\documentstyle[amssymb,amsfonts,psfig,12pt]{article} \input MathMacro.tex \newcommand{\eqn}[1]{(\ref{#1})} \newcommand{\is}{$I^{s}\;$} \newcommand{\mbf}[1]{\mbox{\boldmath ${#1}$}} \newcommand{\bc}{\begin{center}} \newcommand{\ec}{ \end{center}} \newcommand{\page}{\vfil \break} \newcommand{\bi}{\begin{itemize}} \newcommand{\ei}{ \end{itemize}} \newcommand{\benum}{\begin{enumerate}} \newcommand{\eenum}{ \end{enumerate}} \newcommand{\bd}{\begin{description}} \newcommand{\ed}{ \end{description}} \def\rp{{r^\prime}} \def\rpt{{\tilde r^\prime}} \def\rtil{{\tilde r}} \def\qtil{{\tilde q}} \def\kp{{k^\prime}} \def\kt{{\tilde{k}}} \def\qp{{q^\prime}} \def\pp{{p^\prime}} \def\qpt{{\tilde q^\prime}} \def\R{{\hbox{\bf R}}} \def\rn{{\R^n}} \def\Z{{\hbox{\bf Z}}} \def\eps{{\varepsilon}} \def\implies{{\Rightarrow}} \def\RR{{\frak R}} % \newcommand{\lesssim}{\mbox{$\,\stackrel{\textstyle{<}}{\sim}\,$}} \renewcommand {\theequation}{\thesection.\arabic{equation}} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{remark}{Remark}[section] \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}