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\begin{document}
\Large
\vskip 1in
\bc
{\bf \LARGE  Global well-posedness and blowup \\
of one-dimensional wave maps below \\ 
the energy norm}\\
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
Mark Keel, Terence Tao \\
\bigskip
{\it Mathematics Department \\ \bigskip UCLA \\ }  
\bigskip
\ 
\bigskip
{\tt www.math.ucla.edu/$\sim$tao/preprints } 
\ec

\page

\noindent
{\LARGE
The purpose of this talk is to:
}
\bigskip
\bi
 \item State the well-posedness and blowup problem for wave-maps

 \item Prove global well-posedness for large data and blowup for
small data in one dimension and in various norms.  In particular:

 \item Global well-posedness for large $H^s$ data, $s > 3/4$, in the spirit
of some recent work of Bourgain.

 \item Ill-posedness in $\dot H^{1/2}$ or in any rougher space.
\ei

\page

\noindent
What is a wave map?

\bi
\item Let $M$ be a spacetime manifold of dimension $n+1$, and $N$ be
a Riemannian manifold.  For maps $\phi: M \to N$, we consider the energy 
functional
$$ L(\phi) = \int_M \langle\partial_\alpha \phi, \partial^\alpha \phi\rangle.$$
Critical points of this functional are called wave maps.
\item For simplicity we will specialize to the case $M = \R^{n+1}$ (Minkowski
space), and $N = S^{d-1} \subset \R^d$ (the sphere in $\R^d$).  The 
Euler-Lagrange equations for the above functional then become
$$ \Box \phi = - \phi (\partial^\alpha \phi \cdot \partial_\alpha \phi),$$
where $\phi$ is now thought of as a map from $\R^{n+1}$ to $\R^d$.

\item This equation is schematically of the form $\Box \phi = |\nabla \phi|^2$,
but exhibits some extra cancellation, as can be seen by writing in
space-time co-ordinates:
$$ \Box \phi = - \phi ( |\nabla \phi|^2 - |\phi_t|^2 ).$$
\item We consider the Cauchy problem for this equation, with initial data
$$ \phi(0,x) = f(x), \phi_t(0,x) = g(x),$$
where $f$ is in some Sobolev space $H^s$, $g$ is in the derivative space
$H^{s-1}$, and starts on the sphere:
$$|f(x)|^2 = 1, f(x) \cdot g(x) = 0.$$

\item It is easy to show that solutions which start on the sphere
will stay on the sphere as long as the solution has some mild
regularity.

\item We ask the question: for which Sobolev spaces $H^s$ is the
Cauchy problem well-posed?  (i.e. solutions exist in $H^s$, are unique,
and depend continuously on the data in $H^s$).  We can ask for
local well-posedness or global well-posedness.  

\item Scaling considerations suggest $s \geq n/2$ as a necessary
condition.  Thus a natural conjecture would be that one has local
well-posedness for $s \geq n/2$ and ill-posedness for $s < n/2$.
In particular, one expects local well-posedness in the energy
norm $H^1 \times L^2$ in two dimensions.  From energy conservation
one should also expect global well-posedness in this setting.
However this particular problem appears extremely difficult.
(The corresponding problem for harmonic maps is known, though, and is due
to Helein).

\ei
\page

\bigskip
\noindent

\bi

\item This problem (and close variants) has been studied by many authors
(Choquet-Bruhat, Christodolou, Gu, Grillakis, 
Klainerman-Machedon, Kovalyov, Selberg, Shatah, Sideris, Struwe, Tataru, 
Zadeh, ...).  Here are some prior results.

\item The Cauchy problem is locally well-posed in $H^s$ if $s > n/2$.
The critical case $s = n/2$ is extremely subtle, and is largely
open, although well-posedness is known if the Sobolev space
is replaced by a slightly smoother (but still critical) Besov space.

\item For small smooth data whose image lies near a geodesic, one has global
existence.  For large smooth radial data, one also has global
existence.

\item When $n=2$ and the data is rotationally-covariant, one has global
existence for large $\dot H^1$ data.  (The non-covariant case is still
open).

\item When $n=1$ then one has global existence for smooth (or $H^2$)
data.
\ei

\page

Known blowup results

\bi 
\item  In contrast to existence results, blowup or ill-posedness
results are scarce.
The conjecture is that one has examples of blow-up for all $s < n/2$.

\item  In three dimensions, Shatah and Talvidar-Zadeh have constructed
an example of a finite energy wave map which blows up in finite time.
Note that energy is supercritical in three dimensions.  The example
uses the fact that the sphere has positive curvature; things appear
to be slightly better behaved when the curvature is negative.

