number of terms (considered, e.g. in
\cite{BFG,BR}), it is not clear how semiflows generated by high order
parabolic equations are supposed to approximate the flow generated by
equation (\ref{nl}).
In this paper, motivated by the work of Rosenau \cite{Rosenau} and of
Slemrod \cite{Slemrod}, we show that if instead of polynomial approximations
we use Pad\'e approximants, we recover a family of equations that, in
addition to being in some aspects easier to handle than the original
integro-differential equation (\ref{nl}), also have the desired
well-posedness and convergence properties.
We start by going through the usual gradient expansion scheme, following
\cite{BR}. Then we derive our new equations based on Pad\'e
approximants, we discuss the well-posedness and we prove a convergence
property of the solutions to the newly obtained equations.
\section{Truncation scheme}\label{sec:ts}
We aim to derive the truncated gradient flows of (\ref{vdw}) by expanding in
Taylor series the term $ (u(y)-u(x)) $, and then truncate to some order the
new expression.
Since we are expanding the interface part of the free energy, we shall
omit the bulk energy part in the computations below. Thus, consider
\[
L(u)= \frac{\gamma}{4}\int_\mathbb{R} \int_\mathbb{R} J(|x-y|)(u(y)-u(x))^2
\,dy \,dx.
\]
Below we shall use the notation $D^ku$ for the $k$-th derivative of
$u$.
Setting $x=\eta + \xi$ and $y= \xi-\eta$, we have formally:
\begin{equation}
\begin{aligned}
L(u)&=\frac{\gamma}{2}\int_\mathbb{R} \int_\mathbb{R}
J(2|\eta|)[u(\xi-\eta)-u(\xi+\eta)]^2 d\eta d\xi \\
&=2\gamma \int_\mathbb{R} \int_\mathbb{R} J(2|\eta|) \Big[ \sum_{k=1}^\infty
\frac{\eta^{2k-1}}{(2k-1)!} D^{2k-1}u(\xi)\Big]^2 d\xi d\eta
\\
&=2\gamma \int_\mathbb{R} J(2|\eta|) \sum_{k=1}^\infty
\frac{\eta^{2k}}{(2k)!}\Big[ \int_\mathbb{R} \sum_{i=1}^k
C_{2k}^{2i-1}D^{2i-1}u(\xi)D^{2k-2i+1}u(\xi) d\xi\Big]d\eta,
\end{aligned} \label{exTaylor}
\end{equation}
where $C_{2k}^{2i-1}$ is defined by
\[
C_{2k}^{2i-1} = \frac{(2k)!}{(2i-1)!(2k-2i+1)!},\quad k=1, 2, \dots;\;
i=1, 2,\dots, k.
\]
We now truncate to the {\em nth}-order the last expression of $L(u)$ and write
\[
L_n(u) = 2\gamma \int_\mathbb{R} J(2|\eta|)
\sum_{k=1}^n \frac{\eta^{2k}}{(2k)!}
\Big[ \int_\mathbb{R} \sum_{i=1}^k C_{2k}^{2i-1}D^{2i-1}u(\xi)D^{2k-2i+1}
u(\xi) d\xi\Big] d\eta.
\]
Again, proceeding formally, we compute the $L^2$-gradient flow of the
truncated free energy $E_n(u)$, where
$$
E_n(u) = L_n(u) + \int_\mathbb{R} F(u(x))dx.
