Global regularity of wave maps

Terence Tao (UCLA)

www.math.ucla.edu/∼tao/preprints

Abstract

We describe some recent progress on the global regularity problem for wave maps in the small data case.

What is a wave map?

• Wave maps are the natural analogue of free waves when the wave takes values on a Riemannian manifold rather than a Euclidean space.

• Recall that a function ϕ : R × R^{n} → R^{m} obeys the free wave equation if

□ϕ(t, x) = 0

where □ := Δ − ∂_{t}^{2}. Alternatively, one can think of ϕ as a critical point of the Lagrangian

∫ ∫ ∂_{α}ϕ · ∂^{α}ϕ dxdt = ∫ ∫ (|∇ϕ|^{2} − |ϕ_{t}|^{2}) dxdt.

• Now let M be an arbitrary Riemannian m-dimensional manifold. A wave map ϕ : R×R^{n} → M is a critical point of the Lagrangian

∫ ∫ ∂_{α}ϕ · ∂^{α}ϕ dxdt = ∫ ∫ (|∇ϕ|^{2} − |ϕ_{t}|^{2}) dxdt.

If M is an embedded hypersurface in R^{m+1}, we can write the Euler-Lagrange equation as

□ϕ = −Γ(ϕ)(∂_{α}ϕ, ∂^{α}ϕ)

where Γ(ϕ) is the second fundamental form of M at u.

• Wave maps arise as model equations for the Einstein equations in general relativity, and are also a limiting case of the Ginzberg-Landau wave equations. Stationary wave maps are known as harmonic maps and have also been extensively studied.

• In this talk we shall restrict our attention to the case when M = S^{m} ⊂ R^{m+1}, the m-dimensional sphere. In this case the wave map equation becomes

□ϕ = −ϕ(∂_{α}ϕ · ∂^{α}ϕ).

We shall write ϕ(t, x) as an m-dimensional columnvector, and its adjoint ϕ^{†}(t, x) as an m-dimensional row vector, to rewrite the above as

□ϕ = −ϕ∂_{α}ϕ^{†}∂^{α}ϕ.

• In the 0-dimensional case n = 0, this equation be-

comes

ϕ_{tt}(t) = −ϕ(t)|ϕ_{t}(t)|^{2};

this is the equation for geodesic motion of a particle on the sphere. (The non-linearity −ϕ(t)|ϕ_{t}(t)|^{2} can then be regarded as the centripetal force needed to keep u on the sphere.

• In the one-dimensional case n = 1, the equation be-

comes

ϕ_{tt} − ϕ_{xx} = −ϕ(|ϕ_{t}|^{2} − |ϕ_{x}|^{2});

this can be thought of as the free evolution of a onedimensional string on a sphere. Similarly the n = 2 case is the free evolution of a sheet on a sphere, etc.

• It is not completely obvious from the equation, but if a wave ϕ starts on the sphere (i.e. ϕ^{†}(0)ϕ(0) = 1 and ϕ^{†}(0)ϕ_{t}(0) = 0) then it shall stay on the sphere for all time.

• When the target manifold S^{m} is just the circle S^{1} ⊂ C, then one can solve the wave map equation explicitly as ϕ = exp(iu), where u : R^{1+n} → R solves the free wave equation. (This is because the map u → exp(iu) is an isometry from R to S^{1}). But there is no explicit solution in general.

The regularity problem

• We consider the Cauchy problem for the wave map

equation:

□ϕ = −ϕ(∂_{α}ϕ · ∂^{α}ϕ).
ϕ(0) = f; ϕ_{t}(0) = g
f · f = 1; f · g = 0.

• If f and g are smooth, then the classical local wellposedness theory allows one to construct a unique smooth solution to this equation for short times. This solution will stay on the sphere.

• The regularity problem asks whether a smooth solution can be continued indefinitely in time; i.e. singularities do not develop.

