DIMENSION REDUCTION FOR DIPOLAR BOSE-EINSTEIN CONDENSATES IN THE STRONG INTERACTION REGIME WEIZHUBAO,LOÏC LE TREUST, ANDFLORIANMÉHATS Abstract. We study dimension reduction for the three-dimensional Gross- Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction 5 modelingdipolarBose-Einsteincondensationinastronginteractionregime. The 1 casesofdiskshapedcondensates(confinementfromdimensionthreetodimension 0 two)andcigarshapedcondensates(confinementtodimensionone)areanalyzed. 2 Inbothcases,theanalysiscombinesaveragingtoolsandsemiclassicaltechniques. n Asymptoticmodels are derived,with rates of convergencein termsof two small a dimensionlessparameterscharacterizingthestrengthoftheconfinementandthe J strength of theinteraction between atoms. 9 ] P A . 1. Introduction and main results h t a In this paper, we study dimension reduction for the three-dimensional Gross- m Pitaevskii equation (GPE) with a long-range and anisotropic dipole-dipole interac- [ tion (DDI) modeling dipolar Bose-Einstein condensation [11, 14]. In contrast with 1 the existing literature on this topic [1], we will not assume that the degenerate v dipolar quantum gas is in a weak interaction regime. 7 Based on the mean field approximation [3, 9, 13, 18, 19, 20], the dipolar Bose- 7 1 Einstein condensateismodeledby itswavefunction Ψ := Ψ(t,x)satisfyingtheGPE 2 with a DDI written in physical variables as 0 . ~2 1 i~∂ Ψ = ∆Ψ+V(x)Ψ+Ng Ψ 2Ψ+NC U Ψ 2 Ψ, (1.1) t dip dip 0 −2m | | ∗| | 5 where ∆ is the Laplace operator, V(x) denotes the tra(cid:0)pping harm(cid:1)onic potential, 1 : m > 0 is the mass, ~is the Planck constant, g = 4π~2as describes the contact (local) v m interaction between atoms in the condensate with the s-wave scattering length a , i s X N denotes the number of atoms inthecondensate, and thedipole-dipole interaction r kernel U (x) is given as a dip 1 1 3(x n)2/x2 1 1 3cos2(θ) U (x) = − · | | = − , x R3, (1.2) dip 4π x3 4π x3 ∈ | | | | with the dipolar axis n = (n ,n ,n ) R3 satisfying n = n2+n2+n3 = 1. 1 2 3 ∈ | | 1 2 3 Here θ is the angle between the polarization axis n and relative position of two p atoms (that is, cosθ = n x/x). For magnetic dipoles we have C = µ µ2 , · | | dip 0 dip where µ is the magnetic vacuum permeability and µ the dipole moment, and for 0 dip electric dipoles we have C = p2 /ǫ , where ǫ is vacuum permittivity and p dip dip 0 0 dip the electric dipole moment. The wave function is normalized according to Ψ(t,x)2dx = 1. R3| | Z 1 2 W.BAO,L.LETREUST,ANDF.MÉHATS 1.1. Nondimensionalization. We assume that the harmonic potential is strongly anisotropic and confines particles from dimension 3 to dimension 3 d. We shall denote x = (x,z), where