THE WAVE MAPS EQUATION - UCB Mathematicstataru/papers/wmslides.pdf · THE WAVE MAPS EQUATION ......
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THE WAVE MAPS EQUATION
Daniel Tataru
Department of Mathematics
University of California, Berkeley
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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.)
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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
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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
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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 .
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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.
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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.
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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.
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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.
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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).
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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.
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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
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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
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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)
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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
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(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.
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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.
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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
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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
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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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