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Abstract.WeconsidertheSchro¨dingerequationwithnoradialassumptiononrealhyperbolic spacesHn. We obtain in all dimensionsn2 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< γ1+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
Abstract.Nous´etudionsle´quationdeSchr¨odingersurlesespaceshyperboliquesr´eelsHn,sansaucunehypoth`esederadialite´.Nouscommenc¸onspar´etablirunein´egalit´edispersive optimale en toute dimensionn2,ainsiquuneine´galit´edeStrichartzpourunegrandefamilledepairesadmissibles.Nousende´duisonsquel´equationsemilin´eaireestfortementbienpos´eedansL2ou dansH1,pourdesdonn´eesinitialespetitesetpourdesnonlinearite´srelativementg´ene´rales,enparticulierpourtouteslespuis-´sancesouscritiques(contrairementaucaseuclidien)etsanshypoth`esedinvariancepar changement de jauge. Dans ce dernier cas, on a conservation de la charge et lessolutionsL2locales se prolongent en solutionsL2globales;lephe´nom`eneanaloguedansH1reposesurlaconservationdel´energie,quiestv´eri´eepourdesnonlin´earite´s´de´focalisantes.RappelonsqueBanica,CarlesetStalani[4]avaientpre´cdemmentmontr´equel´equationsemilin´eaire´etaitglobalementbienpos´eepourdesnonline´arit´esinvariantesparchangementdejaugeetpourdesdonn´eesradialespetitesdansL2ou ar-bitraires dansH1. Comme seconde application, nous montrons qu’il y a diffusion (scat-tering) dansL2et dansH1,a`nouveausanshypothe`sederadialit´eoudinvarianceparchangement de jauge. Rappelons que dansRnceci n’est possible que pour l’exposantcritiqueγ= 1 +n4etpeuteˆtrefauxpourdesexposantssouscritiques,tandisquesur l’espace hyperboliquesHn, on a existence globale et diffusion pour tout exposant11+n4(et pour des conditions initiales petites dansL2).Danslecasde´focalisant,nouspouvonse´tendreaucasnonradiallesr´esultatsdediusionH1de [4]. Observonsegalementque,surlespacehyperbolique,touteslesnonlin´earite´sdetypepuissance´nontquuneeta`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 tR xRnThis 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))
for the linear inhomogeneous Cauchy problem(iu(tu(t x) + Δu(t x) =F(t x)0 x) =f(x)The estimates (3) hold for any bounded or unbounded time intervalIRand for allpairs (p q)(p˜ q˜)[2]×[2) satisfying theadmissibility condition(4)p2+qn=n2Notice that both endpoints (p q) = (2) and (p q) = (2n2n2) are included in dimensionn3 while only the first one is included in dimensionn=2.The issue of well–posedness for the nonlinear Cauchy problem (1) has been widelystudied, at least for a power nonlinearityF(u) =± |u|γorF(u) =± |u|γ1uand forsuitable ranges of the exponentγ >1 . Here is a brief account of the classical theory. Inthe model caseF(u)=|u|γ, we havelocal well–posedness inL2in the subcritical caseγ <1+n4;global well–posedness inL2in the critical caseγ=1+n4for small data ;local well–posedness inH1in the subcritical caseγ <1+n42;4global well–posedness inH1in the critical caseγ=1+n2for small data.Notice that the value of the critical exponent depends on the dimensionn. On theother hand, in the model caseF(u) =|u|γ1u, the equation (1) isgauge invariantanddefocusing, which impliesL2andH1conservation laws. Thus, in addition to the previousresults, we haveglobal well–posedness inL2in the subcritical caseγ <1+n4;global well–posedness inH1in the subcritical caseγ <1+n42.Global existence for arbitrary data in the critical case remains an open problem, althoughseveral results are available (Bourgain [5], Tao, Visan & Zhang [19], . . . ).The results above are proved essentially by a fixed point argument in a suitable mixedspaceLp(R;Lq(Rn)), using Strichartz estimates in combination with conservation lawswhen available.As a byproduct, this method shows that solutionsu(t x) to (1) are small in a suitableLqxsense ast→ ±∞. Hence, asymptotically, the contribution of the nonlinearity isdominated by the linear part and thenonlinearequation (1) becomes close to thelinearequation (2). This basic observation is at the origin of scattering theory for NLS. ByL2scatteringwe mean that, for every global solutionuC(R L2(Rn)), there existscattering datau±L2(Rn) such thatku(t)eitΔu±kL2x0 ast→ ±∞The definition ofH1scatteringis analogous.The classical scattering theory for NLS, in the defocusing caseF(u) =|u|γ1u, canbe summarized as follows :scattering inL2holds in the critical caseγ=1+4nfor small data ;scattering inH1holds for 1+4n<γ <1+n42;