x R3 d denotes the variable in the confine−d direction(s) − and z Rd denotes the va∈riable in the transversal direction(s). In applications, ∈ we will have either d = 1 for disk-shaped condensates, or d = 2 for cigar-shaped condensates. Similarly, we denote n = (n ,n ) with n R3 d and n Rd. The x z x − z ∈ ∈ harmonic potential reads [2, 15, 16] m V(x) = ω2 x 2+ω2 z 2 2 x| | z| | where ω ω . We introduce three di(cid:0)mensionless para(cid:1)meters z x ≫ 4πN a C s dip ε= ω /ω , β = | |, λ = , x z 0 a 3g 0 | | p where the harmonic oscillator length is defined by [2, 15, 16] ~ 1/2 a = . 0 mω (cid:18) x(cid:19) The dimensionless parameter λ measures the relative strength of dipolar and s- 0 wave interactions. Let us rewrite the GPE (1.1) in dimensionless form. For that, we introduce the new variables t˜, x˜, z˜ and the associated unknown Ψ defined by x z t˜= ω t, x˜ = , z˜= , Ψ(t˜,x˜,z˜)= a3/2Ψ(t,x,z). (1.3) x a a 0 e 0 0 The dimensionless GPE equation reads [2, 15, 16] e 1 1 1 i∂ Ψ = ∆Ψ+ x˜ 2+ z˜2 Ψ+βσ Ψ 2Ψ+3λ β U Ψ 2 Ψ, (1.4) t˜ −2 2 | | ε4| | | | 0 dip∗| | (cid:18) (cid:19) (cid:16) (cid:17) whereeσ = signeas 1,1 . Defineethe diffeereential operators ∂ne= ne and ∈ {− } · ∇ ∂nn = ∂n∂n. Mathematically speaking, the convolution with Udip in equation (1.1) has to be considered in the distributional sense and we have the following identity (see [3]) 1 3(x n)2 1 1 Udip(x) = p.v. 4π x3 1− x·2 = −3δ(x)−∂nn 4π x , x∈ R3, (cid:18) | | (cid:18) | | (cid:19)(cid:19) (cid:18) | |(cid:19) (1.5) with δ being the Dirac distribution. Remark 1.1. Let us define the Fourier transform of a function u L1(R3) by ∈ u(k) = u(x)e ikxdx, x R3. − · R3 ∈ Z From identity (1.5), we get b 1 (k n)2 U (k) = + · , for all k R3. (1.6) dip −3 k 2 ∈ | | We can re-formulatde the GPE (1.4) as the following Gross-Pitaevskii-Poisson system (GPPS) [3, 7] 1 1 1 i∂t˜Ψ = −2∆Ψ+ 2 |x˜|2+ ε4|z˜|2 Ψ+β(σ−λ0)|Ψ|2Ψ−3λ0β(∂nnϕ)Ψ, (cid:18) (cid:19) (1.7) e∆ϕ = eΨ 2, lim ϕ(t˜,x˜) =e 0. e e e −| | x˜ | |→∞ e DIMENSION REDUCTION FOR DIPOLAR BEC 3 Under scaling (1.3), dimension reduction of the above GPPS (1.4) was formally derived from 3D to 2D and 1D in [1, 7, 17] for any fixed β, λ and n when ε 0+. 0 → Rigorous mathematical justification was only given in the weak interaction regime, i.e. when β = (ε) from 3D to 2D and when β = (ε2) from 3D to 1D [1]. It is an O O open problem for the case where β is fixed when ε 0+. → 1.2. New scaling. In order to observe the condensate at the correct space scales, we will now proceed to a rescaling in x and z. Let us denote α = ε2d/nβ 2/n. (1.8) − The scaling assumptions are α 1 and ε 1. ≪ ≪ We define the new variables z˜ t = t˜, z = , x = α1/2x˜, ′ ′ ′ ε whichmeans thatthetypicallengthscalesofthedimensionlessvariables areεinthe z-direction and α 1/2 in the x-direction. The wavefunction is rescaled as follows: − Ψε,α(t,x,z ):= εd/2α n/4Ψ(t˜,x˜,z˜)eit˜d/2ε2. ′ ′ ′ − Notice that the L2 norm of Ψε,α is left invariant by this rescaling, so we still have e Ψε,α(t,x,z)2dxdz = 1. R3| | Z We end up with the following rescaled GPE (for simplicity we omit the primes on the variables): α α2 x 2 iα∂ Ψε,α = Ψε,α ∆ Ψε,α+ | | Ψε,α t ε2Hz − 2 x 2 +α σ Ψε,α 2+3λ βUε,α Ψε,α 2 Ψε,α (1.9) | | 0 dip ∗| | where the transversal Hamilton(cid:16)ian is (cid:17) 1 z 2 d := ∆ + | | z z H −2 2 − 2 ε,α and U is defined by dip x Uε,α(x,z) = U ,εz , (x,z) R3. dip dip √α ∈ (cid:18) (cid:19) Let us remark that k Uε,α(k ,k ) = ε dαn/2U √αk , z , for all (k ,k ) R3. (1.10) dip x z − dip x ε x z ∈ (cid:18) (cid:19) Thanksdto identity (1.6), we cadn remark that U is a bounded function of R3 into dip [ 1, 2]. For γ > 0, we denote by Vγ the tempered distribution whose Fourier −3 3 dip transform is d 1 (γk n +k n )2 γ x x z z V (k ,k ) = + · · (1.11) dip x z −3 γk 2+ k 2 (cid:18) | x| | z| (cid:19) so that Vγ (k ,kd) [ 1/3,2/3] for all (k ,k ) R3 and dip x z ∈ − x z ∈ \ d Uε,α(k ,k )= ε dαn/2V√αε(k ,k ), for all (k ,k ) R3. dip x z − dip x z x z ∈ d 4 W.BAO,L.LETREUST,ANDF.MÉHATS Let us note that (1.8) is equivalent to βε dαn/2 = 1 − so that equation (1.9) becomes α α2 x 2 iα∂ Ψε,α = Ψε,α ∆ Ψε,α+ | | Ψε,α t ε2Hz − 2 x 2 +α σ Ψε,α 2+3λ V√αε Ψε,α 2 Ψε,α. (1.12) | | 0 dip ∗| | Remark 1.2. The spectrum of(cid:16) z is the set of integers N. (cid:17)We define (ωk)k N an orthonormal basis of L2(R3) maHde of eigenvectors of where ω is the g∈round z 0 H state (associated to the eigenvalue 0) ω (z) = π d/4e z2/2. 0 − −| | Remark 1.3. Since (Vγ ) is uniformly bounded in L and dip γ 0 ∞ ≥ d Vγ V0 a.e. dip → dip as γ 0, Lebesgue’s dominated convergence Theorem ensures that → d d Vγ U V0 U in L2(R3) dip∗ → dip∗ for all U L2(R3). Moreover, let us remark that ∈ n2 d V0 U(x,z) = z − U(x,z), (x,z) R3 dip∗ 3d ∈ for all U such that U(x,z) = V(x, z ) for all (x,z) R3. | | ∈ In this paper, we study the behavior of the solution of equation (1.12) as ε 0 → and α 0 independently so that β may be bounded but can also tends to + . → ∞ Our key mathematical assumption will be that the wavefunction Ψε,α at time t = 0 is under the WKB form: Ψε,α(0,x,z) = Ψα (x,z) := A (x,z)eiS0(x)/α, (x,z) R3. (1.13) init 0 ∀ ∈ Here A is a complex-valued function and S is real-valued. 