Wave equations

Non-linear wave equations

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Overview

Let $R^{d+1}$ be endowed with the Minkowski metric

$ds^{2}=dx_{}^{2}-dt^{2}$.

(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

$\Box :=\partial _{a}\partial ^{a}=\Delta -\partial _{t}^{2}$

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 D 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

$\Box f=0,$

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

$\Box f=f$.

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 R^{n+1}. 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

$F(f,Df,D_{}^{2}f)=0$

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 semi-linear], [#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

$\int \partial _{a}f\partial ^{a}fdxdt$

(this being the Lagrangian for the free wave equation). As such the equation usually comes with a divergence-free stress-energy tensor $T^{ab}$, 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 $R^{d+1}$ conformally to a compact subset of $S^{d}xR$ 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 $u=t+x,v=t-x$ which can be used to factorize $\Box$. 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 $(u,v)\rightarrow (F(u),Y(v))$ 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 $L^{2}$-based spaces such as Sobolev spaces H^s 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 $u=t+r,v=t-r$. Indeed one can think of spherically symmetric wave equations as one-dimensional wave equations with a singular drag term $(n-1)f_{r}/r$.

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 $exp(i(k.x+-|k|t))$ 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.

$\Box u=F(U,DU),~\Box _{s}v=G(U,DU)$

where $U=(u,v)$ and $\Box _{s}=s^{2}\Delta -\partial _{t}^{2}$ for some $0<s<1$. 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.

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Semilinear wave equations

[Note: Many references needed here!]

Semilinear wave equations (NLW) and semi-linear Klein-Gordon equations (NLKG) take the form

$\Box f=F(f),\Box f=f+F(f)$

respectively where $F$ is a function only of $f$ and not of its derivatives, which vanishes to more than first order.

Typically $F$ grows like $|f|^{p}$ for some power $p$. If $F$ is the gradient of some function $V$, then we have a conserved Hamiltonian

$\int |f_{t}|^{2}/2+|\nabla f|^{2}/2+V(f)\ dx.$

For NLKG there is an additional term of $|f|^{2}/2$ in the integrand, which is useful for controlling the low frequencies of $f$ . If V is positive definite then we call the NLW defocussing; if $V$ is negative definite we call the NLW focussing. The term "coercive" does not have a standard definition, but generally denotes a potential $V$ which is positive for large values of $f$ .

semilinear NLW

NLW with derivatives

DNLW

DNLW

DDNLW

DDNLW

Yang-Mills

Yang-Mills Equations

Maxwell-Klein-Gordon

Maxwell-Klein-Gordon Equations

Dirac

Dirac Equations

Wave maps

Wave maps are maps f from R^{d+1} to a Riemannian manifold M which are critical points of the Lagrangian

\int f _ a . f ^ a dx dt.

When M is flat, wave maps just obey the wave equation (if viewed in flat co-ordinates). More generally, they obey the equation

Box f = G ( f ) Q_0( f , f )

where G ( f ) is the second fundamental form and Q_0 is the null form [#ddnlw mentioned earlier]. When the target manifold is a unit sphere, this simplifies to

Box f = - f Q_0( f , f )

where f is viewed in Cartesian co-ordinates (and must therefore obey | f |=1 at all positions and times in order to stay on the sphere). The sphere case has special algebraic structure (beyond that of other symmetric spaces) while also staying compact, and so the sphere is usually considered the easiest case to study. Some additional simplifications arise if the target is a Riemann surface (because the connection group becomes U(1), which is abelian); thus S² is a particularly simple case.

This equation is highly geometrical, and can be rewritten in many different ways. It is also related to the Einstein equations (if one assumes various symmetry assumptions on the metric; see e.g. BgCcMc1995).

wave maps

Quasilinear wave equations (QNLW)

In a local co-ordinate chart, quasilinear wave equations (QNLW) take the form

partial_ a g^{ ab }(u) partial_ b u = F(u, Du).

One could also consider equations where the metric depends on derivatives of u, but one can reduce to this case (giving up a derivative) by differentiating the equation. One can also reduce to the case g^{00} = 1, g^{0i} = g^{i0} = 0 by a suitable change of variables. F is usually quadratic in the derivatives Du, as this formulation is then robust under many types of changes of variables.

Quasilinear NLWs appear frequently in general relativity. The most famous example is the [#Einstein Einstein equations], but there are others (coming from relativistic elasticity, hydrodynamics, [#Minimal_Surface_Equation minimal surfaces], etc. [Ed: anyone willing to contribute information on these other equations (even just their name and form) would be greatly appreciated.]). The most interesting dimension is of course the physical dimension d=3.