\item  Another sign of problems when $s \leq n/2$ is provided by

\ 

Proposition.  The Cauchy problem for the 2D wave-map cannot
be {\it analytically} well-posed in $\dot H^1$, when the
target manifold is the unit circle $S^1$.

\ 

Proof.  We have the explicit solution $\phi = e^{iu}$, where $u$
solves the free wave equation $\Box u = 0$, and $S^1$ is embedded in 
the complex plane.  In particular, we have
travelling wave solutions 
$$ \phi(t,x) = e^{i (g(x+t) - g(x-t))/2}$$
with initial data $\phi(0) = 1$, $\phi_t(0) = ig'$, and real
$\dot H^1$ functions $g$.

\item In order for the solution operator to be analytically well-posed
in $\dot H^1$, the exponentiation map $g \mapsto e^{ig}$ must be
an analytic map from $\dot H^1$ to $\dot H^1$.  In particular, every
component of the Taylor series
$$ e^{ig} = 1 + ig - \frac{g^2}{2!} + \ldots$$
must be in $\dot H^1$.   But $\dot H^1(\R^2)$ is not an algebra (it
barely fails to control the $L^\infty$ norm), so
this fails. $\square$

\item  This result can be generalized to show that one does not have
analytic dependence of the data whenever $s \leq n/2$, although it
does not actually show ill-posedness.  This implies that the
critical problem for Sobolev spaces
cannot be solved purely by iterative methods.  (Although
it may be possible to get around this by choosing a clever choice
of co-ordinates or frame bundle, or by transforming the equation
using some differential geometry).

\item On the other hand, there are critical spaces in which well-posedness
can be obtained by analytic methods.  The key seems to be that the space has
to be an algebra.  One example is the Besov space $\dot B^{n/2,1}_2$ for $n>1$.
For $n=1$ the space $L^{1,1}$ of functions with bounded variation is
another example.

\item We have no new results on the wave map in $\dot H^1$ in two dimensions.
However, for the corresponding critical problem in $\dot H^{1/2}$ in
one dimension, we do have ill-posedness.  (More on this later).

\ei

\page

The wave map in $\R^{1+1}$

\bi
 \item In one dimension the wave map equation becomes much simpler to study
(although there are still open problems!)  If we introduce the null
co-ordinate system
$$ u = x+t, v = x-t$$
then the wave map equation becomes
$$ \phi_{uv} = -\phi (\phi_u \cdot \phi_v).$$
with Cauchy data on the line $u=v$.

Some other features of the one-dimensional problem:

\item The wave map equation becomes completely integrable (Pohlmeyer),
and there are infinitely many symmetries and conservation laws.  For
instance, the equation is invariant under the change of variables
$u \to \Phi(u)$, $v \to \Psi(v)$, for {\it arbitrary} diffeomorphisms
$\Phi$, $\Psi$.  The entire theory of inverse scattering also
applies.

\item In certain cases the wave map equation can be transformed
into other completely integrable equations such as the sine-Gordon
equation (Pohlmeyer).

\item The energy norm $\dot H^1$ is now sub-critical rather than critical 
($n=2$) or super-critical ($n>2$).

\item The fundamental solution does not decay in time.  This means
that there are no Strichartz estimates and no local smoothing.

As we shall see, we will have to heavily use the null structure
and the conservation laws in order to compensate for this lack 
of smoothing.

\ei

\page
\noindent

A pointwise conservation law

\bi
\item The wave map equation in null co-ordinates is
$$ \phi_{uv} = - \phi (\phi_u \cdot \phi_v).$$
However, since $\phi$ stays on the sphere, we have $\phi \cdot \phi = 1$
and so (by differentiating) $\phi \cdot \phi_v = 0$.  Thus we can
rewrite the above equation as
$$ \phi_{uv} = R \phi_v$$
where $R$ is the matrix
$$R = \phi_u \phi^t - \phi \phi_u^t.$$
The key observation is that $R$ is {\it anti-symmetric}.  In other
words, the $u$ variation of $\phi_v$ is equal to an infinitesimal rotation
of applied to $\phi_v$.  This implies
$$ \partial_u ( |\phi_v|^2 ) = 0,$$
i.e. that the function $|\phi_v|$ is independent of the $u$ variable,
and depends only on the $v$ variable.  In particular, it is determined
by the initial data.  (This identity was first observed by Pohlmeyer;
this particular proof was motivated by similar arguments by Chang, Yang,
and Wang).