0 0 Let us introduce another parameter γ > 0 to get a better understanding of the different phenomena involved during the limiting procedures. In this paper, we will study instead of equation (1.12) the following one : iα∂tψ = εα2Hzψ− α22∆xψ+ |x2|2ψ+α σ|ψ|2 +3λ0Vdγip∗|ψ|2 ψ, (1.14) ψ(0,x,z) = A (x,z)eiS0((cid:16)x)α, (x,z) R3. (cid:17) 0 ∀ ∈ From now on, we denote by Ψε,α,γ the solution ψ of equation (1.14). Let us insist on the fact that Ψε,α,γ is equal to the solution Ψε,α of equation (1.12) if we assume that γ = ε√α. 1.3. Heuristics. In this section, we derive formally the limiting behavior of the solution of (1.14) as ε (strong confinement limit), α (semiclassical limit) and γ (limit of the dipole-dipole interaction term) go to 0. Our main result, stated in the next section, will be that in fact these limits commute together: the limit is valid as ε, α and γ converge independently to zero. Thus, this gives us as a by-product the behavior of the solution of equation (1.12) as ε and α converge independently to zero. DIMENSION REDUCTION FOR DIPOLAR BEC 5 a) Strong confinement limit : ε 0. Let us fix α (0,1] and γ [0,1]. Following → ∈ ∈ [6], in order to analyze the strong partial confinement limit, it is convenient to begin by filtering out the fast oscillations at scale ε2 induced by the transversal Hamiltonian. To this aim, we introduce the new unknown Φε,α,γ(t, ) = eit z/ε2Ψε,α,γ(t, ). H · · It satisfies the equation α2 x 2 t iα∂ Φε,α,γ = ∆ Φε,α,γ + | | Φε,α,γ +αFγ ,Φε,α,γ t − 2 x 2 ε2 (cid:18) (cid:19) where the nonlinear function is defined by Fγ(θ,Φ) = eiθ z σ e iθ zΦ 2+3λ Vγ e iθ zΦ 2 e iθ zΦ. (1.15) H − H 0 dip∗| − H | − H A fundamental remark(cid:16)is (cid:12)that for (cid:12)all fixed Φ, the function(cid:17)θ Fγ(θ,Φ) is 2π- (cid:12) (cid:12) 7→ periodic, since the spectrum of only contains integers. For any fixed α > 0 and z H λ = 0, Ben Abdallah et al. [6, 5] proved by an averaging argument that we have 0 Φε,α,γ = Φ0,α,γ + (ε2), where Φ0,α,,γ solves the averaged equation O α2 x 2 iα∂ Φ0,α,γ = ∆ Φ0,α,γ + | | Φ0,α,γ +αFγ (Φ0,α,γ), Φ0,α,γ(t = 0) = Ψα t − 2 x 2 av init (1.16) γ where F is the averaged vector field av 1 2π Fγ (Φ) = Fγ(θ,Φ)dθ. (1.17) av 2π Z0 In our study, we consider the case λ R and a similar averaging argument should 0 ∈ give us the same result Φε,α,γ = Φ0,α,γ + (ε2). O b) Semi-classical limit : α 0. Letus remark that equation(1.14) iswritten inthe → semi-classical regime of "weakly nonlinear geometric optics", which can be studied by aWKBanalysis. Hereweareonly interested inthelimiting model, sointhefirst stage of the WKB expansion. Let us introduce the solution S(t,x) of the eikonal equation S 2 x 2 x ∂ S + |∇ | + | | = 0, S(0,x) = S (x) (1.18) t 0 2 2 and filter out the oscillatory phase of the wavefunction by setting Ωε,α,γ = e iS(t,x)/αΨε,α,γ, (1.19) − so that 1 α ∂ Ωε,α,γ + S Ωε,α,γ + Ωε,α,γ∆ S = i ∆ Ωε,α,γ iHzΩε,α,γ (1.20) t ∇x ·∇x 2 x 2 x − ε2 i σ Ωε,α,γ 2+3λ Vγ Ωε,α,γ 2 Ωε,α,γ, − | | 0 dip∗| | (cid:16) (cid:17) where Ωε,α,γ(0,x,z) =A (x,z), for all (x,z) R3. 