The Schrödinger-Poisson system on the sphere

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Patrick Gérard, Florian Méhats

To cite this version:

Patrick Gérard, Florian Méhats. The Schrödinger-Poisson system on the sphere. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2011, 43 (3), pp.1232-1268.

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PATRICK GÉRARD AND FLORIAN MÉHATS In memory of Naoufel Ben Abdallah, with our friendship

Abstract. We study the Schrödinger-Poisson system on the unit sphereS² of R³, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results concern the Cauchy problem for this system. We prove that this problem is regularly well-posed on everyH^s(S²)with s >0, and not uniformly well-posed onL²(S²). The proof of well-posedness re- lies on multilinear Strichartz estimates, the proof of ill-posedness relies on the construction of a counterexample which concentrates exponentially on a closed geodesic. In a second part of the paper, we prove that this model can be obtained as the limit of the three dimensional Schrödinger-Poisson system, singularly per- turbed by an external potential that confines the particles in the vicinity of the sphere.

1. Introduction and main results

Let S² ⊂ R³ be the unit sphere. For functions defined on S², one considers the operatorG defined by

G(f)(x) = 1 4π

Z

S²

1

|x−y|f(y)dσ(y), (1.1) where σ denote the surface measure on S² and | · |is the Euclidean norm on R³.

In this paper, we are interested in the following Schrödinger-Poisson system on S²:

i∂_tu+ ∆_σu=G(|u|²)u, u(t= 0) =u₀. (1.2) Here ∆_σ denotes the Laplace-Beltrami operator on S². This system models the transport of a gas of quantum charged particles confined on a surface, here the sphere, and interacting through the Poisson potential, which is the Coulombian interaction in the Hartree approximation. Our purpose in studying this ideal system is twofold.

First, we are interested in understanding the wellposedness of the Cauchy problem for (1.2): choice of the function space, local in time or global in time solutions, stability of the flow map, . . . From this point of view, this paper comes in the continuity of several recent works concerning the nonlinear Schrödinger equation on Riemannian manifolds or in inhomogeneous media [9, 10, 11, 12, 13, 19, 20].

Second, we wish to justify this system for modeling a quantum gas via some as- ymptotic analysis, starting from a more conventional 3D Schrödinger-Poisson sys- tem with a singular perturbation which stands for a strong confinement potential.

Strongly confined Schrödinger-Poisson systems have previously been studied in Eu- clidean spaces: in [5, 16] for the confinement on a plane and in [3] for the confine- ment on an axis. The idea here is to investigate the influence of the geometry on

1

the confinement procedure. The case of the sphere can be seen as a step before more general manifolds, which can be interesting in some applications in the field of nanoelectronics. Quantum dynamical systems confined on a surface have been studied previously in [15, 17, 28] in linear situations. Starting from a similar scaling on the transverse Hamiltonian, these authors consider the linear Schrödinger equa- tion with a confinement on a general surface and derive an effective Hamiltonian which locally depends on the curvature properties of the surface. Here our approach is mainly concentrated on understanding the nonlinear effects.

Our first result states that this problem is locally well-posed in H^s, s > 0 and globally well-posed in the energy space:

Theorem 1.1. Let s > 0 a real number. For every bounded subset B ⊂H^s(S²), there exists T ∈ (0,+∞] and a subspace X_T of C((−T, T), H^s(S²) such that, for u₀ ∈ B, the Cauchy problem (1.2) admits a unique solution u ∈ X_T. For all 0< T^′ < T, the application u₀ 7→ u∈C([−T^′, T^′], H^s(S²))is Lipschitz continuous on B. Moreover, ifs≥1 one can choose T = +∞ and this global solution satisfies the following two conservation laws:

ku(t)kL² =Q₀, k∇σu(t)k²L² +1 2

Z

S²

G(|u|²)|u|²dσ=E₀. (1.3) Let us now deal with the limit case s = 0: after Theorem 1.1, the question whether this system is well-posed on L²(S²) is natural. In the case of a plane, the answer is positive thanks to Strichartz estimates in dimension 2. Our second result is that, in the case of the sphere, this system is not locally (uniformly) well-posed on S². The precise statement is as follows.

Theorem 1.2. For all ball B of L²(S²) centered on 0 and for all T > 0, the applicationu₀7→u isnot uniformly continuous from B∩H¹(S²), endowed with the L² norm, into C([−T, T], L²(S²)).

