Equations dispersives sur les variétés
Orléans, 9–11 April 2008
On the Schrödinger equation
on Damek–Ricci spaces
Maria Vallarino
(Université d’Orléans)
Joint work in progress with
Jean-Philippe Anker and
Vittoria Pierfelice
(Université d’Orléans)
1. Introduction: euclidean setting Rn
The Nonlinear Schrödinger equation

i ∂ u(t, x) + ∆ u(t, x) = F (u)
t
u(0, x) = f (x)
(1)
has motivated a number of mathematical results: nonlinear optics, Bose-Einstein condensates, . . .
The main tool to study (1) is the dispersive
estimate for the homogeneous Cauchy problem

i ∂ u(t, x) + ∆ u(t, x) = 0
t
u(0, x) = f (x)
(2)
whose solution is
u(t, x) = eit∆f (x) = (f ∗ st)(x)
with kernel
st(x) =
−n
−n
2 π 2
n −n
sign(t) i π
4
|t| 2
e
|x|2
e−i 4t
Dispersive estimate :
keit∆k
L1→L∞ = kstkL∞
. |t|
−n
2
By the dispersive estimate and via the
T T ∗–method (Ginibre–Velo, Keel–Tao) one
studies

i ∂ u(t, x) + ∆ u(t, x) = F (t, x)
t
u(0, x) = f (x)
(3)
whose solution is
1 t i(t−s)∆
it∆
u(t, x) = e
f (x) +
e
F (s, x)ds
i 0
Z
Strichartz estimates :
ku(t, x)kLpLq . kf (x)kL2 + kF (t, x)k
t
x
x
p̃′ q̃ ′
Lt Lx
(4)
∀ (p, q), (p̃, q̃) in the admissible interval
1
n
n
+
=
p
2q
4
1/q
1/p+n/2 1/q=n/4
1/2
1/2−1/n
1/2
In dimension n > 2,
true at the endpoint
1/p
the estimate (4) holds
1 − 1)
,
(1
2 2 n
References: [Ginibre–Velo], [Keel–Tao]
Well posedness for NLS
Well posedness results for the Nonlinear Schrödinger
equation



i∂tu + ∆u = F (u)
u0 = f
Definition. Let s ∈ R. The NLS equation
(1) is locally well-posed on H s(M ) if, for
any bdd subset B of H s(M ), there exist T > 0
and a Banach space XT continuously contained into C([−T, T ]; H s(M )), such that :
i) for any Cauchy data u0(x) ∈ B, (1) has a
unique solution u(t, x) ∈ XT ;
ii) the map u0(x) ∈ B → u(t, x) ∈ XT is continuous.
We say that the equation is globally wellposed if these properties hold with T = ∞.
We consider the model cases F (u) = |u|γ and
F (u) = |u|γ−1u.
F (u) = |u|γ :
• local wellposedness in L2(Rn ) in the sub4;
critical case γ < 1 + n
global wellposedness in L2(Rn ) in the crit4 for small L2 data
ical case γ = 1 + n
• local wellposedness in H 1(Rn ) in the sub4 for small H 1 inicritical case γ < 1 + n−2
tial data;
global wellposedness in H 1(Rn ) in the crit4 for small H 1 data
ical case γ = 1 + n−2
F (u) = |u|γ−1u: in this case we have gauge
invariance hence conservation of L2 mass and
H 1 energy. Same results as above, in addition:
• global existence and uniqueness in L2(Rn )
4;
the subcritical case γ < 1 + n
• global existence and uniqueness in H 1(Rn )
4
the subcritical case γ < 1 + n−2
2. NLS on manifolds
Aim : extend theory from Euclidean space to
Riemannian manifolds


Tn torus




Sn sphere

Hn hyperbolic space




Damek–Ricci space
Motivations :
• understand influence of geometry
◦ compact case :
expect less dispersion
weaker results
[Burq–Gérard–Tzvetkov] [Bourgain]
◦ noncompact case :
expect more dispersion
stronger results
• transfer results (in both directions)
between the flat case and the curved case
NLS on hyperbolic space Hn
◦ [Banica] weighted dispersive estimate


hyperbolic spaces with n ≥ 3



radial data



weight w(x) = sinh |x|
|x|
n
3o
−
−n
2 +|t| 2 kw(x)−1 f (x)k 1
kw(x)u(t, x)kL∞
.
|t|
Lx
x
◦ [Pierfelice] weighted Strichartz estimate


