THE WAVE MAPS EQUATION

Daniel Tataru

Department of Mathematics

University of California, Berkeley

1

Laplace equation:

−∆φ = 0, φ : Rn → RSolutions are critical points for the Lagrangian

Le(φ) =1

2

∫Rn|∇φ(x)|2dx.

Similar if φ : Rn → Rm. Consider now a Rieman-nian manifold (M, g), and functions

φ : Rn →M

Its derivatives are sections of the pullback bundle

∂αφ : Rn → TφM ∂αφ ∈ φ∗(TM),

φ∗(TM) = ∪x∈Rn{x} × Tφ(x)M

Natural Lagrangian:

LeM(φ) =1

2

∫Rn|∇φ(x)|2gdx

Covariant differentiation on the pullback bundle

∇XV = (∇φ∗XV ), X ∈ TRn, V ∈ φ∗(TM).

Euler-Lagrange equations:

−∇α∂αφ = 0 (harmonic maps)

−∆φi = Γijk(φ)∂αφj∂αφ

k (in local coord.)

2

Wave equation:

2φ = 0, φ : R× Rn → R

2 = ∂2t −∆x

Pseudo-riemannian metric in R× Rn:

(ds)2 = −(dt)2 + (dx)2

Lifting indices with respect to this metric:

−∂α∂αφ = 0

Lagrangian:

Lh(u) =1

2

∫Rn−|∂tφ|2 + |∂xφ(x)|2dx

If instead we take

φ : R× Rn →M

then the natural Lagrangian is

LhM(φ) =∫Rn−|∂tφ|2g + |∂xφ(x)|2gdx

Euler-Lagrange equation:

∇α∂αφ = 0 (wave maps)

In local coordinates:

2φi = Γijk(φ)∂αφj ∂αφ

k

3

Isometrically embedded manifolds:

M ⊂ Rm

Second fundamental form:

S : TM × TM → NM

〈S(X,Y ), N〉 = 〈∂XN,Y 〉

Wave maps equation:

2φ = −Sφ(∂αφ, ∂αφ)Special case: M = Sm−1 (sphere)

Sφ(X,Y ) = φ〈X,Y 〉

2φ = −φ〈∂αφ, ∂αφ〉

Special case: M = Hm (hyperbolic space) Wethink of it as the space-like hyperboloid

φ20 = 1 + φ2

1 + · · ·+ φ2m

in the Minkowski space R× Rm.

Sφ(X,Y ) = φ〈X,Y 〉L

2φ = −φ〈∂αφ, ∂αφ〉LHere L is the Lorentzian inner product,

〈X,Y 〉L = −X0Y0 +X1Y1 + · · ·+XmYm

4

Let M1 be a submanifold of M .

Q: Are the wave maps into M1 also wave maps

into M ?

A: Yes, if and only if M1 is a totally geodesic

submanifold of M .

Special case: M1 = γ, a geodesic in M . Then

γ has one dimension and no curvature. Hence,

with respect to the arclenght parametrization,

the wave maps equation into γ is nothing but

the linear wave equation. Thus for any target

manifold M we have at our disposal a large supply

of wave maps associated to the geodesics of M .

5

The Cauchy problem:∇α∂αφ = 0 in R× Rn

φ(0, x) = φ0(x), ∂tφ(0, x) = φ1(x) in Rn

The initial data (φ0, φ1) must satisfy

φ0(x) ∈M, φ1(x) ∈ Tφ0(x)M, x ∈ Rn

Commonly one chooses the initial data in Sobolev

spaces,

φ0 ∈ Hs, φ1 ∈ Hs−1

1. Conserved energy.

E(φ) =1

2

∫Rn|∂tφ|2g + |∂xφ|2gdx

2. Scaling. The wave maps equation is invariant

with respect to the dimensionless scaling

φ(t, x) → φ(λt, λx) λ ∈ R

Note however that the energy is scale invariant

only in dimension n = 2.

6

Local well-posedness in Sobolev spaces

Q: Given initial data

(φ0 ∈ Hs, φ1 ∈ Hs−1)

find T > 0 and an unique solution

u ∈ C(−T, T ;Hs), ∂tu ∈ C(−T, T ;Hs−1)

Scale invariant initial data space: s = n2.

s >n

2

small data

large time⇔ large data

small time

s =n

2

small data

small time⇔ small data

large time

s <n

2

small data

small time⇔ large data

large time

If s > n2 by Sobolev embeddings Hs ⊂ L∞ there-

fore the solutions are expected to be continuous.

Hence one can study them using local coordi-

nates.

If s ≤ n2 then it is not even clear how to define

the Sobolev space Hs of M valued functions.

7

Problem 1. Prove local well-posedness for s > n2.

