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\begin{document}
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{\bf \LARGE  Some problems in nonlinear dispersive PDE}

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Terence Tao (UCLA)\\

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\noindent

Linear evolution equations

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\item An \emph{evolution equation} is an equation - usually, though not always, a partial differential equation (PDE) - which determines the evolution of a field $u(x,t)$ (or perhaps a system of fields, usually real or complex valued) in both space $x \in \R^n$ and time $t \in \R$.  Examples:

\item The equation $\partial_t u = -\kappa u$ for some damping constant $\kappa > 0$ is a damping equation, solutions decay exponentially in time at rate $\kappa$.  

\item The equation $\partial_t u + v \cdot \nabla_x u = 0$ for some velocity $v \in \R^n$ is a transport equation, solutions move in space at
velocity $v$.

\item The one-dimensional wave equation $-\partial_t^2 u + \partial_x^2 u = 0$ can be factored into two transport equations, one travelling at velocity $+1$ and the other at velocity $-1$.

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More linear evolution equations

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\item For more complicated linear evolution equations, one can get some intuitive idea of how they behave by restricting attention to solutions of a certain \emph{frequency} $\xi \in \R^n$ - roughly, this means that for each time $t$, the solution $u(x,t)$ ``looks like'' a multiple of $e^{i x \cdot \xi}$; this can be made rigorous using the Fourier transform.  In particular, applying $\nabla_x$ to such a field should be like multiplication by $i \xi$.  Some examples:

\item The \emph{heat equation} $\partial_t u - \Delta u = 0$.  For solutions of frequency $\xi$, this heuristically becomes
$\partial_t u = - |\xi|^2 u$, which is a damping equation with damping rate $|\xi|^2$.  Thus frequencies, especially high frequencies, dissipate to zero, and we call this equation a \emph{dissipative equation}.

\item The \emph{Schr\"odinger equation} $i \partial_t u + \Delta u = 0$, when applied to solutions of frequency $\xi$, becomes $\partial_t u + \xi \cdot \nabla_x u = 0$, so we expect solutions of frequency $\xi$ to be transported by velocity $\xi$.  (Actually, that is the phase velocity; the group velocity (which is more important) is $2\xi$).  Thus different frequencies get transported at different speeds, and we call this equation a \emph{dispersive equation}.

\item The \emph{wave equation} $-\partial_t^2 u + \Delta u = 0$, for solutions of frequency $(\xi_1, 0, 0)$, becomes 
the one-dimensional wave equation in the $(1,0,0)$ direction, with solutions propagating at velocity $\pm (1,0,0)$.  More generally, solutions with frequency $\xi$ propagate at velocities $\pm \xi/|\xi|$.  So parallel frequencies propagate in the same directions, but non-parallel frequencies do not; this equation is ``partly'' dispersive.

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Non-linear dispersive equations

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\item We can modify a linear dispersive equation into a nonlinear one in many ways.  One of the mildest is by adding
some function $F(u)$ of $u$ on the right-hand side; this gives a \emph{semilinear dispersive equation}.  An example is 
the \emph{nonlinear Schr\"odinger equation (NLS)}
$$ i\partial_t u + \Delta u = \pm |u|^{p-1} u$$
where $p > 1$ is an exponent, and $\pm$ is a sign (the $\pm = +$ sign is the \emph{defocusing case}, the $\pm = -$ sign is the \emph{focusing case}).  Similarly there is the \emph{nonlinear wave equation (NLW)}
$$ -\partial_t^2 u + \Delta u = \pm |u|^{p-1} u.$$

\item A more serious modification is to add a function $F(u, \nabla u)$ to the right-hand side that depends on both $u$ and
the first derivatives of $u$; this gives a \emph{semilinear dispersive equation with derivative non-linearity}.  An example is the \emph{wave maps equation}, which looks roughly like
$$ -\partial_t^2 u + \Delta u = - \Gamma(u)(\partial_\alpha u, \partial^\alpha u)$$
where $\Gamma$ is some geometric function (a Christoffel symbol, actually; this is a generalization of the equation for
geodesic flow).

\item Even more serious still is to replace the constant coefficient dispersive operators on the LHS by variable coefficient dispersive operators, but with the coefficients also depending on the field $u$; this gives a \emph{quasilinear dispersive equation}.  An example is the \emph{Einstein equations}, which in suitable co-ordinates looks vaguely like this:
$$ g^{\alpha \beta}(u) \partial_\alpha \partial_\beta u = Q^{\alpha \beta}(u) \partial_\alpha u \partial_\beta u$$
where $g$ and $Q$ are various geometric quantities.  

\item There are many examples of nonlinear dispersive equations which
are important in physics (Yang-Mills, Korteweg-de Vries, Ginzburg-Landau,
etc.), including the ones listed above.

