Dispersive estimates in Rn (n ≥ 2)
for the Schrödinger and the wave equations
Simon Moulin
Nantes, Laboratoire de Mathématiques Jean Leray
11 avril 2008 - Orléans
Simon Moulin (Nantes, LMJL)
Dispersive estimates
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1
Introduction
2
New results
Wave equation (n ≥ 3)
Schrödinger equation (n ≥ 4)
Dimension n = 2, high frequencies
3
Ideas of the proofs
4
Conclusion
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1
Introduction
2
New results
Wave equation (n ≥ 3)
Schrödinger equation (n ≥ 4)
Dimension n = 2, high frequencies
3
Ideas of the proofs
4
Conclusion
Simon Moulin (Nantes, LMJL)
Dispersive estimates
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Goals
Plan of study for Cauchy problem associated to evolution problems.
To study the linear problem.
To obtain a priori estimates for the linear problem.
To study only some equations and to obtain a priori estimates for
these equations.
To search what properties could be generalized.
To study non-linear problems by perturbative methods.
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A priori estimates
Energy conservation.
Dispersive estimates: Lpx (Rn ) → Lqx (Rn ).
Strichartz estimates.
Strichatz estimates are a priori estimates, for example for one-order PDE:
L2x (Rn ) → Lpt (0, t; Lqx (Rn )).
For two-order PDE 2, one have to have more regularity on initial conditions.
Ingredients to obtain these estimates:
dispersive estimates,
T ∗ T argument,
Sobolev embeddings (for example Hardy-Littlewood-Sobolev inequality).
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Schrödinger equation
Schrödinger equation
i∂t u − ∆u = 0
with initial condition u(0, x) = u0 (x).
Conservation of the L2 norm:
ku(t)kL2 = kû(t)kL2 = kû0 kL2 = ku0 kL2 .
Kernel:
u(t, x) = e−it∆ u0 (x) =
1
(4πit)n/2
Z
ei
|x−y |2
4t
u0 (y) dy.
Rn
The L1 − L∞ estimate:
ku(t)kL∞ ≤ C|t|−n/2 ku0 kL1 ,
t 6= 0.
Interpolation between L2 − L2 estimate and L1 − L∞ estimate.
0
Lp − Lp estimates with 2 ≤ p ≤ +∞ and α = 1 − 2/p:
n
−it∆ e
p0 p ≤ C|t|−α 2 .
L →L
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Equations
Wave equation:
∂t2 u − ∆u = 0,
u(0, x) = u0 (x),
∂t u(0, x) = u1 (x).
Schrödinger equation:
i∂t u − ∆u = 0,
u(0, x) = u0 (x).
Dispersion relation: Schrödinger ω = kξk2 and wave ω = ±kξk.
Group speed: Schrödinger vg = 2ξ and wave vg = ±ξ/kξk.
Phase speed: Schrödinger ξ · (vφ − ξ) = 0 and wave ξ · (vφ ∓ ξ/kξk) = 0.
We try to establish global in time decay estimates of the solution u(t, x)
according to initial data u0 .
Dispersion for Schrödinger ku(t)kL∞ ≤ C|t|−n/2 ku0 kL1 , t 6= 0.
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Equations
Wave equation:
∂t2 u − ∆u = 0,
u(0, x) = u0 (x),
∂t u(0, x) = u1 (x).
Schrödinger equation:
i∂t u − ∆u = 0,
u(0, x) = u0 (x).
Dispersion relation: Schrödinger ω = kξk2 and wave ω = ±kξk.
Group speed: Schrödinger vg = 2ξ and wave vg = ±ξ/kξk.
Phase speed: Schrödinger ξ · (vφ − ξ) = 0 and wave ξ · (vφ ∓ ξ/kξk) = 0.
We try to establish global in time decay estimates of the solution u(t, x)
according to initial data u0 .
Dispersion for Schrödinger ku(t)kL∞ ≤ C|t|−n/2 ku0 kL1 , t 6= 0.
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High and low frequencies
Free operator spectra σ(G0 ) = σess (G0 ) = [0, +∞[.
High frequencies: λ 1, h 1.
Low frequencies: λ 1, h 1.
R0 (λ) = (G0 − λ2 )−1 = h2 (h2 G0 − z 2 )−1 = h2 R0,h (z) with z = hλ.
Cut-off functions:
ηa for low frequencies and χa for high frequencies.
With ηa (G) + χa (G) = Pac .
High frequencies: oscillations, low regularity.
Low frequencies: slow variations, high regularity.
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Dispersion for the wave equation
Dispersive estimates for the free wave equation
L1 − L∞ low frequencies estimates (n ≥ 3, ∀t):
√
it G0 −(n+1)/4+ε
G0
ηa (G0 )
≤ Cε hti−(n−1)/2 ,
e
1
∞
L →L
L1 − L∞ high frequencies estimates (n ≥ 2, ∀t 6= 0):
√
n−1
it G0 −(n+1)/4−ε
G0
χa (G0 )
≤ Cε |t|− 2 ,
e
1
∞
L →L
0 < ε 1.