0 ∈ For all fixed ε > 0, we can expect that Ωε,α,γ = Ωε,0,γ + (α), O 6 W.BAO,L.LETREUST,ANDF.MÉHATS as α 0 where Ωε,0,γ solves the equation → 1 ∂ Ωε,0,γ + S Ωε,0,γ + Ωε,0,γ∆ S (1.21) t x x x ∇ ·∇ 2 = iHzΩε,0,γ i σ Ωε,0,γ 2+3λ Vγ Ωε,0,γ 2 Ωε,0,γ, − ε2 − | | 0 dip∗| | Ωε,0,γ(0,x,z) =A (cid:16)(x,z), for all (x,z) R3. (cid:17) 0 ∈ Remark 1.4. A key point here in this analysis is that the nonlinearities Fγ and Fγ are gauge invariant i.e. for all U L2(R3), all γ [0,1] and for all t, we have av ∈ ∈ Fγ(t,UeiS/α) = Fγ(t,U)eiS/α, Fγ (UeiS/α) = Fγ (U)eiS/α. av av c) Dipole-dipole interaction limit γ 0. We expect that for any (ε,α) (0,1]2 → ∈ Ψε,α,γ = Ψε,α,0+ (γq) O where q > 0 and α α2 x 2 iα∂ Ψε,α,0 = Ψε,α,0 ∆ Ψε,α,0+ | | Ψε,α,0 (1.22) t ε2Hz − 2 x 2 +α σ Ψε,α,0 2+3λ V0 Ψε,α,0 2 Ψε,α,0, | | 0 dip∗| | Ψε,α(cid:0),0(t = 0) = Ψαinit. (cid:1) In this paper, the main difficulty we have to tackle and also the main difference with respect to the previous work of the authors [4] in the case λ = 0, is the study 0 of this limit γ 0. → d) Thesimultaneous study of thethree limits. Weintroduceforany(ε,α,γ) (0,1]3 ∈ Aε,α,γ(t,x,z) = eit z/ε2e iS(t,x)/αΨε,α,γ(t,x,z), for (x,z) R3, H − ∈ which is the solution of the equation 1 iα∆ t ∂ Aε,α,γ + S Aε,α,γ + Aε,α,γ∆ S = xAε,α,γ iFγ ,Aε,α,γ ,(1.23) t ∇x ·∇x 2 x 2 − ε2 (cid:18) (cid:19) Aε,α,γ(0,x,z) = A (x,z). 0 We will also consider the solution Aε,0,γ of (1.23) with α = 0, the solution Aε,α,0 of (1.23) with γ = 0 and the solution A0,α,γ of 1 iα∆ ∂ A0,α,γ + S A0,α,γ + A0,α,γ∆ = xA0,α,γ iFγ A0,α,γ , (1.24) t ∇x ·∇x 2 x 2 − av A0,α,γ(0,x,z) =(cid:0)A (x,z(cid:1)), 0 for all (x,z) R3. As long as the phase S(t, ) remains smooth, i.e. before the ∈ · formation of caustics in the eikonal equation (1.18), we expect to have Aε,α,γ = A0,0,0+ (ε2 +α+γq), O and the solution Ψε,α,γ of equation (1.14) is expected to behave as Ψε,α,γ(t,x,z) = e it z/ε2eiS(t,x)/αA0,0,0(t,x,z)+ (ε2+α+γq) (1.25) − H O for some q > 0. DIMENSION REDUCTION FOR DIPOLAR BEC 7 1.4. Main results. Inthispaper,ourmaincontributionistherigorousstudyofthe dipole-dipole interaction limits γ 0 as well as the study of the three simultaneous → limits ε 0, α 0 and γ 0 involved in the problem. The techniques used for → → → the study of the limits ε 0 and α 0 were developed by the authors in [4]. We → → will recall and use some of the results proved in this first paper. 