Let R^{d+1} be endowed with the Minkowski metric
ds2 = dx2 - dt2.
(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 = D - \partial_t2
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, D2f) = 0
where D denotes differentation in space or time and the
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 f dx dt
(this being the Lagrangian for the free wave equation). As such the equation usually comes with a divergence-free stress-energy tensor T^{a b}, 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 x R 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) -> (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 L2-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 = s2 D - \partial_t2 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.
Solutions to the linear wave equation and its perturbations are either estimated in mixed space-time norms L^q_t L^r_x, or in X^{s,b} spaces, defined by
|| u ||_{s,b} = || <x>^s <|x|-|t|>^b \hat{u} ||_2
Linear space-time estimates are known as Strichartz estimates. They are especially useful for the 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 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].
[Thanks to Chengbo Wang for some corrections – Ed.]
These estimates extend to some extent to the Klein-Gordon equation Box u = m^2 u. A useful heuristic to keep in mind is that this equation behaves like the Schrodinger equation +i u_t + 1/(2m) Delta u = 0 when the frequency x has magnitude less than m, but behaves like the wave equation for higher frequencies. Some basic Strichartz estimates here are in [MsSrWa1980]; see for instance [Na-p], [MacNaOz-p], [MacNkrNaOz-p] for more recent treatments.
For inhomogeneous estimates it is known that a solution with zero initial data and forcing term containing s-1 derivatives in a dual
space L^Q’_t L^R’_x will lie in L^q_t L^r_x if both (q,r) and (Q,R) are admissible in the above sense, if s >= 0, and if one has the scaling condition
1/Q + d/R + 1/q + d/r = d + s + 1. The “s-1” represents a smoothing effect of one derivative, though this full gain is only attainable if one uses
the energy exponents L^1_t L^2_x and L^infty_t L^2_x. It is possible to obtain inhomogeneous estimates in which only one of the exponents
are admissible; this phenomenon was first observed in [Har1990], [Ob1989] (see also [KeTa1998]). More recently in [Fc-p2], inhomogeneous
estimates are obtained with the above scaling condition assuming the weaker conditions 1 <= q,r <= infty and 1/q < (n-1)(1/2-1/r) or (q,r) = (infty,2)
and similarly for Q,R, and if the following additional conditions hold:
· In d=1,2 no further conditions are required;
· When d=3 r, R are required to be finite;
· When d > 3, either 1/q + 1/Q < 1, (n-3)/r <= (n-1)/R, and (n-3)/R <= (n-1)/r, or 1/q + 1/Q = 1, (n-3)/r < (n-1)/R, (n-3)/R < (n-1)/r, r >= q, and R >= Q.
Strichartz estimates extend to situations in which there is a potential or when the metric is variable. For local-in-time estimates and smooth potentials or
metrics this is fairly straightforward (the potential can be treated by iterative methods, and the metric by parametrix methods). More interesting issues
arise for global-in-time estimates with smooth potentials/metrics or local-in-time estimates with rough potentials/metrics (the two types of results are linked by
scaling). For potentials of power-type decay, the global results are as follows:
· For potentials of the form V = a/|x|^2 with d >= 3 and a > -(d-2)^2/4, one has global Strichartz estimates [BuPlStaTv-p]; a simplified proof and more general result dealing with inverse square-like potentials which are not too negative is in [BuPlStaTv-p2]. The condition on a is necessary to avoid bound states. For potentials decaying slower than this, Strichartz estimates can fail (Duyckaerts?)
· For potentials decaying an epsilon faster than 1/|x|^2, and assumed to be nonnegative, dispersive and Strichartz estimates were obtained when d=3 in [GeVis2003].
(More results to be added in future).
See [FcKl2000]. Null forms can also be handled by identities such as
2 Q_0(f, y) = Box(f y).
D^{-2a} (fy) is in L^{q/2}_t L^{r/2}_x
as long as 0 <= a <= d/2 - 2/q - d/r [FcKl-p]. Similar estimates for null forms also exist [Pl2002]; see also [TaVa2000b], [Ta-p4].
There are some isolated examples of multilinear estimates that cannot be obtained from linear and bilinear estimates and Holder's inequality. For instance, the inequality
|\int D^{-a}D_-(u_1 u_2) u_3 u_4 dx dt| <~ ||f_1||_{H^{n/2-a}} ||f_2||_2 ||f_3||_2 ||f_4||_2
is proven for d=2 and 3/4 < a < 1 in [Sb-p].