0 < ε 1.
0
Lp − Lp estimates (n ≥ 3, 2 ≤ p < +∞ et α = 1 − 2/p):
√
n−1
it G0 −α(n+1)/4
G0
Pac 0
≤ C|t|−α 2 .
e
Lp →Lp
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Problem:
Which properties of dispersive estimates are preserved if we perturb
equations with a potential ?
Perturbed equations:
i∂t u − ∆u + V (x)u = 0,
∂t2 u − ∆u + V (x)u = 0,
x ∈ Rn , n ≥ 2.
Obstacles to decay for the perturbed problem:
Eigenvalues.
Waves u(t, x) = eiλt u0 (x). No decay in time.
Resonances.
Results of Jensen-Kato (79) and Rauch (78): decay in hti−1/2 instead of
hti−3/2 for Schrödinger equation with a potential (in 3D) if 0 resonance.
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Problem:
Which properties of dispersive estimates are preserved if we perturb
equations with a potential ?
Perturbed equations:
i∂t u − ∆u + V (x)u = 0,
∂t2 u − ∆u + V (x)u = 0,
x ∈ Rn , n ≥ 2.
Obstacles to decay for the perturbed problem:
Eigenvalues.
Waves u(t, x) = eiλt u0 (x). No decay in time.
Resonances.
Results of Jensen-Kato (79) and Rauch (78): decay in hti−1/2 instead of
hti−3/2 for Schrödinger equation with a potential (in 3D) if 0 resonance.
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How to well-pose the problem ?
Assumptions on V to define eitG et eit
√
G
.
Assumptions on V to have:
no eigenvalues > 0 and no resonances > 0,
σ(G) = [0, ∞[∪{λj ≤ 0}.
Typical assumption:
V ∈ L∞ (Rn ),
|V (x)| ≤ Chxi−δ ,
δ > 1.
(V (δ))
Assumption 0 not eigenvalue nor resonance: 0 regular point.
Simon Moulin (Nantes, LMJL)
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Known results
Schrödinger equation (0 regular)
Same estimations as the free problem under the following assumptions:
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Known results
Schrödinger equation (0 regular)
Same estimations as the free problem under the following assumptions:
Journée, Soffer, Sogge (91): dim ≥ 3, V (δ) avec δ > n + 4 + technical
conditions, V̂ ∈ L1 .
High frequencies: iteration of Duhamel formula and regularization
estimates. δ > n et V̂ ∈ L1 .
Low frequencies: expansions of the resolvant near 0 proved by
Jensen-Kato. δ > n + 4.
Yajima (95): same assumptions with δ > n + 2.
Let W = s − lim e−itG eitG0 the usual wave operator.
t→∞
eitG Pac = WeitG0 W ∗ and if kW kL∞ →L∞ is finite, then
keitG Pac k1→∞ = kWeitG0 W ∗ k1→∞ ≤ C|t|−n/2
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Known results
Schrödinger equation (0 regular)
Same estimations as the free problem under the following assumptions:
Journée, Soffer, Sogge (91): dim ≥ 3, V (δ) avec δ > n + 4 + technical
conditions, V̂ ∈ L1 .
Yajima (95): same assumptions with δ > n + 2.
Schlag (04): dim 2, V (δ) with δ > 3.
Schlag, Rodnianski (04): dim 3, small potential (small Rollnik norm and
Kato norm)
Vodev (05), Yajima (05): dim 3, V (δ) with δ > 5/2.
Vodev: use of Φ(t; h) = eitG φ(h2 G) − eitG0 φ(h2 G0 ) and semi-classical
type estimates with small h.
Simon Moulin (Nantes, LMJL)
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Known results
Schrödinger equation (0 regular)
Same estimations as the free problem under the following assumptions:
Journée, Soffer, Sogge (91): dim ≥ 3, V (δ) avec δ > n + 4 + technical
conditions, V̂ ∈ L1 .
Yajima (95): same assumptions with δ > n + 2.
Schlag (04): dim 2, V (δ) with δ > 3.
Schlag, Rodnianski (04): dim 3, small potential (small Rollnik norm and
Kato norm)
Vodev (05), Yajima (05): dim 3, V (δ) with δ > 5/2.
Golberg (06): dim 3, V ∈ L3/2− (R3 ) ∩ L3/2+ (R3 ), 0 < 1.
Expansion in Born series by iterating R(z) = R0 (z) − R0 (z)VR(z), plus
limiting absorption principle.
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Wave equation (0 regular)
Beals, Strauss (93): dim ≥ 3, V (δ) avec δ > n + 1 et V small enough or
δ > 3n, V ≥ 0 plus conditions on the derivatives.
