Pointwise decay estimates and asymptotics for semilinear wave ...

Nikodem Szpak ESI, 2010-02-16
Pointwise decay estimates and asymptotics
for semilinear wave equations with small and large data
Nikodem Szpak
University Duisburg-Essen
Workshop on Quantitative Studies of Nonlinear Wave Phenomena 2010
Erwin Schrödinger International Institute for Mathematical Physics (ESI), Wien
16 Feb 2010
• Plan of the talk:
– [Review:] Optimal pointwise decay estimates for ✷u = F (u) ∼ u p , p ≥ 3(> 1 + √ 2)
Same decay estimates for ✷u = F (∂u) ∼ (∂u) p with p ≥ 3
– Delicate case ✷u = q ab ∂ a u ∂ b u (p = 2)
→ Structure of terms: Huygensian, null, weak null and singular (blowup in finite time)
– Pointwise decay for defocusing ✷φ = −|φ| p−1 φ with big initial data by conformal method, 3 ≤ p < 5
[+Roger Bieli]
1

Nikodem Szpak ESI, 2010-02-16
Motivation – Einstein equations
• Einstein eqs. around Minkowski metric g µν = η µν + h µν in harmonic gauge ✷ g x µ = 0
˜✷ g h µν ≡ g αβ ∂ α ∂ β h µν = 1 4 ∂ µh α α ∂ vh β β − 1 2 ∂ µh αβ ∂ ν h αβ + Q µν (∂h, ∂h) + G µν (h)(∂h, ∂h)
• Linearized Einstein eqs. (→ gravitational waves)
✷ η h µν = 0 + terms of order O(h 2 )
Solutions of linearized eqs. ≠ solutions of nonlinear eqs.
• Essential for the asymptotics
– semilinear terms ∼ ∂h · ∂h (“self-interaction of waves”)
– quasilinear terms ∼ h · ∂ 2 h (“scattering on non-flat geometry”)
• Other models: wave maps, Yang-Mills equations
✷φ = F (φ)(∂φ, ∂φ)
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Nikodem Szpak ESI, 2010-02-16
Overview of decay estimates
• ✷u = 0
Huygens principle: compactly supported initial data → support on the light-cone, decay 1/t
• ✷u = F (u, ∂u) ∼ u p · (∂u) q for u ≈ 0 and p + q ≥ 3
– in null direction t − r = const: standard 1/t decay
– for t ≫ r nonlinear scattering: tail u(t, x) ∼ 1/t p+q−1
• ✷u = q ab (u) ∂ a u ∂ b u
– Blowup in finite time [John] for ✷u = (∂ t u) 2
– Global existence if null condition [Ch, Kl]: q µν ξ µ ξ ν = 0 on every null vector ξ (η µν ξ µ ξ ν = 0)
and decay |u(t, x)| 〈t + |x|〉 −1 〈t − |x|〉 −1/2
(→ generalized to quasilinear wave equations, systems of wave equations...)
– Weak null condition [Li-Ro]: asymptotic eq. (ignore light-cone derivatives and cubic terms)
(∂ t + ∂ r )(∂ t − ∂ r )(ru) ∼ rA mn (∂ t − ∂ r ) m u(∂ t − ∂ r ) n u + (...)