\item  This law is not restricted to the sphere.  It is related to the
conformal invariance of the equation, and can also be deduced from
the observation that the stress-energy tensor is trace-free in $1+1$ 
dimensions.

\item  Of course a similar law holds with the roles of $u$ and $v$ reversed,
One consequence of this is that the energy
$$ \int \frac{1}{2} |\phi_u|^2 + \frac{1}{2} |\phi_v|^2\ dx$$
is constant in time.  

\item A second consequence is the following.  Suppose
that $\phi_u, \phi_v$ were supported in the interval $[-C,C]$ at time
$t=0$.  Then at time $t=T$, $\phi_u$ will be supported in $[-C-T,C-T]$
and $\phi_v$ will be supported in $[-C+T,C+T]$.  In particular,
for $T > C$, $\phi$ will separate into two travelling waves:

\begin{figure}[htbp] \centering
\ \psfig{figure=profile.eps,height=3in,width=3.6in}
        \end{figure}

In other words, if the initial data is equal to a constant (say
$e_1$) outside of a compact set, then it will eventually scatter
perfectly to a free solution.

\item This observation has an easy application:

\begin{lemma}  There exists arbitrarily small $C^\infty_0$ initial
data (with initial velocity zero) such that the $\dot H^{1/2}$
norm of the solution goes to infinity as $T \to \infty$.
\end{lemma}

\item Based on this lemma, a rescaling argument, and a finite-speed-of
propagation argument, it is easy to show that the wave map equation is
ill-posed in $\dot H^{1/2}$ in one dimension.  

\item The lemma is proven by considering what the limiting free solution
is in the interval $[C-T, -C+T]$.  If it is equal to $e_1$, then the
$\dot H^{1/2}$ norm will stay constant.  Otherwise, it is easy to show
that it will grow logarithmically with time.  (One shows that
the quantity $\int \phi' H\phi\ dx$ goes to infinity as $T \to \infty$,
where $H$ is the Hilbert transform.

\item It remains to show that the limiting free solution can attain a value
other than $e_1$ in the interior of the light cone.  When the target manifold
is a circle $S^1$ this is impossible from the sharp Huygens' principle in odd dimensions, and the fact that the wave map is the exponential of a free solution.

\item However, for the sphere $S^2$ and higher, one can have wave maps for which
the sharp Huygens principle breaks down.  The idea is to take initial data of the form $e_1 + \eps f(x)$ for some small $\eps$, and expand $\phi$ as a power
series in $\eps$.  The first few terms will indeed obey sharp Huygens, but
the $\eps^5$ term does not.

\ei

\page
\noindent
Classical energy estimate: local well posedness in $s > 3/2$.
\bi
 \item The energy method gives local well-posedness up to time
$T$ only for $s > 3/2$, which is above energy $s=1$ or scaling $s=1/2$.
 \item Main estimate for energy method:
$$\| \phi\|_{L^\infty_t H^s} \lesssim \| data\|_{H^s} + 
\int_0^T \| \Box \phi(t)\|_{H^{s-1}}\ dt$$
$$
\lesssim data + \int_0^T \| \nabla^{s-1} (\phi D\phi D\phi)(t) \|_2\ dt$$
$$\lesssim data + \int_0^T \| \phi D\phi D^s \phi\|_2\ dt$$
$$\lesssim data + T \sup_t \|\phi(t)\|_\infty \|D\phi(t)\|_\infty 
\|D^s \phi(t)\|_{H^s}$$
$$\lesssim data + T \| \phi\|_{L^\infty_t H^s}^3,$$
so the $H^s$ norm of the solution is controlled by the data for $s > 3/2$
if $T$ is sufficiently small.
By setting up an iteration scheme and adapting this estimate for differences
we get well-posedness in $H^s$.

 \item This method did not use the null structure of the non-linearity.
If one considered the superficially similar equation
$$ \phi_{uv} = \phi_u \phi_u$$
then the energy method is sharp; there is no well-posedness for $s \leq 3/2$.

\ei

\page

Global existence

\bi
 \item In the classical energy estimate, $\Box^{-1}$ has only
one order of smoothing.  By choosing appropriate norms, it is possible
to make $\Box^{-1}$ have two orders of smoothing, which allows one
to push the regularity results down further.

 \item By exploiting the null structure one can push the local existence
result down to $s > 1/2$.  The argument is similar but relies on
norms such as the $H^{s,s}$ norm
$$ \| \phi\|_{H^{s,s}} = \| D_u^s D_v^s \phi \|_{L^2_{u,v}} + 
\text{lower order terms}$$
This is a special case of the $H^{s,\delta}$ spaces used by
Klainerman, Machedon, and many others.  The key observation
is that $\Box^{-1}$ maps $H^{s-1,s-1}$ to $H^{s,s}$.

