Null form estimates for the cone

Terence Tao (UCLA)

     www.math.ucla.edu/∼tao/preprints

Abstract

Linear and bilinear L^{p} estimates for the wave equation have applications to non-linear wave equations, but the full range of estimates which hold is not completely known. However, Wolff has recently obtained a nearly sharp L^{p} estimate in the bilinear setting, and the author has obtained the endpoint of this estimate.

Introduction

• Consider waves

\frac{∂^{2}u}{∂t^{2}} − Δu = 0

in R^{n} with initial data in a Sobolev space H^{s}×H^{s−1}.

• Q1. Can we control u in some spacetime Lebesgue norm L_{x,t}^{p}?

• Q2. Can we control bilinear expressions of u, e.g. |u_{t}|^{2} − |∇u|^{2} in L_{x,t}^{p}?

• Question 1 is useful in controlling the non-linear effects of the semi-linear wave equation □u = |u|^{p}, while Question 2 addresses the non-linear effects of equations such as the wave map equation

□u = −uQ(u, u)

where

Q(u, v) = u_{t}v_{t} − ∇u · ∇v.

• Q is known as a null form, because Q(u, v) vanishes when u = v is a travelling wave.

• The estimates in Q1 are known as generalized Strichartz estimates, and have been completely classified (around the early 90s, with the final endpoint result obtained last year).

• The estimates in Q2 are known as null form estimates. Although we do not yet have a complete classification, there has been much recent progress, and many near-sharp results have been obtained, at least in the pure spacetime-norm case. (We do not consider mixed space-time norm estimates in this talk).

Fourier transform

• Waves can be studied using the space-time Fourier transform. With the notation of the previous slide, we have

u = { \atop F dσ}

where dσ is induced surface measure on the double light cone Γ_{+} ∪ Γ_{−}, with

Γ_{±} = {(ξ, τ) ∈ R^{n+1} : τ = ±|ξ|}

and F is a function on Γ_{+}∪Γ_{−} satisfying the weighted L^{2} estimate

∫_{R_{n}} |F (ξ, ±|ξ|)|^{2}|ξ|^{2s} dξ ≲ 1

• By dyadic decomposition we can reduce to when F is supported on

Γ_{0} = {(ξ, τ) ∈ R^{n+1} : τ = |ξ|, 1 ≤ |ξ| ≤ 2}.

In this case all spacetime derivatives are essentially irrelevant, and Q1 reduces to

∥{ \atop F dσ}∥_{L_{x,t}^{p}} ≲ ∥F ∥_{2}

• This type of estimate is known as an adjoint restriction estimate. By adapting the T T^{∗} arguments of Tomas and Stein, Strichartz proved the above estimate for all p ≥ \frac{2(n+1)}{n−1}. The key ingredient is the curvature of the surface Γ_{0}, as reflected in the decay of d̂σ:

|d̂σ(x, t)| ≲ (1 + |t|)^{−\frac{n−1}{2}}.

• Strichartz’ estimate is sharp. The example for this is the Knapp example, in which F is a bump function adapted to the sector

{(ξ, τ) ∈ Γ_{0} : ∠ξ, e_{1} ≲ 1/r}

for some r ≫ 1:

6

Bilinear estimates

• We now consider the question of obtaining control on the bilinear expression ∥Q(u, v)∥_{L^{p}(x,t)} for waves u, v and suitable Sobolev control on the initial data.

• We can write u = { \atop F dσ} and v = { \atop Gdσ}.

• As a model case, we that F and G are supported on Γ_{0} [This is not the most typical case]. The estimate then simplifies to

∥Q({ \atop F dσ}, { \atop Gdσ})∥_{p} ≲ ∥F ∥_{2}∥G∥_{2}.

• To understand Q, suppose first that u, v are travelling waves

u = ce^{i(x·ξ+t|ξ|)}, v = c′e^{i(x·η+t|η|)}

with |ξ|, |η| ∼ 1. Since

u_{t} = |ξ|u, v_{t} = |η|v, ∇u = ξu, ∇v = ηv

we have

Q(u, v) = (|ξ||η| − ξ · η)uv ∼ ∠(ξ, η)^{2}uv.

