THE WAVE BESTIARY

TERENCE TAO

Abstract. We give some solutions to the wave equation, and indicate how they can be used to find the (conjectured) sharpest possible wave equation estimates (Strichartz, local smoothing, etc).

1. Linear examples

In this section we study some examples of the problem

□u = 0
u(0, x) = f(x)
u_{t}(0, x) = g(x)

and try to describe the shape of u, f, g. These examples give restrictions on the exponents in such estimates as Strichartz estimates

∥u∥_{L_{x}^{q}L_{t}^{r}} ≲ ∥f∥Ḣ^{s} + ∥g∥Ḣ^{s−1}

or local smoothing estimates

∥u∥_{L^{p}([1,2]×R^{n})} ≲ ∥f∥_{L_{α}^{p}} + ∥g∥_{L_{α−1}^{p}}

as well as the usual variants. We will not be very precise with stationary phase error estimates etc.; the point is more to give caricatures of the various subspecies and genera of the family of wave equation solutions. For instance, we shall be lax about the distinction between “compactly supported” and “rapidly decreasing”.

Our examples shall almost always be unit frequency, which means that their Fourier transforms are supported in the unit annulus. By rescaling one can make these examples have Fourier transforms supported in any annulus, and by adding several of them together one can get examples with more general Fourier support. But the unit frequency case already captures the shape and propagation characteristics of the examples, which is really the most important thing to understand.

We shall not concern ourselves here with endpoint examples - examples which show that the endpoint of a range of estimates fails. For some examples of these, see [8].

In the unit frequency case, derivatives are irrelevant, and f and g play virtually the same role (analogous to the roles of the cosine and sine in trigonometry). As such, we shall ignore g and talk mostly about f.

The most basic example is the flash. This is when f is a bump function adapted to the unit ball^{1} The solution u has size

|u(x, t)| ∼ (1 + |t|)^{−(n−1)/2}

1991 Mathematics Subject Classification. 42B.

1Since we are assuming f to have unit frequency, one cannot actually have f compactly supported on the ball, but it can be rapidly decreasing, which is good enough.

in the cone

{(x, t) : |t| = |x| + O(1)}.

Outside of this cone u is negligible; when the dimension is odd the strong Huygens principle kicks in and in fact u vanishes outside this cone (if f truly had compact support to begin with). In particular, |u| ∼ 1 on a ball of radius O(1) around the origin. (The flash hangs around for unit time before dispersing in an expanding sphere).

The above estimates are a rephrasing of the standard estimate for the decay of the fundamental solution of the wave equation, localized to unit frequency. The flash example predicts the sharp amount of local smoothing and the sharp fixed time estimates for p < 2, but is not a very good example for p > 2. (Estimates for p > 2 are about how waves concentrate; estimates for p < 2 are about how waves disperse. Estimates for p = 2 say neither, thanks to conservation of energy.)

By time shifting the flash upwards by R ≫ 1, we obtain the focus example. Now |f(x)| ∼ R^{−(n−1)/2} on the annulus {x : |x| = R + O(1)}, and the solution u(t, x) focuses when t = R +O(1), x = O(1), at which point |u(t, x)| ∼ 1. The focus example, and its scaled variants, show that Strichartz estimates use the sharp number of derivatives, and also provide a good example for local smoothing estimates for p large. (e.g. in R^{2+1} they should predict the sharp amount of derivative loss (ignoring epsilons) for p ≥ 4). They also predict the sharp fixed-time estimates for p > 2.

Now let R range over a O(1)-separated set from R_{0} to 2R_{0}. By summing up many focus examples together we get a strobe example. Now the initial data f has size |f(x)| ∼ R^{−(n−1)/2} on a large part of the annulus {x : R_{0} ≤ |x| ≤ 2R_{0}} (but the intensity alternates between light and dark with radial wavelength ∼ 1), and the solution u(t, x) has size ∼ 1 on a large part of the set R_{0} ≤ t ≤ 2R_{0}, x = O(1) (at the spatial origin, the wave keeps turning itself on and off). This example is the enemy of estimates which try to estimate u in too low a temporal exponent; for instance, for Strichartz estimates they give the necessary condition q ≥ 2. (They also prevent certain types of local smoothing estimates; unlike the focus example, these waves are not impressed by any sort of averaging in time.)