Asymptotically (light-cones) not free ↔ slower decay than 1/t
Example: ✷u = u∆u [Lindblad, Alinhac] → decay only like ε/t 1−cε or log(t)/t
→
Nonlinear waves may behave essentially different than linear! - depending on the structure of nonlinearities
(Gravitational waves ← structure of the Einstein equations)
More structural conditions
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Nikodem Szpak ESI, 2010-02-16
• ✷u = F with initial data (u, ∂ t u)| t=0 = (f, g)
→ Duhamel representation formula
∫
∫
f(y)
u(t, x) = ∂ t
4πt dµ S(y) +
S(t,x)
Decay estimates (in weighted L ∞ spaces)
S(t,x)
g(y)
4πt dµ S(y) + 1
4π
∫
K(t,x)
F (s, y)
t − s dµ K(s, y)
• Weighted-L ∞ estimate [Asakura: (f, g, F ) ∈ C 3 × C 2 × C 2 → C 2 ; NS: (˜L ∞ ) 3 → ˜L ∞ ]
‖u‖˜L∞
≡ ‖τ + τ q−1
−
u(t, x)‖ L ∞ ‖〈x〉q f‖ L ∞ + ‖〈x〉 q+1 ∇f‖ L ∞ + ‖〈x〉 q+1 g‖ L ∞
+ ‖〈x〉 q τ + τ q−1
−
F (t, x)‖ L ∞
with τ ± := 〈t ± |x|〉 and 〈x〉 := 1 + |x|. Estimates for ‖τ + τ q−1
−
∂α u(t, x)‖ L ∞ follow by analogy
• By iteration ✷u n+1 = F (u n ) converges in ˜L ∞ for
|F (u)| |u| p , |f(x)| ≤ f 0
〈x〉 m, |∇f(x)| ≤ f 1
〈x〉 m+1, |g(x)| ≤ g 0
〈x〉 m+1
and p > 1 + √ 2, m ≥ 2. Then u n → u ∈ ˜L ∞ and
‖u‖˜L∞
≤ C ⇔ |u(t, x)| ≤
C
〈t + |x|〉〈t − |x|〉 q−1, q := min(p − 1, m) 4

Nikodem Szpak ESI, 2010-02-16
Perturbation theory → optimal decay rates
• ✷u = F (u) with data (u, ˙u)| t=0 = (εf, εg) (← natural scale)
u =
∞∑
ε n v n
n=1
→
{ ✷v1 = 0 with data(f, g)
✷v n = F n (v n−1 , ..., v 1 ) with data(0, 0)
where F n (v n , ..., v 1 ) ← collection of powers of ε. [First nontrivial: F p ≠ 0]
• From convergence in ˜L ∞ → pointwise control of the truncation error
n∑
u(t, x) − ε k v k (t, x)
∣ ∣ ε n
〈t + |x|〉〈t − |x|〉 q−1
k=0
• Optimal decay rates ← low orders solvable analytically:
– First order: ✷u 0 = 0 → Huygens
– Second (p-th) order: ✷u p = A(u 1 ) p → already generic decay rate: u p ∼ u !
– All higher orders: too weak to change the asymptotics (← convergence of the perturbation series)
– We can calculate v p (almost) explicitly
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Nikodem Szpak ESI, 2010-02-16
2nd perturbation → expected decay rates
• ✷u n+1 = F (u n , ∂u n ) (assume spherical symmetry for simplicity) leads to
✷u 1 = 0 + data (f, g) → u 1 (t, r) =
✷u 2 = F 2 (u 1 , ∂u 1 ) + data (0, 0) → u 2 (t, r) = 1 r
h(t − r)
r
∫ t
0
(h depends on f, g)
∫ r+(t−t
′ )
dt ′ dr ′ r ′ F 2 (u 1 , ∂u 1 )(t ′ , r ′ )
|r−(t−t ′ )|
Tail (t ≫ r) Wave (t ∼ r) Iteration
Structure F 2 (u 1 , ∂u 1 ) = u 2 (t, r) ∼ = u 2 (t, r) ∼ = scheme
u p 1
At −p+1 Bt −1 for p ≥ 3
Blog(t)/t for p = 2
works
fails → blowup
(∂u 1 ) p At −p+1 Bt −1 for p ≥ 3 works
null → ∂ + u 1 · ∂ − u 1 0 Bt −1 works
weak null → ∂ + u 1 · ∂ + u 1 At −3 Bt −1 works
“singular” → ∂ − u 1 · ∂ − u 1 At −1 Blog(t)/t fails→ blowup
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Nikodem Szpak ESI, 2010-02-16
Modified null derivatives → classification of tails and cancellations
• Modified null deriv’s: D ± := ∂ t ± 1 r ∂ r(r·) = 1 r ∂ ±(r·) → quick decay for D + u (standard for D − u)
→ D ± u(t, r) = 1 ∫
r ′ 1
F dv ∓ → |D + u|
r
r〈t + r〉 p−1, |D 1
−u|
r〈t − r〉 p−1
null
• ✷u = D + u∂u → any decay rate! → Huygen’s principle (no scatter) – iteratively kills the rhs’s...
