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\begin{document}
\title[Global solutions of the equations of elastodynamics]
{Global solutions of the equations of elastodynamics for
incompressible materials}

%\markboth{DAVID G. EBIN}{GLOBAL SOLUTIONS OF THE EQUATIONS OF ELASTODYNAMICS}
\author{David G. Ebin}
\thanks{Partially supported by NSF grant DMS 9304403}
\address{SUNY at Stony Brook, NY 11794-3651}
\email{ebin@math.sunysb.edu}
\date{December 29, 1995}
\commby{James Glimm}
\subjclass{Primary 35L70, 35Q72, 73C50, 73D35}
\keywords{Non-linear hyperbolic, elastodynamics, incompressible,
global existence}

\begin{abstract}
The equations of the dynamics of an elastic material are a 
  non-linear hyperbolic system whose unknowns are functions of space and
  time.  If the material is incompressible, the system has an additional 
  pseudo-differential term.  We prove that such a system has global (classical)
  solutions if the initial data are small.  This contrasts with the case of
  compressible materials for which F. John has shown that such
  solutions may not exist even for arbitrarily small data.
\end{abstract}
\maketitle

%\newpage
\section {Introduction}
\setcounter{equation}{0}
%\newtheorem{theorem}{Theorem}
%\begin{equation}gin{theorem}
%\noindent
%{\bf Theorem:} {\sl
\begin{thmn} The initial value problem for the equations of motion of an
incompressible hyperelastic homogeneous isotropic material has
classical solutions for all time, if the initial displacement and
velocity are small. %}
\end{thmn} %\end{theorem}

The displacement and velocity must be small in the norm of equation
(5.4) below, where $k$ must be at least $9$ for the displacement and at
least $8$ for the velocity.  This norm defines a Hilbert space which
includes all functions which
when differentiated up to $k$ times and multiplied by polynomials of degree
no more than $k$ are in $L^2.$  In particular it includes all smooth
functions of compact support and even all rapidly decreasing
functions.  Thus the theorem applies to all small rapidly decreasing data.

The mathematical meaning of the physical properties of our
material is described in Section 3, where the equations of motion are derived.

Our theorem was proved for the special case of neo-Hookean materials
in \cite{eb}, but the present more general result is important on two counts.
From the standpoint of elasticity it is important because the
neo-Hookean model is not appropriate in most cases.
From the mathematical perspective it is significant
because the non-linearities in the neo-Hookean case are due solely to the
incompressibility, while in the more general case they are a consequence of the
basic constitutive laws as well.  Indeed, in our study of the neo-Hookean case
we were able to rewrite the equations so that the non-linear terms were reduced
to lower order.  Here we are forced to proceed by a different method.

It is interesting to note that this more general theorem is false
for compressible materials as is shown by the counter-example of F. John
\cite{j1}.  Furthermore, our theorem does not apply to motions which are
constant in any one dimension (a case in which shocks may appear)
because the data for such motions are not finite in the norm of (5.4).

For small time, solutions of the above initial value problem are known
\cite{ebsa}, as are solutions of corrresponding initial-boundary value
problems  \cite{hrre}, \cite{ebsi1}, \cite{ebsi2}.

We thank Professors D. Christodoulou, S. Simanca, and S. Sogge for several very
useful conversations.
\section{Background}
\setcounter{equation}{0}
Over the past twenty years there have been a number of papers
devoted to the initial value problem for non-linear wave equations with small
initial data.  Klainerman \cite{k1} showed that solutions exist for all time
if
the number of space dimensions is at least four.  John, on the other hand,
\cite{j2}, \cite{j3}
found, in dimension three, examples with arbitrarily small data which
develop a singularity in finite time.  Then Klainerman \cite{k3} discovered
 a condition on the non-linearities (Klainerman's null condition) which
guarantees solutions in the large for the three-dimensional case as well.

These mathematical ideas have been applied in two areas, relativity
theory and elastodynamics.  In relativity theory one has Einstein equations
which can be written as a system of non-linear wave equations on
$\R^1 \times \R^3.$
One can
consider the initial value problem for this system by prescribing data on
$0 \times \R^3.$
  If the data are zero, one
 gets the trivial solution which is Minkowski space.