1.4.1. Existence, uniqueness and uniform boundedness results. Let us make precise ourfunctionalframework. Forwavefunctions, wewillusethescaleofSobolevspaces adapted to quantum harmonic oscillators: Bm(R3):= u Hm(R3) such that (x m+ z m)u L2(R3) { ∈ | | | | ∈ } for m N. ∈ Remark 1.5. Assuming that m 2, we get that ≥ Bm(R3) ֒ Hm(R3)֒ L (R3). ∞ → → Hm(R3) and Bm(R3) are two algebras. In this paper, we will also make frequent use of the estimate xk∂κu C u , for all u Bm(R3) and k+ κ m (1.26) k| | z kL2 ≤ k kBm ∈ | | ≤ (see [12] and [6] for a more general class of confining potential). For the phase S, we will use the space of subquadratic functions, defined by SQ (R3 d)= f k(R3 d;R)) such that ∂κf L (R3 d), for all 2 κ k . k − { ∈ C − x ∈ ∞ − ≤ | | ≤ } where k N, k 2. In the following theorem, we give existence and uniqueness ∈ ≥ results for equations (1.18), (1.23) and (1.24), as well as uniform bounds on the solutions. Theorem 1.6. Let (ε,α,γ) [0,1]3, A Bm(R3) and S SQ (R3 d), where 0 0 s+1 − ∈ ∈ ∈ m 5 and s m+2. Then the following holds: ≥ ≥ (i) There exists T > 0 such that the eikonal equation (1.18) admits a unique solution S ([0,T];SQ (R3 d)) s([0,T] R3 d). s − − ∈ C ∩C × (ii) There exists T (0,T] independent of ε, α and γ such that the solutions Aε,α,γ ∈ and A0,α,γ of, respectively, (1.23) and (1.24), are uniquely defined in the space C([0,T];Bm(R3)) C1([0,T];Bm 2(R3)). − ∩ (iii) The functions (Aε,α,γ) are bounded in ε,α,γ C([0,T];Bm(R3)) C1([0,T];Bm 2(R3)) − ∩ uniformly with respect to (ε,α,γ) [0,1]3. ∈ 1.4.2. Study of the limits α 0, ε 0 and γ 0. We are now able to study the → → → behavior of Aε,α,γ as α 0, ε 0 and γ 0. → → → Theorem 1.7. Assume the hypothesis of Theorem 1.6 true. Then, for all (ε,α,γ) ∈ [0,1]3, for all q (0,1), we have the following bounds: ∈ (i) Averaging result: Aε,α,γ A0,α,γ Cε2 (1.27) k − kL∞([0,T];Bm−2(R3)) ≤ (ii) Semi-classical result: Aε,α,γ Aε,0,γ Cα (1.28) k − kL∞([0,T];Bm−2(R3)) ≤ 8 W.BAO,L.LETREUST,ANDF.MÉHATS (iii) Dipole-dipole interaction limit result: Aε,α,γ Aε,α,0 C γq (1.29) k − kL∞([0,T];Bm−5(R3)) ≤ q (iv) Global result: Aε,α,γ A0,0,0 C (ε2 +α+γq). (1.30) k − kL∞([0,T];Bm−5(R3)) ≤ q The constants C and C do not depend on α, ε and γ but C does depend on q. The q q estimates related to the original equation (1.12) can be summarized in the following diagram: Aε,α,ε√α O(ε2+(√αε)q) // A0,α,0 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ (α+(√αε)q) ❉❉O❉❉(❉α+ε2+(√αε)q) (α) O ❉❉ O ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ (cid:15)(cid:15) ❉!! (cid:15)(cid:15) Aε,0,0 // A0,0,0 (ε2) O Remark 1.8. The case λ = 0 has already been studied by the authors in [4] where 0 we got estimates that are similar to (1.27) and (1.28). Remark 1.9. Assume that either n = 0 or n = 0. Then, for all (ε,α) (0,1]2, x z ∈ for