Beals (94): dim ≥ 3, V ∈ S(Rn ) + V ≥ 0 or small potential.
(Iteration of Duhamel formula plus estimates on the kernels.)
Cardoso, Cuevas, Vodev (05): dim 2 and 3, V (δ) with δ > (n + 1)/2 + in
dim 2 no resonances > 0, high frequencies estimates.
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Known results in dimension n ≥ 4
Yajima (Schrödinger): δ > n + 2 and V̂ ∈ L1 (+ technical conditions).
Beals, Strauss (Ondes): decay (δ > n + 1) + small potential or decay +
regularity (V ∈ S(Rn )) + V ≥ 0.
Some methods:
Resolvant properties (limiting absorption principle, Bessel function
properties).
Resolvant expansion around 0.
Duhamel formula, iteration of Duhamel formula.
Wave operator (Yajima).
Goldberg, Visan (06)
The usual dispersive estimate for the Schrödinger equation in n ≥ 4 is not
true for a compact support potential V , V ∈ C α , α < (n − 3)/2.
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Known results in dimension n ≥ 4
Yajima (Schrödinger): δ > n + 2 and V̂ ∈ L1 (+ technical conditions).
Beals, Strauss (Ondes): decay (δ > n + 1) + small potential or decay +
regularity (V ∈ S(Rn )) + V ≥ 0.
Some methods:
Resolvant properties (limiting absorption principle, Bessel function
properties).
Resolvant expansion around 0.
Duhamel formula, iteration of Duhamel formula.
Wave operator (Yajima).
Goldberg, Visan (06)
The usual dispersive estimate for the Schrödinger equation in n ≥ 4 is not
true for a compact support potential V , V ∈ C α , α < (n − 3)/2.
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Problems
Problem 1
Which dispersive estimate can we obtain with decay assumption only ?
Problem 2
Which regularity assumption do we have to make to obtain usual dispersive
estimate ?
Problem 3
Which is the minimal assumption that we have to make ?
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High frequencies results
Vodev (06): in dim n ≥ 4, at high frequencies, V (δ) with
δ > (n + 1)/2 for the wave equation,
δ > (n + 2)/2 for the Schrödinger equation,
same decay as the free problem but with a loss of (n − 3)/2 derivatives.
Dispersive estimates (for all a > 0):
√
it G −(n−1)/2−ε
G
χa (G) 1 ∞ ≤ Cε |t|−(n−1)/2 , ∀t 6= 0, ∀0 < ε 1,
e
L →L
itG −(n−3)/4
χa (G)
e G
L1 →L∞
Simon Moulin (Nantes, LMJL)
≤ C|t|−n/2 ,
Dispersive estimates
∀t 6= 0,
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1
Introduction
2
New results
Wave equation (n ≥ 3)
Schrödinger equation (n ≥ 4)
Dimension n = 2, high frequencies
3
Ideas of the proofs
4
Conclusion
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Dispersive estimates (Wave n ≥ 3)
Assumption linked with the resolvant kernel (n ≥ 3):
Z sup
|x − y |−n+2 + |x − y |−(n−1)/2 |V (x)| dx < +∞.
y∈Rn
(VOn)
Rn
Integrability assumption (n = 3): V ∈ L3/2−ε (R3 ),
0 < ε 1. (L3/2−)
Low frequencies
Assumptions (VOn) in dim n ≥ 4 and (VOn)+(L3/2-) in dim n = 3, 0 regular.
There exists a0 > 0 as for all 0 < a ≤ a0 :
√
n−1
it G −(n+1)/4+ε
G
ηa (G) 1 ∞ ≤ Cε hti− 2 , ∀t, ∀0 < ε 1.
e
L →L
Dispersive estimate
Assumptions V(δ) with δ > (n + 1)/2 and 0 regular. For all 0 < ε 1, t 6= 0:
√
it G −(n+1)/4+ε
G
hGi−(n−3)/4−2ε Pac 1 ∞ ≤ Cε |t|−(n−1)/2 ,
e
L →L
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Dispersive estimates (Wave n ≥ 3)
Assumption linked with the resolvant kernel (n ≥ 3):
Z sup
|x − y |−n+2 + |x − y |−(n−1)/2 |V (x)| dx < +∞.
y∈Rn
(VOn)
Rn
Integrability assumption (n = 3): V ∈ L3/2−ε (R3 ),
0 < ε 1. (L3/2−)
Low frequencies
Assumptions (VOn) in dim n ≥ 4 and (VOn)+(L3/2-) in dim n = 3, 0 regular.