• Rule of thumb: If one can show that ϕ stays Hölder continuous for all time, then one should be able to establish smoothness for all time. This is because of two reasons: Hölder continuity combined with finite speed of propagation allows one to localize to a small subset of the sphere, thus allowing one to ignore the geometry and work in local co-ordinates; and secondly, scaling arguments suggest that one can use iteration arguments using the wave map equation to bootstrap Hölder continuity to better regularity.

• This rule of thumb is rigorous in the harmonic map case, but is only a heuristic in the wave map case. However, if one can show that ϕ stays in the Sobolev space H_{loc}^{s} for some s > n/2, then one can obtain regularity (Klainerman-Machedon 1997 in dimensions n ≥ 4; Klainerman-Selberg 1997 for n = 2, 3; Keel-Tao 1998 for n = 1. For s > n/2 + 1 regularity follows from the classical energy method). Note that s > n/2 is exactly the condition required by Sobolev embedding to ensure Hölder continuity. Regularities H^{s}, s > n/2 are commonly referred to as subcritical, s < n/2 as supercritical, and s = n/2 as critical.

• On the other hand, we have conservation of energy

∫ \frac{1}{2}|∇ϕ|^{2} + \frac{1}{2}|ϕ_{t}|^{2} dx

(basically because of time translation invariance of the Lagrangian). Because of this, we know that smooth wave maps stay in H^{1} for all time. In particular, this implies global regularity in the energysubcritical one-dimensional case n = 1. (This result goes back to Gu 1980 and Ladyzhenskaya-Shubov 1984).

• In the energy-supercritical case n ≥ 3, conservation of energy does not ensure regularity, and indeed in this situation one has examples of blow-up in finite time even for smooth symmetric data (Shatah 1988, Cazenave-Shatah-Tahvildar-Zadeh 1998) However all of these examples require large data (not restricted to a small subset of the target manifold).

• The energy-critical case n = 2 is the most interesting. In this case one has regularity if the initial data is rotationally invariant (Christodolou-Tahvildar-Zadeh 1993; Struwe 2000) or rotationally covariant (Shatah-Tahvildar-Zadeh 1992; Struwe 2000), but the general case remains open.

• Under smallness assumptions on the data one can also obtain regularity. For instance, if the initial data stays close to a geodesic in a smooth norm then one has global regularity (Sideris 1989). This result was significantly improved by Tataru (1998 for n ≥ 4, 1999 for n = 2, 3), in which the initial data is only required to be small in a Besov space norm very close to, but not quite Ḣ^{n/2}. (More on this later).

• The purpose of this talk is to discuss the following improvement to Tataru’s results: Theorem. (T., Oct 2000 for n ≥ 5, T., Nov 2000 for n ≥ 2) If n ≥ 2, the target manifold is a sphere and the initial data (f, g) is smooth and small in Ḣ^{n/2} × Ḣ^{n/2−1} norm, then the wave map stays smooth globally in time.

• In particular, in the two-dimensional case we see that small energy implies regularity. From this and energy conservation we will be able to obtain regularity for general data as soon as we show that energy does not concentrate. This is known in the rotationally invariant or covariant cases, and in simpler model problems such as the energy-critical semilinear wave equation, but not for the general equation.

• Very recently (Nov 2000), Klainerman and Rodnianski have extended the above arguments to the case when the target manifold is an arbitrary Lie group (not necessarily compact), in the high-dimensional case n ≥ 5.

• This result appears to be the first critical Sobolev result which can handle derivatives in the non-linearity. (For semilinear evolution equations, in which the non-linearity is a pure power, one can obtain critical regularity results just from Strichartz estimates).

• In addition to the regularity problem, there are other interesting problems such as scattering, and wellposedness at the critical regularity. Our argument does not quite address these issues, although with a little more work it may be possible to do so.

Tataru’s results

• In order to describe Tataru’s regularity result we need some notation. For any integer k, let P_{k} be a Littlewood-Paley projection to spatial frequencies |ξ| ∼ 2^{k}. From orthogonality we have

∥f∥Ḣ^{n/2} ∼ (∑_{k} ∥P_{k}f∥2 ; Ḣ^{n/2})^{1/2} ∼ (∑_{k}(2^{nk/2}∥P_{k}f∥_{2})^{2})^{1/2}.