$$
We have
\begin{equation} \label{varch}
\begin{aligned}
\langle \frac{\delta E_n(u)}{\delta u}, v\rangle
&= \frac{d}{d\theta} E_n(u+\theta v)|_{\theta =0} \\
&=2\gamma \int_\mathbb{R} J(2|\eta|) \sum_{k=1}^n \frac{\eta^{2k}}{(2k)!}
\{ \sum_{i=1}^k C_{2k}^{2i-1}\int_\mathbb{R}
[D^{2i-1}u(\xi)D^{2k-2i+1}v(\xi)\\
&\quad + D^{2i-1}v(\xi)D^{2k-2i+1}u(\xi)]d\xi \} d\eta
+\int_\mathbb{R} f(u(\xi))v(\xi)\,d\xi \\
&=2\gamma \int_\mathbb{R} J(2|\eta|) \sum_{k=1}^n
\frac{\eta^{2k}}{(2k)!} \{ \sum_{i=1}^k C_{2k}^{2i-1}\int_\mathbb{R}
[(-1)^{2k-2i+1}D^{2k}u(\xi) \\
&\quad +(-1)^{2i-1}D^{2k}u(\xi)]v(\xi)d\xi \} d\eta +
\int_\mathbb{R} f(u(\xi))v(\xi)\,d\xi \\
&=-\gamma \int_\mathbb{R} \Big\{ \sum_{k=1}^n\Big[\frac{2^{2k+1}}{(2k)!}
\int_\mathbb{R} J(2|\eta|)\eta^{2k}d\eta \Big]D^{2k}u(\xi) -
f(u(\xi))\Big\}v(\xi) d\xi \\
&=-\gamma \int_\mathbb{R} \Big\{ \sum_{k=1}^n \rho_{2k} D^{2k}u(\xi) -
f(u(\xi))\Big\}v(\xi) d\xi, \quad \mbox{for all }v\in L^2(\Omega),
\end{aligned}
\end{equation}
where $\langle \cdot , \cdot\rangle$ is the $L^2$ inner product and
$\rho_{2k}$ is the non-negative
quantity
\begin{equation}
\rho_{2k} = \frac{2^{2k+1}}{(2k)!}\int_\mathbb{R}
J(2|\eta|)\eta^{2k}\,d\eta
= \frac{1}{(2k)!}\int_\mathbb{R} J(|z|)z^{2k} \, dz,
\quad k=1, 2, \dots
\label{rho}
\end{equation}
Note that in (\ref{varch}) we used integration by parts, and homogeneous
boundary conditions for the derivatives of $u$ of any order.
For the infinite series to be at least formally defined we must assume that
all the moments $\rho_{2k}$ of $J(\cdot)$ are finite.
Thus, the $L^2$-gradient flow derived using $E_n$ is
\begin{equation} \label{tr}
\frac{\partial u}{\partial t}(x, t) = \gamma \sum_{k=1}^n \rho_{2k}
D^{2k}u(x, t) - f(u(x, t)),\quad x\in \mathbb{R}.
\end{equation}
Note that we can also derive formally the $L^2$-gradient flow of the expanded
free energy (\ref{exTaylor}). This is
\begin{equation} \label{inf}
\frac{\partial u}{\partial t} = \gamma \sum_{k=1}^{\infty} \rho_{2k}
D^{2k}u - f(u),
\end{equation}
which can be written in the form
\[
\frac{\partial u}{\partial t} = \gamma \int_\mathbb{R}
J(|z|)\cosh (zD) (u) \,dz - f (u),\quad x \in \mathbb{R},
\]
which is reminiscent of equations derived in \cite{Salje}.
By $\cosh(zD)$ we have defined the differential operator
\[
\cosh(zD)(u) = \frac{1}{(2k)!}\sum_{k=1}^{\infty}z^{2k}D^{2k}u.
\]
The symbol of this operator is then
\[
\cos(z\xi) = \frac{1}{(2k)!}\sum_{k=1}^{\infty}(-1)^k z^{2k}\xi^{2k}.
\]
As a remark, we note that an easier (but also formal) way of obtaining
equation (\ref{inf})
is to observe that the symbol of the integral operator $A$, such that
\[
A u(x) = \int_\mathbb{R} J(|x-y|) (u(y)-u(x)) dy,
\]
is given by
\begin{equation}\label{sym}
\mathcal{S}(A)(k) = \int_\mathbb{R} J(w) (\cos (kw)-1) dw,
\end{equation}
then one can expand $(\cos(kw) -1)$ in Taylor series and integrate term
by term the resulting expression.
Let us now define the operator
$\displaystyle \widetilde{A}_n u = \sum_{k=1}^n \rho_{2k} D^{2k}u$, and for each
$n \in \mathbb{N}$ consider the following initial value problem
in $H^{2n}(\mathbb{R})$:
\begin{equation} \label{tildePn}
\begin{gathered}
u_t = \gamma \widetilde{A}_n u - f(u),\quad (x,t) \in \mathbb{R}
\times (0,\infty),\\
\hspace*{-4.6cm}u(0)=u_0.
\end{gathered}
\end{equation}
Well-posedness of these problems is not obvious. If $J(\cdot) \geq 0$ and
$n$ is an even number, these problems are not well-posed in positive time.