[Note: Many references needed here!]
Semi-linear 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 + |Ñf|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.
To analyze these equations in H^s we need the non-linearity to be sufficiently smooth. More precisely, we will always assume either that F is smooth, or that F is a p^th-power type non-linearity with p > [s]+1.
The scaling regularity is s_c = d/2 - 2/(p-1). Notable powers of p
include the L2-critical power p_{L2} = 1 +
4/d, the H^{1/2}-critical or conformal power p_{H^{1/2}} = 1 + 4/(d-1),
and the H^1-critical power p_{H^1} = 1 + 4/{d-2}.
Dimension d |
Strauss exponent (NLKG) |
L2-critical exponent |
Strauss exponent (NLW) |
H^{1/2}-critical exponent |
H^1-critical exponent |
1 |
3.56155... |
5 |
infinity |
infinity |
N/A |
2 |
2.41421... |
3 |
3.56155... |
5 |
infinity |
3 |
2 |
2.33333... |
2.41421... |
3 |
5 |
4 |
1.78078... |
2 |
2 |
2.33333... |
3 |
The following necessary conditions for LWP are known. Firstly, for focussing NLW/NLKG one has blowup in finite time for large data, as can be seen by the ODE method. One can scale this and obtain ill-posedness for any focussing NLW/NLKG in the supercritical regime s < s_c; this has been extended to the defocusing case in [CtCoTa-p2]. By using Lorentz scaling instead of isotropic scaling one can also obtain ill-posedness whenever s is below the conformal regularity
s_{conf} = (d+1)/4 - 1/(p-1)
in the focusing case; the defocusing case is still open. In the H^{1/2}-critical power or below, this condition is stronger than the scaling requirement.
To make sense of the non-linearity in the sense of distributions we need s >= 0 (indeed we have illposedness below this regularity by a high-to-low cascade, see [CtCoTa-p2]). In the one-dimensional case one also needs the condition 1/2 - s < 1/p to keep the non-linearity integrable, because there is no Strichartz smoothing to exploit.
Finally, in three dimensions one has ill-posedness when p=2 and s = s_{conf}
= 0 [Lb1993].
p (d/4-s) <= 1/2 ( (d+3)/2 - s) (*)
and excluding the double endpoint when (*) holds with equality and s=s_{conf} [Ta1999]. The main tool is two-scale Strichartz estimates.
p ((d+1)/4-s) <= (d+1)/2d ( (d+3)/2 - s); (**)
see [KeTa1998] for the double endpoint when (**) holds with equality and s=s_{conf}, and [LbSo1995] for all other cases. A slightly weaker result also appears in [Kp1994].
GWP and scattering for NLW is known for data with small H^{s_c} norm when p is at or above the H^{1/2}-critical power (and this has been extended to Besov spaces; see [Pl-p4]. This can be used to obtain self-similar solutions, see [MiaZg-p2]). One also has GWP in H^1 in the defocussing case when p is at or below the H^1-critical power. (At the critical power this result is due to [Gl1992]; see also [SaSw1994]. For radial data this was shown in [Sw1988]). For more scattering results, see below.
For the defocussing NLKG, GWP in H^s, s < 1, is known in the following cases:
s > [2(p-1)^2 - (d+2-p(d-2))(d+1-p(d-1))] / [2(p-1)(d+1-p(d-3))]
[MiaZgFg-p]. Note that this is the range of p for which s_conf obeys both the scaling condition s_conf > s_c and the condition (**).
GWP and blowup has also been studied for the NLW with a conformal factor
Box u = (t^2 + (1 - (t^2-x^2)/4)^2)^{-(d-1)p/4 + (d+3)/4} |u|^p;
the significance of this factor is that it behaves well under conformal compactification. See [Aa2002], [BcKkZz2002], [Gue2003] for some recent results.
Scattering theory for semilinear NLW
[Thanks to Kenji Nakanishi for many helpful additions to this section - Ed.]
The Strauss exponent
p_0(d) = [d + 2 + sqrt(d2 + 12d + 4)]/2d
plays a key role in the GWP and scattering theory. We have p_0(1) = [3+sqrt(17)]/2; p_0(2) = 1+sqrt(2); p_0(3) = 2; note that p_0(d-1) is always between the L2 and H^{1/2} critical powers, and p_0(d) is always between the H^{1/2} and H^1 critical powers.
