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\begin{document}

\title[Wave maps]{On wave maps at the critical regularity}

\author{Terence Tao}
\address{Department of Mathematics, UCLA, Los Angeles CA 90095-1555}
\email{ tao@@math.ucla.edu}
\subjclass{35J10}

\vspace{-0.3in}
\begin{abstract}
We informally describe the strategy of Tataru to show well-posedness of wave maps at the critical regularity in two and higher dimensions, provided that the initial data is small in an $l^1$ Besov space.
\end{abstract}

\maketitle

\section{Introduction}

The wave map equation has the schematic form
$$ \Box \phi = \Gamma(\phi) Q_0(\phi, \phi)$$
where $\Gamma$ is some analytic function of $\phi$, and $Q_0$ is the null form
$$ Q_0(\phi, \psi) := \phi_t \psi_t - \nabla \phi \cdot \nabla \psi.$$
The critical regularity for such spaces is $\dot H^{n/2}$.  It is an interesting open problem (especially in two dimensions) whether this equation is well-posed in this critical regularity.  (In one dimension it turns out one has ill-posedness at the critical regularity \cite{tao:counter}, but this example seems to shed no light on the higher-dimensional case).

In \cite{tataru:wave1}, \cite{tataru:wave2}, Tataru showed that these equations are well-posed for $n \geq 2$ in the slightly smaller Besov space $\dot B^{n/2,1}$, defined by
$$ \| f \|_{\dot B^{n/2,1}} := \sum_j \| S_j f \|_{\dot H^{n/2}}$$
where $S_j$ is the Littlewood-Paley projection to frequencies $|\xi| \sim 2^j$.  The purpose of this note is to briefly describe, in informal terms, the central ideas of Tataru's argument.

We shall discard the $\Gamma(\phi)$ term and consider the model equation
$$ \Box \phi = Q_0(\phi, \phi).$$
We may write this equation as
$$ \phi = \phi_0 + \Box^{-1} Q_0(\phi,\phi)$$
where $\phi_0$ is the solution to the free wave equation with the same initial data as $\phi$.  The issue is then to understand the non-linear form $\Box^{-1} Q_0(\phi,\phi)$.  In order to treat this equation by an iteration scheme, we eventually want some estimate of the form
$$ \| \Box^{-1} Q_0(\phi,\phi) \|_X \lesssim \| \phi\|_X \| \phi\|_X$$
where $X$ is some critical regularity space (which must also contain $\phi_0$).

We can break up $\phi$ into Littlewood-Paley pieces $S_j \phi$, and consider the interactions
$$ \Box^{-1} Q_0(S_j \phi, S_k \phi).$$
We shall focus on the case of equal frequency interactions $j=k$.  One must also deal with the unequal frequency interactions $j \neq k$, but these will turn out to be treatable by a modification of the $j=k$ case.  Then one has to sum all the dyadic interactions, but this turns out to be fairly easy because we are in an $l^1$ based Besov space.  (For the original $\dot H^{n/2}$ problem, one only has $l^2$ control of the various frequency pieces of $\phi$, which creates a serious difficulty to this method.  This failure to sum the interactions of unequal frequencies is the main remaining obstacle to solving the wave maps problem).

By scaling we can consider just the interactions of frequencies comparable to 1:
$$ \Box^{-1} Q_0(S_0 \phi, S_0 \phi).$$
Since $S_0 \phi$ has frequencies $\sim 1$, the null form $Q_0(S_0 \phi, S_0 \phi)$ will have frequencies $\lesssim 1$.  It is thus possible to have (+-) interactions in which opposing frequencies from $S_0 \phi$ interact to form a very low frequency mode (with $|\xi| \ll 1$).  However, as it turns out, this type of interaction is increasingly weak as the dimension increases (because it becomes very difficult for two large frequencies to cancel to form a small frequency at high dimension), and we can afford to ignore this effect for $n \geq 2$.  (For $n = 1$ this effect is actually quite significant, and is the cause of the ill-posedness results in \cite{tao:counter}.)  We are thus reduced to considering the form
\be{asok}
S_0 \Box^{-1} Q_0(S_0 \phi, S_0 \phi).
\end{equation}
Since all frequencies are $\sim 1$, derivatives have pretty much become irrelevant, so we shall start talking about $L^2$ instead of $\dot H^{n/2}$, etc.  (This is not to say that the $n/2$ index is totally irrelevant - it was necessary in order to scale down to the $|\xi| \sim 1$ case in the first place!)