Clearly, if $n$ is an odd number the problems \eqref{tildePn} for $n$ are
not well-posed in negative time (various aspects of \eqref{tildePn} for $n$
odd have been considered in \cite{BFG,BR}). This is to be expected, as we
are trying to approximate the flow generated by a bounded operator by
parabolic semiflows. If the usual assumption of nonnegativity of $J(\cdot)$
(which in certain cases does not have any physical basis) is not imposed,
taking polynomial truncations becomes even more contentious. We note that
the limit as $n \rightarrow \infty$ of (\ref{tildePn}) has been considered
by Dubinskii \cite{Dub}.
In the following, we will approximate the flow generated by (\ref{nl}) by
taking operator Pad\'e approximants of \eqref{tildePn}.
Let $\mathcal{S}(\widetilde{A}_{2n})$ be the symbol
of the operator $\widetilde{A}_{2n}$ (a polynomial of degree $4n$). If
$
q_{2n}/r_{2n}\; \mbox{ is the }[2n/2n] \mbox{ Pad\'e approximant of
}\mathcal{S}(\widetilde{A}_{2n})$,
(where $p_{2n}, q_{2n}$ are polynomials of degree $2n$), then we consider the
differential operators $R_n$ and $Q_n$ of order $2n$, such that their symbols are
$q_{2n}$ and $r_{2n}$, respectively.
In this way, the truncation to degree $2n$ of the symbol of $\widetilde{A}_{2n}
R_n$ is the symbol of $Q_n$. For each $n\in \mathbb N$ we define the operator
\begin{equation}\label{An}
A_n = Q_n R_n^{-1}
\end{equation}
acting on $L^{2}(\mathbb{R})$, which is a $[2n/2n]$ Pad\'e-type approximant
of the operator
\[
{\widetilde A}_{\infty} = \sum_{k=1}^{\infty} \rho_{2k} D^{2k}u .
\]
Instead of \eqref{tildePn}, we shall consider now the problem,
\begin{equation} \label{Pn}
\begin{gathered}
u_t = \gamma A_n u - f(u),\quad (x,t) \in \mathbb{R} \times (0,\infty ), \\
\hspace*{-4.6cm} u(0)=u_0\,.
\end{gathered}
\end{equation}
Note the nice commutativity property: $R_n Q_n = Q_n R_n$ on smooth
enough functions (usual property of differential operators with
constant coefficients). Thus, we can rewrite the equation
$$
u_t+f(u) = Q_n R_n^{-1}u
$$
as
$$
R_n(u_t+f(u))=Q_n u.
$$
When $n=1$, problem \eqref{Pn} turns out to be the initial value problem
\begin{equation}
\label{p1}
\Big(\rho_2 I -\rho_4 \frac{\partial^2}{\partial x^2} \Big)
(u_t +f(u)) = \gamma \rho^2_2 \frac{\partial^2 u}{\partial x^2},
\quad x \in \mathbb{R} ,
\end{equation}
where $I$ is the identity operator.
The Cahn-Hilliard equation and the viscous diffusion equation
\cite{NCP} can easily be derived from the conserved order parameter
version of equation (\ref{nl}), which is (see \cite{Stol}):
\begin{equation}
\label{cop}
u_t = \gamma \int_\mathbb{R} J(|x-y|) (A u(x,t) - A u(y,t) -f(u(x,t))+
f(u(y,t))\,dx\,dy\,.
\end{equation}
After the change of variables and expanding in Taylor series, one obtains
\begin{equation}\label{fcop}
u_t = -\gamma {\widetilde A}_{\infty} \circ {\widetilde A}_{\infty}u
+ {\widetilde A}_{\infty}f(u)
\end{equation}
Truncating at the first order term and scaling time, one obtains the
Cahn-Hilliard equation in the scalar form,
\[
u_t = \frac{\partial^2}{\partial x^2} \big( f(u) - \gamma \rho_2
\frac{\partial^2 u}{\partial x^2} \big).
\]
Setting $\gamma=0$ in (\ref{fcop}) and taking the $[2/2]$ Pad{\'e} approximant leads
to
\[
\Big(\rho_2 I-\rho_4 \frac{\partial^2}{\partial x^2} \Big)
u_t = \gamma \rho^2_2 \frac{\partial^2}{\partial x^2} f(u),
\]
which was analyzed in \cite{NCP} in an $L^\infty(\mathbb{R})$ setting.
