\documentstyle[12pt,amsfonts]{article}
\begin{document}
\title{The Inviscid Limit of the Complex Ginzburg-Landau Equation}
\author{Jiahong Wu\thanks{Supported by the NSF grant DMS
9304580 at IAS.} \thanks{E-mail address: jiahong@math.ias.edu}
\\School of Mathematics\\The Institute for Advanced Study
\\Princeton, NJ 08540 }
\date{}
\maketitle
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{define}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{example}[thm]{Example}
\newtheorem{lemma}[thm]{Lemma}
\def\theequation{\thesection.\arabic{equation}}
\newpage
\section{Introduction}
\setcounter{equation}{0}
\label{sec:int}
The complex Ginzburg-Landau (CGL for short) equation plays an important
role in describing
spatial pattern formation and the onset of instabilities in
fluid dynamical systems \cite{CH}. A general form of the CGL equation
without driving is
\begin{equation}\label{CGL}
\partial_t u = (a+i\nu)\Delta u -(b+i\mu)|u|^{2\sigma}u,\quad
(x,t)\in {\Bbb R}^n\times (0,\infty)
\end{equation}
where $u$ is a complex-valued function of a space variable $x\in {\Bbb R}
^n$ and of a time variable $t\in (0,\infty)$, and
$\sigma>0, a>0,b>0, \nu>0,\mu$
are real parameters.
\vspace{.12in}
By taking $a=b=0$ in (\ref{CGL}), the CGL equation formally becomes the
nonlinear Schr\"{o}dinger (NLS for short) equation
\begin{equation}\label{NLS}
i \partial_t v =-\nu\Delta v +\mu|v|^{2\sigma}v
\end{equation}
Naturally the question of inviscid limit arises. Does the solution $u$ of
the CGL equation (\ref{CGL}) tend to (in an appropriate space norm)
the solution $v$ of the NLS equation
(\ref{NLS}) as the parameters $a$ and $b$ tend to $0$? What is the
convergence rate? The answers are not immediate especially when the
initial data for these equations are not smooth.
\vspace{.12in}
Because of its importance in both mathematical theory and physical
applications, the inviscid limit has been extensively investigated for
many partial differential equations such as Burgers'
equation \cite{bo}, the quasi-geostrophic equation \cite{wu1}
and most notably the
Navier-Stokes equations. For smooth initial data and in absence of
boundary, the inviscid limit of the Navier-Stokes equations is the
corresponding Euler equations and the rate is the optimal $O(\nu)$ where
$\nu$ is the viscosity coefficient (\cite{bm},\cite{c},\cite{k}). But the
situation changes if the initial data is not that smooth. It's
shown in \cite{CW1} that the convergence rate for vortex patch type
data is only $O(\sqrt{\nu})$. If the data is even less smooth, the
inviscid limit of the Navier-Stokes equations can be
modified Euler equations or other equations we currently do not know
(\cite{CW2},\cite{CW3}). These inviscid limit results for the
Navier-Stokes equations turn out to be crucial in
proposing corrections to the ``K-41" Kolmogorov theory \cite{bc}.
\vspace{.12in}
The CGL equation, derived from the Navier-Stokes equations via multiple
scaling methods in convection \cite{NW}, has been studied only
recently in problems related to existence and properties of solutions
(\cite{GV},\cite{DGL}).
Although there are claims in physics literature, it seems that the
inviscid limit question has not been addressed before and we see no
mathematical proof existing. In this paper
we are mainly concerned with the
global (in time) inviscid
limit of the CGL equation
(\ref{CGL}) in $L^2, L^{2\sigma+2}$ and $H^1$ spaces while
the initial data $u_0$ is taken in $L^2({\Bbb R}^n)$
or $H^1({\Bbb R}^n)$. The global existence results of
the CGL equation (\ref{CGL})
with $L^2$ or $H^1$ initial data are newly available \cite{GV}.
\vspace{.12in}
The expected inviscid limt, the NLS equation, is known to have
finite time blow-up solutions for critical and supercritical
exponent $\sigma$ in the focusing case ($\mu<0$).
Since our main interest is in global inviscid limit results, the exponent
$\sigma$ is assumed to be subcritical or critical with small initial
data when $\mu<0$. The term ``critical" (resp. ``subcritical", resp.
``supercritical") at the level of $L^p$ indicates $n\sigma=p$ (resp.
$n\sigma
p$) and at the level of $H^r$ indicates
$\sigma(n-2r)=2$ (resp. $\sigma(n-2r)<2$, resp. $\sigma(n-2r)>2$).
\vspace{.12in}
We approach the inviscid limit problem by employing extensive
energy estimates to bound the difference between the solutions of the
CGL equation and the NLS equation in terms of the initial data and the
parameters. The initial data for the NLS equation is taken at least as
regular as in $H^1({\Bbb R}^n)$ so that the energy estimates make
sense (see more details in Remark \ref{rem2}). Furthermore, some assumptions
on the parameters (like (\ref{bm}) in Section \ref{sec:l2})
are necessary in order to
obtain a closed equation for the normed difference. Otherwise, only a
hierarchy of differential equations, the so called ladder structure,
can be developed \cite{DGL}.
\vspace{.12in}
The remainder of this paper is organized as follows. In Section
\ref{sec:ex} we review the existence and regularity of solutions
to the CGL equation and the NLS equation.
In Section \ref{sec:l2} we establish two global $L^2$ inviscid limit results.
The first theorem
states that the $L^2$ difference between solutions of the CGL
equation and the NLS equation is of order
$O(\sqrt{a}) +O(b^\frac{2\sigma+1}{2(2\sigma+2)})$ if the
initial data for the NLS equation is taken in $H^1({\Bbb R}^n)$
(see Theorem \ref{main}).