Our third result concerns the asymptotic analysis. The starting model is the Schrödinger-Poisson system (or Hartree equation) in R³ with a strong potential that confines the particles near the sphere. We consider the following system, on the unknown u^ε(t, x):

i∂tu^ε=−∆u^ε+V_c^εu^ε+V(|u^ε|²)u^ε, u^ε(t= 0) =u^ε₀, (1.4) where V_c^ε and V(|u|²) are respectively the applied confinement potential and the selfconsistent Poisson potential, given by

V_c^ε(x) = 1 ε²V_c

|x| −1 ε

(1.5) and

V(|u|²)(x) = 1 4π

Z 1

|x−x^′||u(x^′)|²dx^′. (1.6) The parameter ε > 0 is the order of magnitude of the width of the confined gas, compared to a typical length, for instance the radius of the sphere. The square ofε also denotes the ratio between the kinetic energy of the particles and the confinement energy. One refers to [5] and [16], where similar singularly perturbed systems were derived and put in dimensionless form, starting from systems in physical variables.

Our aim is to derive a simpler asymptotic system satisfied by u^ε asε→0.

The functionV_cis fixed, independent ofε, and is supposed to go to infinity at the infinity, faster than the harmonic potential, as stated in the following assumption.

Note that we do not solve the asymptotic problem in the precise case of a harmonic confinement potential, due to a lack of a priori estimates (Assumptionα >2is used several times in the proofs of Lemma 3.1 and Proposition 3.5).

Assumption 1.3. The confinement potential is a C∞ function such that

∀z∈R V_c(z)≥C|z|^α, (1.7) where α >2. Moreover, it satisfies the following condition:

∀k∈N, ∃C_k >0 : ∀z∈R

∂^kV_c

∂z^k (z)

≤C_kV_c(z). (1.8) Let us define the energy adapted to our system. We set

B¹ =n

u∈H˙¹(R³) : (V_c^ε)^1/2u∈L²(R³)o , equipped with the norm

kuk²B1 =k∇uk²_L²+k(V_c^ε)^1/2uk²_L². (1.9) The two conservations laws associated to (1.4) are the conservation of mass and the conservation of energy:

ku^ε(t)k²_L² =ku^ε₀k²_L² (1.10) k∇u^ε(t)k²L² +k(V_c^ε)^1/2u^ε(t)k²L² +1

2k∇V(|u^ε(t)|²)k²L²

=k∇u^ε₀k²L² +k(V_c^ε)^1/2u^ε₀k²L² +1

2k∇V(|u^ε₀|²)k²L².

The scaling (1.5) of the confinement potential suggests that, ifku^ε₀k²_L² is of order 1, it will be natural to start with an energy of order _ε¹2. Consequently, the solution of (1.4) will satisfy the following natural bounds:

ku^εkL² ≤C, εku^εk^B1 ≤C.

From Theorem 1.2, one can guess that these bounds will not be sufficient in order to analyze the limit of u as ε goes to zero. Consequently, we shall assume some regularity of the initial data with respect to the angular variable σ. To be more precise, let us introduce the spherical coordinates (r, σ) ∈ R∗

+×S² defined for all x ∈R³\ {0} by r = |x|and σ = _|^x_x|. We recall that the Laplace operator has the following form in spherical coordinates:

∆ = 1

r²∂_r r²∂_r + 1

r²∆_σ.

In particular, a remarkable property is that this operator commutes with ∆σ. This property will be crucial is our nonlinear analysis. In the sequel, we shall seth∇σi= (1−∆_σ)^1/2. We introduce a last notation. The transversal confinement operator is the following operator:

H_r=−1

r²∂_r r²∂_r

+V_c^ε(r), (1.11)

with domain D(H_r) =

u∈L²(R₊, r²dr) : 1

r²∂_r r²∂_ru

∈L²(R₊, r²dr), V_c^εu∈L²(R₊, r²dr)

.

Assumption 1.3 implies that H_ris self-adjoint onL²(R₊, r²dr), with a compact re- solvent. We shall denote by(E_p^ε)p∈N,(ψ_p^ε)p∈Nits eigenvalues and its eigenfunctions.

We make the following assumption on the initial data.