(
✷u = q σλ ∂ σ u∂ λ u = q σλ D σ u − σ ) (
r u D λ u − λ )
r u ≡ Q σλ D σ uD λ u + P λ u
r D λu + R u2
r 2
Order with respect to presence of D + and ∂ − (← to see one more cancellation!)
✷u = Q −− ∂ − u∂ − u
+ Q ++ D + uD + u + Q +− D + u∂ − u + P + u
r D +u
+ P − u
r ∂ −u
}
→ blowup in finite time
}
→ Huygens (no scattering)
}
→ Huygens (cancellation)
+ R u2
r 2 }
→ tail ∼ 1/t
3
Extra cancellation from total derivative:
∫
r u r ∂ −u dv − = 0
7

Nikodem Szpak ESI, 2010-02-16
Asymptotics for big data from positive energy (defocusing nonlinearity)
• Defocusing semilinear wave equations
✷φ = −|φ| p−1 φ (✷ ≡ ∂ 2 t − ∆)
give rise to a conserved positive energy
∫ ( 1
E =
R 3 2 |∂ tφ| 2 + 1 2 |∇φ|2 + 1 )
p + 1 |φ|p+1
d 3 x ≥ 0
• Existing results:
– Jörgens’61: global existence of C 2 solutions in the energy subcritical case 1 < p < 5
(Struwe’88: the spherically symmetric critical p = 5 case; Grillakis’90: the general critical case p = 5)
– Global existence is not known for p > 5
– Pecher’74: uniform boundedness of solutions for 2 < p < 5
– Strauss’68: uniform decay 1/t 1−ɛ and scattering for 3 ≤ p < 5
Bahouri and Shatah’98: finite energy solutions decay to zero for p = 5
Hidano’01: scattering and decay to zero for 2.5 < p ≤ 3
Ginibre and Velo’89: scattering in the energy space for 1 < p < 5, but in n = 3 only for small data
• We go further and study the pointwise behavior of solutions for 3 ≤ p < 5.
Pointwise estimates exist only for small data [Asakura, Strauss-Tsutaya, N.S.] ← perturbation techniques.
Conformal compactification [Choquet-Bruhat, Christodoulou et al] → adjusted to the nonlinearity.
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Nikodem Szpak ESI, 2010-02-16
Asymptotics for big data – defocusing nonlinearity, spherical symmetry
• For null coordinates u = t + r, v = t − r define
conformal mapping K + ∋ (u, v) → (ũ, ṽ) ∈ K −
ũ := − 1
1
up−2, ṽ := −
v p−2
1
t
I
K +
• The conformal factor Ω := ˜r/r
multiplies the transformed solution ˜φ(ũ, ṽ) := Ω −1 φ(u, v)
which satisfies the transformed wave equation in K −
˜✷˜φ = − 1 [
]
ũ − ṽ
p−1
|˜φ| p−1 ˜φ
(p − 2) 2 (−ṽ) 1/(p−2) − (−ũ)
} {{ 1/(p−2) }
α(ũ,ṽ)
• We consider some ”pseudo-energy” flux (Ẽt, Ẽr) in K −
such that div Ẽ ≤ 0
→ boundedness of the flux along null lines ũ = const.
−1
J
1
S
r
• → Uniform boundedness of |˜φ(ũ, ṽ)| ≤ ˜C
H
K − 9
• → Pointwise boundedness of |φ(u, v)| ≤ ˜C · Ω(u, v)

Nikodem Szpak ESI, 2010-02-16
Asymptotics for big data – defocusing nonlinearity, spherical symmetry
• Consider equations
t
˜✷˜φ = α(ũ, ṽ) |˜φ| p−1 ˜φ
K +
with α ≥ 0 and ∂ t α ≤ 0
1
I
• And the corresponding ”pseudo-energy” flux in K −
Ẽ t := r 2 [ 1
2 (∂ t ˜φ) 2 + 1 2 (∂ r ˜φ) 2 + 1
p + 1 α · |˜φ| p+1 ]
,
Ẽ r := −r 2( ∂ t ˜φ · ∂r ˜φ) ,
1
r
• → div Ẽ = 1
p+1 r2 · ∂ t α · |˜φ| p+1 ≤ 0
S
• Boundedness of the null Flux(ũ) := ∫ ũ=const. (Ẽt + Ẽr) dṽ
→ 1
p + 1
∫
ũ=const.
