Stability of equilibria for a Hartree equation for random fields

J. Math. Pures Appl. 137 (2020) 70–100

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Journal de Mathématiques Pures et Appliquées www.elsevier.com/locate/matpur

Stability of equilibria for a Hartree equation for random fields C. Collot a,∗ , A.-S. de Suzzoni b a b

Courant Institute of Mathematical Sciences, New York University, New York, United States of America CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France

a r t i c l e

i n f o

Article history: Received 20 November 2018 Available online 23 March 2020 MSC: 35B35 35B40 Keywords: Hartree equation Random fields Stability Scattering Mots-clés : Équation de Hartree Champs aléatoires Stabilité Diffusion

a b s t r a c t We consider a Hartree equation for a random field, which describes the temporal evolution of infinitely many fermions. On the Euclidean space, this equation possesses equilibria which are not localized. We show their stability through a scattering result, with respect to localized perturbations in the not too focusing case in high dimensions d ≥ 4. This provides an analogue of the results of Lewin and Sabin [22], and of Chen, Hong and Pavlović [11] for the Hartree equation on operators. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger and Gross-Pitaevskii equations. © 2020 Published by Elsevier Masson SAS.

r é s u m é On considère une équation de Hartree pour des champs aléatoires décrivant la dynamique d’un système infini de fermions. Sur l’espace euclidien, cette équation admet des équilibres non-localisés. On prouve la stabilité de ces derniers à travers un résultat de diffusion pour des perturbations localisées et en dimension supérieure à 4, lorsque la non-linéarité est faiblement focalisante. Cela fournit une contrepartie aléatoire aux résultats de Lewin et Sabin [22] et de Chen, Hong et Pavlović [11] qui traitent de l’équation de Hartree pour des opérateurs densités. La preuve s’appuie sur des techniques dispersives utilisées dans l’étude de la diffusion des équations de Schrödinger et de Gross-Pitaevskii. © 2020 Published by Elsevier Masson SAS.

* Corresponding author. E-mail addresses: [email protected] (C. Collot), [email protected] (A.-S. de Suzzoni). https://doi.org/10.1016/j.matpur.2020.03.003 0021-7824/© 2020 Published by Elsevier Masson SAS.

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1. Introduction 1.1. Mean-field dynamics of an infinite number of fermions The present work concerns the following Hartree equation for a random field:   i∂t X = −ΔX + w ∗ E(|X|2 ) X.

(1)

Here X : I × Rd × Ω → C is a time-dependent random field over the Euclidean space Rd, and (Ω, A, dω) is the underlying probability space. The expectation is with respect to this probability space: E(|X|2 )(t, x) :=  |X|2 (t, x, ω)dω. The convolution product is denoted by ∗ and w is a real-valued pair interaction potential. Ω By this, we mean that we only consider interactions between two particles (and no more), and that this interaction is characterized by w. Equation (1) has been introduced in [12] as an effective dynamics for a large, possibly infinite, number of fermions in a mean field regime. Indeed, consider the evolution of a finite number of fermions interacting through the potential w. Under some mean-field hypothesis, as the number of particles tend to infinity, the system is approximated to leading order by the following system of N coupled Hartree equations on R × Rd , for an orthonormal family (uj )1≤j≤N (to be compliant with the Pauli principle):  i∂t uj = −Δuj +

w∗(

N 

 |uk | ) uj , j = 1, ..., N. 2

(2)

k=1

N Let us associate to this orthonormal family the operator γ = 1 |uj uj |, which is the orthogonal projection onto Span{(uj )1≤j≤N }. There exists a large literature about the derivation of this system of equations and about other related approximation results. In particular, it has been shown that, if the wave function of the original fermionic system is close to a Slater determinant, then, in the mean-field limit and under sufficient conditions for w, the associated one-particle density matrix converges to the above operator γ. For the derivation of Equation (2) from many body quantum mechanics we refer to [2–5,13,16]. Note that the socalled exchange term appearing in the Hartree-Fock equation is not present in (2), which is motivated by the fact that it is of lower order in certain regimes, see the aforementioned references. To deal with infinitely many particles, it is customary to use the density matrices framework, which is an operator formalism. Namely, the family (uj )1≤j≤N solves Equation (2) if and only if the operator γ defined above solves the corresponding Hartree equation: i∂t γ = [−Δ + w ∗ ργ , γ].