\section{Well-posedness and convergence}\label{sec:wpc}
Clearly, for each $n \in \mathbb{N}$, the operator $A_n$ defined by
(\ref{An}) is a linear operator, since both $Q_n$
and $R_n$ are linear and the inverse of a linear operator is linear.
Since the symbol of $A$ (equation (\ref{sym})) is a bounded function from
$\mathbb{R}$ into $\mathbb{R}$, so are the symbols of $A_n$,
$n \in \mathbb{N}$. Therefore, by
applying the Plancherel formula in the form
$$
\| A_n u\|_2 = \| \widehat {A_n u}\|_2 = \|\mathcal{S}(A_n) \widehat{u}\|_2,
$$
where $\widehat{u}$ is the Fourier
transform of $u \in L^2(\mathbb{R})$, we see that $A_n$ are bounded
operators in $L^2(\mathbb{R})$. Furthermore, it is not hard to show that if
$J(\cdot) \geq 0$ the
symbol of $A_n$ is negative for each $n$.
We now restrict ourselves to the space $\{ u \in L^2(\mathbb R) ;$
$\mathop{\rm supp }u = \Omega \}$,
where $\Omega$ is a bounded domain in $\mathbb{R}$, and
for each $n \in \mathbb{N}$ we study the following initial-value problem
\begin{equation}\label{IVP}
\begin{gathered}
u_t = \gamma A_n u - f(u),\quad (x,t)\in \Omega \times (0,\infty), \\
\hspace*{-4.6cm}u(0)=u_0.
\end{gathered}
\end{equation}
We would like to prove that this problem generates a flow on a
forward-invariant subset of $L^{2}(\Omega)$, which contains all the steady
state patterns. Here we are guided by the function-theoretic setting in
\cite{GHHMV}. Let
\begin{equation}\label{Z}
Z= L^2(\Omega) \cap \{|u(x)| \leq 1 \hbox{ a.e. in } \Omega \}.
\end{equation}
Then by Theorem 2.16 of Hoh \cite{Hoh}, which builds on the work of
Corr\`ege, the negativity of symbols of $A_n$ implies that $Z$ is
forward-invariant under the flow generated by (\ref{IVP}); for (\ref{nl})
this result has been proved in \cite{Fife2} and used extensively in
\cite{DGS}.
We make the following assumptions:
\begin{itemize}
\item[(A1)] $J(\cdot) \geq 0$; $J \in L^1(\mathbb{R})$ and
there exists $\alpha>0$ such that $\int_\mathbb{R} J(x)e^{\alpha|x|} dx
< \infty$;
\item[(A2)] the function $f: Z \to Z$
is locally Lipschitz continuous.
\end{itemize}
Note that (A1) assures that all the coefficients $\rho_{2k}$, $k \in
\mathbb{N} $, are defined and positive, and that the operator $A_n$
is defined for each $n \in \mathbb{N}$.
For a fixed $n \in \mathbb{N}$, we say that a function
$u:[0,T)\to L^2(\Omega)$ is
a {\em (classical) solution} of (\ref{IVP}) on $[0,T)$ if $u$ is continuous on
$[0,T)$, continuously differentiable on $(0,T)$, and (\ref{IVP}) is satisfied
on $[0,T)$. We have:
\begin{theorem} \label{thm3.1}
Suppose that the hypotheses (A1) and (A2) are satisfied.
Then for each $u_0 \in Z$, $n \in \mathbb{N}$, the
initial-value problem (\ref{IVP}) has a unique global solution
$u_n \in C([0, \infty) \times Z)$.
Moreover, for each $n \in \mathbb{N}$ the mapping $u_0\to u_n$ is
continuous in $L^2(\Omega)$.
\end{theorem}
\begin{proof}
The theory of Lipschitz perturbations of linear evolution equations (see Pazy
\cite{Pazy}) assures the existence and uniqueness of a local solution
$u_{n_0}(x,t,u_0)$, defined on a maximal interval of existence
$[0,\tau^{n_0})$ (with $\tau^{n_0}$ depending on $\|u_0\|_2$), and also
the continuity of $u_{n_0}$ with respect to the initial condition.