Another key power is
p_*(d) = [d+2 + sqrt(d^2 + 8d)]/2(d-1)
which lies between the L^2 critical power and p_0(d-1).
Caveats: the d=1,2 cases may be somewhat different from what is stated here (partly because some of the powers here are not well-defined). Also, in many of the NLW results one needs some additional decay at spatial infinity (e.g. finiteness of the conformal energy), except in the special H^1-critical case. This is because (unlike NLS and NLKG) there is no a priori bound on the L2 norm (even with conservation of energy).
Scattering for small H^1 data for arbitrary NLW:
Scattering for large H^1 data for defocussing NLW:
Scattering for small smooth compactly supported data for arbitrary NLW:
Scattering for small H^1 data for arbitrary NLKG:
Scattering for large H^1 data for defocussing NLKG:
Scattering for small smooth compactly supported data for arbitrary NLKG:
An interesting (and apparently under-explored) problem is what happens to these global existence and scattering results when there is an obstacle. For NLW-5 on R3 one has global regularity for convex obstacles [SmhSo1995], and for smooth non-linearities there is the general quasilinear theory. If one adds a suitable damping term near the obstacle then one can recover some global existence results [Nk2001].
On the Schwarzschild manifold some scattering and decay results for NLW and NLWKG can be found in [BchNic1993], [Nic1995], [BluSf2003]
Non-relativistic limit of NLKG
By inserting a parameter c (the speed of light), one can rewrite NLKG as
u_tt/c2 - D u + c2 u + f(u) = 0.
One can then ask for what happens in the non-relativistic limit c -> infinity (keeping the initial position fixed, and dealing with the initial velocity appropriately). In Fourier space, u should be localized near the double hyperboloid
t = +- c sqrt(c2 + x2).
In the non-relativistic limit this becomes two paraboloids
t = +- (c2 + x2/2)
and so one expects u to resolve as
u = exp(i c2
t) v_+ + exp(-i c2 t) v_-
u_t = ic2 exp(ic2 t) v_+ - ic2 exp(ic2
t) v_-
where v_+, v_- solve some suitable NLS.
A special case arises if one assumes (u_t - ic2 u) to be small at time zero (say o(c) in some Sobolev norm). Then one expects v_- to vanish and to get a scalar NLS. Many results of this nature exist, see [Mac-p], [Nj1990], [Ts1984], [MacNaOz-p], [Na-p]. In more general situations one expects v_+ and v_- to evolve by a coupled NLS; see [MasNa2002].
Heuristically, the frequency << c portion of the evolution should evolve in a Schrodinger-type manner, while the frequency >> c portion of the evolution should evolve in a wave-type manner. (This is consistent with physical intuition, since frequency is proportional to momentum, and hence (in the nonrelativistic regime) to velocity).
A similar non-relativistic limit result holds for the Maxwell-Klein-Gordon
system (in the Coulomb gauge), where the limiting equation is the coupled
Schrodinger-Poisson system
i v^+_t + D v/2 = u v^+
i v^-_t - D v/2 = u v^-
D u = - |v^+|2 + |v^-|2
under reasonable H^1 hypotheses on the initial data [BecMauSb-p]. The asymptotic relation between the MKG-CG fields f, A, A_0 and the Schrodinger-Poisson fields u, v^+, v^- are
A_0 ~ u
f ~ exp(ic2 t) v^+ + exp(-ic2
t) v^-
f_t ~ i M exp(ic2)v^+ - i M
exp(-ic2 t) v^-
where M = sqrt(c^4 - c2 D) (a variant of c2).
[Contributions to this section are sorely needed!]
The sine Gordon equation
Box u = sin(u)
in R^{1+1} arises in the study of optical pulses, or from the Scott model of a continuum of pendula hanging from a wire. It is a completely integrable equation, and has many interesting solutions, including "breather" solutions.
Because the non-linearity is bounded, GWP is easily obtained for L2 or even L^1 data.
[Thanks to Chengbo Wang for some corrections – Ed.]
Quintic NLW/NLKG on R2
The D-NLW equation is given by
Box u = F(u, Du)
where u is scalar or vector valued, and F is at least quadratic. The D-NLKG equation is similar:
Box u = u + F(u, Du).
From a LWP perspective the two equations are virtually equivalent, but the NLKG is slightly better behaved for the GWP (it decays like the NLW at one higher dimension).