The Fourier support of $S_0 \phi$ is the region $\{ (\xi, \tau): |\xi| \sim 1 \}$.  To oversimplify things, let us only consider three components of this region:
$$ C_r := \{ (\xi, \tau): |\xi| \sim 1; \angle \xi, e_1 \leq 1/10; |\tau - |\xi|| \ll 1 \}$$
$$ C_l := \{ (\xi, \tau): |\xi| \sim 1; \angle \xi, -e_1 \leq 1/10; |\tau - |\xi|| \ll 1 \}$$
$$ I := \{ (\xi, \tau): |\xi| \sim 1; ||\tau| - |\xi|| \sim 1 \}.$$

$C_r$ is the right half of the upper unit light cone; $C_l$ is the left half of the upper unit light cone; and $I$ is the intermediate region, well separated from the light cone.  Let $P_{C_r}$, $P_{C_l}$, and $P_I$ denote the Fourier projections to these regions.

Let us consider various components of \eqref{asok}.  We shall ignore the contribution of
$$ Q_0(P_{C_r} \phi, P_{C_r} \phi).$$
This is because $Q_0$ is a null form, and the Fourier symbol of $Q_0$ vanishes when the frequencies of the arguments are parallel.  Since $C_r$ is a relatively narrow sector, the frequencies of $P_{C_r} \phi$ and $P_{C_r} \phi$ are pretty close to parallel, and so we expect the contribution to be small.   (Of course, this is an oversimplification, and the small angle interactions do require a certain amount of technicalities, which result in most of the complications in Tataru's papers.  However, these small angle interactions are quite rare, especially in higher dimensions, and the strong cancellation effect of the $Q_0$ null form also helps substantially).  

We similarly ignore the $C_l$-$C_l$ interaction.  We also ignore the $I$-$I$ interaction because it is rather trivial (at no point does the singularities of $\Box^{-1}$ intervene, either at the inputs or the output).

Now consider a $C_l$-$C_r$ interaction.  The only place for this interaction to go in our simplified analysis is $I$.  On $I$ the operator $\Box^{-1}$ is harmless.  Also, there is no cancellation to be exploited from $Q_0$ here, and the derivatives in $Q_0$ are also harmless, so we reduce (schematically at least) to
\be{clcr}
P_I( (P_{C_l} \phi) (P_{C_r} \phi) ).
\end{equation}

By symmetry, the only other type of interaction to consider is a $C_l$-$I$ interaction.  This interaction cannot go to $C_l$, so it in our simplified analysis it must go to either $I$ or $C_r$.  Of the two, $C_r$ is the most dangerous, because the $\Box^{-1}$ symbol becomes singular.  Thus we focus exclusively on the case when the output is in $C_r$.  In this case we cannot exploit any cancellation in $Q_0$, so we ignore it, and we reduce to
\be{cli}
\Box^{-1} P_{C_r}( (P_{C_l} \phi) (P_I \phi) ).
\end{equation}
Similarly we have the symmetric counterpart
\be{cri}
\Box^{-1} P_{C_l}( (P_{C_r} \phi) (P_I \phi) ).
\end{equation}

Our problem is now to construct function spaces $X_{C_r}$, $X_{C_l}$, $X_I$ for the three components $P_{C_r} \phi$, $P_{C_l} \phi$, $P_I \phi$ such that (a) $X_{C_r}$ and $X_{C_l}$ contain the free solutions $P_{C_r} \phi_0$, $P_{C_l} \phi_0$; and (b) the expressions \eqref{clcr}, \eqref{cli}, \eqref{cri} remain in the spaces $X_I$, $X_{C_r}$, $X_{C_l}$ respectively.