Now suppose we take the flash example and hit it with a Lorentz boost at a velocity near the speed of light, namely, 1 − 2^{−2j} for some large j. Before the boost, the flash lurked inside a unit ball for time O(1) before dispersing. After the boost, it behaves differently. It now lurks in a Lorentz contracted ball, or more precisely a 1 × 2^{−j} disk, for a dilated time of O(2^{j}) rather than O(1). Furthermore, the disk is no longer stationary, but travels at virtually the speed of light. (Well, to be precise, it travels at 1 − 2^{−2j}, but the discrepancy between this velocity and 1 is irrelevant at the given time scale and spatial scales). In equations,

|u(x, t)| ∼ 1 when t = O(2^{j}), \underline{x} = O(1), x_{n} = t + O(2^{−j})

where x = (\underline{x}, x_{n}) ∈ R^{n−1} × R. This example is usually called the Knapp example, and variants appear for other situations such as the Schrödinger equation (although in this case one must use the Gallilean transforms rather than the Lorentz transforms), and for the restriction theory of the sphere. Note that this solution now has a red-shifted frequency of ∼ 2^{j} rather than ∼ 1, and in fact its Fourier transform lies on a 1 × 2^{j} sector of the light cone.

As a general rule of thumb, any exotic numerology in the linear theory of the wave equation, such as the condition

\frac{1}{q} + \frac{(n − 1)/2}{r} ≤ \frac{(n − 1)/2}{2}

for Strichartz estimates, can usually be blamed on the Knapp example. They also show that one cannot gain any derivatives in local smoothing estimates, only lose derivatives; this observation is interesting when p is larger than 2 but is not too large (otherwise the focus example becomes more important - focussing can really mess up an L^{p} norm for very large p).

Recent advances in the study of the wave equation have benefited from the idea that a general solution to the wave equation can be broken up as the linear combination of Knapp examples (also known as “wave packets”). This idea, in conjunction with square function estimates and Kakeya estimates for space-time tubes oriented in null directions, has yielded some of the best estimates to date on the wave equation. See e.g. [1], [6], [11], [5].

As an example of this, readers might amuse themselves by constructing a (rescaled) focus example at frequency 2^{j} via super-imposing about 2^{(n−1)j} Knapp examples, moving in a 2^{−j}-separated set of near-light velocities.

Another example is the wave train - about 2^{2j} parallel Knapp examples stacked together like a roll of life-savers (moving at nearly the speed of light). This creates a function which is mostly comparable to 1 on a spatial 1×2^{j} tube but is oscillating quite rapidly in this tube, with frequency ∼ 2^{j}. The tube then moves along its long axis at nearly the speed of light for time ∼ 2^{j} before dispersing - in other words, it traverses a constant multiple of its own length before it loses coherency.

These wave trains, combined with the Besicovitch set construction, have been used to show the impossibility of non-trivial local smoothing estimates which do not lose any derivatives at all. See [10]. The point is to select a number of wave trains which are disjoint at time t = 0, but which interact within a Besicovitch set for times 1 ≤ t ≤ 2, thus causing the L^{p} norm at times [1, 2] to be substantially larger than the L^{p} norms at time zero even after averaging in time.

More exotic examples of waves exist. One example comes when f is a bump function of height 1 adapted to the “ring”

\{(\underline{x}, x_{n}) : |\underline{x}| = R + O(1), x_{n} = O(1)\}.

For times 2R ≤ t ≤ 3R, one can show that |u(t, \underline{x}, x_{n})| ∼ R^{(n−3)/2} when

|\underline{x}| = O(1), x_{n} = ±\sqrt{t^{2} − R^{2}} + O(1).

This comes from an estimate of the fundamental solution and Pythagoras’s theorem. This rather unusual example does not impact the Strichartz estimate or local smoothing estimates mentioned earlier, but does play a minor role in certain spherical maximal function estimates, and in some mixed norm local smoothing estimates, but only when n ≥ 4.

2. Bilinear examples

In addition to linear estimates, bilinear estimates have been attracting increasing interest in the last decade, both for purely theoretical reasons and for applications to non-linear wave equations. Basically, one now has two waves

□u_{i} = 0

u_{i}(0, x) = f_{i}(x)
(u_{i})_{t}(0, x) = g_{i}(x)

for i = 1, 2, and one is interested in estimating some bilinear expression Q(u_{1}, u_{2}) of u_{1} and u_{2}, such as the pointwise product, or some variant in which various derivative operators (such as □) are introduced. A model type of bilinear expression is

D_{+}^{β_{+}}D_{−}^{β_{−}}(u_{1}u_{2})

where D_{+} and D_{−} are the Fourier multipliers corresponding to |ξ|+|τ| and ||ξ|−|τ|| respectively. (Roughly speaking, these are the two first-order factors of □, the first one being an elliptic derivative, the second one being a derivative with respect to the characteristic cone of the wave equation).

The examples here are more complicated than the linear ones studied earlier, and are generally easiest to handle on the Fourier transform side. The functions u_{1}, u_{2} transform to distributions on the light cone, and the question is to estimate (the inverse Fourier transform of) a weighted convolution of two such distributions. Usually the weight vanishes near the light cone (the “null form condition”), so as to attenuate contributions when u_{1} and u_{2} have coherent frequencies. This has the tendency of suppressing the linear phenomena of the previous section, and encouraging genuinely bilinear examples to surface. Most of these examples are drawn from [2].