Christodoulou and Klainerman \cite{ck} studied this initial value problem
and showed that it satisfies the null condition.  Thus solutions to the problem
with small data exist in the large and are small for all time.  Hence Minkowski
 space is globally stable in the sense that small data give a solution which is
globally close to that of Minkowski space.

We will pursue the second area
 of application --- elastodynamics.  This is a natural area for the following
reasons.  Many problems in elasticity involve small deformation.  Such problems
are sometimes modeled by the equations of elasticity linearized about the
state of no deformation, the solutions of such equations being called
``infinitesimal'' elastic waves.  A more accurate model uses the full
non-linear equations (whose solutions might be called small elastic waves),
but this model is
only good for the length of time for which solutions exist.  Thus it becomes
important to compute this time.  One also wants to know whether the
small elastic waves remain small over long times and whether they are
close to the infinitesimal elastic waves.  Our techniques provide
positive answers to all of these questions in the incompressible case.

Our equations do not satisfy Klainerman's null condition, but we
still use techniques similar to his \cite{k3} to prove existence in the large.
In fact we construct an ``energy'' function $E(t)$ from $L^2$-norms
of the displacement, velocity, and their derivatives and show
that it obeys a certain differential inequality.  Combining this
inequality with Klainerman's inequality \cite{k2}, we can show that if $E(0)$
is
sufficiently small, then it remains finite for all positive $t$, and from
this it follows that our equations have global solutions.

It is interesting to contrast our study with that of John (\cite{j1}).  He
shows that for compressible materials there are waves with arbitrarily small
initial data which blow up (develop shocks) in finite time.  Thus one
has small waves which do not remain small and which diverge from infinitesimal
waves with the same data.

\section{Physical assumptions and derivation of equations}
\setcounter{equation}{0}
We consider a material that fills all of three-space
 and let $\eta(t): \R^3 \rightarrow \R^3 $  denote
the map that takes each material point from
its original position $x$ to the position $\eta(t)(x)$, which it
reaches at time $t$.
Thus the curve of
maps $t \mapsto \eta(t)$ describes the entire motion.  The velocity of this
material point is $ \dot{\eta}(t)(x)$ \footnote{``$\dot{ %\mbox{ }
}$'' means time derivative.}
and the kinetic energy of the motion at time $t$ is
\begin{equation}
 K(\dot{\eta}(t)) = \frac{1}{2} \int_{\R^3}|\dot{\eta}(t)(x)|^{2}dx.
\end{equation}

The hypothesis that our material is homogeneous and hyperelastic
means that the stresses on the material are derived from a potential energy
function $W(D\eta^{\tau}D\eta)$
 where $D\eta$ is the $3 \times 3$
matrix of spatial derivatives of $\eta(t): \R^3 \rightarrow \R^3 $
and $D\eta^{\tau}$ is its
transpose.  We shall denote $D\eta^{\tau}D\eta$ by $C$ below.
\footnote{$C$ is commonly known as the Green deformation tensor or
the right Cauchy-Green tensor \cite{mahu}.}
The additional hypothesis
that the material is isotropic implies that $W$ depends only on the similarity
class of $C$, or equivalently, that $W(C)$ is a symmetric function of the
eigenvalues of $C.$  We shall assume that $W$ is smooth. The final
hypothesis, incompressibility, is expressed
mathematically as the statement that $ %{\rm 
\mathrm{det} (D\eta)$,
 which we shall call $J(\eta)$, is identically one.

With these hypotheses we can derive the equations of motion by using
 Hamilton's principle.  We denote by
\begin{equation} V = \int_{\R^3}W(D\eta^{\tau}(t)(x)D\eta(t)(x))dx \end{equation}
 the potential energy of the configuration
$\eta(t)$ and look for stationary curves $\eta(t)$ of the action:
\begin{equation} {\cal L} = \int_0^T (K-V)\,dt. \end{equation}

We consider a variation $\eta(t, s)$ of $\eta(t) = \eta(t,0) $
where $\eta(0,s) = \eta(0)$ and $\eta(T,s) = \eta(T),$ and we
require that 
\[ %$$
\frac{d}{ds}{\cal L}|_{s=0} = 0,\] %$$
where
\begin{equation} {\cal L}(s) = \int_0^T 
\left(K(\dot{\eta}(t,s))-V(\eta(t,s))\right)\,dt.
\end{equation}