all q such that q = 1 if d= 1, q [1,2) if d= 2, (cid:26) ∈ we get the same conclusion as in Theorem 1.7. The following immediate corollary gives a more accurate approximation of Aε,α,ε√α than A0,0,0. This result can be useful for numerical simulations and has to be related to the ones of Ben Abdallah et al. [5]. Corollary 1.10. Assume the hypothesis of Theorem 1.6 true. Then, for all (ε,α) ∈ [0,1]2, we have the following bound: Aε,α,ε√α A0,0,ε√α C(ε2+α) k − kL∞([0,T];Bm−2(R3)) ≤ where C > 0 does not depend on ε or α. The following proposition concerns the special case of an initial data polarized on one mode of . It generalizes the case studied by Bao, Ben Abdallah and Cai z H [1, Theorems 5.1 and 5.5] where the initial data was taken on the ground state of . z H Proposition1.11. Let k N. Assumethehypothesis ofTheorem 1.6true. Assume ∈ also that A (x,z) = a (x)ω (z), (x,z) R3 0 0 k ∈ where ω is defined in Remark 1.2. Then, the function A0,α,γ stays polarized on the k mode ω i.e. k A0,α,γ(t,x,z) = Bα,γ(t,x)ω (z) for all z Rd. k ∈ DIMENSION REDUCTION FOR DIPOLAR BEC 9 Here, Bα,γ is the solution of 1 iα∆ ∂ Bα,γ + S Bα,γ + Bα,γ∆ S = xBα,γ iGγ,k(Bα,γ), (1.31) t x x x ∇ ·∇ 2 2 − Bα,γ(0,x) = a (x), x R3 d 0 − ∈ where Gγ,k(u)(x) = u(x) ω (z)k σ+3λ Vγ ω u2(x,z) dz. Rd| k | 0 dip∗| k | Z (cid:16) (cid:17) Let q = 1 if d= 1 (q [1,2) if d= 2, ∈ we have moreover the following bound for all α [0,1] : ∈ A0,α,γ A0,α,0 C γq k − kL∞([0,T];Bm−5(R3)) ≤ q where C does not depend on α but depends on q. Hence, we obtain that q Aε,α,ε√α A0,0,0 C(ε2+α+ ε√α q) k − kL∞([0,T];Bm−5(R3)) ≤ and for α (0,1] fixed (cid:0) (cid:1) ∈ Ψε,α,ε√α Ψ0,α,0 Cεq. (1.32) k − kL∞([0,T];Bm−5(R3)) ≤ Remark 1.12. Let us notice that by Remark 1.3, the nonlinearity Gγ,k of equation (1.31) becomes a local cubic nonlinearity when γ = 0 n2 d G0,k(u)(x) = z3−d kωkk4L4(Rd) |u(x)|2u(x), for all x ∈R3−d. (cid:18) (cid:19) The paper is organized as follows. In Section 2, we study some properties of the dipolar term that are needed in the proofs of Theorems 1.6, 1.7 and 1.11 given in Section 3. 2. Study of the dipolar term Let us define for θ R, γ [0,1] and Φ L2(R3) ∈ ∈ ∈ F (θ,Φ)= eiθ z σ e iθ zΦ 2e iθ zΦ , 1 H − H − H | | F (Φ)= 1 2(cid:0)πF (θ,Φ)dθ, (cid:1) 1,av 2π 0 1 (2.1) Fγ(θ,Φ)= 3λReiθ z Vγ e iθ zΦ 2 e iθ zΦ, 2 0 H dip∗| − H | − H Fγ (Φ)= 1 2πFγ(cid:16)(θ,Φ)dθ (cid:17) 2,av 2π 0 2 so that R Fγ = F +Fγ and Fγ = F +Fγ . 1 2 av 1,av 2,av In order to prove the uniform well-posedness of the nonlinear equations (1.23) and (1.24), we will need Lipschitz estimates for Fγ(θ, ) defined by (1.15) and Fγ () av · γ γ · defined by (1.17). We only study here the dipolar terms F and F since the 2 2,av cubic ones F (θ,Φ) and F (Φ) have already been studied in [4, Lemma 2.7.] (see 1 1,av also [6, Proposition 2.5], [10, Lemma 4.10.2] or [8, Lemma 1.24]). 