The expression \eqref{clcr} seems fairly straightforward, and should be able to be dealt with by bilinear wave equation estimates, which are by now very well understood.  The expressions \eqref{cli}, \eqref{cri} however are much worse because of the $\Box^{-1}$ factor, which can create Fourier singularities of the form $1/(\tau - |\xi|)$.  In sub-critical situations one can hope to localize in time to mollify this to something like $1/(|\tau - |\xi|| + \delta)$, which can then be placed in a space such as $X^{s,1/2+}$.  (The $X^{s,1/2+}$ norm may appear large if $\delta$ is small, but recall that we have rescaled frequency to be 1.  If we undo the rescaling, we can actually get quite a small $X^{s,1/2+}$ norm if the data is in $H^s$, $s$ is sub-critical, and $\delta$ is chosen appropriately).  However in the critical case one cannot gain any advantage from time localization (except in the large data case).  In fact, it is by now clear that in the critical case one cannot exclusively work in Fourier space, and one must find some other way to control the $1/(\tau - |\xi|)$ term.  This is sometimes known as the ``division problem''.

Tataru's resolution of the division problem is to work exclusively in physical space, once the above Fourier reductions (and their analogues for small angles, differing frequencies, etc.) have been performed.  

The space $X_I$ is very simple, it is just $L^2_{t,x}$.  The space $X_{C_r}$, $X_{C_l}$ are a bit more complicated.  Roughly speaking, they are those functions which can be pointwise dominated in physical space by averages of free $L^2$ waves with Fourier support in $C_r$ or $C_l$.  A formal definition is
$$ \| \phi \|_{X_{C_r}} := \inf( \int \| f^\alpha \|_2\ d\alpha:
|\phi(x,t)| \leq \int |P_{C_r} S(t) f^\alpha(x)|\ d\alpha \forall (x,t) \in \R^n \times \R )$$
where $d\alpha$ ranges over arbitrary measure spaces, $f^\alpha$ are arbitrary sets of initial data, and $S(t) f^\alpha$ is the free wave with initial data $f^\alpha$.  The norm for $X_{C_l}$ is defined similarly.

There are a couple other ways of defining the spaces $X_{C_r}$; Tataru himself uses a slightly different version of this space in his paper.  The important facts about $X_{C_r}$ are (a) it is a Banach space; (b) it contains $L^2$ free solutions with Fourier support in $C_r$; and (c) it is closed under cutoffs in physical space (e.g. to a half-space $\chi_{t > t_0}$).  Note especially the last property (c).  This property is very hard to achieve in Fourier-based spaces, because cutoffs in time tend to introduce the horrible singularity $1/(\tau - |\xi|)$ alluded to earlier.

It's hard to pin down the space $X_{C_r}$ completely.  A fairly typical element of $X_{C_r}$ can be constructed by taking a free $L^2$ $C_r$ wave, and taking absolute value in physical space.  This completely screws up the Fourier portrait of the function, but all physical space estimates (e.g. Strichartz estimates, or bilinear $L^2$ estimates) for these types of functions can survive
this absolute value procedure completely unscathed (providing that there are no derivatives in the estimate, of course - but we have performed so many frequency localizations by now that derivatives have long since disappeared).