• Define the Besov space Ḃ_{2}^{n/2,1} by

∥f∥Ḃ_{2}^{n/2,1} := ∑_{k} ∥P_{k}f∥Ḣ^{n/2} ∼ ∑_{k} 2^{nk/2}∥P_{k}f∥_{2}.

This space is slightly smaller than Ḣ^{n/2}. However, unlike Ḣ^{n/2}, the Besov space Ḃ_{2}^{n/2,1} embeds by Sobolev embedding into L^{∞} and such functions are even continuous (though not Hölder continuous). It is also closed under multiplication (i.e. it is a Banach algebra).

• Theorem. (Tataru 1998,1999) If n ≥ 2 and the initial data (f, g) is smooth and small in Ḃ_{2}^{n/2,1} × Ḃ_{2}^{n/2−1,1} norm, then the wave map stays smooth globally in time.

• This theorem is achieved by an iteration argument in a very intricate Banach space. Because we have continuity of the wave map, we can use local co-ordinates and forget about the geometry of the equation. For instance, Tataru’s result works equally well for the scalar equation

□ϕ = ϕ∂_{α}ϕ∂^{α}ϕ

as it does for the wave map equation on the sphere

□ϕ = −ϕ∂_{α}ϕ^{†}∂^{α}ϕ.

• However, if one wants to extend this result from the Besov space Ḃ_{2}^{n/2,1} to the Sobolev space Ḣ^{n/2}, one cannot use an iteration argument, no matter how sophisticated the Banach space and estimates. The reason is that the scalar equation above has an explicit solution ϕ = exp(u) where u solves the free wave equation, and this exponential map is ill-behaved on Ḣ^{n/2} (basically because Ḣ^{n/2} does not control L^{∞}):

∥ exp(u)∥Ḣ^{n/2} = ∥∇^{n/2} exp(u)∥_{2} ∼ ∥(∇^{n/2}u) exp(u)∥_{2}

≲ ∥∇^{n/2}u∥_{2}∥ exp(u)∥_{∞} = ∥u∥Ḣ^{n/2} × Trouble!

On the other hand, the wave map equation on the circle S^{1} has an explicit solution exp(iu), and this is well behaved since exp(iu) is automatically bounded. The moral is that the geometry of the sphere becomes important for the Ḣ^{n/2} theory.

Littlewood-Paley decomposition

• Tataru’s Besov space result suggests that one should use Littlewood-Paley decomposition to analyze the wave map equation.

• Split ϕ = ∑_{k} ϕ_{k}, where ϕ_{k} := P_{k}ϕ is the portion of ϕ with Fourier support in |ξ| ∼ 2^{k}. Thus for large k, ϕ_{k} is a high frequency component of ϕ, and vice versa.

• We could then split the non-linearity as

−ϕ∂_{α}ϕ^{†}∂^{α}ϕ^{†} = − ∑_{k_{1},k_{2},k_{3}} ϕ_{k_{1}}∂_{α}ϕ_{k_{2}}^{†}∂^{α}ϕ_{k_{3}}

but this looks complicated. Let’s begin by just decomposing the latter two terms:

∂_{α}ϕ^{†}∂^{α}ϕ = ∑_{k_{1},k_{2}} ∂_{α}ϕ_{k_{1}}^{†}∂^{α}ϕ_{k_{2}}.

We can split this further into three sub-cases: k_{2} ≥ k_{1} + 5, k_{2} ≤ k_{1} − 5, and k_{2} = k_{1} + O(1).

• When k_{2} ≥ k_{1} + 5 we have a low-high interaction: a low frequency component ϕ_{k_{1}} is interacting with a high frequency component ϕ_{k_{2}} to produce a highfrequency output (since the sum of frequencies ∼ 2^{k_{1}} and ∼ 2^{k_{2}} is ∼ 2^{k_{2}}). Thus every frequency component ϕ_{k_{2}} is cumulatively influenced by all the frequencies ϕ_{k_{1}} which are lower than it. This cumulative influence turns out to be manageable in the Besov space case (mainly because the Ḣ^{n/2} norms of ϕ_{k_{1}} are then summable), but causes a possible logarithmic divergence, or “low to high frequency cascade” in the Ḣ^{n/2} case. In the case of the scalar equation given previously, this cascade ultimately leads to singularity formation.

