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\begin{document}
\Large
\vskip 1in
\bc
{\bf \LARGE   Low regularity semi-linear \\ ~ \\ wave equations
}\\
\bigskip
\bigskip
\bigskip
\bigskip
\bigskip
Terence Tao \\
\bigskip
{\it Mathematics Department \\ \bigskip UCLA \\ }  
\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 local existence problem for semilinear wave equations

 \item Discuss some of the known techniques and results

 \item Describe some recent improvements for the low regularity case

\ei

\page

\bigskip
\noindent
{
Let $n \geq 1$, and let $F(u)$ be a power-type non-linearity, i.e. a
function such that
$$ |F(u)| \sim |u|^p, \quad |\nabla F(u)| \sim |u|^{p-1}$$
for some $p>1$.  ($u$ can be scalar or vector valued).  
We consider the following Cauchy problem for the
semilinear wave equation
$$\begin{array}{rl}
\Box u  & = F(u) \\
u(0,\cdot)  & = f \in  H^{\gamma}(\rn) \\
\partial_t u(0,\cdot) &= g \in  H^{\gamma - 1}(\rn),
\end{array}$$
where $u$ is a function on $[0,T] \times \R^n$, $\gamma \geq 0$, and
$\Box =  -\frac{\partial^2}{\partial t^2} + \Delta$ is the D'Alembertian.
}
\page 
\noindent
\bi
 \item We consider the question of which $p$, $\gamma$ does one have
 local existence of solutions $u$ for arbitrary data $f$, $g$ in 
 $H^\gamma \times H^{\gamma-1}$.
 \item Global solutions are unrealistic (without further assumptions on $F$
 or the data) as one can easily construct
 examples of blowup in finite time, based on the ODE $u_{tt} = |u|^p$.
 \item Smooth classical solutions are also unrealistic unless
 $p$ is an integer, so we shall only consider weak solutions.
 \item One can also ask if the problem is locally \emph{well-posed},
 i.e. one has uniqueness and continuous dependence in $H^\gamma$
 of the data in addition to existence.  In practice these properties
 are easy once one obtains existence.
\ei

\page

\bigskip
\noindent
Necessary conditions on $(p, \gamma)$ for local existence
\bi
 \item By scaling the example of blowup in finite time, one can obtain
 blowup in arbitrarily small time for data of fixed norm, unless
 the scaling condition
 $$ p(\frac{n}{2} - \gamma) \geq \frac{n+4}{2} - \gamma\eqno(1)$$
 holds.

 \item For radial data the above scaling condition is known to be sharp;
 that is, one does indeed have local existence whenever (1) holds.
 (Lindblad, Sogge)
 
 \item However, this is false for non-radial data.  By Lorentz-transforming 
 and scaling the finite blowup example, one can make the solution blowup 
 along a light ray in arbitrarily
 small time, for data of fixed norm, unless the concentration condition
 $$ p(\frac{n+1}{4} - \gamma) \leq \frac{n+5}{4} - \gamma\eqno(2)$$
 holds.
\ei

\page
\bi
 \item Finally, Hans Lindblad has an example that shows that one 
 does not have local existence for the equation $ \Box u = - u^2$ in
 $L^2$ for $n=3$.  (This is an endpoint to (2)).  The three-dimensional
 case is pathological in a number of ways.

 \item It is tentatively conjectured that there are no further obstructions
 to local existence.  In other words, one has local existence whenever
 (1), (2) hold and $(n,p,\gamma) \neq (3,2,0)$.
\ei