Moreover, if $\tau^{n_0} < \infty$, then
$\lim_{t \nearrow \tau^{n_0}} \|u(t)\|_2 =\infty$, which is not possible by
the forward invariance of $Z$.
\end{proof}
For each $n \in \mathbb{N}$, we denote by
$\{T_n(t) : Z \to Z,\; t \geq 0\}$ the continuous
semigroup of bounded nonlinear operators
$$
T_n(t)u_0 = u_n(t;u_0), \quad t \geq 0.
$$
Also, let $\{T(t) : Z \to Z,\; t\geq 0\}$
be the continuous semigroup of bounded nonlinear operators generated
by (\ref{nl}).
We would like now to show that solutions to (\ref{IVP}) with $u(x,0)$ given,
converge in the $L^{2}(\Omega)$ -norm to solutions to (\ref{nl}) with the
same initial data, as $n \to \infty$.
In order to prove this, we will use the following lemma:
\begin{lemma} \label{lem3.2}
If $X$ is a Banach space and the sequence
$\{w_n, n \in \mathbb{N}\}\subset C([0, T];X)$ converges to $w$
in the sense of the norm of $C([0, T];X)$, then
\begin{equation}
\lim_{n \to \infty} \int_0^T w_n(r)dr = \int_0^T w(r)dr, \hbox{ in
the } X \hbox{ norm.}
\end{equation}
\end{lemma}
For a proof of the above lemma, see \cite[Theorem 3.3]{Bel}.
We can now prove the following approximation result:
\begin{theorem} \label{thm3.3}
For every $u_0 \in Z$ and each $t > 0$, we have that
\begin{equation} \label{conv}
\| u_n (t; u_0) - u(t; u_0) \|_2 \to 0,\quad \hbox{as }n \to \infty .
\end{equation}
\end{theorem}
\begin{proof}
Denote by $\{S(t); t \geq 0 \}$ and $\{S_n(t); t \geq 0 \}$ the linear
continuous semigroups generated by the linear continuous operators $A$ and
$A_n$ ($n \in \mathbb{N}$), respectively. Since these semigroups are bounded,
we can find some positive constants $M$ and $M_n (n \in \mathbb{N})$ so that
$\|S(t)\|_2 \le M$ and $\|S_n(t)\|_2 \le M_n (n \in \mathbb{N})$.
If we let $g(u) = -f(u)$, then the solutions of (\ref{nl}) and, respectively,
(\ref{IVP}) can be written in the form
\begin{gather*}
u(t; u_0) = S(t)u_0 + \int_0^t S(t-s)g(u(s))\,ds, \quad t \geq 0; \\
u_n(t; u_0) = S_n(t)u_0 + \int_0^t S_n(t-s)g(u_n(s))\,ds, \quad t \geq
0,\; n \in \mathbb{N} .
\end{gather*}
The function $g$ is locally Lipschitz-continuous on $Z$, hence for every positive
constant $c$ there is a constant $L_c > 0$ such that
$$
\| g(u)- g(v) \|_2 \le L_c \|u - v\|_2
$$
holds for all $u, v \in Z$ with $\|u\|_2 \le c$, $\|v\|_2 \le c$.
Since $T$ and $T_n, n\in \mathbb{N}$, are bounded semigroups in $Z$, we can choose
$c$ to be the common $L^2$-upper bound, and thus for all $t > 0$, we have
\begin{align*}
&\| u_n (t) - u(t) \|_2 \\
& \le \| S_n (t)u_0 - S(t)u_0 \|_2 + \int_0^t \| S_n(t-s)g(u_n(s))
-S(t-s)g(u(s)) \|_2 ds \\
& \le \|[S_n (t)-S(t)]u_0 \|_2 +
\int_0^t \|[S_n(t-s)-S(t-s)]g(u(s))\|_2 ds \\
& \quad + \int_0^t \|S_n(t-s)g(u_n(s))-S_n(t-s)g(u(s))\|_2ds \\
& \le \|[S_n (t)-S(t)]u_0\|_2 +
\int_0^t \|[S_n(t-s)-S(t-s)]g(u(s))\|_2 ds \\
& \quad + M_n L_c\int_0^t\|u_n(s)- u(s)\|_2ds,
\end{align*}
for all $n\in \mathbb{N}$. We can rewrite the last inequality as
\begin{align*}
&\frac{d}{dt}\{ e^{-M_nL_{\omega}T} \int_0^t \|u_n(s)-u(s)\|ds \}\\
& \le e^{-M_nL_{\omega}T} \{ \| [S_n (t)-S(t)]u_0\|_2
+ \int_0^t \|[S_n(t-s)-S(t-s)]g(u(s))\|_2 ds \},
\end{align*}
for all $n\in \mathbb{N}$. Then, using the above Lemma, the convergence
(\ref{conv}) is proved if for all $h \in L^2(\Omega)$ we have
\begin{equation} \label{convS}
\| S_n(t)h - S(t)h \|_2 \to 0,\quad \hbox{as }n \to \infty .