Computing $\frac{d}{ds}{\cal L}$ and
performing the usual integration by parts we find that
\begin{equation} \frac{d}{ds}{\cal L}|_{s=0} =
-\int_0^{T} \int_{\R^3}  %\<
\langle \ddot{\eta}(t), \sigma(t) %\>
\rangle  -
 %\<
\langle \Div \frac{\partial W}{\partial D\eta}, \sigma(t) %\>
\rangle dt, \end{equation}
where $\sigma(t) = \partial_{s} \eta(t,s)|_{s=0}$,
$\frac{\partial W}{\partial D\eta} $
is the matrix of
derivatives of $W$ with respect to each $\frac{\partial \eta^i}{\p
x^j}$, and
\begin{equation} \Div \frac{\p W}{\p D\eta} =
\sum_{j=1}^{3} \frac{\p}{\p x^j} \left(\frac{\p W}{\p \frac{\p
\eta^i}{\p x^j}}\right). \end{equation}
Thus the
requirement that $\eta(t)$ be stationary tells us that at any time $t,$
%\mbox{
$\ddot{\eta} - \Div \frac{\p W}{\p D\eta} $ %}
 is perpendicular in
$ L^2(\R^3, \R^3)$ to $\sigma(t).$

One might think that
$\sigma(t):\R^3 \rightarrow \R^3$
is arbitrary so one gets the equation:
\begin{equation} \ddot{\eta} - \Div \frac{\p W}{\p D\eta} = 0. \end{equation}
This is correct for compressible materials, but in our case the requirement
 $J(\eta) \equiv 1$
causes complications.

We note that \[ %$$
\eta(t,s) = \eta(t) + s\sigma(t) + o(s),\] %$$
 which we rewrite as
\begin{equation} \eta(t,s) 
= (id + s\sigma(t) \circ \eta(t)^{-1}) \circ \eta(t) + o(s), \end{equation}
where $ id: {\R^{3}}\longrightarrow {\R^{3}}$ is identity.

Taking Jacobian determinants we get
\begin{equation} J(\eta(t,s)) = \det (Id +s D(\sigma(t) \circ \eta(t)^{-1}))
\circ\eta(t) + o(s), \end{equation}
 where $Id$ means the identity matrix.  Thus the requirement
$J(\eta) \equiv 1$ tells us that
\begin{equation}
1 \equiv J(\eta(t,s)) \equiv \det (Id + s D (\sigma(t) \circ \eta(t)^{-1})
\circ
\eta(t) + o(s) \end{equation}
 and hence that
\begin{equation} 0 = \frac{\partial}{\partial s} J(\eta(t,s))|_{s=0} =
\tr (D(\sigma(t) \circ\eta^{-1} (t))) \circ \eta(t), \end{equation}
where ``$\tr$'' means trace of a matrix.

Equation (3.11) is a restriction on $\sigma(t)$, so we find that
$\ddot{\eta} - \Div \frac{\partial W}{\partial D\eta}$
needs to be perpendicular only to all $\sigma(t)$ satisfying this restriction.

We note that the right side of (3.11) can be written as
\[ %$$
(\div(\sigma(t) \circ \eta(t)^{-1})) \circ \eta(t), \] %$$
 where ``div'' means
divergence.  We will denote this by $ %{\rm 
\mathrm{div}_{\eta(t)}\, \sigma(t).$
In fact if $L$ is any differential operator and $\eta:\R^3 \rightarrow
\R^3 $ is a diffeomorphism, we will use $L_{\eta}f $ to mean
$L(f\circ \eta^{-1})\circ\eta.$
We have need for such $L_{\eta}$ for the following reason: The
functions which we study are usually given as functions of time $t$
and material coordinates $x$.  However, in calculations concerning
incompressibility, we must consider derivatives of such functions with
respect to spatial coordinates $\eta(t)(x).$  If $L$ is an operator involving
differentiations with respect to $x,$ $L_{\eta}$ will have the same
differentiations with respect to $\eta(x)$ and thus will be the desired
operator.