• This logarithmic divergence is similar to the divergence encountered when trying to embed Ḣ^{n/2} into L^{∞}; for each dyadic block the embedding works, but when adding up the blocks we obtain a divergence.

• The case k_{2} ≤ k_{1} − 5 is a high-low interaction and is identical to the low-high interaction.

• The case k_{2} = k_{1} + O(1) is a high-high interaction: two high frequencies come together and produce an output which could be quite low (≲ 2^{k_{1}}), because of cancellation between the frequencies of ϕ_{k_{1}} and ϕ_{k_{2}}. Such a cancellation could in principle cause a “high to low frequency cascade”, which is another potential source of trouble. Indeed, such an effect occurs in one dimension (the energy slowly moves from high frequencies to low frequencies), though this effect is not strong enough to disrupt regularity. However, in higher dimensions the effect becomes weaker (the more dimensions there are, the less likely it is that two frequencies will exactly cancel each other). Even in two dimensions one can control the high-high interactions by using the null structure of the nonlinearity (if ϕ_{k_{1}} and ϕ_{k_{2}} have almost opposite frequencies, then there is some cancellation in the symbol ∂_{α}ϕ_{k_{1}}^{†}∂^{α}ϕ_{k_{2}}). So for heuristic purposes we may ignore the high-high interactions, and only focus on low-high and high-low interactions.

• Now consider the full trilinear expression

− ∑_{k_{1},k_{2},k_{3}} ϕ_{k_{1}}∂_{α}ϕ_{k_{2}}^{†}∂^{α}ϕ_{k_{3}}.

Suppose for the moment that k_{1} > k_{2}, so that ϕ_{k_{2}} was a relatively low frequency. This term should be quite small because the derivative ∂_{α} is falling on a low frequency term instead of a high frequency term. Heuristically,

ϕ_{k_{1}}∂_{α}ϕ_{k_{2}}^{†}∂^{α}ϕ_{k_{3}} = 2^{k_{2}−k_{1}}O(ϕ_{k_{2}}∂_{α}ϕ_{k_{1}}∂^{α}k_{3}).

Thus these “derivative falls on low frequency” terms should be relatively insignificant, and heuristically can be ignored.

• To summarize: Littlewood-Paley decomposition suggests that we should be able to ignore “high-high” and “derivative falls on low frequency” interactions. The main issue is then to deal with the low-high interaction.

• (This is in contrast with the corresponding situation with the harmonic map equation Δϕ = −ϕ∂_{α}ϕ^{†}∂^{α}ϕ. In that equation the low-high interaction can be dealt with by the good smoothing properties of Δ, while the high-high interaction is problematic (unlike the wave map problem, dispersion is not available) but can be resolved by using some geometric structure of the equation, and using the div-curl lemma, which is a tool specifically designed to control high-high interactions.)

Linearization

• We now study the evolution of a single dyadic block ϕ_{k}, which I will rename ϕ_{high}. As discussed earlier, the evolution of this block will be influenced by the cumulative low frequencies ∑_{k′<k−5} ϕ_{k′}, which I will rename ϕ_{low}. If we then project the wave map equation

□ϕ = −ϕ∂_{α}ϕ^{†}∂^{α}ϕ

and ignore “high-high” and “derivative falls on low frequency” terms, we obtain

□ϕ_{high} = −ϕ_{low}∂_{α}ϕ_{low}^{†}∂^{α}ϕ_{high} − ϕ_{low}∂_{α}ϕ_{high}^{†}∂^{α}ϕ_{low}

which simplifies to

□ϕ_{high} = −2ϕ_{low}∂_{α}ϕ_{low}^{†}∂^{α}ϕ_{high}.