Damek–Ricci spaces





radial data
m k
sinh
2|x|
sinh
|x|


weight w(x) =

|x|
2|x|



 same admissible interval as Rn
1−1
kw(x) 2 q u(t, x)kLpLq
t x
. kf (x)kL2
+
x
1−1
kw(x) q̃ 2 F (t, x)k
◦ [Banica–Carles–Staffilani]
application to NLS and scattering
for radial data
p̃′
q̃ ′
Lt Lx
◦
 [Anker–Pierfelice]


hyperbolic spaces






any data
sharp dispersive estimates




Strichartz estimates in an admissible triangle




application to NLS and scattering
◦
 [Ionescu-Staffilani]


hyperbolic spaces





any data




Strichartz estimates

Morawetz inequalities for the solution of NLS with





γ−1

F
(u)
=
u|u|




application to NLS and scattering
AIM:
• generalize [A–P] to Damek–Ricci spaces
• application: Strichartz estimates for

i ∂ u(t, x) + L u(t, x) = F (t, x)
t
u(0, x) = f (x)
L distinguished Laplacian on a DR space
3. Damek–Ricci spaces
• H-type groups
n Lie algebra s.t.
◦
◦
◦
◦
h·, ·i inner product on n
n=v⊕z
[n, z] = {0} and [n, n] ⊆ z
∀Z ∈ z, |Z| = 1 the map JZ : v → v
hJZ X, Y i = hZ, [X, Y ]i
is orthogonal
n H-type algebra
∀X, Y ∈ v
N H-type group
(X, Z) ∈ v × z 7→ exp(X + Z) ∈ N
Dilations on N : ∀a ∈ R+
δa(X, Z) = (a1/2X, aZ)
∀(X, Z) ∈ N
Haar measure dX dZ
Q = (m + 2k)/2, m = dim v, k = dim z
• Damek–Ricci spaces
S = N ⋉ R+
Damek–Ricci space
(X, Z, a)(X ′ , Z ′, a′ ) =
1
= X + a1/2X ′ , Z + a Z ′ + a1/2[X, X ′], a a′
2
Dimension of S:
n=m+k+1
S nonunimodular:
dρ(X, Z, a) = a−1 dX dZ da
right measure
dλ(X, Z, a) = a−(Q+1) dX dZ da left measure
−Q
δ(X, Z, a) = dλ
=
a
modular function
dρ
Left-invariant Riemannian metric on S
λ is the Riemannian measure
Example: if N = Rd then S = Hd+1
∆ Laplace–Beltrami operator on S
Homogeneous Schrödinger equation on S

 i ∂ u(t, x) + ∆u(t, x) = 0
t
 u(0, x) = f (x)
u(t, x) = eit∆f (x) = f ∗ st(x)
Kernel expression : r = |x|
• k even :
st(r) = C
• k odd :
st(r) = C
1
−2
t
1
−2
t
Q2
e−i 4t
2
−i Q4t
e
2
k/2 m/2 i r
D1 D2
e 4t
Z ∞
r
sinh s
ds √
×
cosh s − cosh r
2
(k+1)/2 m/2 i s
× D1
D2
e 4t
where
1 ∂
D1 = − sinh
r r
D2 = − sinh1r/2 ∂r
Kernel estimates :
|st(r)| .

Q

 |t|−n/2 (1 + r) n−1
2 e− 2 r
Q

 |t|−3/2 (1 + r) e− 2 r
⇒ kstkLq (λ) .
∀q>2

 |t|−n/2
 |t|−3/2
if |t| ≤ 1 + r
if |t| > 1 + r
if 0 < |t| ≤ 1
if |t| > 1
Lq (λ) w.r.t. left measure λ
Dispersive estimates : 2 < q, q̃ ≤ ∞
• 0 < |t| ≤ 1 :
it∆
ke
kLq̃′ (λ)→Lq (λ) . |t|
• |t| > 1 :
it∆
ke
1−1 , 1−1} n
− max{ 2
q 2 q̃
kLq̃′ (λ)→Lq (λ) . |t|
−3
2
Proof:
• 0 < |t| ≤ 1 : Interpolation between



keit∆kL2(λ)→L2(λ) = 1


keit∆kL1(λ)→L∞(λ) = kstkL∞(λ) . |t|−n/2



−n/2
keit∆ k 1
=
ks
k
.
|t|
q
q
t
L (λ)
L (λ)→L (λ)
• |t| > 1 : Interpolation between