scattering inH1fails for 11+n2.This paper is a contribution to the study of Strichartz estimates and NLS on a manifoldM. Several results have been obtained for this problem and quite general classes ofmanifolds. The geometry ofMplays obviously an essential role : on a compact orpositively curved manifold, one expects weaker decay properties and hence weaker resultsfor NLS; on the other hand, on a noncompact negatively curved manifold, one expectsbetter dispersion properties than in the Euclidean case and hence stronger well–posednessand scattering results for NLS.ThecompactcasehasbeenstudiedextensivelybyBurq,Ge´rard&Tzvetkov[6]afterearlier results by Bourgain [5] on the torus. In general one obtains Strichartz estimateskeitΔfkLp(I;Lq(M))C(I)kfkH1p(M)which are local in time and with a loss of smoothness in space. As a consequence, theresults for NLS are weaker than onRn. Let us mention in particular the local well–posedness theory inHs(Tn) developed by Bourgain in the early nineties, extended tenyearslatertogeneralcompactmanifoldsbyBurq,G´erard&Tzvetkov,andimprovedinsome special cases such as spheresSn[6] or 4–dimensional compact manifolds [8].In this paper we shall restrict our attention to real hyperbolic spacesM=Hnof di-mensionn2. Actually our results extend straightforwardly to all hyperbolic spacesi.e. Riemannian symmetric spaces of noncompact type and rank one (they extend fur-thermore to Damek–Ricci spaces and this will be the subject of a forthcoming work).Consider the following linear Cauchy problem onHn:(5)(i ∂tu(t) x=)f+(xΔ)Hnu(t x) =F(t x) tR xHnu(0 xOn one hand, Banica [3] (see also [17]) obtained the following weighted dispersive esti-mate, for radial solutions to the homogeneous equation (5) in dimensionn3 :w(x)|u(t x)| ≤C|t|n2+|t|23 Z|f(y)|w(y)1dy HnHerew(x) =sinrhr, whererdenotes the geodesic distance fromxto the origin. On theother hand, Pierfelice [18] obtained the following sharp weighted Strichartz estimate, forradial solutions to the inhomogeneous equation (5) in dimensionn3 :kw211qukLtpLqxCkfkL2x+Ckw1q˜12FkLpt˜Lq˜xHerew(r) =sinrhrn1is the jacobian of the exponential map and (p1q1), (p1˜q1˜) belongto the intervalIn=(p11q)021×0122p+nq=n2. Actually this result wasestablished in the more general setting of Damek–Ricci spaces and it implies unweightedestimates for a wider range of indices, as pointed out by Banica, Carles & Staffilani [4].Our first main result is the following dispersive estimate (Theorem 3.4), which holdsfor general functions (no radial assumption) in dimensionn2 .
Dispersive estimate.Letq q˜(2]. Then, for0<|t|<1, we haveku(t)kLq(Hn)C|t|max{211q21q1˜}nkfkLq˜(Hn)while, for|t|≥1, we haveku(t)kLq(Hn)C|t|23kfkLq˜(Hn)Ifq=q˜=2, we have of courseL2conservationku(t)kL2(Hn)=kfkL2(Hn)for alltR. Oursecond main result is the following Strichartz estimate (Theorem 3.6), which is deducedfrom the previous estimate and holds under the same general assumptions.Strichartz estimate.Assume that(1p1q)and(1p˜1q˜)belong to the trian-gleTn=(p1q1)012×0212p+qnn2(012). ThenqkukLtpLxCkfkLx2+CkFkLpt˜Lxq˜Notice that the setTnof admissible pairs forHnis much wider than the correspondingsetInforRn(which is just the lower edge of the triangle). This striking phenomenon wasalready observed in [4] for radial solutions. It can be regarded as an effect of hyperbolicgeometry on dispersion.1q
121 12n
1 12pFigure 1.Admissible set forHnin dimensionn3Next we apply these estimates to study well–posedness and scattering for the nonlinearCauchy problem(6)(i ∂tu+ Δu=F(u)u(0) =f Throughout our paper, we shall use the following (standard) terminology about thenonlinearityF=F(u) :Fispower–likeif there exist constantsγ >1 andC0 such thatγF(u)| ≤C|u|(7)(||F(u)F(v)| ≤C(|u|γ1+|v|γ1)|uv|
Fisgauge invariantif(8) Im{F(u)u}= 0Fisdefocusingif there exists aC1functionG=G(v)0 such that(9)F(u) =G(|u|2)uNotice that gauge invariance impliesL2conservation ofchargeormass:ZHn|u(t x)|2dx=ZHn|f(x)|2dxt while the defocusing assumption impliesH1conservation ofenergy:ZHn|∇u(t x)|2dxZHnG(|u(t x)|2)dx= constant+If we specialize to the model casesF=± |u|γandF=± |u|γ1u, then the gaugeinvariant nonlinearities areF=± |u|γ1uand the defocusing onesF= +|u|γ1uLet us first summarize our well–posedness results (Theorems 4.2 & 4.4).Well–posedness for NLS.Consider the Cauchy problem(6)with apower--like nonlinearityFof orderγ.Assumeγ1+n4. Then the problem is globally well–posed for smallL2data. For arbitraryL2data, it is locally well–posed ifγ <1+4n.Assumeγ1+n42. Then the problem is globally well–posed for smallH1data. For arbitraryH1data, it is locally well–posed ifγ <1+n42.IfFis gauge invariant andγ <1+4n, the problem is globally well–posed for arbitraryL2data. IfFis defocusing andγ <1+n42, theproblem is globally well–posed for arbitraryH1data.Similar results were obtained in [4] for radial functions and nonlinearitiesF=|u|γ1u.As expected, they are better for hyperbolic spaces than for Euclidean spaces. For in-stance, onHnwe have global well–posedness for smallL2data, for any power 11+4n,while onRnwe must assume in addition gauge invariance. Of course, under this con-dition, we can also handle arbitrarily large data, using conservation laws, as in theEuclidean case.Let us next summarize our scattering results.Scattering for NLS.Consider the Cauchy problem(6)with a power–likenonlinearityFof orderγ.Assumeγ1+n4. Then, for all small datafL2, the unique globalsolutionu(t x)has the scattering property : there existu±L2suchthatku(t)eitΔHnu±kL2(Hn)0ast→ ±∞