There exists a0 > 0 as for all 0 < a ≤ a0 :
√
n−1
it G −(n+1)/4+ε
G
ηa (G) 1 ∞ ≤ Cε hti− 2 , ∀t, ∀0 < ε 1.
e
L →L
Dispersive estimate
Assumptions V(δ) with δ > (n + 1)/2 and 0 regular. For all 0 < ε 1, t 6= 0:
√
it G −(n+1)/4+ε
G
hGi−(n−3)/4−2ε Pac 1 ∞ ≤ Cε |t|−(n−1)/2 ,
e
L →L
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Dispersive estimates (Schrödinger n ≥ 4)
Assumption linked with the resolvant kernel with 0 < ε 1:
Z sup
|x − y |−n+2 + |x − y |−(n−2)/2+ε |V (x)| dx < +∞,
y∈Rn
(VSn)
Rn
Low frequencies
Assumptions (VSn) and 0 regular. There exists a0 > 0 as for all 0 < a < a0 :
itG
6 0.
e ηa (G) 1 ∞ ≤ C|t|−n/2 , ∀t =
L →L
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Dispersive estimates (Schrödinger n ≥ 4)
Dispersive estimate
Assumptions V(δ) with δ > (n + 2)/2 and 0 regular:
itG
e hGi−(n−3)/4 Pac 1 ∞ ≤ C|t|−n/2 ,
L →L
∀t 6= 0.
Dispersive estimate with more regularity
0 regular, V(δ) with δ > n − 1 and V̂ ∈ L1 (Rn ).
itG
e Pac 1 ∞ ≤ C|t|−n/2 ,
L →L
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∀t 6= 0.
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Dispersive estimates (Schrödinger n ≥ 4)
Dispersive estimate
Assumptions V(δ) with δ > (n + 2)/2 and 0 regular:
itG
e hGi−(n−3)/4 Pac 1 ∞ ≤ C|t|−n/2 ,
L →L
∀t 6= 0.
Dispersive estimate with more regularity
0 regular, V(δ) with δ > n − 1 and V̂ ∈ L1 (Rn ).
itG
e Pac 1 ∞ ≤ C|t|−n/2 ,
L →L
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∀t 6= 0.
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High frequencies dispersive estimate in dim 2
Assumption linked with the resolvant kernel:
Z
sup
|x − y |−1/2 |V (x)|dx ≤ C < +∞.
y ∈R2
(V 2)
R2
Wave
Assumption (V2): there exists a0 > 0 as for all a ≥ a0 :
√
it G −3/4−
G
χa (G) 1 ∞ ≤ C |t|−1/2 , t 6= 0, ∀ 0 < ε 1,
e
L →L
Schrödinger
Assumption (V2): there exists a0 > 0 as for all a ≥ a0 :
itG
e χa (G) 1 ∞ ≤ C|t|−1 .
L →L
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1
Introduction
2
New results
Wave equation (n ≥ 3)
Schrödinger equation (n ≥ 4)
Dimension n = 2, high frequencies
3
Ideas of the proofs
4
Conclusion
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Bessel and Hankel function properties
The free resolvant (h2 G0 − z 2 )−1 is defined for ±Im z > 0. Its kernel is of the
form Rh± (|x − y|, z) where
Rh± (σ, z) = ±h−2
iσ −2ν ±
H (σz/h) = h−n R1± (σh−1 , z),
4(2π)ν ν
with ν = (n − 2)/2, Hν± (λ) = λν Hν± (λ), Hν± are the outgoing and incoming
Hankel functions of order ν.
These functions behaves differently around 0 and at infinity.
For large λ, Hν± (λ) = e±iλ bν± (λ),
where bν± (λ) is a symbol of order (n − 3)/2.
±
±
Near λ = 0, Hν± (λ) = aν,1
(λ) + λn−2 log λ aν,2
(λ),
±
±
±
where aν,1 et aν,2 analytic functions, aν,2 ≡ 0 if n odd.
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Kernels associated to the free problem
Thanks to the Stone formula
Z
√
1 ∞ itλ
eit G0 ψ(h2 G0 ) =
e ψ(h2 λ2 )(R0+ (λ) − R0− (λ))λdλ,
iπ 0
√
the kernel of the operator eit G0 ψ(h2 G0 ) is of the form Kh (|x − y |, t) with
Z ∞
σ −2ν
(w)
Kh (σ, t) =
eitλ Jν (σλ)ψ(h2 λ2 )λdλ = h−n K1 (σh−1 , th−1 ),
(2π)ν+1 0
where Jν (z) = z ν Jν (z), Jν (z) = (Hν+ (z) + Hν− (z))/2 is the Besel function of
order ν = (n − 2)/2.