As s decreases toward n2 one gains better infor-

mation concerning the lifespan of solutions. For

s = n2 a local result yields a global result, but

one needs to distinguish between small and large

data.

Problem 2. Under reasonable assumptions on M

prove global well-posedness for small data and

s = n2.

In this case the problem is nonlocal. By looking

at the special solutions on geodesics one sees for

instance that M must be geodesically complete.

Problem 3. Prove ill-posedness for s < n2.

This was recently solved by D’Ancona-Georgiev

using the special solutions which are contained

on geodesics.

8

Global solutions vs. blowup

Q: Are solutions with large data global, or is

there blow-up in finite time ?

Hope: Exploit conserved or decreasing “energy”

functionals. But this can only be done if the

energy (s = 1) is at or above scaling(s = n2). For

wave maps we need to differentiate three cases.

Supercritical (n ≥ 3) Prove that large data so-

lutions for the wave maps equation can blow up

in finite time if n ≥ 3.

Indeed, in this case self-similar blow-up solutions

u(x, t) = u(x

t)

have been constructed by Cazenave, Shatah and

Zadeh.

Subcritical (n = 1)Prove that large data solu-

tions for the wave maps equation are global.

This has been done in work of Keel-Tao.

9

Critical (n = 2) Here the energy is precisely at

scaling, which makes it very difficult to exploit.

Furthermore, numerical evidence (Bizon-Chmaj-

Tabor and Isenberg-Lieblin) indicates that the

outcome seems to depend on the geometry of

M .

For a positive result one needs a nonconcentra-

tion argument, which asserts that energy cannot

focus at the tip of a light cone. Such noncon-

centration arguments are known for other critical

semilinear wave equations.

Open Problem 4.Consider the wave maps equa-

tion in 2 + 1 dimensions. Prove that

a) blow-up of large energy solutions occurs for

certain target manifolds (e.g. the sphere).

b) large energy solutions are global for other tar-

get manifolds (e.g. the hyperbolic space).

10

The second part of the talk is devoted to the

following result:

Theorem 5. Let n ≥ 2. For all “reasonable”

target manifolds M the wave maps equation is

globally well-posed for initial data which is small

in Hn2 × H

n2−1.

I will attempt to explain

(i) What is a “reasonable” target manifold.

(ii) What is the meaning of “well-posed”.

(iii) Why this is a nonlinear problem.

(iv) How can one attempt to approach it.

(v) What is the story of the problem.

(iv) Which are the main ideas in the proof.

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Wave maps as a semilinear equation

Idea: Fixed point argument based on estimatesfor the linear equation

2φ = f, φ(0) = φ0, ∂tφ(0) = φ1

Suppose we want to solve

2φ = N(φ)

with initial data in a Sobolev space X0.

a) X0 is above scaling. Then we need Banachspaces X,Y such that

(linear estimate) ‖χ(t)φ‖X . ‖(φ0, φ1)‖X0+‖f‖Y

(nonlinear mapping) χ(t)N : X → Y

b) X0 is at scaling. Then X,Y should satisfy

‖φ‖X . ‖(φ0, φ1)‖X0+ ‖f‖Y

N : X → Y

A fixed point argument in X yields:- existence of a solution in X,-uniqueness in X,-Lipschitz dependence on the initial data.

12

First try: Energy estimates

‖∇φ‖L∞(Hs−1) . ‖φ0‖Hs + ‖φ1‖Hs−1 + ‖f‖L1Hs−1

This suggests we take X0 = Hs ×Hs−1 and

X = C(Hs) ∩ C1Hs−1, Y = L1Hs−1

The nonlinear mapping property holds for

s >n

2+ 1

Second try: Strichartz estimates

Uses the dispersive effect of the wave equation.

For n ≥ 3, s = n+12 we have

‖∇φ‖L2L∞ . ‖φ0‖Hs + ‖φ1‖Hs−1 + ‖f‖L1Hs−1

Then we modify X,

X = {φ ∈ C(Hs),∇φ ∈ C(Hs−1) ∩ L2L∞}

and get the improved range

s ≥n+ 1

2, n ≥ 3

13

Third try: The null condition. Consider thesimplest bilinear interaction,

2φ = φ1φ2, 2φ1 = 0, 2φ2 = 0

The symbol of the wave operator is

p(τ, ξ) = τ2 − ξ2

and both φ1, φ2 are concentrated in frequencyon the characteristic cone

K = {(τ, ξ) ∈ R× Rn; τ2 − ξ2 = 0}Suppose φ1, φ2 are localized in frequency near(τ1, ξ1) respectively (τ2, ξ2). Then their productis localized in frequency near

(τ, ξ) = (τ1 + τ2, ξ2 + ξ2)

The output φ is largest if (τ, ξ) ∈ K. But thisonly happens if (τ1, ξ1) and (τ2, ξ2) are collinear.This corresponds to waves traveling in the samedirection.

