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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.
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- 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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¨NONLINEAR SCHRODINGER EQUATIONON REAL HYPERBOLIC SPACESJEAN–PHILIPPE ANKER & VITTORIA PIERFELICE
Abstract.WeconsidertheSchro¨dingerequationwithnoradialassumptiononrealhyperbolic spacesHn. We obtain in all dimensionsn≥2 sharp dispersive and Strichartzestimates for a large family of admissible pairs. As a first consequence, we obtain strongwell–posedness results for NLS. Specifically, for small initial data, we proveL2andH1global 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,L2charge is conserved and hence, as in the Euclidean case, itis possible to extend localL2solutions to global ones. The corresponding argument inH1requires conservation of energy, which holds under the stronger condition thatFisdefocusing. Recall that global well–posedness in the gauge invariant case was alreadyproved by Banica, Carles and Staffilani [4], for small radialL2data or for large radialH1data. The second application of our global Strichartz estimates isscatteringfor NLSboth inL2and inH1, with no radial or gauge invariance assumption. Notice that, onEuclidean spacesRn, this is only possible for the critical powerγ=1+4nand can be falsefor subcritical powers while, on hyperbolic spacesHn, global existence and scatteringof smallL2solutions holds for all powers 1< γ≤1+4. If we restrict to defocusingnnonlinearitiesF, we can extend theH1scattering 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
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J.–PH. ANKER & V. PIERFELICE
Abstract.Nous´etudionsl’e´quationdeSchr¨odingersurlesespaceshyperboliquesr´eelsHn,sansaucunehypoth`esederadialite´.Nouscommenc¸onspar´etablirunein´egalit´edispersive optimale en toute dimensionn≥2,ainsiqu’uneine´galit´edeStrichartzpourunegrandefamilledepairesadmissibles.Nousende´duisonsquel’´equationsemi–lin´eaireestfortementbienpos´eedansL2ou dansH1,pourdesdonn´eesinitialespetitesetpourdesnon–linearite´srelativementg´ene´rales,enparticulierpourtouteslespuis-´sancesous–critiques(contrairementaucaseuclidien)etsanshypoth`esed’invariancepar changement de jauge. Dans ce dernier cas, on a conservation de la charge et lessolutionsL2locales se prolongent en solutionsL2globales;lephe´nom`eneanaloguedansH1reposesurlaconservationdel’´energie,quiestv´erifi´eepourdesnon–lin´earite´s´de´focalisantes.RappelonsqueBanica,CarlesetStaffilani[4]avaientpre´cdemmentmontr´equel’´equationsemi–lin´eaire´etaitglobalementbienpos´eepourdesnon–line´arit´esinvariantesparchangementdejaugeetpourdesdonn´eesradialespetitesdansL2ou ar-bitraires dansH1. Comme seconde application, nous montrons qu’il y a diffusion (scat-tering) dansL2et dansH1,a`nouveausanshypothe`sederadialit´eoud’invarianceparchangement de jauge. Rappelons que dansRnceci n’est possible que pour l’exposantcritiqueγ= 1 +n4etpeuteˆtrefauxpourdesexposantssous–critiques,tandisquesur l’espace hyperboliquesHn, on a existence globale et diffusion pour tout exposant1<γ≤1+n4(et pour des conditions initiales petites dansL2).Danslecasde´focalisant,nouspouvonse´tendreaucasnonradiallesr´esultatsdediffusionH1de [4]. Observonsegalementque,surl’espacehyperbolique,touteslesnon–lin´earite´sdetypepuissance´n’ontqu’uneffeta`courteporte´e,contrairementaucaseuclidien.
1.IntroductionThenonlinearSchro¨dingerequation(NLS)inEuclideanspaceRn(1)(i ∂tu(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 appearsin 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 beendeveloped.An essential tool in the study of (1) is thedispersive estimatekeitΔfkL∞(Rn)≤C|t|−2nkfkL1(Rn)for the linear homogeneous Cauchy problem(2)(iu∂(t0u)+=fΔu= 0 t∈R x∈RnThis estimate is classical and follows directly from the representation formula for thefundamental solution. A well known procedure (introduced by Kato [15], Ginibre &Velo [9], and perfected by Keel & Tao [16]) then leads to theStrichartz estimates(3)kukLp(I;Lq(Rn))≤CkfkL2(Rn)+CkFkLp˜′(I;Lq˜′(Rn))