The second theorem improves the convergence rate to the optimal
$O(a)+O(b)$ by taking $v_0\in H^2$ (see Theorem \ref{2th}
for details). Section \ref{sec:hs} treats the $L^{2\sigma+2}$
inviscid limit and the result is given in Theorem \ref{sec}. The main
reason for considering and achieving this type of inviscid limit is the
special form of the nonlinear term in (\ref{CGL}). In Section \ref{sec:h1}
we investigate the $H^1$ inviscid limit and obtain a convergence rate
depending on $a^{-1}\sqrt{b^2+\mu^2}$(see Theorem \ref{h1thm}).
\newpage
\section{Preliminaries}
\setcounter{equation}{0}
\label{sec:ex}
In this section we review the existence results and appropriate properties
concerning solutions of the CGL equation with initial data $u_0$ belonging
to $L^2({\Bbb R}^n)$ or $H^1({\Bbb R}^n)$ and of the NLS equation with data
in $H^1({\Bbb R}^n)$ or $H^2({\Bbb R}^n)$.
\vspace{.13in}
\begin{thm}\label{excgl}
Let $u_0\in L^2({\Bbb R}^n)$.
Then the CGL equation \ref{CGL} with initial
data $u_0$ has a global (in time) solution $u$ satisfying
\begin{equation}\label{cla}
u\in C([0,\infty); L^2)\cap L_{loc}^{2}([0,\infty); H
^{1})\cap L_{loc}^{2\sigma+2}([0,\infty); L^{2\sigma+2})
\end{equation}
with $u(0)=u_0$. Furthermore, $u$ satisfies the energy relation
\begin{equation}\label{energy}
\frac{1}{2}\|u(t)\|_{L^2}^{2} +a \int_{0}^{t}\|\nabla u(t')\|_{L^2}^{2}dt'
+b\int_{0}^{t}\|u(t')\|_{L^{2\sigma+2}}^{2\sigma+2}dt'
=\frac{1}{2}\|u_0\|_{L^2}^{2}
\end{equation}
for any $t\in [0,\infty)$.
\end{thm}
This theorem is stated in \cite{GV} (Proposition 2.1, p.197) and it can
be proved by using either Faedo-Galerkin method or
smoothing approximations.
The solution $u$ is shown to be unique in the class (\ref{cla})
under the assumption
\begin{equation}\label{assu}
\left|1+i\frac{\mu}{b}\right|\le \frac{\sigma+1}{\sigma}
\end{equation}
See \cite{GV}
(Proposition 3.1, p.201) for a proof. The assumption (\ref{assu})
on $b$ and
$\mu$ turns out to be also important in showing inviscid limits in the
subsequent sections.
\vspace{.15in}
The existence result of $H^1$-solutions of the CGL equation is obtained
only very recently in \cite{GV} (Proposition 5.1, p.215) and it states, in
particular, that
\begin{thm}
Assume that either $\mu\ge 0$ or $(n-2)\sigma <2$ and if $n\sigma>2$
$$
\left|1+i\frac{\nu}{a}\right| \le \frac{n\sigma}{n\sigma-2}
$$
Let $u_0\in H^1({\Bbb R}^n)\cap L^{2\sigma+2}
({\Bbb R}^n)$. Then the CGL equation \ref{CGL} has a unique solution $u$
satisfying
$$
u\in C([0,\infty); H^1\cap L^{2\sigma+2})\cap L^{2}_{loc}([0,
\infty); H^2)\cap L^{4\sigma+2}_{loc}([0,\infty); L^{4\sigma+2})
$$
with $u(0)=u_0$, and $u$ satisfies (\ref{energy}).
\end{thm}
\vspace{.15in}
The existence of solutions to the NLS equation with $L^2, H^1$ or
$H^2$ data is now well-documented in monographs
and survey papers (see e.g. \cite{k1},\cite{G}).
We shall only need the results
concerning $H^1$ and $H^2$ solutions.
\begin{thm}\label{H1}
Let $(n-2)\sigma\le 2$ for $n\ge 3$. Then for every
$v_0\in H^1({\Bbb R}^n)$,
there is a $T^*=T^*(\|v_0\|_{H^1})>0$ and a solution
$v$ to the NLS equation \ref{NLS} on $[0,T^*)$ such that
$$
v\in C([0,T^*); H^1)\cap C^1([0,T^*);
H^{-1})\cap L_{loc}^{2\sigma+2}([0,T^*); L^{2\sigma+2})
$$
with $v(0)=v_0$.
Furthermore, for any $t0$ and a unique
solution $u\in C([0,T^*); H^2)$ to the NLS equation \ref{NLS}
with $v(0)=v_0$.
Furthermore, for any $T0$,
$\sigma\le \frac{2}{n-2}$ for $n\ge 3$ if $\mu \ge 0$
and $\sigma \le \frac{2}{n}$ if $\mu<0$.
Let $v_0\in H^1({\Bbb R}^n)$ ($\|v_0\|_{L^2}$
should also be small
if $\mu<0$ and $\sigma=\frac{2}{n}$).
Then the solution of the NLS equation with the initial data $v_0$
satisfies
\begin{equation}\label{gra}
\int |\nabla v|^2 \le {\cal F}(v_0),
\end{equation}
\begin{equation}\label{2+2}
\int |v|^{2\sigma+2} \le {\cal G}(v_0)
\end{equation}
where ${\cal F}(v_0)$ and ${\cal G}(v_0)$ are determined by the initial $v_0
$ but independent of $a$ and $b$ (see their explicit expressions in the proof
below).