10 W.BAO,L.LETREUST,ANDF.MÉHATS 2.1. Some properties of Fγ(θ, ) and Fγ (). Using the fact that Vγ takes its 2 · 2,av · dip values in [ 1/3,2/3] (Remark 1.1), we get the following lemma. − d Lemma 2.1. Introduce the convolution operator Kγ : u Hm(R3) Vγ u Hm(R3) ∈ 7−→ dip∗ ∈ for γ [0,1] and m N where Vγ is defined by (1.11). Then, we get for all ∈ ∈ dip u Hm(R3) ∈ 2 Kγu u . Hm Hm k k ≤ 3k k The following lemma gives Lipschitz estimates for the dipolar terms. Lemma 2.2. For all m 2 and M > 0, there exists C > 0 such that ≥ Fγ (u) Fγ (v) CM2 u v k 2,av − 2,av kBm ≤ k − kBm Fγ(θ,u) Fγ(θ,v) CM2 u v , k 2 − 2 kBm ≤ k − kBm for all u,v Bm(R3) satisfying u M, v M, for all θ R and for all Bm Bm ∈ k k ≤ k k ≤ ∈ γ [0,1]. ∈ Proof. Let us fix γ [0,1], u,v Bm(R3) satisfying u M, v M. To Bm Bm ∈ ∈ k k ≤ k k ≤ begin, assume that θ = 0, then we get that Fγ(0,u) Fγ(0,v) = 3λ Kγ(u2)u Kγ(v 2)v k 2 − 2 kBm 0 | | − | | Bm ≤ 3λ0 Kγ(|u|2)(u−v) Bm(cid:13)(cid:13)+3λ0 (Kγ(|u|2)−Kγ((cid:13)(cid:13)|v|2))v Bm. Lemma 2.1 and Rem(cid:13)ark 1.5 ensure th(cid:13)at (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Kγ(u2)(u v) C Kγ(u2) u v C u2 u v | | − Hm ≤ | | Hmk − kHm ≤ | | Hmk − kHm (cid:13) (cid:13) C(cid:13)u 2 u(cid:13) v CM2 u (cid:13)v (cid:13) . (cid:13) (cid:13) ≤ k(cid:13) kHmk (cid:13)− kHm ≤ k −(cid:13) kB(cid:13)m We also have xmKγ(u2)(u v) C Kγ(u2) xm(u v) | | | | − L2 ≤ | | L∞k| | − kL2 (cid:13)(cid:13) ≤ C Kγ(|u|2) H(cid:13)(cid:13)mku−v(cid:13)(cid:13)kBm ≤ CM(cid:13)(cid:13) 2ku−vkBm. For the second term, w(cid:13)e get (cid:13) (cid:13) (cid:13) (Kγ(u2) Kγ(v 2))u C Kγ(u2 v 2) u | | − | | Hm ≤ | | −| | Hmk kHm (cid:13)(cid:13) ≤ C |u|2−|v|2 Hm(cid:13)(cid:13)kukHm ≤(cid:13)(cid:13)CM2ku−vkBm(cid:13)(cid:13) and (cid:13) (cid:13) (cid:13) (cid:13) xm(Kγ(u2) Kγ(v 2))u C Kγ(u2 v 2) xmu | | | | − | | L2 ≤ | | −| | L∞k| | kL2 (cid:13)(cid:13) ≤ C |u|2−|v|2 Hmkuk(cid:13)(cid:13)Bm ≤ CM(cid:13)(cid:13)2ku−vkBm. (cid:13)(cid:13) This gives us (cid:13) (cid:13) (cid:13) (cid:13) Fγ(0,u) Fγ(0,v) CM2 u v k 2 − 2 kBm ≤ k − kBm where Cdependsonm butisindependent ofγ, uandv. Sincee iθ z areisometries ± H of Bm, we get for θ R ∈ Fγ(θ,u) Fγ(θ,v) = eiθ z Fγ 0,e iθ zu Fγ 0,e iθ zv k 2 − 2 kBm H 2 − H − 2 − H Bm (cid:13) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)(cid:13) Fγ 0,e iθ zu F(cid:13)γ 0,e iθ zv CM2 e iθ z (u v) (cid:13) ≤ 2 − H − (cid:13)2 − H Bm ≤ − H − (cid:13)Bm ≤(cid:13)(cid:13)(cid:13)CM2(cid:16)ku−vkBm(cid:17) (cid:16) (cid:17)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)