To determine the significance of the restriction on $\sigma(t),$
we note that $L^2({\R^{3}}, {\R^{3}})$ can be decomposed as
\begin{equation} L^2({\R^{3}}, {\R^{3}}) = \div^{-1} (0) 
\oplus \nabla {\cal F}, \end{equation}
where the first summand is the space of divergence-free vector fields
and the second is the space of gradients of functions on $\R^3$.
The summands are clearly perpendicular with respect to $L^2$.
A vector field $v$ is decomposed as $w + \nabla f$, where $f$ solves
the equation
\begin{equation} \Delta f =  \div v  \end{equation}
and $w$ is defined to be $v - \nabla f$.

There is of course a ``sub-$\eta$'' version of this decomposition --- namely:
\begin{equation} L^2({\R^{3}}, {\R^{3}}) =  \diveta^{-1}(0) \oplus
\nabla_{\eta}\, {\cal F}. \end{equation}
If $J(\eta) \equiv 1,$ this decomposition is also orthogonal with
respect to $L^2$.

Thus to say that $\ddot{\eta} - \Div \frac{\partial W}{\partial D\eta}$
is orthogonal to all $\sigma$ satisfying
$\diveta(\sigma) = 0 $ is the same as saying:
\begin{equation} \ddot{\eta} -
\Div \frac{\partial W}{\partial D \eta} = \nabla_{\eta} q \end{equation}
for some function $q(t):\R^{3}\rightarrow \R$. Equation (3.15) with the
accompanying condition $J(\eta(t)) \equiv 1 $ is the equation of
elastodynamics for incompressible materials.

\section{ Analysis of equations}
\setcounter{equation}{0}
Since we are interested in small deformations, we want to consider motions
$\eta(t)$ which are close to $id: \R^{3} \rightarrow \R^{3}$.
For this reason it is natural for us to work with the displacement $u$
which we define to be $\eta - id$.  Clearly $\dot{u} = \dot{\eta}$,
$\ddot{u} = \ddot{\eta} ,$ $Du = D\eta - id,$ and $D^{2}u =
D^{2}\eta.$ Also we can write (3.15) as an equation in $u.$

Furthermore, since $W(C)$ is a symmetric function of the eigenvalues of $C$,
it depends only on their elementary symmetric functions, $\tr(C),$
$\tr(C^{2})$, and $\tr (C^{3})$.  But since $J(\eta) \equiv 1$, we know
that $\det(C)\equiv 1 $, and since the symmetric functions obey the equation:
\[ %$$
6 \det (C) = \tr(C)^3 - 3\tr(C) \tr(C^2) +2 \tr(C^3),\] %$$
we find that
$\tr(C^3)$ is a function of $\tr(C),$ and $\tr(C^2)$.  Thus we can
consider $W(C)$ as a function of the two variables $\tr(C)$ and $\tr(C^2)$.

We now proceed to find $ \Div \frac{\partial W}{\partial D\eta}$
in terms of $u.$ Direct computation shows that
$\frac{\partial \tr(C)}{\partial D\eta} = 2 D\eta$ and
$\frac{\partial \tr(C^2)}{\partial D\eta} =
4 D\eta D\eta^{\tau} D\eta$.
Thus letting $a = \tr(C)$ and $b = \tr(C^2)$ we find that
\begin{equation} \frac{\partial W}{\partial D\eta} = 2 \partial_a W D\eta +
4 \partial _b W D\eta D\eta^{\tau} D\eta. \end{equation}

We apply $\Div$ to this expression and keep in mind that since $J(\eta)
\equiv 1$,
\begin{equation} 1+ \div u + Q(Du) = 1, \end{equation}
where $Q(Du)$ is a polynomial in $Du$ of order $2$.\footnote{Throughout the paper $Q$ will stand for any function of
order at least two; that is, any function whose value and first
derivatives vanish at the origin.}
Then doing a lengthly but routine
computation, we find that \footnote{Repeated indices are summed throughout.}
\begin{align*} %eqnarray*} 
\left( \Div \frac{\p W}{\p D\eta}\right)^i & = %&
 c_0 \Delta u^i
+ c_1 \nabla^i
\div(u)+ c_2 \nabla^i \mid Du \mid ^{2} + c_3(\partial_j u^k +
\partial_k u^j) \partial _j \partial _k u^i  \\
& %&
 + c_4 (\partial_i u^j +
\partial_j u^i) \Delta u^j + Q(Du)^{ij}_{k\l} \p_k \p_{\l} u^j,  
\end{align*} %eqnarray*}
where $c_0, \ldots , c_4$ are positive constants
\footnote{$c_n$ will be a sequence of
positive constants throughout the paper.}  and each
$Q(Du)^{ij}_{k\l}$ again stands for
a function in $Du$ of order at least two.  For convenience we scale
units so that $c_0=1.$