\end{prop}
{\bf Proof of Proposition \ref{33}}.\quad
The idea of showing
$(\ref{gra})$ and $(\ref{2+2})$ is to use both the conservation of the
$L^2-$norm and of the energy (see (\ref{energy1})).
But we need to distinguish between the defocusing ($\mu\ge 0$) and the
focusing ($\mu<0$) case.
\vspace{.12in}
For $\mu \ge 0$, both the $H^1$-norm and the
$L^{2\sigma+2}$-norm are easily controlled by using the conservation laws.
In fact, using Lemma \ref{gl},
$$
\|\nabla v\|_{L^2}^{2}\le \frac{2}{\nu} E(v_0)
$$
$$
\int |v|^{2\sigma+2} \le C(\sigma)\|\nabla v\|_{L^2}^{n\sigma}
\|v\|_{L^2}^{2\sigma+2-n\sigma}
$$
$$
\le C(\sigma)\left(
\frac{E(v_0)}{\nu}\right)^{\frac{n\sigma}{2}}\|v_0\|_{L^2}^{
2\sigma+2-n\sigma}
$$
where $C(\sigma)$ is a constant depending on $\sigma$.
\vspace{.13in}
For $\mu<0$, the bounds can still be obtained in the $L^2-$subcritical
case ($\sigma<\frac{2}{n}$) and in the $L^2-$critical case ($\sigma=
\frac{2}{n}$) provided $\|v_0\|_{L^2}$ is small enough.
Indeed, using Lemma \ref{gl},
$$
E(v_0)=\frac{\nu}{2}\int |\nabla v|^2 +\frac{\mu}{2\sigma+2}\int |v|^{
2\sigma+2}
$$
\begin{equation}\label{ev0}
\ge \frac{\nu}{2}\int |\nabla v|^2 + C(\sigma)\mu
\left(\int |\nabla v|^{2}\right)^{n\sigma/2}\|v\|_{L^2}^{2\sigma+2-n\sigma} \end{equation}
If $n\sigma=2$ and $\|v_0\|_{L^2}$ is small enough, say,
$$
\frac{\nu}{2} + C(\sigma)\mu \|v_0\|_{L^2}^{2\sigma+2-n\sigma}>0
$$
Then it follows from (\ref{ev0}) that
$$
\int |\nabla v|^2\le \left(
\frac{\nu}{2} + C(\sigma)\mu \|v_0\|_{L^2}^{2\sigma+2-n\sigma}\right)^{-1}
E(v_0)
$$
If $n\sigma<2$, we use the following simple lemma
\begin{lemma}\label{simple}
Let $P,Q$ and $\beta<2$ are all positive numbers. If $y\ge 0$ satisfies
$$
y^2-P y^{\beta}\le Q
$$
Then $y$ is bounded by
$$
y\le \max\left\{ (2P)^{\frac{1}{2-\beta}}, \sqrt{2Q}\right\}
$$
\end{lemma}
Applying Lemma \ref{simple} to (\ref{ev0})
$$
\int |\nabla v|^2 \le \max\left\{ \left(C(\sigma)\nu^{-1}(-\mu)
\|v_0\|_{L^2}^{2\sigma+2-n\sigma}\right)^{\frac{2}{2-n\sigma}}
,4\nu^{-1}E(v_0)\right\}
$$
The proof of this proposition is concluded if we denote by ${\cal F}(v_0)$
and ${\cal G}(v_0)$ the bounds for $\int |\nabla v|^2$ and $\int |v|^{
2\sigma+2}$ in either the defocusing case or the focusing case. \qquad $\Box$
\vspace{.13in}
\noindent{\bf Proof of Lemma \ref{simple}}\quad The proof is easy. Suppose
not. Then
$$
y> (2P)^{\frac{1}{2-\beta}},\qquad y> \sqrt{2Q}
$$
But $y> (2P)^{\frac{1}{2-\beta}}$ implies $Py^{\beta-2}<1/2$ and thus
$$
y^2\le Py^{\beta}+Q \le Py^{\beta-2} y^2 +Q < \frac{1}{2} y^2 +Q
$$
which contradicts $y>\sqrt{2Q}$. \qquad $\Box$
\vspace{.15in}
We are now ready to prove Theorem \ref{main}.
\vspace{.12in}
\noindent{\bf Proof of Theorem \ref{main}.}\quad
Let $u$ satisfy the CGL equation \ref{CGL}
and $v$ satisfy the NLS equation \ref{NLS}. Then the
difference $w=u-v$ satisfies
\begin{equation}\label{diff}
\partial_t w=(a+i\nu) \Delta w + a\Delta v- (b+i\mu) (f(u)-f(v))- bf(v)
\end{equation}
where $f(u)=|u|^{2\sigma}u$.
\vspace{.13in}
We take a nonnegative, smooth cutoff function $\phi$, identically
equal to 1 for $|x|\le 1$ and to 0 for $|x|\ge 2$. We multiply the
equation (\ref{diff}) by $2{\bar w}\phi_{R}^{2}$ where
$\phi_{R}(x)=\phi(\frac{x}{R})$ and $R>0$. Integrating in space we obtain:
$$
\partial_t \int \phi^{2}_{R}|w|^2 =2Re \int \phi^{2}_{R}
\partial_t w \bar {w}
$$
$$
=2Re \left((a+i\nu) (\phi^{2}_{R}\Delta w,w)\right) +2a Re(\phi^{2}_{R}
\Delta v, w)
$$
\begin{equation}\label{len}
- 2Re \left((b+i\mu) (\phi^{2}_{R}(f(u)-f(v)), w)\right) -2bRe(\phi^{2}_{R}
f(v), w)
\end{equation}
where $(F,G)=\int_{{\Bbb R}^n} F\bar{G}$, $\bar{G}$ is the
complex conjugate of $G$
and $Re$ denotes the real part.