(3)

Above, [, ] denotes the commutator, and ργ (x) = γ˜ (x, x) is the density of particles, that is the diagonal of the integral kernel γ˜ (x, y) of the operator γ. An infinite number of particles can then be modeled by a solution of (3) which is not of finite trace (the trace of the operator being, by the derivation of the model, the number of particles). Solutions of (3) with an infinite number of particles were studied previously in [7–9,26] for example, and more recently in [10,11,22,23]. In [12], the second author proposed (1) as an alternative equation to (3). It generalizes Equation (2) for a finite number of particles in the following sense. To an orthonormal family (uj )1≤j≤N , one can associate N the random variable X(x, ω) = 1 uj (x)gj (ω), where (gj )1≤j≤N is any orthonormal family in L2ω . The family (uj )1≤j≤N then solves (2) if and only if the random variable X solves (1). Equation (1) is also in close correspondence with the Hartree equation for density matrices (3). Indeed, for infinite numbers of particles, taking X to be a Gaussian random field with covariance operator γ, we have that X solves (1) if and only if γ solves (3) (provided γ is taken in an appropriate metric space). We refer to [12] for how to

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relate the solutions of the two Equations (1) and (3) and their corresponding equilibria. One reason behind the study of (1) is that this equation shares more direct resemblances with the commonly studied nonlinear Schrödinger equation. The existence of solutions of Equation (1) in L2ω Hxs is investigated in [12]. The local well-posedness is established in the case of localized initial data, as well as in the case of localized perturbations of the equilibria described below. In particular, almost everywhere in the probability space, the random variable solves the corresponding Schrödinger equation in integrated formulation. Though the results are stated and proved in the case of a Dirac potential w = δ{x=0} , their adaptation to the present case of regular interaction potentials is straightforward. Moreover, as for the defocussing nonlinear Schrödinger equation, scattering for large but localized solutions is expected for Equation (1), at least in energy critical and subcritical regimes. This has been showed in dimension 3 in the case of a Dirac potential in [12]. Nontrivial equilibria are thus non-localized, which corresponds to being not of trace-class in the framework of density matrices. 1.2. Statement of the result The equilibria at stake in the present paper are the following. For Equation (3), any non-negative Fourier multiplier γ : u → F −1 (|f (ξ)|2 Fu) with symbol |f |2 is a stationary solution, where F denotes the Fourier transform. Note that density operators are non-negative, we write the symbol in the form |f |2 to be able to give an analogous equilibrium in the random framework. The analogous equilibrium for Equation (1) are given by Wiener integrals

2

f (ξ)eiξ.x−it(m+|ξ| ) dW (ξ),

Yf (t, x, ω) :=

(4)

ξ∈Rd

for a distribution function f : Rd → C (note that this equilibrium has the same law if we replace f by |f | so that we can assume f : Rd → [0, +∞)). Above dW (ξ) denotes infinitesimal complex Gaussians characterized by