Assumption 1.4. The initial data u^ε₀ satisfies

kh∇σiu^ε₀kL²+εku^ε₀k^B1 ≤C, (1.12) where C >0is independent of ε∈(0,1]. Moreover, we have

lim sup

ε→0 k¹ε²Hr>Rh∇σiu^ε₀kL² +k¹_h∇σi>Rh∇σiu^ε₀kL²

→0 as R→+∞. (1.13) Let us comment on these assumptions. In (1.12), theL² bound ofh∇σiu₀ means a supplementary decay at the infinity for this quantity, compared to a H¹ norm which would only control theL² norm of ¹_rh∇σiu^ε₀. The second assumption means thath∇σiu^ε₀ is partially relatively compact inL² with respect to the σ variable ε²- oscillatory with respect to the operatorH_r defined by (1.11) (see [18] for an abstract definition of ε-oscillatory sequences with respect to a self-adjoint operator).

Our third theorem is the following one.

Theorem 1.5. Under Assumptions1.3and1.4on the confinement potential and the initial data, for all ε >0, (1.4) admits a unique global solution u^ε∈C(R, L²∩B₁).

Moreover, the following equation admits a unique global solution v^ε ∈ C⁰(R,B¹) such that h∇^σiv^ε∈C⁰(R, L²):

i∂tv^ε=Hrv^ε−∆σv^ε+G |v^ε|²

v^ε, v^ε(t= 0) =u^ε₀, (1.14) whereG is the following generalization of the operator Gdefined in(1.1): for func- tions defined on R³,

G(f)(rσ) = 1 4π

Z _+∞

0

Z

S²

1

|σ−σ^′|f(r^′σ^′)r^′2dr^′dσ^′. (1.15) Finally, we have the approximation of(1.4)by the limit system(1.14): for allT >0,

εlim→0kh∇σi(u^ε−v^ε)kL^∞([−T,T],L²)= 0.

Notice that, in (1.14), the nonlinear Schrödinger dynamics operates only in the σ variable. Indeed, the function ω^ε defined by

ω^ε(t, r, σ) =e^itHrv^ε(t, r, σ) solves the system

i∂tω^ε=−∆σω^ε+G |ω^ε|²

ω^ε, ω^ε(t= 0) =u^ε₀, (1.16) which is a "mixed-state" version of the scalar equation (1.2). Indeed, let us decom- pose the functionω^ε on the eigenvectors of the confinement operator Hr, setting

ω^ε(t, r, σ) =

+∞

X

p=0

ω_p^ε(t, σ)ψ^ε_p(r),

i.e.

v^ε(t, r, σ) =

+∞

X

p=0

e^−itEpεω^ε_p(t, σ)ψ_p^ε(r).

Then each componentω^ε_p satisfies i∂_tω_p^ε=−∆_σω_p^ε+G





+∞

X

q=0

|ω_q^ε|²



ω_p^ε, ω_p^ε(t= 0) = Z +∞

0

u₀ψ_p^εr²dr .

Notice that the componentsω_p^εare only coupled through the selfconsistent potential.

In particular, if the initial data is polarized on a single eigenmode, i.e. u^ε₀ =v_p0ψ_p^ε₀, then the solution of (1.14) remains polarized on the same mode p₀, for all time:

v^ε(t, r, σ) =e^{−itEεp0/ε2}ω^ε_p0(t, σ)ψ_p^ε₀(r) and ω^ε_p0 satisfies (1.2).

2. The Schrödinger-Poisson system on the sphere

This section is devoted to the analysis of the Schrödinger-Poisson system (1.2) on the two-dimensional sphereS². As announced in the introduction, we will prove that this system is locally well-posed inH^s(S²)for all s >0, but not for the critical case L²(S²). To make precise this statement, let us recall the notion of well-posedness that is meant here.

Definition 2.1. Let s∈R. We shall say that the equation (1.2) is, locally in time, uniformy well-posed in H^s(S²) if, for any bounded subsetB of H^s(S²), there exists T >0and a Banach spaceX_T continuously embedded intoC([−T, T], H^s(S²)), such that

(i) For every Cauchy data u₀ ∈B, (1.2) admits a unique solution u∈X_T, (ii) If u₀ ∈H^σ(S²) for σ > s, then u∈C([−T, T], H^σ(S²)),

(iii) The map u0 ∈B7→u∈XT is uniformly continuous.