\vspace{.13in}
For simplicity of notation, we denote by $I,II,III,IV$ the four terms on the
RHS of (\ref{len}) and now estimate them separately.
$$
I=-4Re((a+i\nu)(\phi_{R}\nabla w, w\nabla \phi_{R})-2Re((a+i\nu)(\phi^{2}
_{R}\nabla w
, \nabla w))
$$
and therefore
$$
|I|\le a\|\phi_{R} \nabla w\|_{L^2}^{2} +4
\frac{a^2+\nu^2}{a}\|w\nabla \phi_{R}\|
_{L^2}^{2} - 2a \|\phi_{R}\nabla w\|_{L^2}^{2}
$$
The second term can be estimated similarly
$$
|II|\le (a+\epsilon) \|\phi_R \nabla v\|_{L^2}^{2}
+ 4a^2 \epsilon^{-1}
\|w\nabla \phi_R\|_{L^2}^{2} +a\|\phi_R\nabla w\|_{L^2}^{2}
$$
where $\epsilon>0$ is small.
Adding the estimates for $I$ and $II$ and using \newline
$\|\nabla \phi_R\|_{L^\infty}
\le R^{-1}\|\nabla \phi\|_{L^\infty}$,
\begin{equation}\label{i+ii}
|I|+|II| \le (a+\epsilon)\|\phi_R \nabla v\|_{L^2}^{2}+\left(4\frac{a^2
+\nu^2}{a}+4a^2 \epsilon^{-1}\right)
R^{-2}\|\nabla \phi\|_{L^\infty}^{2}\|w\|_{L^2}^{2}
\end{equation}
\vspace{.12in}
Under the assumption
$$
\left|1+i\frac{\mu}{b}\right| \le \frac{\sigma+1}{\sigma}
$$
we can show
\begin{equation}\label{III}
III=-2Re \left((b+i\mu) (\phi^{2}_{R}(f(u)-f(v)), w)\right)\le 0.
\end{equation}
In fact, noticing $f(u)=|u|^{2\sigma}u$ and using
$$
f(u)-f(v)=\int_{0}^{1}\left[(\sigma+1)(u-v)|Z|^
{2\sigma} +\sigma (\bar{u}-\bar{v})Z^2|Z|^{2\sigma-2}\right]d\lambda
$$
where $Z=\lambda u+(1-\lambda)v$, we rewrite $III$ as
$$
III=-2Re \left((b+i\mu)\int_{0}^{1}d\lambda\int
\phi^{2}_{R}\left[(\sigma+1)|w|^2 |Z|^{2\sigma}+\sigma
\bar{w}^2Z^2|Z|^{2\sigma-2}
\right]dx\right)
$$
$$
\le 2\sigma b \max\left\{ 0, \quad \left|1+i\frac{\mu}{b}\right|
-\frac{\sigma+1}{\sigma}\right\}
\int_{0}^{1}\int \phi_{R}^{2}|w|^2|Z|^2 =0
$$
\vspace{.12in}
For the term IV, we use the Young inequality $AB\le \frac{A^\rho}{\rho}+
\frac{B^\varrho}{\varrho}$ for $A,B\ge 0$ and
$\frac{1}{\rho}+\frac{1}{\varrho}=1$
to obtain (noticing that $f(v)=|v|^{2\sigma}v$)
$$
|IV|= 2b|(\phi^{2}_{R} f(v), w)|
$$
\begin{equation}\label{IV}
\le \frac{2\sigma+1}{\sigma+1}b^\frac{2\sigma+1}{2\sigma+2}\int
\left(\phi^{\frac{2}{2\sigma+1}}_{R} |v|\right)^{2\sigma+2}
+ \frac{1}{\sigma+1}b^{\frac{2\sigma+1}{2\sigma+2}+1}
\int |w|^{2\sigma+2}
\end{equation}
\vspace{.18in}
Collecting the above estimates (\ref{i+ii}), (\ref{III}), (\ref{IV})
and letting $R\to \infty$ and $\epsilon\to 0$, we obtain
$$
\partial_t\int |w|^2 \le a\|\nabla v\|_{L^2}^{2}
$$
\begin{equation}\label{to1}
+ \frac{2\sigma+1}{\sigma+1}b^\frac{2\sigma+1}{2\sigma+2}\int
|v|^{2\sigma+2}+ \frac{1}{\sigma+1}b^{\frac{2\sigma+1}{2\sigma+2}+1}
\int |w|^{2\sigma+2}
\end{equation}
\vspace{.13in}
Using the estimates in Proposition \ref{33}
and integrating in $t$
$$
\|w\|_{L^2}^{2}\le \|u_0-v_0\|_{L^2}^{2} + a {\cal F}(v_0) t
+ C_1(\sigma)b^{\frac{2\sigma+1}{2\sigma+2}}(1+b){\cal G}(v_0) t
$$
\begin{equation}\label{last}
+ C_2(\sigma) b^{\frac{2\sigma+1}{2\sigma+2}}\left(b\int_{0}^{t}\|u(t')\|_
{L^{2\sigma+2}}^{2\sigma+2} dt'\right)
\end{equation}
where $C_1, C_2$ are constants depending on $\sigma$ only. The expected
estimate (\ref{imp}) is obtained after applying
(\ref{energy}) to the last term of (\ref{last}).