\page

\bigskip
\noindent
{\LARGE The high regularity case: $\gamma > \frac{n}{2}$}
\bi
 \item When the data is very smooth ($\gamma > \frac{n}{2}$) and
 the non-linearity is sufficiently smooth, one can obtain local existence
 via the classical energy method.
 \item The energy method relies on two estimates.  The first is a linear
 estimate (the ``energy estimate'')
 $$ \| u \|_{L^\infty_t H^\gamma} \lesssim \|f\|_{H^\gamma} +
 \|g\|_{H^{\gamma-1}} + \| F\|_{L^1_t H^{\gamma-1}}$$
 which holds for all $f$, $g$, $F$, where $u$ is the solution 
 to the linear problem
$$\begin{array}{rl}
\Box u  & = F \\
u(0,\cdot)  & = f  \\
\partial_t u(0,\cdot) &= g,
\end{array}$$
 \item The second estimate is a non-linear estimate (Schauder's lemma),
 which states that if $u$ is in $H^\gamma$ for $\gamma > \frac{n}{2}$
 and $F$ is sufficiently smooth, then $F(u)$ is also in $H^\gamma$
 (and thus in $H^{\gamma-1}$).
\item With these two estimates local existence can be obtained by the
contraction mapping principle (aka Picard iteration).  In other
words, we show that the non-linear problem is a small perturbation
of the linear problem in some Banach space.
\ei

\page

\bi
\item For function $v$, consider the solution $u$ to the problem
$$\begin{array}{rl}
\Box u  & = F(v) \\
u(0,\cdot)  & = f  \\
\partial_t u(0,\cdot) &= g,
\end{array}$$

\item To find a solution to the non-linear problem we need to find a fixed
point of the map $v \mapsto u$.  It suffices to find a Banach
space $X$ such that $v \mapsto u$ is a contraction on $X$.  In practice
it's enough to show that $v \mapsto u$ maps a ball in $X$ to
itself, since proving a contraction only requires a little more computation.

\item For the energy method we choose the space $X = L^\infty_t H^\gamma$.
If $v$ is bounded in this space, then by Schauder's lemma $F(v)$ is in the
space $L^\infty_t H^{\gamma-1}$, and so (if time $T$ is small)
$F(v)$ will be small in $L^1_t H^{\gamma-1}$.  But by the energy estimate
this implies that $u$ is bounded in $L^\infty_t H^\gamma = X$.  
By taking a little more care one can show that $v \mapsto u$ is actually
a contraction on $X$, and so we get our solution.

\item From the contraction mapping theorem one can also get uniqueness
and continuous dependence on the data.
\ei

\page
{\LARGE The general case.}

\bi 
\item Virtually all local existence results for hyperbolic equations are
based on the above contraction mapping principle.  There are variants,
but basically we need three ingredients:

\item A Banach space $X$ to contain the solution,
and a supplementary space $Y$ to contain the inhomogeneity.  (For the
energy method, $X = L^\infty_t H^\gamma$ and $Y = L^1_t H^{\gamma-1}$).

\item A linear estimate
$$ \| u \|_X \lesssim \|f\|_{H^\gamma} + \|g\|_{H^{\gamma-1}} + \|F\|_Y;$$

\item A non-linear estimate, which has the qualitative form
$$ u \hbox{ is small in } X \quad\implies \quad F(u) \hbox{ is small in } Y.$$

\item Once we have the above ingredients, we can set up an iteration scheme
to find a solution, and the finer details (actually
obtaining a contraction, showing that the solution $u$ has the same
regularity as the data, etc.) tend to be pretty routine.
\ei

\page

{\LARGE Example: the conformal wave equation for $n=3$}

\bi
\item We illustrate the above formalism with the equation
$$ \Box u = |u|^3$$
in three dimensions, with data in $H^{1/2} \times H^{-1/2}$.

\item This equation is somewhat special, being invariant under conformal
transformations.  The choice of data regularity is also ``conformal''
as it is essentially independent of the choice of co-ordinates.  It
lies on the intersection of the two conditions (1), (2).

\item We can use the above machinery to obtain local existence for
this problem (indeed, one can even get global existence for small data).
We choose the solution space to be $X = L^4_{t,x}$ and the inhomogeneity
space to be $Y = L^{4/3}_{t,x}$.