The proof of Theorem 1.1 uses essentially the techniques developed in the works of Burq, Gérard, Tzvetkov [9, 10, 11, 12]. It can be decomposed into two main steps:

the proof of well-posedness under the assumption of a multilinear estimate and the proof of the multilinear estimate itself. More precisely, we follow the work of Gérard and Pierfelice [20], who have adapted the methodology to the particular structure of Hartree-type nonlinearities as in (1.2). The analysis relies in a crucial way on a quadrilinear estimate. For the sake of completeness and readability, we sketch all the steps of this proof, although the new contribution in this proof essentially concerns the quadrilinear estimate proved in subsection 2.2.

More precisely, Theorem 1.1 can be seen as a direct corollary of the following Propositions 2.2 and 2.7, completed with the conservation laws (1.3) (which are very classical are will not be proved here).

2.1. Well-posedness via quadrilinear estimate. The following proposition re- duces the study of wellposedness for (1.2) to a quadrilinear estimate.

Proposition 2.2. Suppose that, for every χ∈C∞

0 (R) and for all ε∈(0,¹₂), there exists C_ε>0 such that, for any f₁, f₂,f₃, f₄ ∈L²(S²) satisfying

1√

1−∆σ∈[Nj,2Nj](f_j) =f_j, j= 1,2,3,4, (2.1)

one has the following quadrilinear estimate:

sup

τ∈R

Z

R

Z

S2

χ(t)e^itτG(u1u2)u3u4dσdt

≤Cε(m(N1, N2, N3, N4))^ε

4

Y

i=1

kfjkL²(S²)

(2.2) where u_j(t) =S(t)f_j for j= 1,2,3,4, S(t) =e^it∆σ denoting the free evolution, and where m(N₁, N₂, N₃, N₄) denotes the product of the smallest two numbers among N₁, N₂, N₃, N₄. Then the Cauchy problem (1.2) is locally uniformly well-posed in H^s(S²) for any s >0.

Proof. We follow the steps of the proof of Theorem 1 in [10], see also [12, 20].

Proposition 2.2 will be proved thanks to a fixed point procedure formulated in the Bourgain spacesX^s,b.

Step 1. Reformulation of the problem in the Bourgain spaces.

Following the definitions in [7] and [12], we introduce the following family of Hilbert spaces

X^s,b =n

u∈S^′(R×S²) : 1 +|i∂_t+ ∆_σ|²b/2

(1−∆_σ)^s/2u∈L²(R×S²)o , (2.3) for s, b ∈ R. For any T > 0, we also denote by X_T^s,b the space of restrictions of elements ofX^s,bto(−T, T)×S². We gather in the following lemma some interesting properties of these spaces.

Lemma 2.3. The Bourgain spaceX^s,b satisfies the following properties.

(i) ∀f ∈H^s(S²), ∀b >0, the function S(t)f belongs to X^s,b. (ii) ∀b > ¹₂, X^s,b֒→C(R, H^s(S²)).

(iii) Let b, b^′ such that0≤b^′ < ¹₂, 0≤b <1−b^′. There exists C >0 such that, if T ∈(0,1] and w(t) =R_t

0S(t−τ)f(τ)dτ, then kwk_X^s,b

T ≤CT^1−b−b′kfk_X^s,−b′

T

.

Proof. The first property (i) stems directly from the definition (2.3) of X^s,b. Re- marking that

kukX^s,b =kvkH^b(R,H^s(S²)), where v(t) =S(−t)u(t), (2.4) the second statement (ii) is obvious and the last statement (iii) appears to be a consequence of the following elementary result, proved e.g. in [21]:

if g(t) = Z _t

0

h(τ)dτ then kgkH^b(−T,T)≤CT^1−b−b′kfk_H^−b′_(−T,T).

As a consequence of this lemma, it is easy to prove by a standard contraction argument that the Cauchy problem (1.2) is locally uniformly well-posed inH^s(S²), s >0, as soon as the following result holds true:

Lemma 2.4. Assume that the quadrilinear estimate of Proposition 2.2 holds true.

