Prepublications membres permanents

Second - Order Elliptic

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Prepublications (membres permanents) [AP1] Albert I., Grenier E., Denis J.B., Rousseau J. Quantitative Risk As- sessment from Farm to Fork and Beyond : a global Bayesian approach concerning food-borne diseases. 2007. [AP2] Alibaud N. and Imbert C. A non-local perturbation of first order Hamilton-Jacobi equations with unbounded data. 2007. [AP3] Ardon R., Cohen L.D., Yezzi A. A new implicit method for surface seg- mentation by minimal paths in 3D images [AP4] Barles G. and Imbert C. Second-Order Elliptic Integro-Differential Equations : Viscosity Solutions Theory Revisited. Annales IHP : to ap- pear, 2007. [AP5] Barles G., Chasseigne E. and Imbert C. The Dirichlet problem for second-order elliptic integro-differential equations. 2007. [AP6] Barles G., Da Lio F., Lions P.-L. and Souganidis P.E. On the homogeni- zation and ergodic problems for fully nonlinear equations in half-space type domains. 2007. [AP7] Basile G., Bernardin C., Olla S. Thermal Conductivity for a Momentum Conserving Model. Preprint [hal-00017718 - version 2] (05/04/2006) and Ceremade 2006-3.

• no-photon quantum


• diffusion equation via entropy estimates


• dirac operators via hardy- dirac inequalities


• bayesian regularized


• structure interaction problem


• optional user-interaction


• global bayesian


• mogeneous boltzmann equation

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Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 24
• Langue: English

Extrait

¨ NONLINEAR SCHRODINGER EQUATION ON REAL HYPERBOLIC SPACES JEAN–PHILIPPE ANKER & VITTORIA PIERFELICE

Abstract.WeconsidertheSchro¨dingerequationwithnoradialassumptiononreal hyperbolic spacesH n . We obtain in all dimensionsn≥2 sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong well–posedness results for NLS. Specifically, for small initial data, we proveL 2 andH 1 global well–posedness for any subcritical power (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearityF. On the other hand, ifFis gauge invariant,L 2 charge is conserved and hence, as in the Euclidean case, it is possible to extend localL 2 solutions to global ones. The corresponding argument in H 1 requires conservation of energy, which holds under the stronger condition thatFis defocusing. Recall that global well–posedness in the gauge invariant case was already proved by Banica, Carles and Staffilani [4], for small radialL 2 data or for large radial H 1 data. The second application of our global Strichartz estimates isscatteringfor NLS both inL 2 and inH 1 , with no radial or gauge invariance assumption. Notice that, on Euclidean spacesR n , this is only possible for the critical powerγ=1+ 4 n and can be false for subcritical powers while, on hyperbolic spacesH n , global existence and scattering of smallL 2 solutions holds for all powers 1< γ≤1+ 4 . If we restrict to defocusing n nonlinearitiesF, we can extend theH 1 scattering results of [4] to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearities : the geometry of hyperbolic spaces makes every power–like nonlinearity short range.

2000Mathematics Subject Classification.Primary 35Q55, 43A85 ; Secondary 22E30, 35P25, 47J35, 58D25. Key words and phrases.NonlinearSchro¨dingerequation,hyperbolicspace,dispersiveinequality, Strichartz estimate, wellposedness, scattering. 1

2

J.–PH. ANKER & V. PIERFELICE

Abstract.Nous´etudionsl’e´quationdeSchr¨odingersurlesespaceshyperboliquesr´eels H n ,sansaucunehypoth`esederadialite´.Nouscommenc¸onspar´etablirunein´egalit´e dispersive optimale en toute dimensionn≥2,ainsiqu’uneine´galit´edeStrichartzpour unegrandefamilledepairesadmissibles.Nousende´duisonsquel’´equationsemi– lin´eaireestfortementbienpos´eedansL 2 ou dansH 1 ,pourdesdonn´eesinitialespetites etpourdesnon–linearite´srelativementg´ene´rales,enparticulierpourtouteslespuis-´ sancesous–critiques(contrairementaucaseuclidien)etsanshypoth`esed’invariance par changement de jauge. Dans ce dernier cas, on a conservation de la charge et les solutionsL 2 locales se prolongent en solutionsL 2 globales;lephe´nom`eneanalogue dansH 1 reposesurlaconservationdel’´energie,quiestv´erifi´eepourdesnon–lin´earite´s ´ de´focalisantes.RappelonsqueBanica,CarlesetStaffilani[4]avaientpre´cdemment montr´equel’´equationsemi–lin´eaire´etaitglobalementbienpos´eepourdesnon–line´arit´es invariantesparchangementdejaugeetpourdesdonn´eesradialespetitesdansL 2 ou ar- bitraires dansH 1 . Comme seconde application, nous montrons qu’il y a diffusion (scat- tering) dansL 2 et dansH 1 ,a`nouveausanshypothe`sederadialit´eoud’invariancepar changement de jauge. Rappelons que dansR n ceci n’est possible que pour l’exposant critiqueγ= 1 + n 4 etpeuteˆtrefauxpourdesexposantssous–critiques,tandisque sur l’espace hyperboliquesH n , on a existence globale et diffusion pour tout exposant 1<γ≤1+ n 4 (et pour des conditions initiales petites dansL 2 ).Danslecasde´focalisant, nouspouvonse´tendreaucasnonradiallesr´esultatsdediffusionH 1 de [4]. Observons egalementque,surl’espacehyperbolique,touteslesnon–lin´earite´sdetypepuissance´ n’ontqu’uneffeta`courteporte´e,contrairementaucaseuclidien.

1.Introduction ThenonlinearSchro¨dingerequation(NLS)inEuclideanspaceR n (1) ( i ∂ t u(t x) + Δu(t x) =F(u(t x)) u(0 x) =f(x) has motivated a number of mathematical results in the last 30 years. Indeed, this equa- tion (especially in thecubiccaseF(u)=±u|u| 2 ) seems ubiquitous in physics and appears in many different contexts, including nonlinear optics, the theory of Bose–Einstein con- densates and of water waves. In particular a detailed scattering theory for NLS has been developed. An essential tool in the study of (1) is thedispersive estimate ke itΔ fk L ∞ (R n ) ≤C|t| − 2 n kfk L 1 (R n ) for the linear homogeneous Cauchy problem (2) ( i u ∂ ( t 0 u ) + =f Δ  u= 0 t∈R x∈R n This estimate is classical and follows directly from the representation formula for the fundamental solution. A well known procedure (introduced by Kato [15], Ginibre & Velo [9], and perfected by Keel & Tao [16]) then leads to theStrichartz estimates (3)kuk L p (I;L q (R n )) ≤Ckfk L 2 (R n ) +CkFk L p˜ ′ (I;L q˜ ′ (R n ))