The control of \eqref{clcr} is now straightforward.  One wishes to put
$$ (P_{C_l} \phi) (P_{C_r} \phi)$$
in $L^2_{t,x}$.  But $P_{C_l} \phi$ is pointwise dominated by an average of $L^2$ $C_l$ free solutions, and similarly for $P_{C_r} \phi$.  By Minkowski's inequality (and the fact that we are now working exclusively in physical space), it thus suffices to show that the product of an $L^2$ $C_l$ free solution and an $L^2$ $C_r$ free solution stay in $L^2_{t,x}$.  But this is a standard bilinear estimate.

It remains to tackle \eqref{cli}, which is a bit trickier.  (Of course, \eqref{cri} is symmetric to \eqref{cli}).  The issue is to understand how the space $X_{C_r}$ can absorb the nasty $\Box^{-1}$ symbol.  A big clue comes from Duhamel's formula:
$$ \Box^{-1} F(t) = \int \chi_{t > t'} S(t-t') F(t')\ dt'.$$
This formula expresses $\Box^{-1} F$ as the integral of free solutions $S(t-t') F(t)$, multiplied by the cutoff to the half-space $\chi_{t > t'}$.  This fits suspiciously well with our construction of the space $X_{C_r}$.  In fact, we see that if $F$ is in $L^1_t L^2_x$, then Duhamel's formula is able to place $\Box^{-1} P_{C_r} F$ in $X_{C_r}$.

The reader at this point might enthusiastically say that we must be close to being done, that all we need to do is place $(P_{C_r} \phi) (P_I \phi)$ in $L^1_t L^2_x$.  But $P_I \phi$ is a generic $L^2_{x,t}$ function, which means that we have to put $P_{C_r} \phi$ in $L^2_t L^\infty_x$.  Strichartz estimates let us do this, but only when $n \geq 4$.  (Recall that we are frequency localized, and so we can pile on as many derivatives as are necessary in order for Strichartz estimates to work).  So we only have a partial success.  (This explains why Tataru's first paper \cite{tataru:wave1} only covers the $n \geq 4$ case).

More recent bilinear estimates \cite{borg:cone}, \cite{wolff:cone}, \cite{tao:cone} can manage to place $P_I \phi$ in a slightly better space than $L^2_{x,t}$; in fact, one can get $L^{(n+3)/(n+1)}_{x,t}$.  This is enough to get the $n=3$ case, but the $n=2$ case remains well out of reach, as the Strichartz estimate remains uncooperative.  (We need $L^{5/2}_t$ decay on $P_{C_r} \phi$, but Strichartz will only concede $L^4_t$ decay to us).  This is rather disappointing, considering that $n=2$ is the most interesting case.

One could try to then push for more advanced bilinear estimates to crack the $n=2$ case - basically, we need an estimate which gives $L^{4/3}_t$ decay in time or better.  (We don't really care about decay in $x$; $L^\infty_x$ bounds will suffice).  Such an estimate may well be true.  However, Tataru pursues a different approach, using null frames.

The basic point is that Duhamel's formula isn't just restricted to the standard $(x,t)$ co-ordinate frame.  In fact, every frame $(x',t')$ in which $t'$ is timelike and $x'$ is spacelike will have some form of Duhamel's principle, and we should be able to place $\Box^{-1} P_{C_r} F$ in $X_{C_r}$ whenever $F$ is in $L^1_{t'} L^2_{x'}$.

What about if $t'$ is null, and $x'$ contains a null direction?  The conventional wisdom is that we are now in serious trouble, because some characteristics now travel along constant-$t'$ slices, and one should not expect any sort of solvability at all for the Cauchy problem (without additional consistency conditions), let alone a Duhamel formula.  However, remember that we've got this projection $P_{C_r}$ to exploit.  The operator $\Box^{-1}$ - which is our main enemy - can only terrorize the right half $C_r$ of the light cone.  This means that the only characteristics which matter are those which are normal to $C_r$.  The upshot of all this is that the Duhamel formula survives more or less intact providing that $t'$ is not normal to $C_r$.  In particular, if $t'$ is a null direction in $C_l$, then we can still put $\Box^{-1} P_{C_r} F$ in $X_{C_r}$ whenever $F$ is in $L^1_{t'} L^2_{x'}$.