Then for all s >0, one can find b, b^′ satisfying0< b^′ <1/2< b <1−b^′ and such that we have the estimate

kG(u₁u₂)u₃kX^s,−b′ ≤Cku₁kX^s,bku₂kX^s,bku₃kX^s,b, (2.5)

kG(|u|²)uk_X^σ,−b′ ≤Ckuk²_X^s,bkukX^σ,b for σ ≥s. (2.6) The end of this section consists in the proof of this lemma.

Step 2. Quadrilinear estimate in X^s,b.

Let us first give without proof two estimates on the operator G. These results can be obtained straightforwardly using Hardy-Littlewood-Sobolev inequalities in dimension 2.

Lemma 2.5. The operator Gdefined by (1.1)satisfies the following estimates:

kG(f)kL^q(S²)≤CkfkL^p(S²), for 1< p <2 and 1 q = 1

p −1

2, (2.7) kG(f)kL^∞(S2)≤Ckfk^θL^p(S²)kfk¹_L⁻¹₍^θS²), for p >2 and θ= p

2p−2. (2.8) We now reformulate the quadrilinear estimate (2.2) in the X^s,b spaces.

Lemma 2.6. Under the same assumption as in Proposition 2.2, let u₁, u₂, u₃,u₄ satisfy

1√

1−∆σ∈[Nj,2Nj](uj) =uj, j = 1,2,3,4. (2.9) Then, for every s >0, there exists0< b^′ <1/2 such that

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt

≤C m(N₁, N₂, N₃, N₄)^s

4

Y

i=1

ku_jk_X^0,b′. (2.10) Proof. The desired estimate (2.10) can be obtained by interpolation between the two following inequalities:

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt

≤C m(N₁, N₂, N₃, N₄)^1/2

4

Y

i=1

ku_jkX^0,1/4 (2.11) and, for any b >1/2 and0< ε <1/2,

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt

≤C_εm(N₁, N₂, N₃, N₄)^ε

4

Y

i=1

ku_jkX^0,b. (2.12) Let us prove the first estimate (2.11). By symmetry and self-adjointness of G, only two cases have to be considered: m(N₁, N₂, N₃, N₄) = N₁N₂ and m(N1, N2, N3, N4) = N1N3. In the first case, we deduce from Hölder and from (2.8) that

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt

≤ kG(u₁u₂)kL²(R,L^∞(S²))ku₃u₄kL²(R,L¹(S²))

≤Cku₁u₂k^2/3_L2(R,L⁴(S2))ku₁u₂k^1/3_L2(R,L¹(S2))ku₃u₄kL²(R,L¹(S²))

≤CN₁^1/2N₂^1/2

4

Y

i=1

ku_jkL⁴(R,L²(S²)),

since ku_ikL⁸(S²) ≤CN_i^3/4ku_ikL²(S²) due to the spectral localization. Hence, to get (2.11) in this case, it suffices to remark that

ku_jkL⁴(R,L²(S²)) =kS(−t)u_jkL⁴(R,L²(S²))≤CkS(−t)u_jkH^1/4(R,L²(S²))=Cku_jkX^0,1/4,

where we used the isometric action ofS(t), a Sobolev embedding in dimension one and (2.4).

The second case m(N₁, N₂, N₃, N₄) = N₁N₃ can be treated as follows. Using (2.7), we have

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt

≤ kG(u₁u₂)kL²(R,L⁴(S²))ku₃u₄kL²(R,L^4/3(S²))

≤Cku1u2kL²(R,L^4/3(S²))ku3u4kL²(R,L^4/3(S²))

≤Cku1kL⁴(R,L⁴(S²))ku2kL⁴(R,L²(S²))ku3kL⁴(R,L⁴(S²))ku4kL⁴(R,L²(S²))

≤CN₁^1/2N₃^1/2

4

Y

i=1

ku_jkL⁴(R,L²(S2)) ≤CN₁^1/2N₂^1/2

4

Y

i=1

ku_jkX^0,1/4,

where we usedku_ikL⁴(S2) ≤CN_i^1/2ku_ikL²(S2).