We have the following estimates for all σ > 0, t 6= 0 and h > 0:
(w)
|K1 (σ, t)| ≤ Chti−s hσis−(n−1)/2 ,
s ≥ 0,
for n ≥ 2.
|Kh (σ, t)| ≤ C|t|−s h−(n+1)/2 σ −(n−1)/2+s ,
for n ≥ 2,
Moreover if 0 ≤ s ≤ (n − 1)/2,
(w)
(S)
|Kh (σ, t)| ≤ Chs−(n−1)/2 |t|−s−1/2 σ −(n−1)/2+s ,
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Perturbative methods
Resolvant formulaes:
R ± (λ) − R0± (λ) = R0± (λ)VR ± (λ) = R ± (λ)VR0± (λ).
If inverse exists:
R(z) = (1 + R0 (z)V )−1 R0 (z) = R0 (z)(1 + VR0 (z))−1 ,
R + (λ) − R − (λ) = (1 + R0+ (λ)V )−1 R0+ (λ) − R0− (λ) (1 + R0− (λ)V )−1 .
Duhamel formula:
eitG = eitG0 + i
Z
t
ei(t−τ )G Veiτ G0 dτ,
0
e
it
√
√
G
=e
it
G0
√
√
Z t
p
sin(t G0 ) √
sin((t − τ ) G0 ) iτ √G
√
+i √
( G − G0 ) −
Ve
dτ.
G0
G0
0
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Assumption 0 regular
Assumption 0 regular:
the operator T = (1 − V ∆−1 )−1 : L1 → L1 is bounded.
We have (n ≥ 3)
±
VR (λ) − VR ± (0) 1 1 ≤ Cλβ , 0 < λ ≤ 1, β > 0.
0
0
L →L
As 1 + VR0± (0) = 1 − V ∆−1 is invertible on L1 , 1 + VR0± (λ) is invertible on L1
for 0 < λ ≤ λ0 .
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Low frequencies comparators (n ≥ 3)
Ψ(t; h) = eitG ψ(h2 G) − T ∗ eitG0 ψ(h2 G0 )T (Schrödinger)
√
√
Φ(t; h) = eit G ψ(h2 G) − T ∗ eit G0 ψ(h2 G0 )T (wave)
Estimates on ψ(h2 G0 ) et ψ(h2 G).
Estimates on perturbation terms:
Z ∞
itG0
Ve ψ(h2 G0 ) 1 1 dt,
L →L
−∞
Z
∞
−∞
itG
Ve ψ(h2 G)
∞
Z
−∞
Z
L1 →L
dt,
1
∞
−∞
√
it G0
ψ(h2 G0 )
Ve
L1 →L1
it √G
ψ(h2 G)
Ve
L1 →L1
dt,
dt.
Estimates on comparators.
Estimates on propagators.
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Use of low frequencies (h ≥ 1)
A important point in the proofs is the following estimate (n ≥ 2):
Z ∞
√
it G0
h
ψ(h2 G0 )f 1 dt ≤ γn Cn (V )h−(n−3)/2 kf kL1 ,
Ve
L
−∞
h > 0,
where ψ ∈ C0∞ ((0, +∞)), γn > 0 is a constant independant of V , h and f , and
Z
|V (x)|dx
Cn (V ) := sup
< +∞.
(n−1)/2
n
y∈R
Rn |x − y |
Our approch is based on the fact that if
Cn (V )h−(n−3)/2 1,
then under raisonnable assumptions on the potential, we have
Z ∞
it √G
e n (V )h−(n−3)/2 kf k 1 .
ψ(h2 G)f 1 dt ≤ C
h
Ve
L
−∞
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More details in the proof: wave n ≥ 3
Proposition 3 (dim n ≥ 3)
kΦ(t; h)kL1 →L∞ ≤ Ch−(n+1)/2−β |t|−(n−1)/2 ,
Proof of the low frequency estimate (dim n ≥ 3):
Z ∞
We write σ −(n+1)/4 ηa (σ) =
ψ(σθ)θ(n+1)/4−1 dθ,
h ≥ h0 .
σ > 0.
a−1
√
√
−(n+1)/4
Then eit G G−(n+1)/4 ηa (G) − T ∗ eit G0 G0
ηa (G0 )T 1 ∞
L →L
Z ∞
√
≤
Φ(t, θ) 1 ∞ θ(n+1)/4−1 dθ
L →L
a−1
≤ C|t|−(n−1)/2
Z
∞
θ−1−β/2 dθ ≤ C|t|−(n−1)/2 .
a−1
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More details in the proof: wave n ≥ 3
Proposition 3 (dim n ≥ 3)
kΦ(t; h)kL1 →L∞ ≤ Ch−(n+1)/2−β |t|−(n−1)/2 ,
Proof of the low frequency estimate (dim n ≥ 3):
Z ∞
We write σ −(n+1)/4 ηa (σ) =
ψ(σθ)θ(n+1)/4−1 dθ,
h ≥ h0 .