• Thus Q is only large when ∠(ξ, η) ∼ 1, in which case Q(u, v) is like uv. We are thus led to study the estimate

∥{ \atop F dσ_{1}}{ \atop Gdσ_{2}}∥_{p} ≲ ∥F ∥_{2}∥G∥_{2} (∗)

where dσ_{1} and dσ_{2} are surface measure on disjoint sectors in Γ_{0} of angular width ∼ 1 and separation ∼ 1; if one can establish this estimate, one has a good chance of proving null form estimates for this value of p.

• From Strichartz and Hölder’s inequality, (*) holds for p ≥ (n + 1)/(n − 1). However, one does not expect this to be sharp, because the Knapp examples on dσ_{1} and dσ_{2} are so different:

• Klainerman and Machedon observed that (*) also holds for p = 2. This is because we can use Plancherel to rewrite (*) as

∥(F dσ_{1}) ∗ (Gdσ_{2})∥_{2} ≲ ∥F ∥_{2}∥G∥_{2}

and use a Cauchy-Schwarz argument, exploiting the transversality between dσ_{1} and dσ_{2} via the estimate

∥dσ_{1} ∗ dσ_{2}∥_{∞} ≲ 1

• This L^{2} estimate (and the null form estimates it generates) have been crucial in the low-regularity local existence theory of non-linear wave equations.

• Using some bilinear Knapp variants, Machedon noted that the condition p ≥ (n + 3)/(n + 1) is necessary for (∗), and conjectured that this was also sufficient.

• In 1995 Bourgain showed that when n = 2, that (*) could be pushed down to p > 2 − 13/2408. This was improved by Vargas and the author to p > 2−8/121 last year.

• The full conjecture was mostly proved by Wolff early in this year; he obtained the estimate for all p > (n + 3)/(n + 1). The endpoint p = (n + 3)/(n + 1) has since been obtained by the author.

Red and blue waves

• We now undo the Fourier transform and rewrite our simplified conjecture back in terms of waves.

• We say that a wave u is red if û is supported on the sector

{(ξ, |ξ|) : 1 ≤ |ξ| ≤ 2, ∠(ξ, e_{1}) < 1/10}

and blue if û is instead supported in

{(ξ, |ξ|) : 1 ≤ |ξ| ≤ 2, ∠(ξ, −e_{1}) < 1/10}.

• Red and blue waves propagate along different sets of characteristics; red waves propagate along null directions close to (−e_{1}, 1), and blue waves along null directions close to (e_{1}, 1).

• For any red wave u, define the energy E(u) by

E(u) = ∫ |u(x, t)|^{2} dx

for any t ∈ R; this definition is independent of t. (Because frequency is localized, one can dispense with the derivatives when defining energy). Similarly define E(v). The previous results can then be rewritten as • Theorem. (Wolff, T.) For all p ≥ (n + 3)/(n + 1),

all red waves u and all blue waves v we have

∥uv∥_{p} ≲ E(u)^{1/2}E(v)^{1/2}.

• We now indicate some of the ideas used in the proof.

Spatial localization

• Following standard techniques, we prove the global spacetime estimate by first proving local spacetime estimates.

• Let p_{0} = (n+3)/(n+1), and for all R ≫ 1 let A(R) be the best constant such that

∥uv∥_{L^{p_{0}}(Q)} ≤ A(R)E(u)^{1/2}E(v)^{1/2}

for all red and blue waves u and v, and all spacetime cubes Q of side-length R.

• Wolff showed that A(R) = O(R^{ε}) for all ε > 0; a technical argument of Bourgain then shows that (∗) holds for all p > p_{0}. Wolff’s estimate can be tightened to A(R) = O(1), which yields the endpoint p = p_{0}.

• One new innovation in Wolff’s argument is that bounds on A(R) are obtained inductively, by first proving

A(R) ≲ (log R)^{C}A(R^{1−ε}) + R^{Cε},

The endpoint argument is similar but is based on the more precise inequality

A(R) ≤ (1 − ε)A(R/2) + C.

Frequency localization

• Fix R, and let Q be a cube of radius R.

• Let us first bound ∥uv∥_{L^{p}(Q)} under a very strong assumption on u, namely that û is supported in a frequency sector of angular width 1/\sqrt{R}. This means that u has very little angular dispersion.

• In this case, we have the strong L^{2} estimate

∥uv∥_{2} ≲ R^{−(n−1)/4}E(u)^{1/2}E(v)^{1/2};

without the low angular dispersion hypothesis we could not obtain the R^{−(n−1)/4} factor. This estimate goes back at least to Mockenhaupt. The gain comes from the observation that ∥dσ_{1} ∗ dσ_{2}∥_{∞} is not just bounded by 1 as before, but is now much smaller, about R^{−(n−1)/2}, when dσ_{1} is restricted to a sector of width R^{−1/2}.