We now sketch the proof of the second estimate (2.12), for allb >1/2, which is directly inspired from [12, 20]. Let us start by proving in a first step the result for the special case where theu_j are supported in time in an interval of size 1, say(0,1) after translation. Let vj(t) = S(−t)uj(t) and let fj(τ) = ˆvj(τ) denote the Fourier transform in time ofv_j. Applying the inverse Fourier transform, we get

Z

S²

G(u₁u₂)u₃u₄dσ= 1 (2π)²

ZZZZ

R⁴

e^{it(τ1+τ2+τ3+τ4)}×

×G(S(t)f₁(τ₁)S(t)f₂(τ₂))S(t)f₃(τ₃)S(t)f₄(τ₄)dσdτ₁dτ₂dτ₃dτ₄.

Hence, the quadrilinear estimate (2.2), applied pointwise inτ₁,τ₂, τ₃,τ₄ and after having chosenχ∈C∞

0 (R) such that χ= 1 on(0,1), gives

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt

≤C_εm(N₁, N₂, N₃, N₄)^ε

4

Y

i=1

Z

Rkvˆ_jkL²(S2)(τ)dτ

≤C_εm(N₁, N₂, N₃, N₄)^ε

4

Y

i=1

khτi^bvˆ_jkL²(R×S2)

=C_εm(N₁, N₂, N₃, N₄)^ε

4

Y

i=1

ku_jkX^0,b,

where the Cauchy-Schwarz inequality could be applied sinceb >1/2.

To treat the general case, it suffices now to introduce a function ψ ∈ C∞ 0 (R) supported in (0,1) such that P

n∈Zψ(t−ⁿ₂) = 1, to decompose uj = P

n∈Zuj,n, withu_j,n=ψ(t−ⁿ₂)u_j(t), j= 1,2,3,4, and to apply the result of the first step to the functions uj,n. The conclusion follows from the almost orthogonality property satisfied by the Bourgain spaces

X

n∈Z

ku_j,nk²_X^0,b ≤Cku_jk²_X^0,b.

Step 3. Proof of Lemma 2.4by dyadic decomposition.

We have now the tools to prove Lemma 2.4. We shall only prove (2.5), the proof of (2.6) being similar. By duality, we have to show that, for allu₄∈X^−s,b′,

Z

R

Z

S²

G(u₁u₂)u₃u₄dσdt ≤C





3

Y

j=1

ku_jkX^s,b



ku₄k_X^−s,b′. (2.13) Let us perform a dyadic decomposition of the u_j’s, setting u_j,N =

1√

1−∆σ∈[N,2N](u_j). We have ku_jk²_X^s,b ≃X

N

N^2sku_j,Nk²_X^0,b ≃X

N

ku_j,Nk²_X^s,b (2.14) where N are dyadic integers. Proving (2.13) is equivalent to estimating the sum

X

N1,N2,N3,N4

J(N1, N2, N3, N4),

where

J(N₁, N₂, N₃, N₄) = Z

R

Z

S²

G(u_1,N1u_2,N2)u_3,N3u_4,N4dσdt.

We now remark that in the above sum, all the terms withN₄>4 max(N₁, N₂, N₃) are zero. To prove this point, we first claim that the operator G is a function of

∆σ, more precisely, that we have for allf G(f) = 1

4π Z

S²

1

|x−y|f(y)dσ(y) = (1−4∆_σ)^−1/2(f). (2.15) The proof of the claim is done below. Moreover, from spectral localization u_j,Nj is the sum of spherical harmonics of degrees between p

N_j/2 and p 2N_j. One can deduce from properties of products of spherical harmonics that the product G(u_1,N1u_2,N2)u_3,N3 is a sum of spherical harmonics of degrees less than p2 max(N₁, N₂, N₃), so orthogonal tou₄ if N₄ >4 max(N₁, N₂, N₃).

Hence, by symmetry, it suffices to consider the terms with N₁ ≤N₂ ≤N₃ and N4 ≤4N3. Pick s^′ such that 0< s^′ < s. By Lemma 2.6, there exists 0 < b^′ <1/2 such that

X

N1,N2,N3,N4

J(N₁, N₂, N₃, N₄)≤C X

N1,N2,N3

X

N4≤4N3

N₁^s′N₂^s′

4

Y

i=1

ku_j,NjkX^0,b′,

≤C



 Y

j=1,2

X

N_j

N_j^s′−sku_j,Njk_X^s,b′



 X

N3

X

N4≤4N3

N₃ N4

₋s

ku_3,N3k_X^s,b′ku_4,N4k_X^−s,b′

where we used (2.14). The series inN₁ and N₂ converges easily. DenotingN₄ = 2ⁿ andN₃/N₄ = 2^m, the series inN₃,N₄ can be bounded by a simple Cauchy-Schwarz argument. Indeed, this series reads

X

m≥−2

2^−ms X

n≥max(0,−m)

ku_3,2m+nk_X^s,b′ku_4,2ⁿk_X^−s,b′

≤



 X

m≥−2

2^−ms



 X

n

ku_3,2ⁿk²_Xs,b′

!1/2

X

n

ku_4,2ⁿk²_X⁻s,b′

!1/2

≤Cku₃kX^s,b′ku₄kX^−s,b′.