Thus we derive:
\begin{align*} %eqnarray*} 
\ddot{u}^i &= %&
 \Delta u^i  + c_1 \nabla^i
\div(u)+ c_2 \nabla^i \mid Du \mid ^{2} + c_3(\partial_j u^k +
\partial_k u^j) \partial _j \partial _k u^i  \\
& %&
 + c_4 (\partial_i u^j +
\partial_j u^i) \Delta u^j + Q(Du)^{ij}_{k\l} \p_k\p_{\l} u^j
+ \nabla_{\eta}^i q.
\end{align*} %eqnarray*}
However, since $ \div u = Q(Du) $, this simplifies to

\begin{align} %eqnarray}
\ddot{u}^i &= %&
 \Delta u^i  + \nabla ^i Q(Du) +
c_3 (\partial _j u^k + \partial_k u^j) \partial _j \partial _k u^i  \\
& %&
 + c_4 (\partial_i u^j + \partial_j u^i) \Delta u^j
+ Q(Du)^{ij}_{k\l} \p_k\p_{\l} u^j + \nabla_{\eta}^i q.  \notag
\end{align} %eqnarray}
Now we must investigate $\nabla_{\eta} q$, which is the second summand
of
$\ddot{\eta} - \Div \frac{\partial W}{\partial D\eta}$ in the decomposition
(3.13).  $q$ satisfies:

\[ %$$
 \Delta_{\eta} q = %{\rm 
\mathrm{div}_{\eta}\left(\ddot{\eta} -
\Div \frac {\partial W}{\partial D\eta} \right). \] %$$

Taking a time derivative of $J(\eta) \equiv 1$, we find that
\begin{equation} \diveta\dot{\eta} = 0. \end{equation}  
Taking a second time derivative we get
\[ %$$
 \diveta\ddot{\eta} = [v \cdot \nabla, \div]_{\eta} \dot{\eta}, \] %$$
where $v$ is defined as $\dot{\eta} \circ \eta^{-1} $ and $[\:,\:]$ means
commutator.
Thus $q$ is the solution of
\[ %$$
\Delta_{\eta} q = \left([v \cdot \nabla, \div]_{\eta} \dot{\eta}
- \diveta \Div \frac {\partial W}{\partial D\eta}\right),\] %$$
so $q$ depends only on $\eta$ and $\dot{\eta}$ or equivalently on
$(u, \dot{u}),$ but not on $\ddot{\eta}$.
Hence the right side of (4.3)
involves only spatial derivatives of $u$ and the first time derivative
in the gradient term $\nabla_{\eta} q .$
Combining the gradient terms we define $F$ to be
$ \n Q(Du) + \nabla_{\eta} q$ so (4.3) becomes:

\begin{align} %eqnarray} 
\ddot{u}^i & = %&
 \Delta u^i +
c_3 (\partial_j u^k + \partial_k u^j) \partial_j \partial_k u^i\\
& %&
 + c_4 (\partial_i u^j + \partial_j u^i) \Delta u^j
+ Q(Du)^{ij}_{k\l} \p_k\p_{\l} u^j
+ F^i. \notag \end{align} %eqnarray}

Now we proceed to discuss the initial data for our equation.  $\eta(0)$
should be a diffeomorphism of $\R^3$ which is near the identity and has
Jacobian identically one.  Thus $u(0) = \eta(0) - id $ is small and
satisfies equation (4.2).  Similarly $\dot{\eta}(0) = \dot{u}(0) $ is
small and it satisfies equation (4.4).

We shall show that if $ \|u(0)\| _{LS,9} $ and $\| \dot{u}(0) \|
_{LS,8}$ \footnote{These norms are defined in equation (5.4) below.}
are sufficiently small, then equation
(4.5) has a solution for all $t$.