Note that this equation is now linear in ϕ_{high}, because all the terms quadratic in the high frequencies are high-high interactions and hence heuristically negligible. (This philosophy also appears in other recent work on global existence of non-linear equations, such as the work of Bourgain and others on global well-posedness below the energy norm).

• At this point we are stuck, because we still have the low-high interactions left to deal with, and iteration techniques will not handle this non-linearity. (We still are faced with the scalar equation counterexample). However, at this point the geometry of the equation comes into play.

Littlewood-Paley theory on the sphere

• Recall that ϕ lies on the sphere, so we have ϕ^{†}ϕ = 1 and ϕ^{†}∂_{α}ϕ = 0.

• What does this imply about ϕ_{low} and ϕ_{high}? If we take low-frequency components of both sides of ϕ^{†}ϕ = 1 and ignore high-high interactions we obtain

ϕ_{low}^{†}ϕ_{low} ≈ 1

i.e. ϕ_{low} also stays on the sphere. If we take highfrequency components we obtain

ϕ_{high}^{†}ϕ_{low} + ϕ_{low}^{†}ϕ_{high} ≈ 0

or in other words that ϕ_{low} is essentially orthogonal to ϕ_{high}. “high frequencies lie in the tangent space of low frequencies”. Similarly, if we take low-frequency components of ϕ^{†}∂_{α}ϕ = 0 we obtain

ϕ_{low}^{†}∂_{α}ϕ_{low} ≈ 0

(another indication that ϕ_{low} stays on the sphere) and if we take high frequency components and ignore “high-high” and “derivative falls on low” components we get

ϕ_{low}^{†}∂_{α}ϕ_{high} ≈ 0.

So the high frequencies and its derivatives are all essentially orthogonal to the low frequencies.

Parallel transport

• Using these geometric identities we can rewrite our linearized evolution

□ϕ_{high} = −2ϕ_{low}∂_{α}ϕ_{low}^{†}∂^{α}ϕ_{high}

in a much better form:

□ϕ_{high} = −2(ϕ_{low}∂_{α}ϕ_{low}^{†} − ∂_{α}ϕ_{low}ϕ_{low}^{†})∂^{α}ϕ_{high}.

Denote the expression inside the parentheses as A_{α}. The point is that A_{α} is an anti-symmetric matrix, or (in fancier language) A_{α} is a connection on the trivial R^{m}-bundle on R^{1+n}. Our equation then becomes

(∂_{α}∂^{α} + 2A_{α}∂^{α})ϕ_{high} = 0

or, what is essentially the same thing modulo “derivative falls on low” terms,

D_{α}D^{α}ϕ_{high} = 0

where D_{α} := ∂_{α} + A_{α}.

• Thus ϕ_{high} solves the wave equation relative to the connection A_{α}. This connection has a natural interpretation: it is parallel transport on the sphere along ϕ_{low} (recall that ϕ_{high} and its derivatives essentially lie on the tangent plane of the sphere at ϕ_{low}).

• In other words: As the low frequencies evolve across the sphere, they carry the high frequencies along with them by parallel transport (with the high frequencies also continuing to evolve via the free wave equation).

• These facts are only available in the true wave map equation. Other models such as the scalar equation described earlier do not have this structure, and so we are now free of those counterexamples.

The gauge transformation

• Equations of the form D_{α}D^{α}ϕ = 0 naturally arise in gauge field equations (e.g. in the Maxwell-Klein-Gordon and Yang-Mills equations), and have been studied extensively. An important observation is that the above equation is invariant under gauge transformations

ϕ ↦ Uϕ; A_{α} ↦ UA_{α}U^{†} + ∂_{α}U

whenever U is a field taking values in SO(m), the space of orthogonal matrices.

• Ideally, we would like to find a gauge transform that would transform the connection A_{α} to zero, since D_{α}D^{α}ϕ = 0 then transforms to the free wave equation □ϕ = 0. But this is only possible if the connection is flat, or more precisely if the curvature

F_{αβ} = ∂_{α}A_{β} − ∂_{β}A_{α} + [A_{α}, A_{β}]

vanishes.