\item The question still remains of whether one has global well-posed
ness in $H^s$ for $1/2 < s < 1$.  Since the $H^s$ norm is not conserved
one cannot simply iterate the local well-posedness argument.

\item Theorem (K.,T.): One has global well-posedness for large data
in $H^s$, $s > 3/4$.

\ei

\page

Bourgain's approach

\bi

\item Recently, Bourgain showed how one could combine an $H^1$ conservation
law with smoothing effects to obtain global well-posedness for certain
equations in $H^s$, for some $s < 1$.  The rough ideas are as follows.

\item Suppose we have a semi-linear wave equation $\Box \phi = F(\phi)$
with a conserved Hamiltonian $H(\phi)$ which is essentially
equivalent to the $H^1$ norm.
We may rewrite this equation as
$$ \phi = \phi_0 + \Box^{-1} F(\phi)$$
where $\phi_0$ is the free solution.  If the data is in $H^1$ then
one has global existence from conservation of the Hamiltonian.  The
issue is to get global existence if the data is only in
$H^s$ for some $s = 1 - \eps$.

\item  Since the initial data is in $H^s$, $\phi_0$ is in $H^s$.
Thus we cannot expect $\phi$ to have any better regularity than $H^s$.
However, it is possible for the non-linear portion of $\phi$,
$\Box^{-1} F(\phi)$, to still be relatively smooth; for instance,
it could be in $H^1$.  This can happen because of the smoothing
properties of $\Box^{-1}$, at least if the non-linearity $F(\phi)$
does not contain too many derivatives.

Let $N$ be a large number, and let $P_{low}$ and $P_{high}$ be
frequency cutoffs to $|\xi| < N$ and $|\xi| > N$ respectively.
We decompose $\phi = P_{low} \phi + P_{high} \phi = \phi_{low} + \phi_{high}$;
$\phi_{low}$ is smooth and therefore has finite (albeit large) hamiltonian,
while $\phi_{high}$ is rough but is small in many norms (e.g. $L^2$ norm).

\item Assume informally that the low frequencies and high frequencies
decouple, and that we get two separate equations that look something like
$$ \Box \phi_{low} = F(\phi_{low})$$
$$ \Box \phi_{high} = F(\phi_{high}).$$
Then $\phi_{low}$ would exist globally, since $\phi_{low}$ has finite
energy and we have conservation of the Hamiltonian.  Also, $\phi_{high}$
will exist for a long time (e.g. $N^{0.001}$), since it is small
in many norms.  So, letting $N \to \infty$, we see that $\phi$ would
exist for arbitrarily large periods of time.

\item Of course, one does not get perfect decoupling of frequencies
because of non-linear effects.  However, the contribution of the non-linearity
is something like
$$ \Box^{-1} ( F(\phi_{high}+\phi_{low}) - F(\phi_{low}) - F(\phi_{high}) ).$$
If there is enough smoothing, then this quantity may remain in $H^1$
even if $\phi_{high}$ is not in $H^1$.  So the effect of the non-linearity
can be absorbed into $\phi_{low}$ at the cost of increasing the Hamiltonian
slightly.  If $s$ is close enough to one, the size of this error term
can be adequately controlled, and one can get existence up to an arbitrary
time by letting $N \to \infty$.

\item For instance, in the context of the NLS
$$ i \phi_t - \Delta \phi = |\phi|^2 \phi$$
in two dimensions, Bourgain combined the $H^1$ conservation law
and smoothing effects to show global well-posedness in $H^s$
for $3/5 < s$.  For the NLW
$$ \Box \phi = |\phi|^2 \phi$$
in three dimensions, Kenig, Ponce, and Vega have similarly shown
well-posedness for $3/4 < s$.

\item Unfortunately, Bourgain's argument is not directly applicable
to the wave map situation, because of the lack of smoothing; the 
error term in the $\phi_{high}$ equation is not negligible, and the error 
term in the $\phi_{low}$ equation is not small in $H^1$.
However, we can compensate for this by the extremely strong conservation laws 
available (i.e. Pohlmeyer's identities), and show that the $H^1$ norm of
the low frequencies is still under control.