3

−
it∆


ke
kL1(λ)→Lq (λ) = kstkLq (λ) . |t| 2



−3
it∆
ke
kLq′ (λ)→L∞(λ) = kstkLq (λ) . |t| 2



3

−
it∆

k e
kLq′ (λ)→Lq (λ) . |t| 2
The crucial estimate is
it∆
ke
kLq′ (λ)→Lq (λ) . |t|
−3
2
∀q > 2
(5)
It follows from:
Lemma S Damek–Ricci space
T f = f ∗ k convolution operator
with radial kernel k
kT kLq′ (λ)→Lq (λ) =
Z ∞
0
|k(r)|q/2 φ0(r) dr
2/q
where φ0 spherical function
−Q
2r
φ0(r) ≤ C (1 + r) e
Applying to T = eit∆ and k = st we get (5)
Remark: very similar proof in hyperbolic spaces
[Anker–Pierfelice] except for the last crucial
estimate.
In hyperbolic spaces the Kunze–Stein phenomenon holds, but in Damek–Ricci spaces
it fails !


Solution :
i∂tu + ∆u = F (t, x)
u(0, x) = f (x)

1 t
u(t, x) = e
f (x) +
ds ei(t−s)∆F (s, x)
i 0
Strichartz estimates :
Z
it∆
ku(t, x)kLpLq (λ) . kf kL2(λ) + kF (t, x)k
t
x
x
p̃′ q̃ ′
Lt Lx (λ)
∀ (p, q), (p̃, q̃) in the admissible triangle Tn
n 1 1
−
2 2 q
!
1
1
≤ ≤
p
2
and
1 1
1
1
− < <
2 n
q
2
1/q
1/2
1/2−1/n
1/2
1/p
Well posedness results for NLS on S



i∂tu + ∆u = F (u)
u0 = f
(6)
The same results as in the flat case are true.
In addition:
Theorem Assume |F (u)| ≤ C|u|γ . Then for
4 problem (6) is globally wellposed
all γ ≤ 1+ n
4 it
for small L2 data, and for all γ ≤ 1 + n−2
is globally wellposed for small H 1(S) data.
[Anker–Pierfelice: hyperbolic spaces]
4. The Laplacian L
E0 unit vector in R
{E1, . . . , Em} orthonormal basis of v
{Em+1, . . . , En−1} orthonormal basis of z
X0, X1, . . . , Xn−1 left invariant vector fields
on S which agree with E0, E1, . . . , En−1 at the
identity
L=−
n−1
X
Xi2
i=0
left-invariant Laplacian s.a. on L2(ρ)
Relationship between L and ∆
Q2 −1/2
1/2
L=δ
∆−
δ
4
Recall: δ modular function δ(X, Z, a) = a−Q
Q homogeneous dimension of N
⇒ eitL is a convolution operator with kernel
2
1/2 −i Q4t
σt = δ
e
st
Estimate from below of σt:
−Q
1
/2
−n/2 n−1
2
|σt(x)| ≥ C δ (x) |t|
r
e 2r
if r = |x| ≥ C (1 + |t|)
⇒ σt ∈
/ L∞(ρ)
⇒ eitL is not bounded from L1(ρ) to L∞(ρ)
⇒ there is no dispersive L1(ρ) − L∞(ρ)
effect
The following weighted Strichartz
estimates hold:



i∂tu + Lu = F (t, x)
u(0, x) = f (x)
kukLpLq (δq dρ) . kf kL2(ρ) + kF kLp̃′ Lq̃′ (δ dρ)
q̃ ′
(p, q), (p̃, q̃) ∈ Tn
δq = δ
weight:
In coordinates:
1− 2q
q
−Q 1− 2
δq (X, Z, a) = a
Proof:
• we transfer the Strichartz estimates
obtained for the equation associated with ∆
to the equation associated with L using
2
i Q4t
1/2
σt = δ
e
st
• we apply two properties of the modular
function:
(i) f ∗ (δ 1/2g) = δ 1/2 [(δ −1/2f ) ∗ g]
(ii) kδ −1/2f kLq (λ) = kf kLq (δq ρ)
1≤q<∞
Indeed, δ = dλ
dρ
Z
|δ −1/2f |q dλ =
=
=
=
Z
Z
Z
Z
δ −q/2 |f |q dλ
δ −q/2 |f |q δdρ
|f |q δ 1−q/2dρ
|f |q δq dρ
5. Further developments :
◦ NLW in Damek–Ricci spaces
◦ NLS and NLW in higher rank symmtric
spaces
◦ Discrete setting (trees, buildings)