\section{Energy estimates -- outline of proof}
\setcounter{equation}{0}
We have shown (with Saxton) in \cite{ebsa} that our equation (3.14) (or
equivalently (4.5)) has a unique solution on some time interval $[0,T)$,
where $T \leq \infty$ depends on the initial data.  Furthermore, if $T$
is finite and maximal, then the pair $(u(t), \dot{u}(t))$ must become
arbitrarily large as $t \rightarrow T$.  One can prove that if the
initial data are small, then $(u(t), \dot{u}(t))$ do not get large and
thus $T = \infty ,$ or the solution exists for all time.  We shall now
outline this proof.

Given $v(t,x)$ an $\R^3$-valued function on $\R \times \R^3$ we
(following Klainerman \cite{k3}) define an energy function:
\begin{equation*} 
%eqnarray*} 
E^0(t)(v)=\int_{\R^3}\left\{ \frac{1}{2}(1+t^2+|x|^2)(|\n v|^2 +
|\dot{v}|^2) + 2tx^i %\<
\langle \p _i v, \dot{v} %\>
\rangle + 2t  %\<
\langle \dot{v}, v %\>
\rangle -|v^2|\right\} dx.
\end{equation*} %eqnarray*}

If $ \square v = f $, a direct computation shows that
\begin{equation} \frac{d}{dt} E^0(t)(v) = \int_{\R^3}  %\<
\langle  Lv, f %\>
\rangle dx, \end{equation}
where $Lv = (1 + t^2 + |x|^2 )\dot{v} + 2tx^i \p_i v +2tv. $

Furthermore, the energy $E^0 (t)$ is equivalent to a certain $L^2$-norm
of $v$ and its derivatives, which we now define.

Let $\Gamma_0 = \p_t $ and let $\Gamma_i = \p_i $ on $\R \times \R^3,
i=1,2,3$.

Let $\Gamma_4, \Gamma_5, \Gamma_6$ be the infinitesimal rotations
$\Gamma_{3+k} = x^i \p_j - x^j \p_i$, where $\{ijk\}$ are cyclic permutations
of $\{123\}.$  Also let $\Gamma_7, \Gamma_8, \Gamma_9$ be the infinitesimal
Lorentz transformations $\Gamma_{i+6} = t\p_i + x^i \p_t.$  Finally let
$\Gamma_{10} = t\p_t + x^i \p_i.$  Then we define the first Lorentz-Sobolev
norm of $v$ by
\begin{equation} \|v\|^2_{LS,1} = \int_{\R^3} \left(|v|^2 +
\sum_{\l=0}^{\l=10} |\Gamma_{\l} v|^2 \right)dx.
\end{equation}
In \cite{k3} it is shown that there are positive constants $c_{\l}$
and $c_u$ such that
\begin{equation} c_{\l} \| v\|^2 _{LS,1} 
\leq E^0 (t)(v) \leq c_u \|v \|^2 _{LS,1}.
\end{equation}
Higher Lorentz-Sobolev norms can be defined by
\begin{equation} \|v\|^2 _{LS,k} = \sum_{|\alpha| 
\leq k} \int_{\R^3} |\Gamma^{\alpha} v|^2,
\end{equation}
where $\alpha$ is a multi-index $(\alpha_1, \ldots, \alpha_{10})$ and
$\Gamma^{\alpha} = \prod_{i=0}^{10} \Gamma_i^{\alpha_i}. $  Similarly we
define higher energy functions:
\begin{equation} E^0_s(t)(v) = \sum_{|\alpha| 
\leq s} E^0(t)(\Gamma^{\alpha} v). \end{equation}