• On the other hand, from the definition of energy we

have

∥u∥_{L^{2}(Q)} ≤ R^{1/2}E(u)^{1/2}

so by Hölder we have

∥uv∥_{L^{1}(Q)} ≤ RE(u)^{1/2}E(v)^{1/2}.

Interpolating between the L^{1} and L^{2} estimate we get

∥uv∥_{L^{p_{0}}(Q)} ≤ E(u)^{1/2}E(v)^{1/2};

note that 1 < p_{0} < 2 for all dimensions n.

General frequencies

• To handle the general case, we split u and v into low frequency pieces, and try to argue that each piece of u interacts with at most one piece of v.

• We can partition the red portion of Γ_{0} into about R^{(n−1)/2} sectors of angular width 1/\sqrt{R}, which induces a decomposition of u into many smaller red waves, each of which has low frequency dispersion.

• Suppose one of these pieces has frequency support in a sector centered about the null vector (ω, 1) for some |ω| = 1. Then one can exploit the uncertainty principle and (essentially) partition this piece further into pieces which are physically concentrated on \sqrt{R} × R tubes oriented in the dual direction (−ω, 1).

• Combining these two decompositions, we obtain a partition

u = ∑_{T} u_{T}, v = ∑_{T} v_{T}

where u_{T} has low frequency dispersion and is spatially concentrated in a \sqrt{R} × R tube T oriented in a red null direction, and similarly for v.

• Wolff’s argument then proceeds by analyzing the geometry of the tubes T, locating those slightly smaller cubes (of side-length R^{1−ε}) where the tubes T are concentrating in a “bush”:

• It turns out that one can isolate a quite small number (only (log R)^{C}) of high density cubes this way.

• Outside these cubes we exploit the low frequencydispersion estimate. The point is that every tube v_{T} can only interact with at most O(R^{Cε}) tubes u_{T′} coming out of each bush:

• The contribution inside the bad cubes is O((log R)^{C}A(R^{1−ε}a)), so we have

A(R) ≲ (log R)^{C}A(R^{1−ε}) + R^{Cε}.

The endpoint case - an incomplete sketch

• This is an oversimplified, and slightly inaccurate account of how the endpoint is obtained.

• By going through Wolff’s argument carefully, one can separate the cube Q into N R/2 cubes which are “bad” (1 ≤ N ≤ 2^{n+1}) and tighten the inequality to

A(R) ≤ N^{−ε}A(R/2) + C.

• If N > 1 then N^{−ε} < 1, and we can use this to obtain uniform bounds on A(R).

• However, there is still the case in which Q has only one “bad” R/2 sub-cube. In this case we take the R/2 cube and subdivide it further into R/4 cubes. If we again have one bad cube, we divide it further. Eventually we end up with a bad r-cube Q_{r} for some r < R which does split up into more than one bad r/2-cube, so that

∥uv∥_{L^{p}(Q_{r})} ≤ (1 − ε)A(r/2) + C.

• To deal with the remaining portion Q\Q_{r} we use the fact that u and v must be highly concentrated near Q_{r}, and that u and v propagate in different directions, to obtain

∥uv∥_{L^{p}(Q\Q_{r})} ≤ εA(R) + C

• If one chooses ε small enough, then the two estimates

together give

∥uv∥_{L^{p}(Q)} ≤ (1 − ε/2)A(R) + C, which is enough to obtain uniform bounds on A(R).

Remarks

• Similar bilinear estimates have been conjectured for other equations (Klein-Gordon, Schrödinger, Helmholtz equations), but only partial results are known. The techniques used for the cone fail completely, because of the breakdown in the strong Huygens principle. In particular, the crucial fact that a blue tube can only intersect one red tube in a bush is false for any other equation.

• For instance, in R^{2+1} the analogous bilinear estimate for the Schrödinger equation is expected to hold for p ≥ 5/3, but the best result is p > 15/8 (Vargas, T.) by techniques somewhat different from the ones here.

• Our model bilinear estimate can be generalized to the more difficult (but more useful) null form estimates that are within an epsilon of being sharp. (The epsilon loss is caused by various dyadic decompositions). Hopefully, these estimates should have application to non-linear wave equations.