Klainerman (85): The bilinear form ∂αφ1∂αφ2satisfies a cancellation condition, i.e. it kills theinteraction of parallel waves. Algebraically:

τ1τ2 − ξ1ξ2 = 0 if (τ1, ξ1) ‖ (τ2, ξ2) ∈ K

14

The Xs,b spaces are associated to the wave op-

erator as the Sobolev spaces are associated to

∆.

‖u‖Xs,b = ‖u(1 + |ξ|+ |τ |)s(1 + ||ξ| − |τ |)θ‖L2

Mapping properties:

2 : Xs,b → Xs−1,b−1

‖χ(t)u‖Xs,b . ‖f‖Xs−1,b−1 + ‖(u0, u1)‖Hs×Hs−1

Optimal choice of spaces

X = Xs,12, Y = Xs−1,−12

Nonlinear estimates for Γ(φ)∂αφ∂αφ:

‖∂αφ∂αφ‖Xs−1,−1

2. ‖φ‖2

Xs,12

Xs,12 ·Xs,12 → Xs,12, Xs,12 ·Xs−1,−12 → Xs−1,−1

2

True for

s >n

2Klainerman-Machedon, Klainerman-Selberg.

(94-96)

15

The critical case s = n/2. Nonlinearity:

N(φ) = Γ(φ)∂αφ ∂αφ

Bilinear estimates for the spaces X,Y :

∂αX · ∂αX → Y, X · Y → Y, X ·X → X

X,Y must be compatible with scaling. Dyadic

pieces and Littlewood-Paley decomposition:

Xj = {φ ∈ X, supp φ ⊂ {|τ |+|ξ| ∈ [2j−1,2j+1]}}

‖φ‖2X ≈∑j

‖Sjφ‖2Xj

Dyadic bilinear interactions:

a) High freq. × high freq. → low freq.

Pj(Xk ·Xk) → Xj,

Pj(Xk · Yk) → Yj, j ≤ k

Pj(∂αXk · ∂αXk) → Yj

b) High freq. × low freq. → high freq.

Xj ·Xk → Xk,

Xj · Yk → Yk, Xk · Yj → Yk j < k

∂αXj · ∂αXk → Yk

16

(division) 2−1 : Yj → Xj

(i) The division problem: Find dyadic spaces Xj,Yj which satisfy the dyadic estimates.

(ii)The summation problem: Sum up all the dyadicestimates. The summation in k is trivial, thetrouble comes from j.

Naive choice of spaces:

X = Xn2,

12, Y = X

n2−1,−1

2

T. (98-00): Solution for the division problem.Also there is a gain 2−ε|j−k| in the high-high in-teractions.Theorem 6. The wave maps equation is well-posed for initial data which is small in the homo-

geneous Besov space Bn2,21 × B

n2−1,21 .

Good news: Scale invariant =⇒ global.

Bad news: Smaller space than the Sobolev spaceH1 × L2; does not see the geometry of M .

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Wave maps as a nonlinear equation

Nonlinear wave equation:

(NLW ) P (φ, ∂φ, ∂2φ) = 0, φ(0) = φ0, ∂tφ(0) = φ1

Definition 7. (NLW) is well-posed in Hs0×Hs0−1

if for each M > 0 there is some T > 0 so that:(i) (a-priori bound for smooth solutions) Foreach smooth initial data (φ0, φ1) with

‖(φ0, φ1)‖Hs0×Hs0−1 ≤M (1)

there is an unique smooth solution φ in [−T, T ]which satisfies uniform bounds for all s ≥ s0:

‖φ‖C(−T,T ;Hs)∩C1(−T,T ;Hs−1) . ‖(φ0, φ1)‖Hs×Hs−1

(ii) (weak stability) For some s < s0 and any twosuch smooth solutions φ, ψ in [−T, T ] we have

‖φ−ψ‖C(Hs)∩C1(Hs−1) . ‖(φ0−ψ0, φ1−ψ1)‖Hs×Hs−1

(iii) (rough solutions as limits of smooth ones)For any initial data (φ0, φ1) which satisfies (1)there is a solution ψ ∈ C(Hs0) ∩ C1(Hs0−1) in[−T, T ], depending continuously on the initial data,which can be obtained as the unique limit ofsmooth solutions.