• Unfortunately, F_{αβ} does not vanish - that would be too miraculous. However, a lesser miracle occurs, in that F_{αβ} vanishes to top order. More precisely, since

A_{α} = ϕ_{low}∂_{α}ϕ_{low}^{†} − ∂_{α}ϕ_{low}ϕ_{low}^{†}

we have

∂_{α}A_{β} = ϕ_{low}∂_{αβ}ϕ_{low}^{†}−∂_{αβ}ϕ_{low}ϕ_{low}^{†}+∂_{α}ϕ_{low}∂_{β}ϕ_{low}^{†}−∂_{β}ϕ_{low}∂_{α}ϕ_{low}^{†}

∂_{β}A_{α} = ϕ_{low}∂_{αβ}ϕ_{low}^{†}−∂_{αβ}ϕ_{low}ϕ_{low}^{†}+∂_{β}ϕ_{low}∂_{α}ϕ_{low}^{†}−∂_{α}ϕ_{low}∂_{β}ϕ_{low}^{†} [A_{α}, A_{β}] = −ϕ_{low}(∂_{α}ϕ_{low}^{†}∂_{β}ϕ_{low}−∂_{β}ϕ_{low}^{†}∂_{α}ϕ_{low})ϕ_{low}^{†}

+(∂_{α}ϕ_{low}^{†}∂_{β}ϕ_{low} − ∂_{β}ϕ_{low}^{†}∂_{α}ϕ_{low}). Some of these terms are quadratic in the first derivatives, while others are linear in the second derivatives. But the terms which are linear in the second derivatives cancel! So we only have terms quadratic in the first derivatives, which are much better to work with. (In high-low interactions, which are always our enemy, there will be one high derivative and one low derivative, as opposed to two high derivatives, which is much worse). Thus the curvature is “smaller” than one might expect, although it is not completely zero (except in the circle case S^{m} = S^{1}).

• In other words, the parallel transport connection does not have a very large curvature.

• Because of this small curvature, we can find a good gauge transformation U, either by explicitly constructing U in order to make UA_{α}U^{−1} + ∂_{α}U small (T.) or by switching to the Coulomb gauge (Klainerman-Rodnianski). The explicit construction of U is obtained in a recursive manner, building up one dyadic frequency block of U at a time, and discarding any “high-high” or “derivative falls on low” interactions as error terms.

• Once one applies this gauge transform (or “renormalization”), the evolution for ϕ_{high} is then essentially the free wave equation. The dangerous low-high interaction has been almost completely eradicated, and one can finally use iteration arguments to proceed. (One then has to invert the gauge transform, but in principle this is straightforward since U is orthogonal so U^{−1} = U^{†}).

Remarks

• The gauge transform U is heavily adapted to the connection A_{α}, and hence to the low frequencies ϕ_{low}. It appears to be somewhat unstable: small perturbations in ϕ_{low} could lead to large perturbations in U. Because of this, we were unable to extract a wellposedness or scattering result from this argument (though perhaps one just needs to refine the argument further).

• The connection structure of the wave map equation has been observed by many other authors. However, it appears that the ability to gauge transform this structure to something simpler is only possible once one analyzes the wave map equation using the Littlewood-Paley decomposition. It seems that separating the evolution into high and low frequency components is an invaluable tool for studying the global behaviour of these types of equations.

• (For the experts) The above techniques are also related to global well-posedness theory below the energy norm pioneered by Bourgain, and extended by many authors. In this theory the high-high interactions are almost always small. In Bourgain’s original approach one assumes that the high-low interaction is also small (i.e. there is a “smoothing effect” in the equation). However even when the high-low interaction is not negligible, one can often use the finer structure of the equation to show that this interaction does not cause very much energy to be transferred between the low and high frequencies (cf. recent work by Keel, T., Colliander, Staffilani, Takaoka).