\end{equation}
By the Trotter approximation theorem \cite{Pazy}, in order to have
(\ref{convS}) it suffices to prove the following convergence in
the $L^{2}(\Omega)$ norm, for the corresponding resolvents:
\begin{equation} \label{cr}
\text{For every $h \in L^2(\Omega)$ and some $\lambda >0$,
$R(\lambda , A_n)h \to R(\lambda , A)h$ as $n \to \infty$,}
\end{equation}
where $R(\lambda ,A) = (\lambda I -A)^{-1}$ and
$R(\lambda ,A_n) = (\lambda I-A_n)^{-1}$, $n \in \mathbb{N}$.
Since $A$ and $A_n (n \in \mathbb{N})$ are infinitesimal generators of the
uniformly continuous semigroups $\{S(t), t \geq 0\}$ and, respectively,
$\{S_n(t), t \geq 0 \}$ ($n \in \mathbb{N}$),
then the resolvent sets $\rho (A)$ and $\rho (A_n) (n \in \mathbb{N})$ contain
$(0, \infty)$ and
$$
\|R(\lambda , A)\|_2 \le M/\lambda ,\|R(\lambda ,A_n)\|_2 \le
M_n/\lambda \quad
\hbox{for } \lambda > 0, \; n=1,2,\dots
$$
We have then
\begin{equation}
\begin{aligned}
\| R(\lambda , A_n)h - R(\lambda , A)h\|_2
& = \| R(\lambda , A_n)\{(\lambda I - A) - (\lambda I - A_n)\}
R(\lambda , A)h \|_2 \\
& = \| R(\lambda , A_n)[A_n - A] R(\lambda , A)h \|_2 \\
& \le \frac{M_n}{\lambda } \| [A_n - A]R(\lambda , A)h \|_2, \quad
(\lambda > 0)
\end{aligned} \label{res}
\end{equation}
for all $h \in L^2(\Omega)$. On the other side, for each
$n \in \mathbb{N}$ the symbol $\mathcal{S}(A_n)$ is the $[2n/2n]$
Pad{\' e} approximant of $\mathcal{S}(A)$. This fact
and the Plancherel formula implies
\begin{equation} \label{fourier}
\begin{aligned}
\|(A_n - A)\xi\|_2
&= \| \mathcal{F}[(A_n - A)\xi] \|_2\\
&= \|[\mathcal{S}(A_n) - \mathcal{S}(A)]\mathcal{F}\xi \|_2\\
&\le \|\mathcal{S}(A_n) - \mathcal{S}(A)\|_2 \|\xi\|_2 \to 0,
\end{aligned}
\end{equation}
as $n \to \infty$ for all $\xi \in L^2(\mathbb R)$,
where $\mathcal{F}\xi$ denotes the Fourier transform.
Now (\ref{fourier}) and (\ref{res}) imply (\ref{cr}), and this completes the
proof.
\end{proof}
\subsection{Conclusion}\label{sec:concl}
By expanding the nonlocal term in the expression of the free energy
(\ref{vdw}) in Taylor series and truncating the result, one ends up with
equations which are not always well-posed, the well-posedness depending on
the order of truncation and the direction of time chosen. It is not clear
whether the solutions to the unbounded flows can in any sense approximate
the solution to the bounded flow given by (\ref{nl}). In this paper we
proposed a set of new equations, which can be put forward as possible models
for phase transitions in solids. (Our method of proof relies on having
$J(\cdot) \geq 0$, which is a reasonable assumption to make in this
context.) By using Pad\'e approximation, we approximated the flow generated
by (\ref{nl}) by some bounded flows. The new equations have the advantage of
being well-posed for all orders of the Pad\'e approximation.
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\end{document}