\item The non-linear estimate is trivial:
$$ u \hbox{ is small in } L^4_{t,x} \quad\implies \quad 
|u|^3 \hbox{ is small in } L^{4/3}_{t,x}.$$

\item The linear estimate
$$ \| u \|_{L^4_{t,x}} \lesssim 
\|f\|_{H^{1/2}} + \|g\|_{H^{-1/2}} + \|F\|_{L^{4/3}_{t,x}}$$
is due to Strichartz and is proven next.
\ei

\page


{\LARGE Sketch of proof of Strichartz' estimate
}

\bi
\item By linearity we can assume that either $F=0$ (homogeneous
case), or $f=g=0$ (inhomogeneous case).
One can obtain the homogeneous estimate from the inhomogeneous
estimate by the $TT^*$ method, so we concentrate on the inhomogeneous
estimate $\|u\|_4 \lesssim \|F\|_{4/3}$.

\item Duhamel's formula states that
$$ u(t) = \int_0^t \frac{\sin(s\sqrt{-\Delta})}{\sqrt{-\Delta}} F(t-s)\ ds.$$
Thus it suffices to prove the fixed-time bounds
$$ \| \frac{\sin(s\sqrt{-\Delta})}{\sqrt{-\Delta}} f \|_{L^4_x} \lesssim 
\frac{1}{|s|^{1/2}} \|f\|_{L^{4/3}_x},$$
since the Hardy-Littlewood fractional integration inequality states that
$$ \| g * \frac{1}{|s|^{1/2}} \|_{L^4_t} \lesssim \| g \|_{L^{4/3}_t}.$$

\ei\page\bi

\item To prove
$$ \| \frac{\sin(s\sqrt{-\Delta})}{\sqrt{-\Delta}} f \|_{L^4_x} \lesssim 
\frac{1}{|s|^{1/2}} \|f\|_{L^{4/3}_x}$$
we interpolate between
the energy estimate
$$ \| \sin(s\sqrt{-\Delta}) f \|_{L^2_x} \lesssim 
\|f\|_{L^2_x},$$
proven by Plancherel's theorem, and the decay estimate
$$ \| \frac{\sin(s\sqrt{-\Delta})}{-\Delta} f \|_{L^\infty_x} \lesssim 
\frac{1}{|s|} \|f\|_{L^{1}_x},$$
proven by a stationary phase computation of the kernel corresponding
to the multiplier $\frac{\sin(s\sqrt{-\Delta})}{-\Delta}$.  

\item Strictly
speaking, the decay estimate is false, but it is true when you localize
in frequency, and you can fix matters up using Littlewood-Paley theory.

\item The decay estimate can be viewed as a statement on the decay
and regularity of the fundamental solution to the linear wave
equation.
\ei

\page
{\LARGE What about other powers and regularities?}
\bi

\item For data which is smoother than conformal $H^{1/2}$ but
rougher than energy $H^{n/2}$ one can adapt the above estimates by
placing derivatives in the Banach spaces $X$ and $Y$.  The linear
estimate commutes with differentiation, so adding the derivatives
there is trivial.  The non-linear estimate gets a little more complicated
and you have to use Leibnitz' rule for fractional differentiation.
This method turns out to be sharp, and one can conclude
that one has local well-posedness as long as
the scaling condition is satisfied.

\item For data which is rougher than $H^{1/2}$ but not too rough
one can modify the above argument by splitting the time and space
integrations, using mixed space-time norms $L^q_t L^r_x$
instead of pure norms $L^p_{t,x}$.  The non-linear estimate remains
trivial, and the linear Strichartz estimate is proven in much the
same way as the original Strichartz estimate.  This technique
is (somewhat surprisingly) also sharp, and one can conclude
that one has local well-posedness as long as
the concentration condition is satisfied and the data is not too rough.