This gives us a bit more freedom, but it's not clear how to exploit this.  After all, if we want to put $(P_{C_l} \phi) (P_I \phi)$ in $L^1_{t'} L^2_{x'}$, we'd have to put $P_{C_l} \phi$ in something like $L^2_{t'} L^\infty_{x'}$, and now we're back to the problem of not having enough Strichartz estimates, and with the additional complication of now being in a null frame to boot.  So we don't seem to have gained anything yet.

So now Tataru pulls out a final trick - a decomposition of $P_{C_l} \phi$ into travelling waves.

We first treat the case when $P_{C_l} \phi$ is an $L^2$ $C_l$ free wave.  In other words, the Fourier transform of $P_{C_l} \phi$ is an $L^2$ measure on the surface $C_l$.  We now split $C_l$ into light rays.  The $L^2$ measure on the surface $C_l$ can then be written as an $L^2$ average of $L^2$ measures on light rays.  (If you like, we are writing the measure in polar co-ordinates, and using Fubini's theorem).  Since $C_l$ is compact, an $L^2$ average is an $L^1$ average.  To summarize: $P_{C_l} \phi$ is an average of inverse Fourier transforms of $L^2$ measures on light rays on $C_l$.

But by Plancherel, the inverse Fourier transform of an $L^2$ measure on a light ray (through the origin) is a function which is $L^2$ in the direction of the light ray, and constant on orthogonal directions.  In particular, such a function is in $L^2_{t'} L^\infty_{x'}$, where $(t',x')$ is the null frame corresponding to this light ray.

To summarize: $P_{C_l} \phi$ is an average of $L^2_{t'} L^\infty_{x'}$ functions, where $(t', x')$ range over null frames such that $t'$ points in a null direction of $C_l$.  This analysis was only for $L^2$ $C_l$ free solutions, but it is clear that it also works for general $X_{C_l}$ functions, by the construction of $X_{C_l}$.

And now we are done, because each $L^2_{t'} L^\infty_{x'}$ function which comes up in this average multiplies with the $L^2_{t',x'}$ function $P_I \phi$ to form an $L^1_{t'} L^2_{x'}$ function.  Since $t'$ is a n ull direction in $C_l$, the contribution to $\Box^{-1} P_{C_r} ((P_{C_l} \phi) (P_I \phi))$ is thus in $X_{C_r}$ by our previous discussion.  The claim then follows by averaging.

That's it!  (Well, okay.  To really kill off the well-posedness problem, one has to get back into the space $\dot B^{n/2,1}$ at all later times $T$.  I'll leave it to you to convince yourself why the spaces $X_{C_l}$, $X_{C_r}$, $X_I$ have this property.  It's actually quite easy, given that all frequencies are 1 and so derivatives (and Besov summations) are irrelevant).

\begin{thebibliography}{10}

\bibitem{borg:cone}
J. Bourgain, \emph{Estimates for cone multipliers},
Operator Theory: Advances and Applications, \textbf{77} (1995), 41--60.

\bibitem{tao:counter}
T. Tao, \emph{Ill-posedness of one-dimensional wave maps at the critical regularity}, to appear, Amer. J. Math.

\bibitem{tao:cone}
T. Tao, \emph{Endpoint bilinear restriction theorems for the cone, and some sharp null
 form estimates}, submitted, Math Z.

\bibitem{tataru:wave1}
D. Tataru, \emph{Local and global results for wave maps I},
Preprint, 1997.

\bibitem{tataru:wave2}
D. Tataru, \emph{Local and global results for wave maps II},
Preprint, 1999.

\bibitem{wolff:cone}
T.~H. Wolff, \emph{A sharp bilinear cone restriction estimate}, to appear, Annals of Math.

\end{thebibliography}


\end{document}
\endinput