Classically one has LWP for H^s when s > d/2+1 HuKaMar1977, but the [#dnlw semilinear theory] suggests that we should be able to improve this to s > s_c = d/2 with a null condition, and to s > d/2 + max(1/2, (d-5)/4) without one (these results would be sharp even in the semilinear case). In principle Strichartz estimates should be able to push down to s > d/2 + 1/2, but only partial results of this type are known. Specifically:

• When d=2 one has LWP in the expected range s > d/2 + 3/4 without a null condition [SmTt-p]
  • For s > d/2 + 3/4 + 1/12 this is in [Tt-p5] (using the FBI transform).
  • For s > d/2 + 3/4 + 1/8 this is in BaCh1999 (using FIOs) and Tt2000 (using the FBI transform).
• When d=3,4,5 one has LWP for s > d/2 + 1/2 [SmTt-p] (using parametrices and the equation for the metric); in the specific case of the [Einstein%20equations Einstein equations] see [KlRo-p3], [KlRo-p4], [KlRo-p5] (using vector fields and the equation for the metric)
  • For s > d/2 + 1/2 + 1/7 (approx) and d=3 this is in [KlRo-p2] (vector fields and the equation for the metric)
  • For s > d/2 + 1/2 + 1/6 and d=3 this is in [Tt-p5] (using the FBI transform).
  • For s > d/2 + 1/2 + 1/5 (approx) and d=3 this is in [Kl-p2] (vector fields methods).
  • For s > d/2 + 1/2 + 1/4 and d\geq 3 this is in BaCh1999 (using FIOs) and Tt2000 (using the FBI transform). See also BaCh1999b.

A special type of QNLW is the cubic equations, where g itself obeys an elliptic equaton of the form Delta g = |Du|^2, and the non-linearity is of the form Dg Du. For such equations, we have LPW for s > d/2 + 1/6 when d \geq 4 [BaCh-p], BaCh2002. This equation has some similarity with the differentiated wave map equation in the Coulomb gauge.

For small smooth compactly supported data of size e and smooth non-linearities, the GWP theory for QNLW is as follows.

• If the non-linearity is a null form, then one has GWP for d\geq3; in fact one can take the data in a weighted Sobolev space H^{4,3} x H^{3,4} Cd1986.
  • Without the null structure, one has almost GWP in d=3 Kl1985b, and this is sharp Jo1981, Si1983
    • In the semi-linear case and when the nonlinearity is quadratic in the derivatives, this is also true outside of a compact non-trapping obstacle [KeSmhSo-p2]. This has been generalized to the quasi-linear case in [KeSmhSo-p3] (and non-linear Dirichlet wave equations are also treated there, as are multiple speeds).
  • With a null structure and outside a star-shaped obstacle with Dirichlet conditions and d=3, one has GWP for small data in H^{9,8} x H^{8,9} which are compatible with the boundary [KeSmhSo-p]. Earlier work in this direction is in Dt1990.
    • For radial data and obstacle this was obtained in Go1995; see also Ha1995, Ha2000.
    • In the semilinear case, the non-trapping condition was removed in [MetSo-p], even in the multiple speed case, provided one has an exponential decay result near the obstacle (this is true, for instance, if the obstacle is a union of a finite number of sufficiently separated strictly convex bodies).
  • For d>3 or for cubic nonlinearities one has GWP regardless of the null structure refernces:KlPo1983 KlPo1983, Sa1982, Kl1985b.
    • In three dimensions with a null structure, for systems with multiple wave speeds, one has GWP So2001
    • In the exterior of an nontrapping obstacle with Dirichlet conditions, with multiple speeds, one has GWP for sufficiently smooth and decaying data obeying the usual compatibility conditions at the boundary [MetSo-p2], if the quasilinear terms obey some symmetry conditions and the semilinear terms are quadratic in Du
      1. When the obstacle is a ball this is in Ha1995.
      2. For d \geq 6 outside of a starshaped obstacle this is in ShbTs1984, ShbTs1986.

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Einstein equations

[Note: This is an immense topic, and we do not even begin to do it justice with this very brief selection of results. For more detail, we recommend the very nice survey on existence and global dynamics of the Einstein equations by Alan Rendall. Further references are, of course, always appreciated. We thank Uwe Brauer, Daniel Pollack, and some anonymous contributors to this section.]

The (vacuum) Einstein equations take the form

R_{ a b } = C R g_{ a b }

where g is the metric for a 3+1-dimensional manifold, R is the Ricci curvature tensor, and C is an absolute constant. The Cauchy data for this problem is thus a three-dimensional Riemannian manifold together with the second fundamental form of this manifold (roughly speaking, this is like the initial position and initial velocity for the metric g). However, these two quantities are not completely independent; they must obey certain constraint equations. These equations are now known to be well behaved for all s > 3/2 [Max-p], [Max2005] (see also earlier work in higher regularities in RenFri2000, Ren2002).

Einstein Equations

Minimal surface equation

This quasilinear equation takes the form

partial _a [ (1 + f_bf^b )^{-1/2} f^a ] = 0

where f is a scalar function on R^{n-1}xR (the graph of a surface in R^n x R ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see Hp1994.

• This is a [#Quasilinear quasilinear wave equation], and so LWP in H^s for s > n/2 + 1 follows from energy methods, with various improvements via Strichartz possible. However, it is likely that the special structure of this equation allows us to do better.
• GWP for small smooth compactly supported data is in [Lb-p].

Wave estimates

Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms $L_{t}^{q}L_{x}^{r}$, or in $X_{}^{s,b}$ spaces, defined by

$\|u\|_{X^{s,b}}=\|<\xi >^{s}<|\xi |-|\tau |>^{b}{\hat {u}}(\tau ,\xi )\|_{2}$

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 $X_{}^{s,b}$ spaces are used primarily for [#bilinear bilinear estimates], although more recently [#multilinear 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.

Linear wave estimates

Bilinear wave estimates

Multilinear wave estimates