\ei

\page

More precisely, we have

\begin{theorem}[Kapitanski, Lindblad, Sogge, Keel, T.] The
conjecture holds if $n\leq 3$.  For $n > 3$ the conjecture
holds assuming the technical condition
$$p(\frac{n+1}{4} - \gamma) \leq \frac{n+1}{2n} (\frac{n+3}{2} -\gamma).
$$
\end{theorem}

\bi
\item The argument for $(n,p,\gamma)=(3,2,0)$ fails because the corresponding
Strichartz estimate fails:
$$ \| u \|_{L^\infty_t L^2_x} \not\lesssim \|f\|_{H^1} + \|g\|_{L^2} +
\|F\|_{L^2_t L^1_x}$$

\item This theorem settles the conjecture stated earlier for $n \leq 3$.
For $n>3$ there is a gap for very low regularity problems, because one
begins to run out of Strichartz estimates.  (More on this later).

\item Our main result is that we have been able to narrow this gap,
relaxing the above condition to
$$
p(\frac{n}{4} - \gamma) \leq \frac{1}{2} (\frac{n+3}{2} -\gamma),
$$
except for an endpoint.
\ei
\page

\centering\ \psfig{figure=rough3.eps,height=2.5in,width=3.3in}
\centerline{Local well-posedness results for $n=3$.  }
\ 

\centering\ \psfig{figure=rough.eps,height=2.5in,width=3.3in}
\centerline{Local well-posedness results for $n=4$.  }
	
\page

{\LARGE What is the problem for very low regularities?}

\bi

\item The above arguments relied on linear estimates of the form
$$ \| u\|_{L^q_t L^r_x} \lesssim \|f\|_{H^\gamma} + \|g\|_{H^{\gamma-1}}
+ \|F\|_{L^\qpt_t L^\rpt_x}$$
for some exponents $q$, $r$, $\qtil$, $\rtil$; the power $p$ of
nonlinearity that this estimate can handle is based on the gap between
$L^q_t L^r_x$ and $L^\qpt_t L^\rpt_x$.

\item There are several limitations as to the type of estimates above that
can occur.  Firstly, there is a dimensional analysis condition that links the
regularity $\gamma$ of the data to the choice of exponents $q$, $r$, $\qtil$,
$\rtil$:
$$ \frac{1}{q} + \frac{n}{r} = \frac{n}{2}-\gamma = \frac{1}{\qpt} + 
\frac{n}{\rpt}-2.$$

\item Translation invariance considerations give
$$ q, \qtil, r, \rtil \geq 2.$$

\item Finally the Knapp example gives more constraints which are independent
of the regularity of the data:
$$\frac{1}{q}+\frac{(n-1)/2}{r}, \frac{1}{\qtil}+\frac{(n-1)/2}{\rtil} \leq \frac{(n-1)/2}{2}.$$

\ei\page
\bi
\item As $\gamma$ gets small, these constraints become increasingly incompatible,
and one cannot make the gap between $L^q_t L^r_x$ and $L^\qpt_t L^\rpt_x$
as wide as one might hope.

\item In other words, the less regular the data becomes, the fewer $L^q_t L^r_x$
spaces one can hope to place the solution in, and eventually the options
become too limited to obtain optimal results.

\ei

\page

{\LARGE How to get around this problem?}

\bi
\item For high regularity data, one placed the solution in a high-regularity
space such as $L^\infty_t H^s$ for $s > \frac{n}{2}$.  For medium
regularity data one placed the solution in a medium regularity space 
$L^q_t L^r_x$.  What do we do for low regularity?

\item One obvious idea is to place negative derivatives on the solution
space, e.g. trying to place $u$ in a negative-order
Sobolev space $L^r_{-\alpha}$.  The linear estimate carries through fine,
but the non-linear estimate is problematic.  Basically, if $u$ is
only assumed to be in a space such as $L^r_{-\alpha}$, there is no way
to control $F(u)$.

\item For instance, take $F(u) = |u|^2$ and let $u(x) = 
e^{i x \cdot \xi} \psi(x)$ for some cutoff $\psi$ and some very high
frequency $\xi$.  As $\xi \to \infty$, $u$ tends to zero in
$L^r_{-\alpha}$, but $F(u)$ remains constant.