The proof of Lemma 2.4 is complete, so Proposition 2.2 is proved.

Proof of the claim (2.15). We use a very old argument that can be found, e.g., in Poincaré’s works [25]. Let us compute explicitely the action of G on a spherical harmonicY_ℓ^m. We define the following function on R³:

u(r, σ) =r^ℓY_ℓ^m(σ)for r ≤1, u(r, σ) =r^−ℓ−1Y_ℓ^m(σ) for r≥1.

It is readily seen that u is harmonic on R³\S², so by computing explicitely the jump of ∂_ru onS² we obtain

−∆u= (2ℓ+ 1)Y_ℓ^mdσ

in the distribution sense, wheredσis the surface measure onS². Sinceuis decreasing at the infinity, this means that

u(r, σ) = 1 4π

Z

S²

1

|rσ−σ^′|(2ℓ+ 1)Y_ℓ^mdσ^′, thus

u(1, σ) = (2ℓ+ 1)G(Y_ℓ^m).

Denotingλ=ℓ(ℓ+ 1), we have 2ℓ+ 1 =√

1 + 4λ, so we have G(Y_ℓ^m) = 1

√1 + 4λY_ℓ^m

and (2.15) is proved.

2.2. Quadrilinear estimate. In this section, we prove the quadrilinear estimate (2.2) which is, thanks to Proposition 2.2, the core of the proof of local existence result in Theorem 1.1. The main result of this section is the following proposition.

Proposition 2.7. There exists a constant C > 0 such that for any f₁, f₂, f₃, f₄ ∈L²(S²) satisfying

1√

1−∆σ∈[Nj,2Nj](f_j) =f_j, j= 1,2,3,4, (2.16) and for all ε∈]0,¹₂[one has the following quadrilinear estimate

sup

τ∈R

Z

R

Z

S2

χ(t)e^itτG(u₁u₂)u₃u₄dσdt

≤C_ε(m(N₁, N₂, N₃, N₄))^ε

4

Y

i=1

kf_jkL²(S²)

(2.17) where u_j(t) =S(t)f_j for j= 1,2,3,4, S(t) =e^it∆σ denoting the free evolution, and χ∈C∞

0 (R) is arbitrary.

Proof. Let us decompose the f_j’s on spherical harmonics:

f_j =

2Nj

X

nj=Nj/2

H_n^j_j with ∆_σH_n^j_j+n_j(n_j+ 1)H_n^j_j = 0, so that

u_j(t) =

2Nj

X

nj=Nj/2

e^{−itnj(nj+1)}H_n^j_j.

The quantity to estimate can be written as Z

R

Z

S²

χ(t)e^itτG(u₁u₂)u₃u₄dσdt= X

n1,n2,n3,n4

ω_n1,n2,n3,n4(τ)I(n₁, n₂, n₃, n₄)

where

ω_n1,n2,n3,n4(τ) = ˆχ



τ −

4

X

j=1

ε_jn_j(n_j + 1)



 withε_j = (−1)^j+1, ˆ

χ being the Fourier transform ofχ, and I(n₁, n₂, n₃, n₄) =

Z

S²

G(H_n¹₁H_n²₂)H_n³₃H_n⁴₄dσ .

By symmetry and self-adjointness of G, assume for instance that N₃ = min(N1, N2, N3, N4). Let us estimate I(n1, n2, n3, n4) by adapting the proof of multilinear estimates in [10, 11, 12, 20]. One can decomposeH_n^j_j using a microlocal partition of the unity with semiclassical cut-off of the formχ_j(x, h_jD) (with small support):

H_n^j_j =ϕj +X

k

χ^k_j(x, hjD)H_n^j_j,

whereϕ_j is regular (for alls,k|D^s|ϕ_jkL² ≤C_skHn^jjkL²) and where the sum is finite.