σ > 0.
a−1
√
√
−(n+1)/4
Then eit G G−(n+1)/4 ηa (G) − T ∗ eit G0 G0
ηa (G0 )T 1 ∞
L →L
Z ∞
√
≤
Φ(t, θ) 1 ∞ θ(n+1)/4−1 dθ
L →L
a−1
≤ C|t|−(n−1)/2
Z
∞
θ−1−β/2 dθ ≤ C|t|−(n−1)/2 .
a−1
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As the we have the resolvant formula
±
±
2
R±
h (z)(1 + h V R0,h (z)) = R0,h (z),
our proof follows the steps
to study properties of Hankel functions Hν± (z),
to deduce properties on R±
0,h (z),
to deduce properties on h2 V R±
0,h (z),
to deduce properties on R±
h (z),
to deduce properties on the different projectors.
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We can show (n ≥ 3, h ≥ 1):
±
R (σ, z) ≤ Ch−2 σ −n+2 + σ −(n−1)/2 .
h
Then:
kh2 V R0,h (z)kL1 →L1 ≤ C sup
y∈Rn
Z
|V (x)| |x − y |−n+2 + |x − y|−(n−1)/2 dx.
Rn
Notice that, in dimension n = 2, we can show (0 < h < 1):
Z
kh2 V R0,h (z)kL1 →L1 ≤ Ch1/2 sup
|V (x)||x − y |−1/2 dx.
y∈Rn
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Rn
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In dimension n ≥ 4, we obtain:
2
h V R0,h (z) − h2 V R0,h (0) 1 1 ≤ Ch−1/2 .
L →L
In dimension n = 3, we obtain:
2
h V R0,h (z) − h2 V R0,h (0) 1 1 ≤ Ch−1 sup
L →L
y∈R3
Z
|V (x)|h(x − y)/hi−1 .
R3
We have to make an other assumption V ∈ L3/2− (R3 ), 0 < 1, to obtain
2
h V R0,h (z) − h2 V R0,h (0) 1 1 ≤ Ch−γ , n = 3.
L →L
We also have
±
R0,h (z)
L1 →L1
≤ C|Im z|−q ,
h > 0, Im z 6= 0,
and the same estimate for R±
h with restrictions on h.
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We then prove in dimension n ≥ 3:
Proposition 1
ψ(h2 G0 ) 1 1 ≤ C, h > 0,
L →L
2
ψ(h G) 1 1 ≤ C, h ≥ h0 ,
L →L
2
2
ψ(h G) − ψ(h G0 )T 1 1 ≤ Ch−β , h ≥ h0 .
L →L
Proposition 2 (dim ≥ 4)
√
it G0
ψ(h2 G0 )f ≤ Ch−1−(n−1)/2 |t|−(n−1)/2 kf kL1 , h > 0,
e
L1
Z ∞
√
it G0
ψ(h2 G0 )f dt ≤ Ch−1−(n−3)/2 kf kL1 , h > 0,
Ve
1
L
−∞
Z
∞
−∞
it √G
ψ(h2 G)f 1 dt ≤ Ch−1−β kf kL1 ,
Ve
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L
Dispersive estimates
h ≥ h0 .
11 avril 2008 - Orléans
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We then prove in dimension n ≥ 3:
Proposition 1
ψ(h2 G0 ) 1 1 ≤ C, h > 0,
L →L
2
ψ(h G) 1 1 ≤ C, h ≥ h0 ,
L →L
2
2
ψ(h G) − ψ(h G0 )T 1 1 ≤ Ch−β , h ≥ h0 .
L →L
Proposition 2 (dim ≥ 4)
√
it G0
ψ(h2 G0 )f ≤ Ch−1−(n−1)/2 |t|−(n−1)/2 kf kL1 , h > 0,
e
L1
Z ∞
√
it G0
ψ(h2 G0 )f dt ≤ Ch−1−(n−3)/2 kf kL1 , h > 0,
Ve
1
L
−∞
Z
∞
−∞
it √G
ψ(h2 G)f 1 dt ≤ Ch−1−β kf kL1 ,
Ve
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L
Dispersive estimates
h ≥ h0 .
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The first two estimates
√ follows from (VOn) and estimates on the kernel
Kh (|x − y|, t) of eit
0 ≤ s ≤ (n − 1)/2
G0
ψ(h2 G0 ), that is for all σ > 0, h > 0 and
|Kh (σ, t)| ≤ Ch−(n+1)/2 |t|−s σ s−(n−1)/2 ,
Z
∞
|t|s |Kh (σ, t)| dt ≤ Ch−(n−1)/2 σ s−(n−1)/2 .
−∞
For the third estimate, we can write
Veit
√
ψ(h2 G) = (iπh)−1
G
X
±
Z
±
0
t
TVPh± (t − τ )Uh± (τ )dτ,
∞
where Ph± (t) =
Z
Uh± (t) =
Z
eitλ ϕ
eh (λ) R0± (λ) − R0± (0) dλ,
0
0
∞
eitλ ϕh (λ)T 1 + (VR0± (λ) − VR0± (0))T
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−1
dλ.