\ei

\page \bi
\item As the above example shows, the enemy is the fact that the high
frequencies of $u$ can corrupt the low frequencies of $F(u)$.  (The
low frequencies of u can also corrupt the high frequencies of $F(u)$,
but this is an obstruction for the high-regularity problem rather
than the low regularity problem).  If one could somehow prevent
the frequencies from interacting with each other, then one could
use negative order Sobolev spaces with impunity, and get sharp
results (down to concentration and scaling) for all regularities.

\item So we have to limit the corruption effect of the high frequencies
of $u$.  \textbf{Key observation:} the high frequencies of $u$ tend to be 
supported on ``thin'' sets.
For instance, if you consider the linear problem $\Box u = 0$, $u(0) = \delta$,
then $u(t)$ is mainly supported on the sphere of radius $t$ (Huygens's 
principle), and so the $2^j$-frequency portion of $u$ is concentrated in
an annulus of width $2^{-j}$ around that sphere.  So even though $u$ is
spread out in most directions, in the radial direction $u$ is quite thin.

\ei
\page

{\LARGE Why does ``thin-ness'' help?}
\bi

\item In the counterexample $u = e^{i x \cdot \xi} \psi(x)$, $u$
had very high frequency but was ``fat'': spread out over a set of
dimension $1$ in all dimensions.  Thus $F(u)$ was also fat, and low-frequency.
If $u$ was supported on a thinner set, then $F(u)$ would also have thinner
support, and $F$ would consist mainly of high-frequency modes instead.
Thus: if $u$ is thin, then $F(u)$ does not have very many low-frequency
components.  In other words, the frequency-corruption effect is not as bad
as one might think (assuming we can show that $u$ is thin).

\item To make this thin-ness quantitative, we need to estimate $u$ not
in a Lebesgue spaces $L^r_x$, but in two-scale spaces $X^{r,p}_0$ defined by
$$ \|f\|_{X^{r,p}_0} = (\sum_Q \|f\|_{L^p(Q)}^r)^{1/r},$$
where $Q$ ranges over a partition of $\R^n$ into unit cubes.  If $u$ is fat
then the $X^{r,p}_0$ norm is pretty much the same as the $L^r$ norm, but if
$u$ is thin and $p<r$ then one expects to gain something in the $X^{r,p}_0$
norm.

\item One also has to define norms $X^{r,p}_k$ for spatial scales $2^{-k}$
other than the unit scale.

\ei

\page

\bi

\item To formally obtain an improvement on local well-posedness results,
we need an improved linear estimate and an improved non-linear estimate.

\item The improved linear estimate states that we can replace the $L^r_x$
space in the standard Strichartz estimate
$$ \| u\|_{L^q_t L^r_x} \lesssim \|f\|_{H^\gamma} + \|g\|_{H^{\gamma-1}}
+ \|F\|_{L^\qpt_t L^\rpt_x}$$
by the spaces $X^{r,p}_k$ and gain something, especially if $f$, $g$, $F$
are high frequency.  These improved estimates are still proven by
the same techniques used to prove the ordinary Strichartz estimates
(i.e. $TT^*$, energy estimate, decay estimate, interpolation).  The
gain ultimately stems from the fact that the fundamental solution of the wave
equation is ``thin''.

\item To close the argument we need non-linear estimates which state,
roughly, that
$$ \sqrt{-\Delta}^{-\alpha} u \hbox{ is small in } X^{r,p}_k
\quad\implies\quad$$
$$\sqrt{-\Delta}^{-p \alpha} |u|^p \hbox{ is small in } L^{r/p}_x.$$
This is achieved by Littlewood-Paley theory; if $u$ is in $X^{r,p}_k$, then
$|u|^p$ is locally integrable at the scale of $2^{-k}$, and so
the $2^k$ frequency piece of $|u|^p$ is well controlled.

\ei
\end{document}