The equation (4.5) for $u$ is a wave equation with additional non-linear
terms, so we modify $E^0(t) $ accordingly.
We define 
\begin{multline*} %eqnarray*} 
E(t)(v) = %&
 E^0 (t)(v) + \int_{\R^3} \frac{1}{2} (1+t^2 +|x|^2)
\Big( c_3 (\p_i u^j + \p_j u^i) \p_i v^k \p_j v^k  \\
 %&
 + c_4 (\p_{\l} u^k + \p_k u^{\l}) \p_i v^k \p_i v^{\l} +
Q(Du)_{i\l}^{jk} \p_j v^i \p_k v^{\l} \Big)\ dx
\end{multline*} %eqnarray*}
so that if $v$ satisfies
\begin{equation*} %eqnarray*} 
\ddot{v}^i = %&
 \Delta v^i + c_3(\p_j u^k + \p_k u^j) \p_j \p_k v^i 
%&
 + c_4 (\p_i u^j + \p_j u^i) \Delta v^j + Q(Du)^{ij}_{kl}
\p_k \p_{\l} v^j
+ F, \end{equation*} %eqnarray*}
then
\begin{equation} \frac{d}{dt}E(t)(v) = \int_{\R^3}  %\<
\langle Lv,F %\>
\rangle + I + II + III, \end{equation}
where:
\[ %$$
 I  =  \int_{\R^3} (1 + t^2 + |x|^2) c_3 (1/2(\p_i \dot{u}^j
+ \p_j \dot{u}^i) \p_j v^i \p_k v^i
- (\p_j^2 u^k + \p_j \p_k u^j)
\p_k v^i \dot{v} ^i) dx, \] %$$
\[ %$$
 II = \int_{\R^3} (1 + t^2 + |x|^2) c_4 (1/2(\p_i \dot{u}^j
+\p_j \dot{u}^i) \p_k v^i \p_k v^j  - (\p_k \p_i u^j + \p_k \p_j u^i)
\p_k v^i \dot{v}^j) dx,
\] %$$
and
\[ %$$
 III = \int_{\R^3} (1 + t^2 + |x|^2)( 1/2  %\<
\langle Q(Du\dot{)}Dv, Dv %\>
\rangle
-\p_k Q(Du)_{i\l}^{jk} \p_j v^i \dot{v} ^{\l}) dx. \] %$$

Clearly $I$ and $II$ are bounded by a constant times the sup-norms
of $D^2 u$ and $ D\dot{u}$ times $E(t)(v).$
Also $III$ is bounded by a constant times the square of the sup-norms
times $E(t)(v).$  To estimate higher derivatives we introduce
\[ %$$
 E_s(t)(v) =  \sum_{|\alpha| \leq s} E(t)(\Gamma^{\alpha} v). \] %$$
In order to estimate the sup-norms above, we use Klainerman's inequality:
\begin{equation} \sup_{x \varepsilon \R^3} \{  w(t,x) \} \leq \frac{c_5}{1+t}
 \|w\|_{LS,2}.
\end{equation}

Also we note that one can get an improved estimate for derivatives of a
function in light-like directions.  Let $\p_r = \frac{1}{|x|} x^i \p_i $
denote differentiation in the (purely spatial) radial direction and let
$\p_{r^\perp}$ denote a spatial derivative in a perpendicular direction
$r^{\perp}.$

Then $\p_r + \p_t $ and $\{\p_{r^{\perp}}\}$ span the derivatives in the
light-like directions.  We fix a point $(t,x)$ in $\R^1 \times \R^3 $
and consider light-like derivatives computed at this point.  Clearly
we can pick spatial coordinates so that $x=(0,0,|x|).$  Then $\p_r =
\p_3 $ so
\begin{equation} \p_t + \p_r = \p_t + \p_3 = \frac{1}{t + |x|}(\Gamma_{10} +
\Gamma_7). \end{equation}

Without loss of generality, we can assume that $\p_r^{\perp} =
\p_1.$  Then
\begin{equation} \p_{r^{\perp}} = \frac{1}{t+|x|} (\Gamma_5 + \Gamma_7)
\end{equation}

From (5.8) and (5.9) we find that Klainerman's inequality can be
strengthened for light-like derivatives.  If $\p_{\l}$ is such a
derivative, we get
\begin{equation}
\sup_{x \epsilon \R^3} \|\p_{\l} w(t,x)\| \leq \frac{c_6}{(1+t)^2}
\|w\|_{LS,3}. \end{equation}

There is a similar inequality for the sup-norm.  Letting
\[ %$$
 |||w|||_{LS,k} = \sup_{x \epsilon \R^3}
\{\sum_{\alpha \leq k } | \Gamma^{\alpha} w(t,x)|\} \] %$$
we find that
\begin{equation} \sup_{x\epsilon \R^3} \|\p_{\l} w(t,x)\| \leq \frac{c_7}{1+t}
|||w|||_{LS,1}.
\end{equation}

The final ingredient needed for our result is an estimate for the
inhomogeneous wave
equation: $\square w = F, $ with  zero initial data.