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The kernel of the operator Ph± (t) is of the form A±
h (|x − y |, t) where
Z ∞
−1
A±
(2π)−ν σ −n+2
eitλ ϕ
eh (λ) Hν± (σλ)) − Hν± (0) dλ
h (σ, t) = ±i4
0
which satisfied the following estimate: for all σ > 0 et h ≥ 1:
Z ∞
±
A (σ, t) dt ≤ Ch−1/2 σ −n+5/2 + σ −(n−1)/2 .
h
−∞
We also have for some h0 > 0 and for all h ≥ h0 :
Z Z ∞
±
U (t)f (x) dtdx ≤ Ckf kL1 .
h
Rn
−∞
This follows from Uh± (t) = T ϕ
bh (t) −
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Z
0
t
TVPh± (t − τ )Uh± (τ )dτ .
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Proposition 3
kΦ(t; h)kL1 →L∞ ≤ Ch−(n+1)/2−β |t|−(n−1)/2 ,
h ≥ h0 .
We make use of the Duhamel formula
√
√
√
√
√
p Z t sin((t − τ ) G0 )
sin(t G0 ) √
it G
it G0
√
√
e
=e
+i
G − G0 −
Veiτ G dτ,
G0
G0
0
which allows us to obtain the following decomposition:
Φ(t; h) = Φ1 (t; h) + Φ2 (t, h).
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Proposition 3
kΦ(t; h)kL1 →L∞ ≤ Ch−(n+1)/2−β |t|−(n−1)/2 ,
h ≥ h0 .
We make use of the Duhamel formula
√
√
√
√
√
p Z t sin((t − τ ) G0 )
sin(t G0 ) √
it G
it G0
√
√
e
=e
+i
G − G0 −
Veiτ G dτ,
G0
G0
0
which allows us to obtain the following decomposition:
Φ(t; h) = Φ1 (t; h) + Φ2 (t, h).
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√
Φ1 (t; h) = ψ1 (h2 G) − T ∗ ψ1 (h2 G0 ) eit G ψ(h2 G)
√
+T ∗ ψ1 (h2 G0 )eit G0 ψ(h2 G) − ψ(h2 G0 )T
p −iT ∗ ψ1 (h2 G0 ) sin t G0 ψ(h2 G) − ψ(h2 G0 )T
p e 2 G) − ψ(h
e 2 G0 )T .
+iT ∗ ψe1 (h2 G0 ) sin t G0 ψ(h
The already proved estimates allows us to obtain:
kΦ1 (t; h)f kL∞ ≤ Ch−(n+1)/2−β |t|−(n−1)/2 kf kL1 + Ch−β kΦ(t; h)f kL∞ .
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√
Φ1 (t; h) = ψ1 (h2 G) − T ∗ ψ1 (h2 G0 ) eit G ψ(h2 G)
√
+T ∗ ψ1 (h2 G0 )eit G0 ψ(h2 G) − ψ(h2 G0 )T
p −iT ∗ ψ1 (h2 G0 ) sin t G0 ψ(h2 G) − ψ(h2 G0 )T
p e 2 G) − ψ(h
e 2 G0 )T .
+iT ∗ ψe1 (h2 G0 ) sin t G0 ψ(h
The already proved estimates allows us to obtain:
kΦ1 (t; h)f kL∞ ≤ Ch−(n+1)/2−β |t|−(n−1)/2 kf kL1 + Ch−β kΦ(t; h)f kL∞ .
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On the other hand:
Z
Φ2 (t; h) = −h
t
√
p T ∗ ψe1 (h2 G0 ) sin (t − τ ) G0 Veiτ G ψ(h2 G)dτ.
0
We can have thanks to the already proved estimates:
t (n−1)/2 kΦ2 (t; h)f kL∞ ≤ Ch−(n+1)/2−β kf kL1
√
+Ch−β sup τ (n−1)/2 eiτ G ψ(h2 G)f t/2≤τ ≤t
L∞
.
By summing up the two contributions:
t (n−1)/2 kΦ(t; h)f kL∞ ≤ Ch−(n+1)/2−β kf kL1 + Ch−β t (n−1)/2 kΦ(t; h)f kL∞
+Ch−β sup τ (n−1)/2 kΦ(τ ; h)f kL∞ .
t/2≤τ ≤t
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On the other hand:
Z
Φ2 (t; h) = −h
t
√
p T ∗ ψe1 (h2 G0 ) sin (t − τ ) G0 Veiτ G ψ(h2 G)dτ.
0
We can have thanks to the already proved estimates:
t (n−1)/2 kΦ2 (t; h)f kL∞ ≤ Ch−(n+1)/2−β kf kL1
√
+Ch−β sup τ (n−1)/2 eiτ G ψ(h2 G)f t/2≤τ ≤t
L∞
.