Among the more intensively studied derivative NLWs are the quadratic DNLWs (which include the Yang-Mills and Maxwell-Klein-Gordon equations) and the DDNLWs (which include equations of wave maps type).
From energy estimates one can always obtain LWP in H^s for s > n/2 + 1. (or s > n/2 if the non-linearity is at most linear in Du). However, this is rarely best possible.
In many cases the non-linearity F has a null structure. The precise meaning of this is hard to quantify exactly, but roughly speaking this means that if u is a plane wave then the highest order terms in the non-linearity vanish. In other words, the self-interaction of plane waves is relatively small. The presence of a null structure usually makes the LWP and GWP theory significantly better.
The GWP theory for small data is usually accomplished by vector fields
methods (or similar methods which try to capture the decay, and proximity to
the light cone, of the global solution), or via conformal
compactification. The method of normal forms is also often useful, as it can
eliminate the worst terms in a non-linearity.
As a general principle, the small data GWP theory becomes better whenever the
order of the non-linearity increases (because this makes the non-linearity even
smaller) or when the dimension increases (because there is more decay).
There is rarely any need to distinguish between u and Du in the small data GWP
theory. In many cases the theory is robust enough to carry over to the quasilinear case.
For small smooth compactly supported data of size e and smooth non-linearities, the GWP theory for D-NLW is as follows.
For small smooth compactly supported data of size e and smooth non-linearities, the GWP theory for D-NLKG is as follows.
For small smooth data of size e on the circle T, and smooth non-linearities, the GWP theory for D-NLKG is as follows.
For non-smooth non-linearities of order p, one has blow-up examples for D-NLKG from small smooth compactly supported data whenever p <= 1+2/d [KeTa1999], although in certain cases (esp. coercive Hamiltonian systems) one still has GWP [Ca1985]. One also has failure of scattering for this range of powers [Gl1981b]. It would be interesting to see if one could obtain GWP for D-NLKG the p>1+2/d case (though in the non-smooth non-linearity case one probably is restricted to d=1,2,3 in order to keep the non-linearity sufficiently smooth).
The equation
\Box u = u2 u_t
in three dimensions is known to be locally well-posed in any sub-critical regularity s>1, and has scattering in H3 [Smh-p]. It would be interesting to see whether one has local well-posedness in the critical energy regularity H^1.
By Quadratic DNLW we mean equations with the schematic form
\Box f = f Df.
This equation has the same scaling as cubic NLW, but is more difficult technically because of the derivative in the non-linearity. In practice one can always add a cubic term f3 to the non-linearity without disrupting any of the well-posedness theory, as f3 is usually much easier to estimate than f Df.
Important examples of this type of equation include the Maxwell-Klein-Gordon and Yang-Mills equations (in the Lorentz gauge, at least), as well as the simplified model equations for these equations. The Yang-Mills-Higgs equation is formed by coupling equations of this type to a semi-linear wave equation. The most interesting dimensions are 3 (for physical applications) and 4 (since the energy regularity is then critical). The two-dimensional case appears to be somewhat under-explored. The Yang-Mills and Maxwell-Klein-Gordon equations behave very similarly, but from a technical standpoint the latter is slightly easier to analyze.
In d dimensions, the critical regularity for this equation is s_c = d/2 - 1. However, there are no instances of Quadratic DNLW for which any sort of well-posedness is known at this critical regularity (except for those special cases where the equation can be algebraically transformed into a simpler equation such as NLW or the free wave equation).
Energy estimates give local well-posedness for s > s_c + 1. Using Strichartz estimates this can be improved to s > s_c + 3/4 in two dimensions and s > s_c + 1/2 in three and higher dimensions [PoSi1993]; the point is that these regularity assumptions together with Strichartz allow one to put f into L2_t L^¥_x, hence in L^1_t L^¥_x, so that one can then use the energy method.
Using X^{s,\theta} estimates [FcKl2000] instead of Strichartz estimates, one can improve this further to d > s_c + 1/4 in four dimensions and to the near-optimal s > s_c in five and higher dimensions. In six and higher dimensions one can obtain global well-posedness for small critical Besov space B^{s_c}_{2,1} [Stz-p3], and local well-posedness for large Besov data. In four dimensions one has a similar result if one imposes one additional angular derivative of regularity [Stz-p2].