This concludes the proof of Theorem \ref{main}. \qquad $\Box$
\vspace{.18in}
The inviscid limit results for the Navier-Stokes equations
(\cite{bm},\cite{CW1},\cite{CW3})
and for the quasi-geostrophic equation \cite{wu1} suggest that
the convergence become faster if the initial data
become smoother. The $L^2$ inviscid limit result of
Theorem \ref{main} can indeed be
improved to the optimal rate $O(a)+O(b)$ if the initial data for the
NLS equation $v_0\in H^
2({\Bbb R}^n)$. This can be shown by modifying
the proof of Theorem \ref{main}.
\vspace{.13in}
If $v_0\in H^2$, then the solution $v$ of the NLS equation with data $v_0$
is in $C([0,T); H^2)$ and satisfies the estimate (\ref{control})
according to
Theorem \ref{H2}.
This fact allows us to estimate $II$ and $IV$ differently.
\begin{equation}\label{secterm}
|II| =2a |Re(\phi_{R}^{2}\Delta v, w)|
\le \int \phi_{R}^{2} |w|^2 + a^2\int \phi_{R}^{2} | \Delta v|^2
\end{equation}
$$
|IV|= 2b \int \phi_{R}^{2}|f(v)||w| \le \int \phi_{R}^{2} |w|^2
+b^2\int |v|^{4\sigma+2}
$$
Applying Gagliardo-Nirenberg's inequality
to $\int|v|^{4\sigma+2}$
$$
\|v\|_{L^{4\sigma+2}}\le C(\sigma)\|\Delta v\|_{L^2}^{\theta}\|v\|_{
L^2}^{1-\theta}
$$
where $C(\sigma)$ is a constant depends on $\sigma$,
$0\le \theta$ is given by
$$
\theta= \frac{n}{2}\left(\frac{1}{2}-\frac{1}{4\sigma+2}\right)
$$
and $\theta\le 1$ if we assume $\sigma\le \frac{2}{n-4}$.
The estimates for $I$, $III$ remain unchanged. Collecting the
estimates and letting $R\to \infty$, we obtain
\begin{equation}\label{hell}
\partial_t \int |w|^2 \le 2 \int |w|^2 + a^2\int |\Delta v|^2
+ b^2C(\sigma) \|\Delta v\|_{L^2}^{n\sigma}\|v_0\|_{L^2}^{4\sigma+2-n\sigma}
\end{equation}
To bound $\|\Delta v\|_{L^2}$, we use (\ref{control}). For $t\le T$,
\begin{equation}\label{biga}
\|\Delta v(t)\|_{L^2}^{2}\le K^2\|v_0\|_{H^2}^{2}
\equiv {\cal A}(v_0)
\end{equation}
where $K$ is a bound in terms of $T$ and ${\cal F}(v_0)$.
Letting
\begin{equation}\label{big b}
{\cal D}(v_0)\equiv C(\sigma){\cal A}(v_0)^{\frac{n\sigma}{2}}\|v_0\|_{L^2}
^{4\sigma+2-n\sigma}
\end{equation}
and integrating (\ref{hell}) in $t$, we obtain
\begin{thm}\label{2th}
Assume $\sigma>0$,
$\sigma \le \frac{2}{n-4}$ for $n\ge 5$ if $\mu\ge 0$
and $\sigma \le \frac{2}{n}$ if $\mu<0$.
Assume $b,\mu$ satisfy
the condition (\ref{bm}).
Let $u_0\in L^2({\Bbb R}^n)$ and $v_0\in H^2({\Bbb R}^n)$ ($\|v_0\|_{L^2}
$ should also
be small if $\mu<0$ and $\sigma=\frac{2}{n}$).
Consider
the difference
$$
w(x,t)=u(x,t)-v(x,t)
$$
between a solution $u$ of the CGL equation \ref{CGL} with $u(x,0)=u_0(x)$
and a solution $v$ of the NLS equation \ref{NLS}
with $v(x,0)=v_0$. Then $w$ obey
the estimate
$$
\|u(t)-v(t)\|_{L^2}^{2} \le \|u_0-v_0\|_{L^2}^{2}e^{2t} +
\frac{1}{2} a^2 {\cal A}(v_0)(e^{2t}-1)+ \frac{1}{2}b^2 {\cal D}(v_0)(e^{2t}
-1)
$$
for $t0$ such that
$$
|1+i\nu/a| \le \frac{\sigma+1-\epsilon-\delta}{\sigma}
$$
and divide $I$ into $I_1$ and $I_2$ with
$$
I_1= -Re(a+i\nu)\left(\int
\phi_{R}^{2} \left[
(\sigma+1-\epsilon-\delta)|w|^{2\sigma}|\nabla w|^2 + \sigma|w|^{2
\sigma-2}(\bar{w}
\nabla w)^2\right]\right)
$$
It is easy to check that
$$
|I_1|\le \sigma a\max\left\{0, |a+i\nu|-\frac{\sigma+1-\delta-\epsilon}
{\sigma}\right\}\int \phi_{R}^{2} |w|^{2\sigma}|\nabla w|^2
=0
$$
$$
I_2 =- a(\delta+\epsilon) \int \phi_{R}^{2} |w|^{2\sigma}|\nabla w|^2
-Re(a+i\nu)\int
\phi_{R}|w|^{2\sigma}\bar{w}\nabla w\nabla \phi_R
$$
Applying Young's inequality to the second term in $I_2$,
$$