 E dW (η)dW (ξ) = δη−ξ dηdξ,   and the scalar m is given by m := Rd w(x)dx Rd |f (ξ)|2 dξ. We refer to [25], Chapter 1, and [1], Part I, for more information on random Gaussian fields. The function Yf is a solution of Equation (1). It is not a steady state but it is an equilibrium, for as a random field its law is invariant by time and space translations (and in particular is not localized). In the seminal work [22], the authors show the stability of the above equilibria for the Equation (3) for density matrices in dimension 2. Important tools are dispersive estimates for orthonormal systems [14,15]. This work has been extended to higher dimension in [11]. Note that in higher dimension, some structural hypothesis is made on the interaction potential w, to solve some technical difficulties about a singularity in low frequencies of the equation that we will identify precisely in the sequel. The stability result corresponds to a scattering property in the vicinity of these equilibria: any small and localized perturbation evolves asymptotically into a linear wave which disperses. We mention equally [10,23] about problems of global well-posedness for the equation on density matrices. The problem of the stability of the equilibria (4) for Equation (1) shares similarities with the stability of the trivial solution for the Gross-Pitaevskii equation i∂t ψ = −Δψ + (|ψ|2 − 1)ψ. In both problems the linearized dynamics has distinct dispersive properties at low and high frequencies, making the nonlinear stability problem harder, especially in low dimensions. The proof of scattering for small data for the GrossPitaevskii equation was done in [19,18,20,21]. We here use spaces with different regularities at low and high frequencies, inspired by [21].

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The result of the present work is the stability of the equilibria (4) for Equation (1), via the proof of scattering of perturbations to linear waves in their vicinity. Our techniques however differ from those used in [11,22], see the strategy of the proof after Theorem 1.1. We hope that the present proof provides different insights than the ones in the framework of operators, as well as relaxing some of the hypotheses on the potential w. In what follows, ξ = (1 + |ξ|2 )1/2 denotes the usual Japanese bracket. Given s ∈ R, s denotes the usual ceiling function applied to s, and (s)+ = max(s, 0) and (s)− = max(−s, 0) are the nonnegative and nonpositive parts of s. We write with an abuse of notation f (ξ) = f (r) with r = |ξ|, if f has spherical symmetry. The space L2ω , H s is the set of measurable functions Z : Rd × Ω → C such that Z(·, ω) ∈ H s almost surely and ˆ ω)|2 dξdω < +∞. ξ2s |Z(ξ, Rd ×Ω

Theorem 1.1 (Stability of equilibria for Equation (1)). Let d ≥ 4. Let s = d2 − 1. Let f be a spherically symmetric function in L2 (Rd ) ∩ L∞ (Rd ). Assume the following hypotheses hold true: • Assumptions on the equilibrium: (i) ξs f ∈ L2 (Rd ),  (ii) Rd |ξ|−1 |f f | < ∞, (iii) ∂r |f |2 < 0 for r > 0, (iv) writing h the inverse Fourier transform of |f |2 , x2 ∂ α h ∈ L∞ (Rd ) for all α ∈ N d with |α| ≤ 2 s , (v) (|x|1−d + |x|2−d )(h + h) ∈ L1 (Rd ). • Assumptions on the potential: w ∈ W s,1 is even with ξw ˆ ∈ L2(d+2)/(d−2) and (w) ˆ − L∞ + w(0) ˆ + ≤ C(f ), where C(f ) > 0 is a constant depending on f . 2d/(d+2)

Then there exists  = (f, w) such that for any Z0 ∈ L2ω H d/2−1 ∩ Lx

L2ω with

Z0 L2ω H d/2−1 + Z0 L2d/(d+2) ≤ , L2 x ω

the solution of (1) with initial datum X0 = Yf (t = 0) + Z0 is global. Moreover, it scatters to a linear solution in the sense that there exists Z− , Z+ ∈ L2ω H d/2−1 such that X(t) = Yf (t) + ei(Δ−m)t Z± + oL2ω H d/2−1 (1) as t → ±∞.

(5)

Remark 1.1. Note that since w is even, its Fourier transform is real, and thus its positive and negative parts (w+ and w− ) are well-defined. Relating the framework of random fields to that of density operators, from the above Theorem 1.1 one obtains a scattering result for the operator:

 γ = E(|XX|) := u → x → E(X(x)X, uL2 (Rd ) ) , with respect to the one associated to the equilibrium Yf : γf = E(|Yf Yf |),

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which is the Fourier multiplier by |f |2 (ξ). This convergence holds in Hilbert-Schmidt Sobolev spaces (where below S2 is the standard Hilbert-Schmidt norm): γHα = ∇α γ∇α S2 . Corollary 1.2. Under the hypotheses of Theorem 1, setting: γ± = E(|Z± Z± |) + E(|Z± Yf (t = 0)|) + E(|Yf (t = 0)Z± |)

(6)

there holds γ± ∈ Hd/2−1 and the convergence: γ = γf + eiΔt γ± e−iΔt + o

d

H 2 −1

(1) as t → ±∞.

