Equations dispersives sur les varietes Orleans April

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Equations dispersives sur les varietes Orleans, 9–11 April 2008 On the Schrodinger equation on Damek–Ricci spaces Maria Vallarino (Universite d'Orleans) Joint work in progress with Jean-Philippe Anker and Vittoria Pierfelice (Universite d'Orleans)

  • xt continuously con

  • nonlinear schrodinger

  • well posedness results

  • equations dispersives sur les varietes orleans

  • globally well

  • con- tinuous


Publié le : lundi 18 juin 2012
Lecture(s) : 38
Source : univ-orleans.fr
Nombre de pages : 22
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Equationsdispersivessurlesvari´et´es Orl´eans,911April2008
On the Schrodinger equation ¨ on Damek–Ricci spaces
Maria Vallarino (Universite´dOrle´ans)
Joint work in progress with Jean-Philippe Anker and Vittoria Pierfelice (Universit´edOrle´ans)
1.Itnroudction:euclideansetting
R n
The NonlinearSchro¨dingerequation iu( t 0 u ( xt )  x =) f +( x Δ) u ( t x ) = F ( u )(1)
has motivated a number of mathematical re-sults: nonlinear optics, Bose-Einstein con-densates,   
The main tool to study (1) is the dispersive estimate for the homogeneous Cauchy prob-lem
t u ( t x ) + Δ u ( t x ) = 0 iu (0  x ) = f ( x )
(2)
whose solution is u ( t x ) = e it Δ f ( x ) = ( f s t )( x ) with kernel s t ( x ) = 2 n π n 2 e sign( t ) i π 4 n | t | 2 n e i | x 4 | t 2
Dispersive estimate : k e it Δ k L 1 L = k s t k L . | t | 2 n
By the dispersive estimate and via the T T –method (Ginibre–Velo, Keel–Tao) one studies i ∂ t u ( t x ) + Δ u ( t x ) = F ( t x ) u (0  x ) = f ( x )
(3)
whose solution is u ( t x ) = e it Δ f ( x ) + 1 i Z 0 t e i ( t s F ( s x ) ds
Strichartz estimates : q ˜ k u ( t x ) k L tp L qx . k f ( x ) k L x 2 + k F ( t x ) k L tp ˜ L x (4) ( p q ) ( p ˜  q ˜) in the admissible interval
1/q
1/2
1/2−1/n
1+ n = n p 2 q 4
1/p+n/2 1/q=n/4
1/2
1/p
In dimension n > 2, the estimate (4) holds true at the endpoint ( 12 12 n 1 )
References: [Ginibre–Velo], [Keel–Tao]
Well posedness for NLS
WellposednessresultsfortheNonlinearSchr¨odinger equation i∂ t u + Δ u = F ( u ) u 0 = 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 X T continuously con-tained into C ([ T  T ]; H s ( M )), such that : i) for any Cauchy data u 0 ( x ) B , (1) has a unique solution u ( t x ) X T ; ii) the map u 0 ( x ) B u ( t x ) X T is con-tinuous. We say that the equation is globally well-posed if these properties hold with T = .
We consider the model cases F ( u ) = | u | γ and F ( u ) = | u | γ 1 u .
F ( u ) = | u | γ :
local wellposedness in L 2 ( R n ) in the sub-4 ; cglriotibcaallwceallspeo γ se < dn1es+si n n L 2 ( R n ) in the crit-ical case γ = 1 + n 4 for small L 2 data
local wellposedness in H 1 ( R n ) in the sub-critical case γ < 1 + n 4 2 for small H 1 ini-tial data; global wellposedness in H 1 ( R n ) in the crit-ical case γ = 1 + 4 2 for small H 1 data n
F ( u ) = | u | γ 1 u : in this case we have gauge invariance hence conservation of L 2 mass and H 1 energy. Same results as above, in addi-tion:
global existence and uniqueness in L 2 ( R n ) the subcritical case γ < 1 + n 4 ;
global existence and uniqueness in H 1 ( R n ) the subcritical case γ < 1 + n 42
2. NLS on manifolds
Aim : extend theory from Euclidean space to Riemannian manifolds
T n torus S n sphere H n hyperbolic space Damek–Ricci space
ewakerreuslts
Motivations : understand influence of geometry compact case : expect less dispersion  [BurqG´erardTzvetkov][Bourgain] noncompact case : expect more dispersion  transfer results (in both directions) between the flat case and the curved case
strongerresults
NLS on hyperbolic space H n [Banica] weighted dispersive estimate hyperbolic spaces with n 3 radial data weight w ( x ) = sin | h x || x | n k w ( x ) u ( t x ) k L x . n | t | 2 + | t | 32 o k w ( x ) 1 f ( x ) k L 1 x
[Pierfelice] weighted Strichartz estimate Damek–Ricci spaces rwaediigahltda w t(a x ) = sin | x h || x | m sin2h | x 2 || x | k same admissible interval as R n
1 1 k w ( x ) 2 q u ( t x ) k L tp L qx . k f ( x ) k L x 2 1 1 + k w ( x ) q ˜ 2 F ( t x ) k L tp ˜ L qx ˜
[Banica–Carles–Staffilani] application to NLS and scattering for radial data
[Anker–Pierfelice] hyperbolic spaces any data sharp dispersive estimates Strichartz estimates in an admissible triangle application to NLS and scattering
[Ionescu-Stalani] hyperbolic spaces any data SMtroircahwaerttzziensetiqmuaaltiteisesforthesolutionofNLSwith F ( u ) = u | u | γ 1 application to NLS and scattering
AIM: generalize [A–P] to Damek–Ricci spaces application: Strichartz estimates for i ∂ t ( t x ) u (0 ux ( t )  x =) f +( x L ) u ( t x ) = F
L distinguished Laplacian on a DR space
ebras.t.LiealglaegrbNa-HyteprgH-typep(X+Z)Nuop7xe
Q = ( m + 2 k ) 2, m = dim v , k = dim z
Dilations on N : a R + δ a ( X Z ) = ( a 1 2 X aZ ) ( X Z ) N
Haar measure dX dZ
n
( X Z ) v × z
3. Damek–Ricci spaces
H-type groups
n
◦ h i inner product on n n = v z [ n z ] = { 0 } and [ n n ] z ◦ ∀ Z z , | Z | = 1 the map J Z : v v h J Z X Y i = h Z [ X Y ] i ∀ X Y v is orthogonal
Damek–Ricci spaces
S = N ⋉ R + Damek–Ricci space ( X Z a )( X  Z  a ) = = X + a 1 2 X  Z + a Z +21 a 1 2 [ X X ]  a a
Dimension of S : n = m + k + 1
S nonunimodular: d ρ ( X Z a ) = a 1 d X d Z d a right measure d λ ( X Z a ) = a ( Q +1) d X d Z d a left measure δ ( X Z a ) = ddρλ = a Q modular function
Left-invariant Riemannian metric on S λ is the Riemannian measure
Example: if N = R d then S = H d +1
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