|I_2|
\le -a\delta\int \phi_{R}^{2} |w|^{2\sigma}|\nabla w|^2
+ \frac{|a+i\nu|^2}{4a\epsilon}\int |w|^{2\sigma+2} |\nabla \phi_R|^2
$$
\vspace{.12in}
Integration by parts in $II$ gives
$$
II=-a Re\int \phi_{R}^{2}\left[(\sigma+1)|w|^{2\sigma}\nabla v\nabla \bar{w}
+ \sigma |w|^{2\sigma-2}\bar{w}^{2}\nabla v\nabla w\right]
$$
$$
-2a Re\int \phi_{R}|w|^{2\sigma}\nabla \phi_{R}\nabla v\bar{w}
$$
Applying Young's inequality,
$$
|II|\le C_1(\sigma)\int \phi_{R}^{2}|w|^{2\sigma+2} + C_2(\sigma)
(a\delta^{-1})^{\sigma+1}\int|\nabla v|^{2\sigma+2}+ a\delta \int
\phi_{R}^{2}|w|^{2\sigma}|\nabla w|^{2}
$$
$$
+ a\int |\nabla \phi_{R}||w|^{2\sigma+2} + a \int |\nabla \phi_{R}|
|\nabla v|^{2\sigma+2}
$$
Noting $\|\nabla \phi_{R}\|_{L^\infty}\le R^{-1} \|\nabla \phi\|_{
L^\infty}$
$$
|I+II| \le C_1(\sigma)\int \phi_{R}^{2}|w|^{2\sigma+2} +
\left(C_2(\sigma)(a\delta^{-1})^{\sigma+1}+ aR^{-1}\|\nabla \phi\|_{L^\infty}\right)\int|\nabla v|^{2\sigma+2}
$$
\begin{equation}\label{es1}
+\left(\frac{|a+i\nu|^2}{4a\epsilon}
R^{-2} \|\nabla \phi\|_{L^\infty}^{2}+ aR^{-1}\|\nabla \phi\|_{L^\infty}
\right) \int |w|^{2\sigma+2}
\end{equation}
\vspace{.13in}
The term $III$ can be dealt with similarly as in the proof of Theorem
\ref{main} and the conclusion is
\begin{equation}\label{es2}
III\le 0, \qquad \mbox{if} \quad |1+i\mu/b|\le \frac{\sigma+1}{\sigma}
\end{equation}
\vspace{.13in}
Now we turn to term $IV$.
$$
IV =-b Re\left(\int \phi_{R}^{2} |w|^{2\sigma} |v|^{2\sigma}v \bar{w}\right)
$$
Using the Young inequality,
\begin{equation}\label{es3}
|IV| \le \frac{2\sigma+1}{2\sigma+2}\int |w|^{2\sigma+2}
+\frac{1}{2\sigma+2}b^{2\sigma+2}\int |v|^{(2\sigma+1)(2\sigma+2)}
\end{equation}
Applying the Gagliardo-Nirenberg inequality
\begin{equation}\label{might}
\int |v|^{(2\sigma+1)(2\sigma+2)} \le C_3(\sigma)\left(
\|\Delta v\|_{L^2}^{\theta}\|v\|_{L^2}
^{1-\theta}\right)^{(2\sigma+1)(2\sigma+2)}
\end{equation}
\begin{equation}\label{junk}
\int |\nabla v|^{2\sigma+2} \le C_4(\sigma)\left(
\|\Delta v\|_{L^2}^{\theta_1}\|v\|_{L^2}^{1-\theta_1}\right)^{(2\sigma+2)}
\end{equation}
where $C_3,C_4$ are constants depending only on $\sigma$ and $\theta,\theta_1
\ge 0$
are given by
$$
\theta= \frac{n}{2}
\left(\frac{1}{2}-\frac{1}{(2\sigma+1)(2\sigma+2)}\right)
,\qquad \theta_1= \frac{n}{2}\left(\frac{1}{2}+\frac{1}{n}-\frac{1}{
2\sigma+2}\right)
$$
By the assumptions on $\sigma$, $\theta,\theta_1\le 1$.
Using (\ref{biga}) of Section \ref{sec:l2},
\begin{equation}\label{es4}
\int |v|^{(2\sigma+1)(2\sigma+2)} \le
{\cal H}(v_0)
,\qquad \int |\nabla v|^{2\sigma+2}\le {\cal K}(v_0)
\end{equation}
with ${\cal H}(v_0), {\cal K}(v_0)$ given by
\begin{equation}\label{g}
{\cal H}(v_0) \equiv
C_3(\sigma) {\cal A}(v_0)^{\theta(2\sigma+1)(\sigma+1)}\|v_0\|_{L^2}
^{(1-\theta)(2\sigma+1)(2\sigma+2)}
\end{equation}
\begin{equation}\label{k}
{\cal K}(v_0)\equiv C_4(\sigma)
{\cal A}(v_0)^{\theta_1(2\sigma+2)}\|v_0\|_{L^2}
^{(1-\theta_1)(2\sigma+2)}
\end{equation}
\vspace{.13in}
Collecting the estimates
(\ref{es1}),(\ref{es2}),(\ref{es3}),(\ref{might}),(\ref{junk})
(\ref{es4}),(\ref{g}),(\ref{k}), integrating in $t$,
and letting $R\to \infty$ and then $\epsilon\to 0$,
we obtain
$$
\int |w|^{2\sigma+2} \le e^{C(\sigma)t} \int |u_0-v_0|^{2\sigma+2}
$$
$$
+C_5(\sigma) b^{2\sigma+2}{\cal H}(v_0) \left(e^{C(\sigma)t}-1\right)
+C_6(\sigma) (a\delta^{-1})^{\sigma+1}{\cal K}(v_0)
\left(e^{C(\sigma)t}-1\right)
$$
where $C,C_5,C_6$ are constants depending only on $\sigma$.