\ei

\page

Heuristic considerations

\bi 
\item Pohlmeyer's identity states that $(|\partial_u \phi|^2)_v = 0$.
This implies the following estimate:
$$ \| \phi_u \|_{L^2_u L^\infty_v} \lesssim \| f\|_{H^1} + \|g\|_{L^2}.$$
Note that we have an $L^2_u L^\infty_v$ estimate, which is stronger
than energy conservation (which is basically a $L^\infty_v L^2_u$
estimate).

\item If the data is only in $H^s$ for $s < 1$, then this estimate
appears useless.  However, suppose we could localize it to the
low frequencies:
$$ \| P_j \phi_u \|_{L^2_u L^\infty_v} \lesssim \| P_j f\|_{H^1} + 
\|P_j g\|_{L^2},$$
where $P_j$ is a frequency cutoff of to ($u$-) frequencies $\lesssim 2^j$.  
Such a 
localization is reasonable to hope
for by the philosophy of Bourgain's approach.  Assuming such
an estimate, we thus have
\begin{equation}\label{1}
 \| P_j \phi_u \|_{L^2_u L^\infty_v} \lesssim 2^{j(1-s)}
\end{equation}
for each $j$, which would show that $\phi$ does not blow up and thus
guarantee global existence.

\item Unfortunately, this estimate becomes harder to prove as the time
$T$ increases, and it turns out that we can only prove (0.1) 
for those $j$ for which $2^j \gg T^N$.  But
this is (essentially)
enough to give polynomial growth in $H^s$ norm (i.e. no blowup).

\item The estimate (0.1) is basically a statement that the high
frequency portions of $\phi$ (say frequency $\gg T^N$)
do not significantly affect the energy of the low frequency portions
of $\phi$, at least up to time $T$.  This allows one to have
frequency-localized energy conservation even if the total energy of
$\phi$ is infinite.
\ei
\page

Sketch of formal argument

$$ \| P_j \phi_u \|_{L^2_u L^\infty_v} \lesssim 2^{j(1-s)}$$
\bi
\item To prove that estimate we use the continuity method; that is, we
assume the above estimate holds for all $2^j \gg T^N$, and show that
this implies the estimate holds with a better constant.

\item We use the Pohlmeyer identity
\begin{equation}\label{2}
 \phi_{uv} = R\phi_u
\end{equation}
where $R = \phi \phi_v^t - \phi \phi_v^t$
is an antisymmetric matrix.  
The idea is to localize this identity in frequency to obtain the above estimate.

\item We compute: 
 $$
\| P_j \phi_u \|_{L^2_u L^\infty_v}^2 
= \| P_j \phi_u^t P_j \phi_u^t \|_{L^1_u L^\infty_v}$$
$$= 2 \| D_v^{-1} P_j \phi_u^t P_j \phi_{uv} \|_{L^1_u L^\infty_v}$$
$$= 2 \| D_v^{-1} P_j(\phi_u^t) P_j(R \phi_u) \|_{L^1 L^\infty}.$$

\item If we try to estimate this by the assumptions, then we obtain
$(C + T)^N 2^{2j(1-s)}$; this will imply our hypothesis only when 
$T \ll 1$, which is no good (only gives local existence).

\item However, we can use the conservation law.  If $R$ was constant
then we could take it out of the projection and the above expression would 
vanish.
More generally, if $R$ is low frequency ($\ll j$) then
the quantity $P_j(\phi_u^t) P_j(R\phi_u)$ is well 
approximated by $P_j(\phi_u^t) R P_j(\phi_u)$, which
also vanishes.  So the low frequency portions of $R$ yield
a gain (about $2^{-\eps j}$).

\item The high frequency portion of $R^{i k}$ is very small because
$R$ has the schematic form $R = \phi \phi_v$, and we have decent control
on the regularity of $\phi$ and $\phi_v$.

\item  Informally, $\phi$ has $s$ degrees of regularity in the $v$
variable in $L^2$, and $\phi_v$ has $s-1$ degrees of regularity, so
$\phi \phi_v$ should have $s-1$ degrees of regularity in $L^2$
since $s > 1/2$.  This may look bad, as $s-1$ is negative, but
this is actually OK because of the $D_v^{-1}$ smoothing term in our
expression.

\item  If $s$ is close enough to 1, or more precisely if $s > 3/4$,
then the contribution of the high frequency portions of $R^{ij}$ are
small enough to yield a gain (of about $2^{-\eps j}$).   So our net estimate
becomes $T^N 2^{-\eps j} 2^{2j(1-s)}$, which implies our hypothesis when
$T \lesssim 2^{\eps j}$.

\item Since $j$ is arbitrary, we can obtain global existence.

\ei

\end{document}