˜r 2 · α · |˜φ| p+1 dṽ ≤ Flux(ũ) ≤ E 0 ,
−1
H
J
• [Pecher’74] Hölder ineq. + integral repr. of the wave eq.
→ Uniform boundedness of |˜φ(ũ, ṽ)| ≤ ˜C
K − 10

Nikodem Szpak ESI, 2010-02-16
Asymptotics for big data – defocusing nonlinearity, spherical symmetry
• Finally, the inverse transformation (ũ, ṽ) → (u, v) and φ = Ω · ˜φ provides the desired estimate:
the global pointwise bound
|φ(u, v)| ≤ ˜C · Ω = ˜C
∣ 1
r
∣
(t − r) − 1 ∣∣∣
≤
p−2 (t + r) p−2
for t ≥ 1 and 0 ≤ r ≤ t
• Optimal Numerical simulations done by Bizoń et al suggest (generically) yes.
C
(1 + t + r)(1 + t − r) p−2
• The same estimate as in the small data regime where it is optimal. We have for t ≫ r
φ(t, r) =
C
t p−1 + O(t−p )
• The power p − 2 in the conformal mapping cannot be increased
→ stronger pointwise decay, but α(t, r) becomes singular
• Spherical symmetry essential for the method to yield optimal decay for p > 3.
On R × R 3 the obvious analogue of the map (u, v) → (ũ, ṽ) is no longer conformal, unless p = 3
(radial and angular parts of the wave operator transform diversely → the resulting equation is not semilinear).
Apply the transformation with p = 3 (conformal) to an equation with nonlinearity of power p > 3
→ gives a global 1/[(1 + t + r)(1 + t − r)] decay, however not optimal. Improve...
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Nikodem Szpak ESI, 2010-02-16
Asymptotics for big data – defocusing nonlinearity, no symmetry
• Conformal mapping K + ∋ (u, v, θ, ϕ) → (ũ, ṽ, ˜θ, ˜ϕ) ∈ K −
ũ := −1/u, ṽ := −1/v, ˜θ := θ, ˜ϕ := ϕ
• ˜φ(ũ, ṽ) := Ω −1 φ(u, v) with the conformal factor Ω := ˜r/r = (t 2 − |x| 2 ) −1 satisfies (in K − )
˜✷˜φ = −|˜φ| p−1 ˜φ
• Energy flux (E t , E r ) in K − such that div Ẽ = 0 → Flux(ũ) Bounded → |˜φ(ũ, ṽ)| ≤ ˜C
• The inverse transformation (ũ, ṽ, ˜θ, ˜ϕ) → (u, v, θ, ϕ) and φ = Ω · ˜φ give the pointwise bound
|φ(u, v)| ≤ ˜C · Ω ≤
C
(t + r)(t − r)
• Improve it by inserting into the wave equation, inverting ✷ (Duhamel formula) and estimating
|φ(t, x)| ≤ ✷ −1 |φ| p ✷ −1 1
(uv) p 1
(1 + t + |x|)(1 + t − |x|) p−2.
for t ≥ 1 and 0 ≤ r ≤ t. (Initial data of compact support do not contribute to the tail)
• Optimal (= spherical symmetry ↔ numerics & small data results)
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Nikodem Szpak ESI, 2010-02-16
Overview of nonlinear asymptotics
Equations (initial data ∼ ε) linearized nonlinear
✷u = F (u) ∼ |u p | no tail tail ∼ ε p /t p−1
tail
✷u + V (x)u = F (u) ∼ |u p |, |V (x)| ∼ 1/|x| k u ∼ ε/t k tail ∼ ε p /t p−1
waves
✷u = (∂ − v) 2 , ✷v = 0 u, v ∼ ε/t u ∼ ε 2 log(t)/t
waves
✷u = u∆u u ∼ ε/t u ()
∼ ε 2 log log(t)/t
ε 2 /t 1−ε
asymptotic form of the Einstein eqs.[L.-R., harmonic coord.] tail
✷φ 1 = Q(∂φ, ∂φ) + (∂φ 2 ) 2 + φ 3 ∂ 2 φ 1 φ i = 0 φ 1 t −1
✷φ 2 = Q(∂φ, ∂φ)
waves
✷φ 3 = Q(∂φ, ∂φ) φ i ∼ ε/t φ 1 ε 2 log(t)/t
→ Weyl scalars (ψ 4 ), algebraic invariants...