This concludes the proof of Theorem \ref{sec}. \qquad$\Box$
\newpage
\section{$H^1$ inviscid limit}
\setcounter{equation}{0}
\label{sec:h1}
Motivated by the inviscid limit results concerning the derivatives of
solutions to the Navier-Stokes equations \cite{CW2},
we consider in this section
the $H^1$ inviscid limit of the CGL equation.
\vspace{.12in}
We first state the main result.
\begin{thm}\label{h1thm}
Let $n\sigma\le 2$.
Assume $u_0\in H^1({\Bbb R}^n)$ ($\|u_0\|_{L^2}$ is small if $n\sigma=2$)
and $v_0\in H^{2}({\Bbb R}^n)$ ($\|v_0\|
_{L^2}$ is small if $\mu<0$ and $n\sigma=2$). Consider
the difference
$$
w(x,t)=u(x,t)- v(x,t)
$$
between a solution $u$ of the CGL equation \ref{CGL}
with $u(0)=u_0$ and $v$ of the
NLS equation \ref{NLS} with $v(0)=v_0$. Then $w$ obeys
for any $T<\infty$ and $t0)$ be the cutoff function as defined in
the proof of Theorem \ref{main}. It is easy to see that
$$
\partial_t\int \phi_{R}^{2}|\nabla u|^2 =2 Re \int \phi_{R}^{2}
(\partial_t\nabla u)\nabla \bar{u}= I+II
$$
where $I$ and $II$ are given by
$$
I= 2Re(a+i\nu)\int \phi_{R}^{2}\Delta(\nabla u)\nabla \bar{u}
$$
$$
II=-2Re(b+i\mu)\int \phi_{R}^{2} \nabla(|u|^{2\sigma}u)\nabla \bar{u}
$$
Integrating by parts,
$$
I=-2a\int \phi_{R}^{2} |\Delta u|^2 -4Re(a+i\nu)\int (\phi_{R}
\Delta u) (\nabla \phi_{R} \nabla \bar{u})
$$
$$
II= 2Re(b+i\mu)\int \phi_{R}^{2} |u|^{2\sigma}u \Delta \bar{u} +
4Re(b+i\mu)\int \phi_{R}|u|^{2\sigma}u\nabla \phi_{R}\nabla \bar{u}
$$
Using Young's inequality to split the terms
in $I$ and $II$ as in the proof of Theorem \ref{main}, adding them
and letting $R\to\infty$,
$$
\partial_t \int|\nabla u|^2 \le -a\int|\Delta u|^2
+ \frac{8(b^2+\mu^2)}{a}\int |u|^{4\sigma+2}
$$
Applying Gagliardo-Nirenberg's inequality
to $\int|u|^{4\sigma+2}$
$$
\|u\|_{L^{4\sigma+2}}\le C(\sigma)\|\Delta u\|_{L^2}^{\theta}\|u\|_{
L^2}^{1-\theta}
$$
where $C(\sigma)$ is a constant depends on $\sigma$ and
$0\le \theta$ is given by
$$
\theta= \frac{n}{2}\left(\frac{1}{2}-\frac{1}{4\sigma+2}\right)
$$
and $\theta\le 1$ because of the assumption $\sigma\le \frac{2}{n-4}$.
Therefore,
$$
\partial_t \int|\nabla u|^2 +a\|\Delta u\|_{L^2}^{2}-\frac{
C(\sigma)(b^2+\mu^2)}{a}\|\Delta u\|_{L^2}^{n\sigma}\|u_0\|_{L^2}
^{4\sigma+2-n\sigma}\le 0
$$
We obtain (\ref{delta1}) after integrating in $t$.
(\ref{delta2}) follows easily from (\ref{delta1}). (\ref{delta3}) is
obtained from (\ref{delta1}) and using Lemma \ref{simple}.
\qquad $\Box$
\vspace{.20in}
We now prove Theorem \ref{h1thm}.
\vspace{.13in}
\noindent {\bf Proof of Theorem \ref{h1thm}}.\quad
The difference $w=u-v$ satisfies the equation
$$
\partial_t w=(a+i\nu) \Delta w + a\Delta v- (b+i\mu) (f(u)-f(v))- bf(v)
$$
where $f(u)=|u|^{2\sigma}u$.
Let $\phi_R(x)$ be the cutoff function as before and we find
$$
\partial_t\int \phi_{R}^{2}|\nabla w|^2
=2Re(a+i\nu)\int \phi_{R}^{2}\Delta(\nabla w) \nabla \bar{w}+2a Re
\int \phi_{R}^{2}\Delta (\nabla v) \nabla \bar{w}
$$
$$
-2Re(b+i\mu)\int \phi_{R}^{2}\nabla(f(u)-f(v))\nabla\bar{w}
-2b Re\int \phi_{R}^{2}\nabla f(v)\nabla \bar{w}
$$
We mark the four terms on the RHS as $I,II,III,IV$ and estimate
them separately.
