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Non-linear wave equations


Let be endowed with the Minkowski metric

(In many papers, the opposite sign of the metric is used, but the difference is purely notational). We use the usual summation, raising, and lowering conventions. The D'Lambertian operator

is naturally associated to this metric, the same way that the Laplace-Beltrami operator is associated with a Riemannian metric.

Space and time have the same scaling for wave equations. We will often use to denote an unspecified derivative in either the space or time directions.

All relativistic field equations in (classical) physics are variants of the free wave equation

where f is either scalar or vector-valued. One can also consider add a mass term to obtain the Klein-Gordon equation

In practice, this mass term makes absolutely no difference to the local well-posedness theory of an equation (since the mass term f is negligible for high frequencies), but often plays a key role in the global theory (because of the improved decay and dispersion properties, and because the Hamiltonian controls the low frequencies more effectively).

There are several ways to perturb this equation. There are linear perturbations, which include the addition of potential terms, connection terms, and drag terms, as well as the replacement of the flat Minkowski metric with a more general curved metric, or by placing obstacles or otherwise changing the topology of the domain manifold . There is an extensive literature on all of these perturbations, but we shall not discuss them in depth, and concentrate instead on model examples of non-linear perturbations to the free wave equation. In the fullest generality, this would mean studying equations of the form

where D denotes differentation in space or time and the Taylor expansion of F to first order is the free wave or Klein-Gordon equation. Such fully non-linear equations, though, are very difficult to study, and have only really been analyzed in the one-dimensional case (in which case it can be subsumed into the general theory of 1+1-dimensional hyperbolic systems). In higher dimensions the only known tool to analyze this case is to differentiate the equation, turning it into a quasi-linear system. As such we do not discuss fully non-linear wave equations here. Instead, we consider three less general types of equations, which in increasing order of complexity are the #Semilinear wave equations, [#dnlw semi-linear with derivatives], and [#Quasilinear quasi-linear] equations.

Non-linear wave equations are often the Euler-Lagrange equation for some variational problem, usually with a Lagrangian that resembles

(this being the Lagrangian for the free wave equation). As such the equation usually comes with a divergence-free stress-energy tensor , which in turn leads to a conserved Hamiltonian E(f) on constant time slices (and other spacelike surfaces). There are a few other conserved quantites such as momentum and angular momentum, but these are rarely useful in the well-posedness theory. It is often worthwhile to study the behaviour of E(Df) where D is some differentiation operator of order one or greater, preferably corresponding to one or more Killing or conformal Killing vector fields. These are particularly useful in investigating the decay of energy at a point, or the distribution of energy for large times.

It is often profitable to study these equations using conformal transformations of spacetime. The Lorentz transformations, translations, scaling, and time reversal are the most obvious examples, but conformal compactification (mapping conformally to a compact subset of known as the Einstein diamond) is also very useful, especially for global well-posedness and scattering theory. One can also blow up spacetime around a singularity in order to analyze the behaviour near that singularity better.

The one-dimensional case n=1 is special for several reasons. Firstly, there is the very convenient null co-ordinate system which can be used to factorize Also, the stress-energy tensor often becomes trace-free, which leads to better conformal invariance properties. There are a vastly larger number of conformal transformations, indeed anything of the form is conformal. Also, the one-dimensional wave equation has no decay, local smoothing, or dispersion properties, and its solutions are essentially travelling waves. Finally, there are a much larger range of spaces beyond Sobolev spaces which are available for well-posedness theory, because the free wave evolution operator preserves all translation-invariant spaces. (In two and higher dimensions only -based spaces such as Sobolev spaces are preserved, because waves can focus at a point (or defocus from a point)).

The higher-dimensional case is usually quite different from the one-dimensional case, although in spherically symmetric situations one can obtain similar behaviour, especially when viewed in the null co-ordinates Indeed one can think of spherically symmetric wave equations as one-dimensional wave equations with a singular drag term

A very basic property of wave equations is finite speed of propagation: information only propagates at the speed of light (which we have normalized to 1) or slower. Also, singularities only propagate at the speed of light (even for Klein-Gordon equations). This allows one to localize space whenever time is localized. Because of this, there is usually no distinction between periodic and non-periodic wave equations. Another application is to convert local existence results for large data to that of small data (though in sub-critical situations this is often better achieved by scaling or similar arguments). Also, the behaviour of blowup at a point is only determined by the solution in the backwards light cone from that point; thus to avoid blowup one needs to show that the solution cannot concentrate into a backwards light cone. One can also use finite speed of propagation to truncate constant-in-space solutions (which evolve by some simple ODE) to obtain localized solutions. This is often useful to demonstrate blowup for various focussing equations.

The non-linear expressions which occur in non-linear wave equations often have a null form structure. Roughly speaking, this means that travelling waves do not self-interact, or only self-interact very weakly. When one has a null form present, the local and global well-posedness theory often improves substantially. There are several reasons for this. One is that null forms behave better under conformal compactification. Another is that null forms often have a nice representation in terms of conformal Killing vector fields. Finally, bilinear null forms enjoy much better estimates than other bilinear forms, as the interactions of parallel frequencies (which would normally be the worst case) is now zero.

An interesting variant of these equations occur when one has a coupled system of two fields u and v, with v propagating slower than u, e.g.

where and

for some This case occurs physically when u propagates at the speed of light and v propagates at some slower speed. In this case the null forms are not as useful, however the estimates tend to be more favourable (if the non-linearities F, G are "off-diagonal") since the light cone for u is always transverse to the light cone for v. One can of course generalize this to consider multiple speed (nonrelativistic) wave equations.

Wave estimates

Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms or in spaces, defined by

Linear space-time estimates are known as #linear Strichartz estimates. They are especially useful for the #semilinear semilinear NLW without derivatives, and also have applications to other non-linearities, although the results obtained are often non-optimal (Strichartz estimates do not exploit any null structure of the equation). The spaces are used primarily for #Bilinear estimates, although more recently #Multilinear estimates have begun to appear. These spaces first appear in one-dimension in [RaRe1982] and in higher dimensions in [Be1983] in the context of propagation of singularities; they were used implicitly for LWP in [KlMa1993], while the Schrodinger and KdV analogues were developed in [Bo1993], [Bo1993b].

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