By summing up the two contributions:
t (n−1)/2 kΦ(t; h)f kL∞ ≤ Ch−(n+1)/2−β kf kL1 + Ch−β t (n−1)/2 kΦ(t; h)f kL∞
+Ch−β sup τ (n−1)/2 kΦ(τ ; h)f kL∞ .
t/2≤τ ≤t
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On the other hand:
Z
Φ2 (t; h) = −h
t
√
p T ∗ ψe1 (h2 G0 ) sin (t − τ ) G0 Veiτ G ψ(h2 G)dτ.
0
We can have thanks to the already proved estimates:
t (n−1)/2 kΦ2 (t; h)f kL∞ ≤ Ch−(n+1)/2−β kf kL1
√
+Ch−β sup τ (n−1)/2 eiτ G ψ(h2 G)f t/2≤τ ≤t
L∞
.
By summing up the two contributions:
t (n−1)/2 kΦ(t; h)f kL∞ ≤ Ch−(n+1)/2−β kf kL1 + Ch−β t (n−1)/2 kΦ(t; h)f kL∞
+Ch−β sup τ (n−1)/2 kΦ(τ ; h)f kL∞ .
t/2≤τ ≤t
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Some ideas in the proof for high frequencies
Estimates for the free operator L1 → L∞ and L2w → L∞ :
←− kernel explicit formula.
Estimates for the perturbation terms with G0 and with G:
←− resolvant estimates and R ± = R0± (1 + VR0± )−1 .
Z ∞
2
−s itG
hxi e ψ(h2 G)f dt ≤ Chkf k2L2 , ∀f ∈ L2 , ∀s > 1/2, 0 < h ≤ 1.
L2
−∞
Proof bases on resovant estimates as
−s ±
hxi R (λ)hxi−s 2 2 ≤ Cλ−1 ,
L →L
λ ≥ λ0 , ∀s > 1/2, λ0 > 0.
Estimates for the comparators L2 → L2 , L1 → L∞ and L2w → L∞ :
←− Duhamel formula.
Projectors estimates:
kφ(h2 G) − φ(h2 G0 )kL1 →L1 ≤ Ch2 ,
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∀0 < h ≤ 1.
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1
Introduction
2
New results
Wave equation (n ≥ 3)
Schrödinger equation (n ≥ 4)
Dimension n = 2, high frequencies
3
Ideas of the proofs
4
Conclusion
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L1 − L∞ wave estimates (n ≥ 3)
Free operator:
low frequencies: |t|−(n−1)/2 + (n + 1)/4 − ε loss.
high frequencies: |t|−(n−1)/2 + (n + 1)/2 + ε loss.
Perturbed operator: V(δ), δ > (n + 1)/2 and 0 regular
Same estimates as the free operator,
+ (n − 3)/2-derivative loss at high frequencies.
L1 − L∞ Schrödinger estimates (n ≥ 4)
Free operator: |t|−n/2 decay.
Perturbed operator: V(δ), δ > (n + 2)/2 and 0 regular
|t|−n/2 decay,
+ (n − 3)/2-derivative loss at high frequencies,
without loss if we add V̂ ∈ L1 and δ > n − 1.
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Perspectives
Can we adapt the proof of the dispersive estimate for Schrödinger with more
regularity (V̂ ∈ L1 and (V(δ)) with δ > n − 1) to the dimension 2 ?
Which are the minimal regularity assumptions to make to obtain the usual
dispersive estimates for Schrödinger equation (without loss of derivatives at
high frequencies) ?
We hope an assumption like V ∈ C α , α ≥ (n − 3)/2, with still a decay
assumption (V(δ)).
Which are the minimal regularity assumptions for the wave equation ?
...
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Some ideas (Cardoso, Cuevas, Vodev)
T (h) := (Id + ψ1 (h2 G0 ) − ψ1 (h2 G))−1 = Id + O(h2 ),
F0 (t, h) = ψ1 (h2 G0 )eitG0 ψ(h2 G),
W0 (t, h) = eitG0 ψ1 (h2 G0 ).
We can derived from Duhamel formula
Z t
eitG ψ(h2 G) = T (h)F0 (t, h) +
iT (h)W0 (t − τ, h)Veiτ G ψ(h2 G)dτ.
0
Iterating m times leads to the identity
eitG ψ(h2 G) =
m
X
F̃j (t, h) + Rm+1 (t, h),
j=0
where
Z
t
W̃0 (t − τ, t)V F̃j−1 (τ, h)dτ,
F̃j (t, h) =
0
itG
2
R0 (t, h) = e ψ(h G),
Z
Rm+1 (t, h) =
t
W̃0 (t − τ, h)V Rm (τ, h)dτ.
0
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