$$
I=-2a\int \phi_{R}^{2}|\Delta w|^2 -4Re (a+i\nu)\int
\phi_{R}\Delta w (\nabla \phi_{R}\nabla \bar{w})
$$
$$
II= -2a Re\int \phi_{R}^{2}\Delta v \Delta \bar{w} -4a Re
\int \phi_{R} \Delta v (\nabla \phi_{R} \nabla \bar{w})
$$
Using Young's inequality to split the terms in $I$ and $II$ in the same
way as in the proof of Theorem \ref{main}, we obtain
$$
|I|+|II| \le -a \int \phi_{R}^{2}|\Delta w|^2
+(2a+\epsilon) \|\phi_{R}\Delta v\|_{L^2}^{2}
$$
\begin{equation}\label{h1i+ii}
+ 8\left(\frac{a^2 +\nu^2}{a}+a^2 \epsilon^{-1}\right)
R^{-2}\|\nabla \phi\|_{L^\infty}^{2}\|\nabla w\|_{L^2}^{2}
\end{equation}
where $\epsilon>0$ is small. Integrating by parts,
$$
III=2Re(b+i\mu)\int \phi_{R}^{2}(f(u)-f(v))\Delta \bar{w}
$$
$$
+ 4Re
(b+i\mu)\int \phi_{R}(f(u)-f(v))\nabla\phi_{R}\nabla\bar{w}
$$
$$
IV=2b Re\int \phi_{R}^{2}f(v)\Delta\bar{w}
+ 4bRe\int \phi_{R}f(v)\nabla \phi_{R}\nabla\bar{w}
$$
Breaking the terms in $III$ and $IV$ we find that
$$
|III+IV|\le a\int \phi_{R}^{2}|\Delta w|^2
$$
$$
+2 a^{-1}(b^2+\mu^2)\int \phi_{R}^{2}|f(u)|^2
+2a^{-1}(2b^2+\mu^2)\int \phi_{R}^{2}|f(v)|^2
$$
\begin{equation}\label{h1sec}
+ 8|b+i\mu|
R^{-1}\|\nabla \phi\|_{L^\infty}\left(\int (|f(u)|^2+|f(v)|^2)\right)
^{\frac{1}{2}}\left(\int |\nabla w|^2\right)^{\frac{1}{2}}
\end{equation}
\vspace{.14in}
Collecting the estimates (\ref{h1i+ii}), (\ref{h1sec}), integrating in $t$
and letting $R\to \infty$ and $\epsilon\to 0$,
$$
\int |\nabla (u(t)-v(t))|^{2}\le \int |\nabla (u_0-v_0)|^2
+ 2a\int_{0}^{t}\|\Delta v(t')\|_{L^2}^{2} dt'
$$
\begin{equation}\label{trunk}
+ 4a^{-1}(b^2+\mu^2)\int_{0}^{t}\|v(t')\|^{4\sigma+2}_{L^{4\sigma+2}}dt'
+ 2a^{-1}(b^2+\mu^2)\int_{0}^{t}\|u(t')\|^{4\sigma+2}_{L^{4\sigma+2}}dt'
\end{equation}
\vspace{.12in}
As shown in the proof of Theorem \ref{2th},
\begin{equation}\label{leaf}
\|\Delta v\|_{L^2}^{2}\le {\cal A}(v_0),\qquad
\|v\|^{4\sigma+2}_{L^{4\sigma+2}}\le {\cal D}(v_0)
\end{equation}
with ${\cal A}(v_0)$ and ${\cal D}(v_0)$ given in (\ref{biga}),(\ref{big
b}).
The estimate for $\int |u|^{4\sigma+2}$ is already given in the proof of
Proposition \ref{prop3} :
$$
\int |u|^{4\sigma+2} \le C(\sigma) \|\Delta u\|_{L^2}^{n\sigma}
\|u\|_{L^2}^{2-(n-4)\sigma}
$$
where $C$ depends on $\sigma$ only. Therefore,
\begin{equation}\label{branch}
\int_{0}^{t}\|u(t')\|_{L^{4\sigma+2}}^{4\sigma+2}dt'
\le C(\sigma)\|u_0\|_{L^2}^{2-(n-4)\sigma} \left(\int_{0}^{t}\|\Delta
u(t')\|_{L^2}^{2}dt'\right)^{\frac{n\sigma}{2}}t^{1-\frac{n\sigma}{2}}
\end{equation}
It then follows from Proposition \ref{prop3} that if $n\sigma=2$ and $\|u_0
\|_{L^2}$ is small
$$
\int_{0}^{t}\|u(t')\|_{L^{4\sigma+2}}^{4\sigma+2}dt'
\le C(\sigma)\|u_0\|_{L^2}^{2-(n-4)\sigma}\|\nabla u_0\|_{L^2}^{2}
$$
\begin{equation}\label{sp1}
\hspace{.2in} \times \left(a-C(\sigma)a^{-1}(b^2+\mu^2)\|u_0\|_{L^2}
^{4\sigma+2-n\sigma}\right)^{-1}
\end{equation}
and if $n\sigma<2$
$$
\int_{0}^{t}\|u(t')\|_{L^{4\sigma+2}}^{4\sigma+2}dt'
\le \max\left\{ C(\sigma)a^{-\frac{n\sigma}{2}}\|\nabla u_0\|_{L^2}^{n\sigma}
\|u_0\|_{L^2}^{2-(n-4)\sigma}t^{1-\frac{n\sigma}{2}}\right. ,
$$
\begin{equation}\label{sp2}
C(\sigma)\left(a^{-2}(b^2+\mu^2)\right)^{\frac{n\sigma}{2-n\sigma}}
\|u_0\|_{L^2}^{2+\frac{8\sigma}{2-n\sigma}}t \left\}\right.
\end{equation}
The bounds in (\ref{sp1}) and (\ref{sp2}) will be denoted by the
notation ${\cal L}(a,b,v_0,t)$. Clearly if $b^2+\mu^2=O(a^2)$ for small
$a$, ${\cal L}(a,b,v_0,t)$ is of order $a^{-\frac{n\sigma}{2}}$.
\vspace{.13in}
The proof of this theorem is completed after inserting
(\ref{leaf}), (\ref{branch}) with (\ref{sp1}) and (\ref{sp2})
into (\ref{trunk}). \qquad $\Box$
\vspace{.3in}
\noindent{\bf Acknowledgements}
\vspace{.15in}
I thank Professor T. Spencer for his advice and Dr. W. Wang for discussions.
\newpage
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