From H\"{o}rmander \cite{h}, Section 6, we have:
\begin{equation} |w(t,x)| \leq \frac{c_8}{1+t+ |x|} \int_0^t \int_{\R^3}
\sum_{|\alpha | \leq 2} (\Gamma^{\alpha} F(s,y))/(1+s+|y|)\,dy\,ds. 
\end{equation}

Now that we have assembled the items needed for our theorem, we proceed
to outline the proof.  Our scheme follows that of
H\"{o}rmander \cite{h}, Section 6.

First we note that if $u_1$ is the solution of
the wave equation with the same initial data as $u$, then $u_1$
satisfies
\begin{equation}   
|||u_{1}(t)|||_{LS,4} \leq \frac{c_9 \varepsilon}{1+t}. \end{equation}
This is because the initial data are of size
$\varepsilon.$
Now we write $u$, the solution of (4.5), as $u_1 + u_2$, where $u_1$ is as
above and $u_2$ has zero initial data.

Thus
\begin{align} %eqnarray} 
\square u_2^i &= %&
 c_3 (\partial_j u^k +
\partial_k u^j) \partial_j \partial_k u^i +
c_4 (\partial_i u^j + \partial_j u^i) \Delta u^j\\
& %&
 + Q(Du)^{ij}_{k\l} \p_k\p_{\l} u^j
+ F .\notag \end{align} %eqnarray}

We write the terms on the right side of (5.14) as $F_1 + F_2 + F_3$, where
$F_1$ consists of terms that are quadratic in $u$ and its derivatives,
$F_2$ consists of terms that are at least cubic, and $F_3 = F$.

The terms of $F_1$ have the form $ (\p_j u^k + \p_k u^j)\p_j\p_k u^i$ or
$(\p_i u^j + \p_j u^i)\Delta u^j $.  Each of these terms contains at
least one light-like derivative (actually a $\p_{r^{\perp}}$ derivative)
except the terms of the form $\p_r u^j
\p_r\p_r u^j.$  However, if we subtract from $F_1$ the expression $\frac{1}{2}
\n (\n u^j \n u^j),$ we eliminate terms of this form.  We alter
$F_1$ in this way and add the same term to $F_3$, so as to keep
equation (5.14) unchanged.  Thus all the terms of $F_1$ are of the form
$ \frac{1}{t+|x|}Du\Gamma Du$ or
$ \frac{1}{t+|x|}\Gamma u D^2 u$.

The terms of $F_2$ have the form $(Du)^m D^2 u$, where $m$ is at least
2.  The terms of $F_3$, being gradients, can be bounded by their
divergences,  which have the same form as the terms of $F_1$ or $F_2$.

Thus applying $\Gamma^{\alpha}$ to (5.14)  (with $|\alpha|\leq 4$) and
using the inequality (5.12), we find that
 \begin{equation} |||u_2 |||_{LS,4} \leq c_{10} E_{8}/(1+t)^2. \end{equation}
Thus we get
 \begin{equation} |||u |||_{LS,4} \leq \frac{c_{11}}{1+t}
\left(\varepsilon+\frac{E_8 (t)}{1+t}\right). \end{equation}


On the other hand, by taking a time derivative of $E_8(t) \; (= E_8(t)(u))$
and doing the
usual energy norm type calculations, we get
\begin{equation*} %eqnarray*} 
\frac{d}{dt} E_8(t) \leq %&
 c_{12} |||u(t)|||_{SL,4} E_8 (t)  
\leq %&
 c_{13} \left( \frac{\varepsilon}{1+t} + \frac{E_8 (t)}{(1+t)^2}\right)
E_8(t).
\end{equation*} %eqnarray*}
Integrating this inequality we find that for small $\varepsilon$
\begin{equation} E_8 (t) \leq (1+t)^{c_{14} \varepsilon} E_8(0). \end{equation}

Thus $E_8 (t)$ remains bounded on any finite time interval.  Furthermore, 
using
(5.7) and noting that $E(0)$ is of size $\varepsilon^2$ we see that
\begin{equation} |||u(t)|||_{LS,4} \leq c_{15} \varepsilon (1+t)^{c_{14} 
\varepsilon -1}, \end{equation}
so that if $\varepsilon$ is small, then $|||u(t)|||_{LS,4}$ actually decays for
large time.  Thus (4.5) has a solution for all time, which remains
small.

%\bigskip
%\bigskip

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\nopagebreak

\end{document}