Thus, we only have to estimate some terms of the form I˜=

Z

S2

G(V1V2)V3V4dσ,

where V_j is of the form V_j = χ_j(x, h_jD)H_n^j_j. The key point, from Burq, Gérard, Tzvetkov [10, 11, 12] is that, using semiclassical Strichartz estimates in dimension one, for each choice of(χ_j)_j=1,2,3,4 one can find a local system of linear coordinates (x, y) such that, for all p, q∈[2,∞]satisfying ²_p +¹_q = ¹₂, we have

kV_jkL^pxL^qy ≤CN_j^1/pkH_n^j_jkL²(S²), for j= 1,2,3,4. (2.18) The proof of this result uses the fact that the functionHn^jj satisfies

h²_j∆_σH_n^j_j+H_n^j_j = 0, withh²_j = 1

n_j(n_j+ 1) ∼N_j²,

which can be locally seen as an evolution equation where the well-chosen variablex plays the role of a time.

In local coordinates we have

|I˜| ≤CkG(V₁V₂)kL^∞_xL^s_ykV₃k_L^1/ε_{x L}r

ykV₄kL²_xL²_y

withs= _2ε¹ and ¹_r = ¹₂ −2ε. Since

|G(V₁V₂)| ≤C

Z 1

|x−x^′|+|y−y^′||V₁V₂|(x^′, y^′)dx^′dy^′, a Hardy-Littlewood Sobolev inequality gives

kG(V₁V₂)kL^∞_x L^s_y ≤Csup

x

Z 1

|x−x^′|^1−2εkV₁V₂(x^′,·)kL¹dx^′

≤C_εkV₁kL^∞_x L²_ykV₂kL^∞_x L²_y,

where we used the fact that all the functions are compactly supported. Then we apply (2.18) with the pairs (p, q) = (∞,2) and(¹_ε, r):

|I˜| ≤C_εkV₁kL^∞_x L²_ykV₂|L^∞_x L²_ykV₃k_L^1/ε_{x L}r

ykV₄kL²_xL²_y

≤C_ε(N₃)^ε

4

Y

j=1

kH_n^j_jkL².

To conclude, it remains to estimate uniformly with respect toτ the quantity Q(τ) = X

n1,n2,n3,n4

|ωn1,n2,n3,n4(τ)|

4

Y

j=1

kH_n^j_jkL²

where, in the sum,n₁, n₂, n₃, n₄ satisfy

∀j∈ {1,2,3,4} Nj

2 ≤n_j ≤2N_j. (2.19)

This quantity has already been estimated in [20], but we reproduce again this proof here for the sake of completeness.

Denote byΛ(k) the set of (n₁, n₂, n₃, n₄) satisfying (2.19) and

4

X

j=1

ε_jn_j(n_j+ 1) =k.

We deduce from the decay ofχˆ at the infinity that Q(τ) ≤CX

ℓ∈Z

1 1 +ℓ²

X

Λ([τ]+ℓ) 4

Y

j=1

kH_n^j_jkL²

≤Csup

k∈Z

X

Λ(k) 4

Y

j=1

kH_n^j_jkL².

Let us denote by α, β the two indices such that m(N1, N2, N3, N4) = (Nα, N_β) and by γ, δ the two other indices: {γ, δ} ={1,2,3,4} \ {α, β}. Then we introduce the set

Γ(k, i, j) =

(n_i, n_j) : N_i

2 ≤n_i≤2N_i, N_j

2 ≤n_j ≤2N_j and ε_in_i(n_i+ 1) +ε_jn_j(n_j + 1) =k

. Now, denoting

S(k, i, j) = X

Γ(k,i,j)

kH_nⁱ_ikL²kH_n^j_jkL², we split the sum as follows:

Q(τ)≤Csup

k∈Z

X

k^′∈Z

S(k, α, γ)S(k−k^′, β, δ).

Then we apply the following elementary result of number theory (see e.g. [10]).

Lemma 2.8. Let σ = ±1. For all ε > 0, there exists C_ε > 0 such that, given M ∈Zand N ∈N∗,

#

(k₁, k₂)∈N² : N ≤k₁ ≤2N, k²₁+σk₂²=M ≤C_εN^ε.