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in H^1 [Lb1993], although one can recover well-posedness in the Besov space B^1_{2,1} [Na1999], or when an epsilon of radial regularity is imposed [MacNkrNaOz-p]. It would be interesting to determine what the situation is in the other low dimensions.
If the non-linearity f Df has a null structure then one can improve upon the previous results. For instance, the model equations for the Yang-Mills and Maxwell-Klein-Gordon equations are locally well-posed for s > s_c in three [KlMa1997] and higher [KlTt1999] dimensions. It would be interesting to determine what happens in two dimensions for these equations (probably one can get down to s > s_c + 1/4). In one dimension the model equation trivially collapses to the free wave equation.
Let A be a connection on R^{d+1} which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the Yang-Mills equation if it is a critical point for the Lagrangian functional
\int F^{a b} F_{a b}
where F:=dA + [A,A] is the curvature of the connection A. The Euler-Lagrange equations for this functional have the schematic form
\Box A + Ñ (\div_{x,t} A) = [A, Ñ A] + [A, [A,A]]
where \div_{x,t} A = \partial_a A^a is the spacetime divergence of A. A more succinct (but less tractable) formulation of this equation is
\partial_a F^{ab} = 0.
It is often convenient to split A into temporal and spatial components as A = (A_0, A_i).
As written, the Yang-Mills equation is under-determined because of the gauge invariance
A -> U^{-1} dU +
U^{-1} A U
F -> U^{-1} F U
in the equation, where U is an arbitrary function taking values in G. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones:
Temporal gauge: A^0 =
0
Coulomb gauge: \partial_i A_i = 0
Lorentz gauge: div_{x,t} A = 0
There are also several other useful gauges, such as the Cronstrom gauge [Cs1980] centered around a point in spacetime.
The Lorentz gauge has the advantage of being invariant under conformal transformations, but it appears that the Yang-Mills equation is not well-behaved in this gauge for rough data. (For smooth data one can obtain local well-posedness in this gauge by energy estimates). The Coulomb gauge is the simplest to work with technically, and in this gauge the bilinear expression [A, Ñ A] acquires a null structure [KlMa1995] which allows for a satisfactory analysis of the equation. Unfortunately there are often difficulties in creating a global Coulomb gauge, and one often has to rely instead on local Coulomb gauges pieced together using finite speed of propagation; see [KlMa1995]. The Temporal gauge is fairly close to the Coulomb gauge, and one can develop a parallel theory for this gauge. The temporal gauge has the advantage of being easy to establish globally, but the null form structure is less obvious (one needs to partition the connection into divergence-free and curl-free components). See e.g. [Ta-p3].
In the Coulomb or Temporal gauges, one can create a model equation for the Yang-Mills system by ignoring cubic terms and any contribution from the "elliptic'' portion of the gauge (A_0 in the Coulomb gauge, or the curl-free portion of A_i in the Temporal gauge). The resulting model equation is
\Box A = Ñ^{-1} Q(A,A) + Q(Ñ^{-1}A, A)
where Q(A,A') is some null form such as
Q(A,A') := \partial_i A \partial_j A' - \partial_j A \partial_i A'.
The results known for the model equation are slightly better than those known for the actual Yang-Mills or Maxwell-Klein-Gordon equations.
The Yang-Mills equations come with a positive definite conserved Hamiltonian
\int |F_{0,i}|2 + |F_{i,j}|2 dx
which mostly controls the H^1 norm of A and the L2 norm of A_t. However, there are some portions of the H^1 x L2 norm which are not controlled by the Hamiltonian (in the Coulomb gauge, it is \partial_t A_0; in the Temporal gauge, it is the H^1 norm of the curl-free part of A_i). This causes some technical difficulties in the global well-posedness theory.
The Yang-Mills equations can also be coupled with a g-valued scalar field f, with the Lagrangian functional of the form
\int F^{a b} F_{ab} + D_a f . D^af + V(f)
where D_a = \partial_a + [Aa, .] are covariant derivatives and V is some potential function (e.g. V(f) = |f|^{k+1}). The corresponding Euler-Lagrange equations have the schematic form
\Box A + Ñ (\div_{x,t} A) = [A, Ñ A] + [A, [A,A]] + [f, D f]
D_a D^a
f = V'(f)
and are generally known as the Yang-Mills-Higgs system of equations. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. The Maxwell-Klein-Gordon system is a special case of Yang-Mills-Higgs.