13

Nikodem Szpak ESI, 2010-02-16
Self-gravitating scalar field in spherical symmetry
Parametrization of the metric
ds 2 = e 2α(t,r) ( −e 2β(t,r) dt 2 + dr 2) + r 2 dΩ 2 ,
Define the mass function m(t, r) = (1 − e −2α )r d−2 . Then
m ′ = κ r d−1 ( ˙φ2 + φ ′ 2 ) ,
ṁ = 2κ r d−1 ˙φφ ′ ,
β ′ =
(d − 2)m
r d−1 ,
✷φ = 2β ¨φ + ˙β ˙φ + β ′ φ ′ .
No clear asymptotics for φ. After rearrangement of the nonlinear terms on the r.h.s.
✷φ = − 1 r β∂ −φ + 2∂ + β∂ + φ + 2 r ∂ −(rβ∂ − φ) + 2 r β∂2 + (rφ) ∼ = − 1 r β∂ −φ.
The leading asymptotics can be calculated exactly
φ(t, r) = C(φ 0, φ 1 )
t 3
( ) 1
+ O .
t 4
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Nikodem Szpak ESI, 2010-02-16
Conclusions
• Properties of the linearized waves vs. nonlinear waves
– in case of Einstein equations: gravitational waves
– decay ← linear and nonlinear terms
– propagation of waves ← null/weak null structure
• Solution with a finite weighted-L ∞ norm:
– Global existence
– Nonlinear stability
‖u‖ L ∞
t,x
(R 1+3 ) ≤ C
initial data (εf, εg) → ‖u‖ L ∞
t,x
(R 1+3 ) ≤ εC
– Pointwise decay
|(∂ n C
)u(t, x)| ≤
〈t + |x|〉〈t − |x|〉 q−1
q describes the rate of scattering (even for comp. supp. data)
Perturbation theory → optimal decay rates.
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Nikodem Szpak ESI, 2010-02-16
• Null condition [Klainerman]:
– for semilinear wave equations
Null and weak null structures
✷u = Q(u)(∂u, ∂u) = q µν (u) ∂ µ u ∂ ν u
and q µν ξ µ ξ ν = 0 on every null vector ξ (→ q µν ∼ η µν )
– for quasilinear wave equations
✷ g(u,∂u) u ≡ g µν (u, ∂u)∂ µ ∂ ν u = 0, g µν = ĝ µνα ∂ α u + ...
and ĝ µνα ξ µ ξ ν ξ α = 0 on every null vector ξ
– For systems of wave equations
✷u A = q A µν BC
∂ µ u B ∂ ν u C
and q BCµν A ξ µ ξ ν = 0 on every null vector ξ
→ Global existence [Christodoulou, Klainerman] and decay 〈t + |x|〉 −1 〈t − |x|〉 −1/2 ∼ 1/t (lightcone)
• Weak null condition [Lindblad & Rodnianski]: asymptotic eq.
(∂ t + ∂ r )(∂ t − ∂ r )(ru) ∼ rA mn (∂ t − ∂ r ) m u(∂ t − ∂ r ) n u + (...)
– ignored: derivatives tangential to the lightcone and cubic terms (Here: null condition ↔ A mn = 0)
Asymptotically (light-cones) not free ↔ slower decay (than 1/t)
• → Global existence + decay estimate 1/t or less.