The basic example occurs when û_{1} and û_{2} are bump functions adapted to opposite sides of the upper unit light cone, e.g. on \{(ξ, |ξ|) : ξ_{n} ∼ 1, |\underline{ξ}| ∼ 1\} and \{(ξ, |ξ|) : −ξ_{n} ∼ 1, |\underline{ξ}| ∼ 1\} respectively. This makes u_{1} and u_{2} like two halves of a flash example, one propagating roughly in the e_{n} direction and the other propagating roughly in the −e_{n} direction. The product is only large near the flash point, i.e. in a unit spacetime ball around the origin. Also, both the D_{+} and D_{−} multipliers pretty much ignore u_{1}u_{2} in this case, as the Fourier transform of u_{1}u_{2} stays a unit distance away from both the origin and the light cone.

One can modify this in a number of ways. For instance, one can make u_{2} much higher frequency (but still have the same amount of frequency dispersion), e.g. û_{1} could be a bump function adapted to {(ξ, |ξ|) : ξ = 2^{j}e_{n} + O(1)} for some large j, with u_{2} unchanged. This basically has the effect of increasing the frequency of u_{1} and u_{1}u_{2} by the same amount, while leaving the essentially support of u_{1} and u_{2} virtually unchanged. The distance from the light cone also remains comparable to 1. One can also fatten up u_{1} by adding up several slightly different copies of the above example for u_{1} together, the copies having separated Fourier supports, and this sometimes leads to a better example, see Example 14.10 of [2].

One can also Lorentz transform the above examples to get some other interesting examples; the Lorentz transform affects not only u_{1} and u_{2}, but also the weights applied to the u_{1} and u_{2}. One can sometimes time reverse one of u_{1}, u_{2} to good effect.

The Knapp example used in the linear case is pretty much useless in the bilinear setting, because if u_{1} and u_{2} are Knapp examples oriented in different directions, then their product will have extremely small support and so would be a poor counterexample. To make these examples more effective, we have to “fatten up” u_{1} and u_{2} in order to make their intersection sizeable. On the Fourier level, this corresponds to shrinking the Fourier supports. For instance, rather than having û_{1} and û_{2} supported on opposite 1 × 2^{j} sectors, one could shrink their supports to parallel 2^{−j} × 2^{j} slits, which has the effect of making u_{1} and u_{2} concentrate on the same 2^{j} × 2^{j} × 2^{−j} × … × 2^{−j} square. This important condition places a lower bound on the spacetime L^{p} space one can hope to place Q(u_{1}, u_{2}) in, namely p ≥ (n + 3)/(n + 1), at least if one assumes L^{2}-type conditions on the initial data. (This bound is now known to be sharp, see [11], [5]). There are other ways to fatten up u_{1} and u_{2}; one is the “stretched cap” construction in [7] and [11], in which one takes a sequence of spherical caps and places a variant of the Knapp example^{2} on each one, causing the resulting physical waves u_{1}, u_{2} to be supported on the union of tubes arranged in a square like the square mentioned above. Another way, when n = 2, is to use ruled surfaces such as hyperboloids, see [6]. One can even Lorentz transform these examples to get further interesting examples, see Example 14.8 in [2].

There are even more exotic bilinear examples, in which a function has Fourier support near a conic section of the light cone, see Example 14.12 of [2]. It is clear that we do not yet have a good intuitive grasp of what the real obstructions are to proving bilinear estimates.

References

[1] J. Bourgain, Estimates for cone multipliers, Operator Theory: Advances and Applications, 77 (1995), 41–60.

[2] D. Foschi, S. Klainerman, Homogeneous L^{2} bilinear estimates for wave equations, preprint.

[3] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis II, International Press, 1995.

[4] C. D. Sogge, Propogation of singularities and maximal functions in the plane, Invent. Math. 104 (1991), 349–376.

[5] T. Tao, Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates, submitted, Math Z.

[6] T. Tao, A. Vargas, A bilinear approach to cone multipliers I. Restriction Estimates, to appear, GAFA.

[7] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), pp. 967–1000.

[8] T. Tao, Counterexamples to endpoints of n = 3 wave equation Strichartz, expository note.

[9] T. H. Wolff, A Kakeya type problem for circles, Amer. J. Math. 119 (1997), 985–1026.

[10] T. Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996), 129–162, Amer. Math. Soc., Providence, RI, 1999.

[11] T. H. Wolff, A sharp bilinear cone restriction estimate, to appear, Annals of Math.

Department of Mathematics, UCLA, Los Angeles, CA 90024

  E-mail address: tao@math.ucla.edu

2Actually, to be historically accurate, the Knapp example was initially a bump function adapted to a cap rather than a sector, and in the context of a sphere rather than a cone.