• One can try to extend the above arguments to other equations, such as the Yang-Mills or Maxwell-Klein-Gordon equations. Most of the steps carry over, except that the vital miracle of the curvature being small does not seem to hold for these equations. I believe that in these equations, the high frequencies cannot be gauge transformed to evolve like the free wave equation, but instead evolve in a genuinely curved fashion. Nevertheless, it may still be possible to obtain enough estimates on this evolution to close some sort of iteration argument.

Function spaces

• In the remainder of this talk I’ll discuss some technical issues relating to the function spaces one must iterate in to make the above argument rigorous.

• In the above argument many terms in the non-linearity were described as being “small” and “negligible”. In practice, what this means is that they have small norm in some space N_{k}, where k is the frequency under consideration (the same k used to define ϕ_{k} = ϕ_{high}).

• In the high dimensional case n ≥ 5 one can choose N_{k} to just be L_{t}^{1}L_{x}^{2} (or more precisely, L_{t}^{1}Ḣ_{x}^{n/2−1}). The reason is that Strichartz estimates allow one to place the wave map ϕ in many Lebesgue spaces such as L_{t}^{∞}L_{x}^{2}, L_{t}^{2}L_{x}^{4} and L_{t}^{2}L_{x}^{∞} (with the appropriate number of derivatives indicated by scaling), and because of this one can easily place the non-linearity in L_{t}^{1}L_{x}^{2} just from Hölder’s inequality, provided that one is in one of the two “good” cases: either “high-high” or “derivative falls on low”. (There are derivatives floating around the place, but one of the advantages of Littlewood-Paley decomposition is that derivatives essentially become constants, e.g. ∇ϕ_{k} is essentially

2^{k}ϕ_{k}).

• In the high-dimensional case, one does not need to exploit any cancellation coming from the null structure in the bilinear expression ∂_{α}ϕ^{†}∂^{α}ϕ. The symbol of this expression is ξ · ξ′ − ττ′, which can become small when (ξ, τ) and (ξ′, τ′) are nearly parallel and close to the light cone (a “small angle interaction”). However in high dimensions small angles are rare (the bulk of interactions will have angle ∼ 1) and so one does not need to exploit the null structure. (This fact holds for wave equations in general, and is not specific to the wave map equation).

• (For the experts) It may be surprising that Strichartz estimates are able to handle a critical regularity problem when derivatives are present, but this is because the renormalization takes away the dangerous highlow interaction case, for which Strichartz estimates fail dramatically.

• In the low dimensional case one must use far more sophisticated spaces than Lebesgue spaces, and one must exploit the null form structure more fully, especially in two dimensions. The reason is that there is far less decay in the wave equation in low dimensions, which translates to far fewer Strichartz estimates. (When n = 4 one loses the L_{t}^{2}L_{x}^{4} estimate; when n = 3 one also loses the L_{t}^{2}L_{x}^{∞} estimate; and when n = 2 one even loses the L_{t}^{4}L_{x}^{4} estimate, which is crucial to handle the semi-linear analogue □ϕ = ϕ^{3} of this equation. In principle we have a better situation than the semi-linear problem because of the null form, but it takes some work to put this null structure to good use).

• Fortunately, this problem was successfully resolved (in principle at least) by Tataru, who introduced both X^{s,b} spaces (which are a kind of spacetime Sobolev space adapted to the wave equations), as well as some null frame spaces L_{t_{ω}}^{q}L_{x_{ω}}^{r}. Tataru’s observation was that even if one does not have good Strichartz estimates (such as L_{t}^{2}L_{x}^{∞} estimates) in the standard Euclidean frame, one can recover these estimates if one switches to a null frame (or more precisely, an average of null frames). The basic idea is to decompose a wave ϕ into a superposition of travelling waves F (x − ωt), each of which obeys L_{t_{ω}}^{2}L_{x_{ω}}^{∞} estimates in the null direction (ω, 1).

• Because we need to combine all these norms together and then prove various multilinear estimates on the combined Banach space, the estimates can get quite lengthy. (The proof of regularity for small energy in the two-dimensional case is 98 pages!). Unfortunately I do not know of a way to get around using these spaces.