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Nonlinearity can cause blowup!

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\item One major difference between linear and nonlinear dispersive equations is that solutions to the former equations tend to not develop singularities (assuming they were smooth at the initial time $t=0$), whereas solutions to the latter equations can.  A very simple ODE example already shows the problem:

\item The linear ODE $u_t = \kappa u$ with initial data $u(0) = 1$ will have a solution $u(t) = e^{\kappa t}$, which stays smooth for all $t$ no matter what $\kappa$ is.

\item However, the ``focusing'' ODE $u_t = \frac{1}{2} |u|^2 u$ with initial data $u(0)=1$ will have a solution $u(t) =
(1-t)^{-1/2}$, which is smooth until time $t=1$, at which point it blows up to infinity.  In contrast, the
``defocusing'' ODE $u_t = - \frac{1}{2} |u|^2 u$ with the same initial data has solution $(1+t)^{-1/2}$, which does not blow up as $t \to +\infty$, and indeed converges to zero.

\item More generally, we expect ``focusing'' nonlinear dispersive equations to have solutions that exist for short times, but develop singularities in large times; in contrast, we expect ``defocusing'' nonlinear dispersive equations to have solutions that stay smooth for all times, and decay at infinity.  

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Open problems

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\item I am interested in rigorous quantifications of the above heuristics, for various equations on various domains and for various types of assumptions on the initial data.  A typical problem: do smooth, finite energy solutions to the defocusing NLS on $\R^3$ stay smooth for all time?  It turns out that the answer depends on the value of $p$.  For $p < 5$ (subcritical case) the answer is yes (proven in late 1980s).  For $p=5$ (critical case) the answer is also yes (proven February 2004, proof is 83 pages long).  Furthermore in this case the solution is known to decay to zero at infinity.  For $p > 5$ (supercritical case) the answer is unknown, and is of difficulty comparable to
the Navier-Stokes problem (a Clay prize problem, worth \$ 1 million).  Roughly speaking, in a subcritical problem the linear part of the equation is ``stronger'' than the non-linear part; in a critical problem they are exactly balanced; and for a supercritical problem the non-linear effects dominate.

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\item In the focusing case, singularities from smooth, finite energy data can form when $p \geq 7/3$ (late 1970s) but cannot form when $p < 7/3$ (late 1980s).  The long-term behavior when singularities do not form is still a very interesting open problem (which concerns an interesting class of solutions
known as \emph{solitons}, which we will not discuss here).

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Techniques

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\item Progress on these PDE problems have come by learning how to use many types of (quite different) tools, including:

\item {\bf Energy estimates and conservation laws}.  These equations come with a number of conservation laws, notably conservation of energy.  Also there are ``almost conservation laws'' for other quantities which are not quite conserved in time, but somehow have ``small'' derivative.  Locating and exploiting as many of these laws as possible is crucial.

\item {\bf Perturbation theory}  To understand the nonlinear equation we often approximate it by a simpler equation, often a linear equation.  We then use perturbation techniques (e.g. non-linear versions of Neumann series, or infinite-dimensional versions of the inverse function theorem) to pass from the linear approximation to the non-linear equation.  This often requires the full machinery of harmonic analysis - Sobolev spaces, the Fourier transform, etc. - and, as the name suggests, is only useful when the non-linear equation is in some sense a ``small'' perturbation of its linear
counterpart.  To deal with large perturbations one has to combine this theory with the other methods listed here.

\item {\bf Gauge transformations}  Special algebraic transformations, often motivated by geometry, can be used to recast the PDE in a form which may be more complicated but is more amenable to analysis.  Some of the recent gauge transformations are quite deep.

\item {\bf Monotonicity formulae}  Not only are certain quantities of the solution conserved, there are other quantities which are always increasing in time or decreasing in time.  These are also extremely useful for a number of reasons.

\item {\bf High frequency / low frequency dynamics} For dispersive equations it is often helpful to split the solution into high frequency and low frequency components, and see how the components evolve on their own and also how they interact with each other.  In particular there are ``cascades of energy'' from high frequencies to low, or vice versa, which need to be studied.

\item {\bf Particle-like behavior}.  Dispersive equations model waves, but every so often waves exhibit particle-like behavior, and this can often be quantified rigorously by means of Fourier analysis.  This allows one to bring the machinery of dynamical systems to the problem.

\item {\bf Variational methods}.  Many of the equations are associated with a Lagrangian, which can then be varied to learn more about the structure of the evolution.  More recently, ``induction on energy'' methods have arisen to take advantage of variational structure in a different way.

\item Many of these techniques are being clarified by current research - it is an exciting time to be working on these problems!

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\end{document}