18

Space for solutions: X. Also set Xs = |D|s0−sX.For a-priori bounds one needs a bootstrap Lemma:Lemma 8. a) Suppose M is sufficiently small.For all smooth solution φ to (NLW) in [−T, T ]subject to (1) we have

‖φ‖X ≤ 2 =⇒ ‖φ‖X ≤ 1

b) In addition for s > s0 and large α > 0, if

‖φ‖X ≤ 2, ‖φ‖Xs ≤ 2α‖(φ0, φ1)‖Hs×Hs−1

then

‖φ‖Xs ≤ α‖(φ0, φ1)‖Hs×Hs−1

For stability estimates one needs the linearizedequations,

P lin(φ)ψ = 0, ψ(0) = ψ0, ∂tψ(0) = ψ1

Lemma 9.Suppose M is sufficiently small. Thenthere is s < s0 so that the linearized equations areuniformly well-posed in Hs−1×Hs for all smoothsolutions φ to (NLW) in [−T, T ].

Linear, but with rough coefficients. Estimate:

‖ψ‖Xs . ‖(ψ0, ψ1)‖Hs×Hs−1

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The paradifferential calculus.

Transform the nonlinear equation (NLW) into an

infinite system of linear equations,

P lin(φ[<j])φ[j] = error

where φ[<j], respectively φ[j] loosely denote the

part of φ which is at frequency less than 2j−2,

respectively ≈ 2j. The term “error” means an

acceptable error, i.e. which is small in an appro-

priate sense:

error ∈ Yj, P lin(φ[<j])−1 : Yj → Xj

Advantage: We need to study linear, frequency

localized equations.

Equivalent to saying that the low frequency con-

tributions of the high-high frequency interactions

are negligible (true for wave maps).

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Can it be done for wave maps ?

The key work is done by Tao(00-01) who proved

(i) when the target is a sphere for n ≥ 2.

Shortly afterward, similar ideas are used for more

general target manifolds: Klainerman-Rodnianski

(n ≥ 5), Shatah-Struwe and Nahmod-Stefanov-

Uhlenbeck (n ≥ 4), Krieger (n = 3, hyperbolic

space). The low dimensions n ≥ 2 and parts (ii),

(iii) of well-posedness are settled in very recent

work of T.

The spaces. The space Y is essentially the

same as in the earlier work of T. However, the

space X is relaxed (enlarged) slightly. This makes

it easier to prove that

X ∩ L∞ is an algebra

which already removes one (out of three) loga-

rithmic divergence in the high-low interaction.

21

Paradifferential calculus and the trilinear es-

timate

Spherical target: the paradiff. equations are

2φi[k] = −2φi[<k]∂αφ

j[<k]∂αφ

j[k] + error

Not quite satisfactory. Instead from |φ|2 = 1 get

φj<k∂αφ

j[k] = error

2φi[k] = 2(φi[<k]∂αφ

j[<k]−φ

j[<k]∂

αφi[<k])∂αφj[k]+error

Antisymmetric gradient potential:

(Aα<k)ij = 2(φi[<k]∂

αφj[<k] − φ

j[<k]∂

αφi[<k])

The reduction to paradifferential form cannot be

done using bilinear estimates; as it turns out, in

addition one needs a trilinear estimate. From the

dyadic bilinear estimates one gets a trilinear one,

Pk(Xk1∂

αXk2∂αXk3

)→ Yk

plus an additional gain unless k = max{k1, k2, k3}.Tao proves that there is an additional gain unless

k1 < min{k2, k3}.22

The gauge transformation

The right hand side of the paradifferential equa-

tions cannot be treated as an error term. The

idea used by Tao, inspired from similar work of

Helein on the harmonic maps, is to eliminate it

using a gauge transformation

φ[k] → U<kφ[k]

To cancel the Aα<k, U<k should satisfy

U[<k]Aα<k = ∂αU[<k]

Solving this requires the compatibility conditions:

∂βAα<k − ∂αAβ<k = [Aα<k, A

β<k]

Not true ! But we do have a good control over

the curl of A so we can get some good approx-

imate solutions using a paradifferential type for-

mulation,

U[k] = 2U[<k](φi[<k]φ

j[k] − φ

j[<k]u

i[k])

The main estimate is

2(U<kφ[k]) = error

23

Embedded manifolds and Moser type esti-mates (T. (02))

Equations:

2φi = −Sijk(φ)(∂αφj, ∂αφ

k)

Paradifferential formulation:

2φ[k] = −2Aα<k∂αφ[k] + error

(Aα<k)ij = ([Sijl(φ)][<k] − [Sjil(φ)][<k])∂αφl[<k]

To work with nonlinear functions of φ one needsMoser type estimates. The classical Moser esti-mates have the form

‖f(u)‖Hs ≤ c(‖u‖L∞)(1 + ‖u‖Hs)

for smooth f . In our case,

‖f(φ)‖X ≤ c(1 + ‖φ‖NX)

where φ is smooth, bounded, with bounded deriva-tives.

Stability estimates and the linearized equa-tions (T. (02)) Likely I have run out time beforethis .....

∇α∇αψ = R(ψ, ∂αφ)∂αφ

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