↩→ True decay rates: on the lightcone (t ∼ |x|) and inside (t ≫ |x|)
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Nikodem Szpak ESI, 2010-02-16
Precise asymptotics from scaling
Simplified for the talk → pure power nonlinearity & spherical symmetry
✷u = u p , u(0, r) = εf(r), ∂ t u(0, r) = εg(r)
The linear(ized) version of the equation with removed scale factor ε
✷u 0 = 0, u 0 (0, r) = f(r), ∂ t u 0 (0, r) = g(r)
has solution
u 0 (t, r) =
h(t − r) − h(t + r)
, h(x) := − x r
2 f(x) − 1 2
∫ ∞
x
yg(y)dy
Introduce the difference
and scale it
In the limit ε → 0
w(t, r) := u(t, r) − εu 0 (t, r) → ✷w = (w + εu 0 ) p
W ε (t, r) := ε −b w(ε −a t, ε −a r), b = p + a(p − 1), a > 0
[ ] p
ε −a(p+1) u 0 (ε −a t, ε −a δ(t − r) − δ(t + r)
r) → Cp , C
r p−k p :=
∫
h p (x)dx.
17

Nikodem Szpak ESI, 2010-02-16
We get a limiting equation
✷W (t, r) = C p
δ(t − r)
r p , lim
ε→0
‖W ε − W ‖ L ∞
1,p−1
= 0.
We can solve exactly
W (t, r) = A p
Θ(t − r)
r
[
]
1
(t − r) − 1
.
p−2 (t + r) p−2
with A p := 2 p−3 C p /(p − 2).
Finally, we reconstruct the asymptotics of u for small ε. Set a = 1 for convenience.
u(t, r) = εu 0 (t, r) + ε 2p−1 W ε (εt, εr)
[
]
= εu 0 (t, r) + ε p Θ(t − r) 1
A p
r (t − r) − 1
p−2 (t + r) p−2
+ o(ε 2p−1 ).
with the error term o(ε 2p−1 ) controlled in the L ∞ 1,p−1
It gives (note that u 0 is localized near t = r)
norm → guarantees at least the same falloff.
u(t, r) ∼ 1/t p−1 for t ≫ r, u(t, r) ∼ 1/r for t, r ≫ 1, t − r = const.
18

Nikodem Szpak ESI, 2010-02-16
Applications (semilinear w/o derivatives)
• → Semilinear wave equations without derivatives (on r.h.s.)
✷u = F (u, x)
• Wave map M 1+3 → S 3 , spherical symmetry, corotational
∂ 2 t u − ∂2 r u − 2 r ∂ ru = − sin(2u) ∼
2 4u3
= −
r 2 r2u +
3r + 2 O(u5 )
→ |u(t, r)| ≤
Cr
〈t + |x|〉 2 〈t − |x|〉 3
• Yang-Mills M 1+3 → SU(2), spherical symmetry
∂ 2 t u − ∂2 r u − 2 r ∂ ru + 2 r 2u = −u3 − 3u2
r
→ |u(t, r)| ≤
Cr
〈t + |x|〉 2 〈t − |x|〉 2 19

Nikodem Szpak ESI, 2010-02-16
Problem with derivatives → spherical symmetry
• In ✷u = q µν ∂ µ u∂ ν u one needs to control 1st derivatives: ✷∂u = ∂F → ‖∂u‖˜L∞ ‖∂F ‖˜L∞
∂u(t, x) = 1
4π
∫ t
0
∫
ds
S(x,t−s)
∂F (s, y)
t − s
d2 y
Problem: ∂F contains ∂ 2 u → no control (loss) of derivatives (in the iteration)
• In spherical symmetry – no problem ← gain of regularity
→
u(t, r) = 1 2r
∫
∂ ± u(t, r) = 1 r
null
∫ t
0
ds
∫ r+(t−s)
|r−(t−s)|
r ′ F (s, r ′ )dr ′ =: 1 r
∫ ∫
r ′ F dv + dv −
r ′ F dv ∓ ∓ 1 ∫ ∫
r ′ F dv
r 2 + dv − → ‖∂ ± u‖˜L∞
‖F ‖˜L∞
• Modified null deriv’s: D ± := ∂ t ± 1 r ∂ r(r·) = 1 r ∂ ±(r·) → quick decay for D + u (standard for D − u)
→
D ± u(t, r) = 1 r
∫
null
r ′ F dv ∓ → |D + u|
1
r〈t + r〉 p−1,
|D −u|
1
r〈t − r〉 p−1 20