The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise.
Attention has mostly focussed on the three and four dimensional cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to A_tt = 0). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data [CaSaTv1998] (see also [Biz-p]). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension [BizTb2001], [Biz-p].
The Yang-Mills equations can also be coupled with a spinor field. In the U(1) case this becomes the Maxwell-Dirac equation.
The Yang-Mills equations in dimension n have many formal similarities with the wave maps equation at dimension n-2 (see e.g. [CaSaTv1998] for a discussion)
[Thanks to Jacob Sterbenz for corrections - Ed.]
MKG and Yang-Mills in R^n, n>4
The Maxwell-Klein-Gordon equation
The Maxwell-Klein-Gordon is the special case of the Yang-Mills-Higgs equation when the Lie group G is just the circle U(1), and there is no potential energy term V(f). Thus A is now purely imaginary, and f is complex.
Despite the name, the Maxwell-Klein-Gordon equation is not really related to the (massive) Klein-Gordon equation. Rather, it is the Maxwell equation coupled with a massless scalar equation (i.e. a free wave equation). If the scalar field f is set to 0, the equation collapses to the linear Maxwell equations, which are basically a vector-valued variant of the free wave equation.
As with Yang-Mills, the three standard gauges are Lorentz gauge, Coloumb gauge, and Temporal gauge. The Lorentz gauge is most natural from a co-ordinate free viewpoint, but is difficult to work with technically. In principle the temporal gauge is the easiest to work with, being local in space, but in practice the Coloumb gauge is preferred because the null form structure of Maxwell-Klein-Gordon is most apparent in this gauge.
In the Coulomb gauge MKG has the schematic form
D A_0 = ff_t
+ F3
\Box A = Ñ^{-1} Q(f, f)
\Box f = Q(Ñ^{-1} A, f) + (A_0)_t f + A_0 f_t
+ F3
where F3 denotes terms that are cubic in (A_0, A, f). Unfortunately, the equation for the A_0 portion of the Coloumb gauge is elliptic, which generates some low frequency issues. However, if we ignore the A_0 terms and the cubic terms then we reduce to the model equation
\Box A = Ñ^{-1} Q(f,
f)
\Box f = Q(Ñ^{-1} A, f)
which is slightly better than the corresponding model for Yang-Mills.
MKG has the advantage over YM that the Coulomb gauge is easily constructed globally using Riesz transforms, so there are less technical issues involved with this gauge.
[Thanks to Jacob Sterbenz for corrections - Ed.]
[More info on this equation would be greatly appreciated. - Ed.]
This equation essentially reads
D_A y = - y
\Box A + Ñ (\div_{x,t} A)= y y
where y is a spinor field (solving a coupled massive Dirac equation), and D is the Dirac operator with connection A. We put y in H^{s_1} and A in H^{s_2} x H^{s_2 - 1}.
In the nonrelativistic limit this equation converges to a Maxwell-Poisson system for data in the energy space [BecMauSb-p2]; furthermore one has a local well-posedness result which grows logarithmically in the asymptotic parameter. Earlier work appears in [MasNa2003].
[More info on this equation would be greatly appreciated. - Ed.]
This equation essentially reads
D y = f y - y
Box f = y y
where y is a spinor field (solving a
coupled massive Dirac equation), D is the Dirac operator and f is a scalar (real) field. We put
y in H^{s_1} and (f, f_t)
in H^{s_2} x H^{s_2 - 1}.
The energy class is essentially (s_1,s_2) = (1/2,1), but the energy density is not positive. However, the L^2 norm of y is also positive and conserved..
Nonlinear Dirac equation
This equation essentially reads
D y - m y = l(g y, y) y
where y is a spinor field, m > 0 is the mass, l is a complex parameter, g is the zeroth Pauli matrix, and (,) is the spinor inner product.
We use DDNLW to denote a semi-linear wave equation whose non-linear term is quadratic in the derivatives, i.e.
\Box f = G(f) Df Df.
A fairly trivial example of such equations arise by considering fields of the form f = f(u), where f is a given smooth function (e.g. f(x) = exp(ix)) and u is a scalar solution to the free wave equation. In this case f solves the equation
\Box f = f''(f) Q_0(f,f)
where Q_0 is the null form
Q_0(f, y) := \partial_af \partial^a y = Ñf . Ñy - f_t y_t.