Remark 1.2. The conditions on f are satisfied by thermodynamical equilibria for bosonic or fermionic gases at a positive temperature T : |f (ξ)|2 =

1 e

|ξ|2 −μ T

−1

, μ<0

and |f (ξ)|2 =

1 e

|ξ|2 −μ T

+1

, μ ∈ R,

respectively, but it is not the case of the fermionic gases at zero temperature: |f (ξ)|2 = 1|ξ|2 ≤μ , μ > 0. Remark 1.3. The smallness assumption on (w) ˆ − L∞ , corresponds to the fact that the equation is not too focusing. The one on (w(0)) ˆ + enables the equation (1) linearized around Yf to have enough dispersion. Note that these assumptions appear both in [22] and [11]. Remark 1.4. The smallness of the initial datum in L2ω , H d/2−1 cannot be improved as d/2 − 1 is the critical 2d/(d+2) regularity. The smallness of the initial datum in Lx , L2ω is related to the smallness of 2ReE(Y¯ S(t)Z0 ) in ΘV . Of course, if Z0 is orthogonal to the Wiener process W , this term is null. This hypothesis on the initial datum may thus be improved. The optimal space for the initial datum is unclear to us, but taking Z0 in low integrability Lebesgue spaces is related to taking the initial datum in [11,22] in low Schatten spaces. Remark 1.5. Let us compare briefly with the related works [11,22]. The framework used in the present article is that of random fields, which provides a new point of view, and strengthen the understanding of equation (1). The comparison between the aforementioned results and ours is not straightforward, the connexion between the frameworks being explained in the introduction. First, the dimension treated differs, [22] providing with a complete proof in dimension 2, [11] with one for all dimensions d ≥ 3, and the present paper for d ≥ 4 (see the remark on lower dimensions below). All works use a linear stability result that is the invertibility of the linear response operator (see Proposition 5.7). Additional dispersive effects are used in [11,22] by means of new Strichartz estimates in the framework of density matrices [6,11,14,15]. We solely use here standard Strichartz estimates, but find that the response of the equilibrium also enjoys an improved decay at low frequencies due to randomness (Lemmas (5.3) and 5.4). This improved decay does not appear in [11,22], and in [11] a singularity that appears at low frequencies is handled by making stronger assumptions on their interaction potential. Our analysis eventually yields a scattering result at critical regularity, as in [22], whereas [11] holds for supercritical regularity.

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Remark 1.6. The stability is not a consequence of a strong convergence to 0 of the perturbation in H d/2−1 almost everywhere in probability, but of the dispersion for the linear dynamics implying in particular local convergence. The solution converges back to the same equilibrium Yf . In particular, it does not trigger modulational instabilities. Remark 1.7. In [21], which solves scattering for the Gross-Pitaevskii equation in dimension greater than 4, a normal form is needed to close the argument and in particular to deal with the low-frequency singularity. We do not require the use of a normal form transformation because of random cancellations. First, exact computations of the linearized equation around the equilibrium enable us to lose only 1/2 derivatives on V := E(|X|2 ) − E(|Yf |2 ) in the low frequencies, while in [21] it is required to lose 1 full derivative on the nonlinearity of the Gross-Pitaevskii equation, which contains quadratic and cubic terms. The second issue is that V , even if it contains a linear and a quadratic term in the perturbation X − Yf , behaves in terms of Lebesgue integration as |X − Yf |2 , which makes the analysis easier. Note that the potential of interaction plays a role to gain derivatives in high frequencies, we do not use it to gain derivatives in low frequencies. Remark 1.8. The strategy of the present paper does not directly apply to dimensions 2 and 3, as dispersive effects are weaker. The proof fails at Proposition 6.1. In dimension 2, the first two Picard iterations display a singularity at low frequency, and would have to be treated separately. This has been successfully done in [22]. In dimension 3, we believe that only the first iteration would have such singularity. In [11], a related singularity at low frequency is compensated by hypotheses on the interaction potential. We believe that our strategy could be modified along the lines of [22] for d = 3. Remark 1.9. The function f may be referred as momentum distribution function for the equilibrium. In the case of |f |2 (ξ) = 1/(e(|ξ|