The above equation can be viewed as the wave equation on the one-dimensional manifold f(R), with the induced metric from R. The higher-dimensional version of this equation is known as the wave map equation.
DDNLW behaves like DNLW but with all fields requiring one more derivative of
regularity. One explicit way to make this connection is to differentiate
DDNLW and view the resulting as an instance of DNLW for the system of fields (f, Df).
The reader should compare the results below with the corresponding
results for quadratic DNLW.
The critical regularity is s_c = d/2. For subcritical regularities s
> s_c, f has some Holder continuity,
and so one heuristically expects the G(f) terms
to be negligible. However, this term must play a crucial role in the
critical case s=s_c. For instance, Nirenberg [ref?] observed that the
real scalar equation
\Box f = -f Q_0(f, f)
is globally well-posed in H^{d/2}, but the equation
\Box f = f Q_0(f, f)
is ill-posed in H^{d/2}; this is basically because the non-linear operator f -> exp(if) is continuous on (real-valued) H^{d/2}, while f -> exp(f) is not.
Energy estimates show that the general DDNLW equation is locally well-posed in H^s for s > s_c + 1. Using Strichartz estimates this can be improved to s > s_c + 3/4 in two dimensions and s > s_c + 1/2 in three and higher dimensions; the point is that these regularity assumptions together with Strichartz allow one to put f into L2_t L^¥_x (or L^4_t L^¥_x in two dimensions), so that one can then use the energy method.
Using X^{s,b} estimates [FcKl2000] instead of Strichartz estimates, one can improve this further to s > s_c + 1/4 in four dimensions and to the near-optimal s > s_c in five and higher dimensions [Tt1999].
Without any further assumptions on the non-linearity, these results are sharp in 3 dimensions; more precisely, one generically has ill-posedness in H2 [Lb1993], although one can recover well-posedness in the Besov space B2_{2,1} [Na1999], or with an epsilon of radial regularity [MacNkrNaOz-p]. It would be interesting to determine what the situation is in the other low dimensions.
If the quadratic non-linearity (Ñf)2 is of the form Q_0(f,f) (or of a null form of similar strength) then the LWP theory can be pushed to s > s_c in all dimensions (see [KlMa1997], [KlMa1997b] for d >= 4, [KlSb1997] for d >= 2, and [KeTa1998b] for d=1).
If G(f) is constant and d is at least 4, then one has GWP outside of convex obstacles [Met-p2]
A special case of two-speed DDNLS arises in elasticity (more on this to be added in later).
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 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]).
The critical regularity is s_c = d/2. Thus the two-dimensional case is especially interesting, as the equation is then energy-critical. The sub-critical theory s > d/2 is fairly well understood, but the s_c = d/2 theory is quite delicate. A big problem is that H^{d/2} does not control L^¥, so one cannot localize to a small co-ordinate patch (or perform algebraic operations properly).
The positive and negative curvature cases are suspected to behave differently, especially at the critical regularity. Intuitively, the negative curvature space spreads the solution out more, thus giving a better chance for LWP and GWP. More recently, distinctions have arisen between the boundedly parallelizable case (where the exists an orthonormal frame whose structure constants and derivatives are bounded), and the isometrically embeddable case. For instance, hyperbolic space is in the former category but not in the latter; smooth compact manifolds such as the sphere are in both.
The general LWP/GWP theory (except for the special one-dimensional and two-dimensional cases, which are covered in more detail below) is as follows.
For further references see [Sw1997], [SaSw1998], [KlSb-p].
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 equations, but there are others (coming from relativistic elasticity, hydrodynamics, 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 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:
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 >= 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.
[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]).
Because of the diffeomorphism invariance of the Einstein equations, these equations are not hyperbolic as stated. However, this can be remedied by choosing an appropriate choice of co-ordinate system (a gauge, if you will). One popular choice is harmonic co-ordinates or wave co-ordinates, where the co-ordinate functions xa are assumed to obey the wave equation Boxg xa = 0 with respect to the metric g. In this case the Einstein equations take a form which (in gross caricature) looks something like
Boxg g = G(g) Q(dg, dg) + lower order terms
where Q is some quadratic form of the first two
derivatives. In other words, it becomes a quasilinear
wave equation. One would then specify initial
data on the initial surface x0 = 0; the co-ordinate x0
plays the role of time, locally at least.
This quasilinear equation takes the form
partiala [ (1 + fbfb)^{-1/2} fa ] = 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].