2

−μ)T

+ 1)

where T is the temperature and μ the chemical potential, it may be referred as the Fermi-Dirac distribution. Even in the defocussing case, some other equilibria than the ones described by a momentum distribution function can have instabilities. Two plane waves, which are orthogonal in probability, propagating in opposite directions, are linearly unstable, which is showed In Section 9. We do not claim nonlinear instability as we did not prove that a non trivial codimensional stability cannot arise. However, we believe that it would follow from the analysis of the linearized around the equilibrium equation since high and low frequencies interact through the cubic term of the nonlinearity. Note that the equilibria of Theorem 1.1 can be seen as a superposition of infinitely many plane waves propagating in different directions, hence this shows the importance of regularity of the underlying function f . The strategy of the proof is the following. First note that the dynamics of Equation (1) near the above equilibria is somewhat similar to that of the Gross-Pitaevskii equation. We use a more direct fixed point argument which does not involve iterations of the wave operator as in [11,22], and dispersion properties at low and high frequencies which are inspired from [21]. We start by reducing the proof to finding a correct functional framework for our contraction argument. Namely, instead of solving an equation for the perturbation Z = X − Yf , we solve a fixed point for the perturbation and the induced potential Z, V where V = E(|Z|2 ) + 2ReE(Y¯f Z)), at the same time. The idea behind this is that even if V contains a linear term in Z, it behaves more like a quadratic term in Z (in the sense of the Lebesgue spaces to which they belong) and thus, we can put it in better spaces regarding dispersion. The fixed point is solved in a classical way by finding the right Banach space Θ for Z, V and proving suitable estimates on the linear and nonlinear terms. The Lipschitz-continuity of the quadratic part is

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treated in a classical fashion, in the sense that it requires that Θ is included in classical Lebesgue, Besov, or Sobolev spaces. The difficulty comes from the linear term. It can be written L=

0 L2 0 L1

,

see (12), where L1 corresponds to the analogue of the linear term in [11,22]. The invertibility and continuity of 1 − L1 had been dealt with in both papers and their treatment is more or less sufficient for our argument. Note that this term is the linear response of the equilibrium, related to the so-called Lindhard function [17,24]. But L2 is where singularities in low frequencies occurs. To get the continuity of L2 , V needs to be in a space that compensates this singularity, namely inhomogeneous Besov spaces, with two levels of regularity, one for the low frequencies and one for the high frequencies. But V contains E(|Z|2 ) and we cannot close the fixed point argument for Z in a space that compensates singularities in low frequencies. This is where we use bilinear estimates on inhomogeneous Besov spaces coming from the scattering for Gross-Pitaevskii literature, [21]. The paper is organized as follows. In Section 2, we state a few definitions and known results that we use in the rest of the paper. In Section 3, we set up the fixed point argument. In Section 4, estimates related to some direct embeddings are given. The linear terms are studied in Section 5 where explicit formulas, continuity estimates and invertibility conditions are obtained. Estimates for the quadratic terms are proven in Section 6, and estimates for the source terms are showed in Section 7. Theorem 1.1 is then proved in Section 8. The last Section 9 is devoted to an instability result in a defocussing case when the momentum distribution function is not smooth. In the appendix A we prove Corollary 1.2. Notations Notation 1.3 (Fourier transform). We define the Fourier transform with the following constants: for g ∈ S,

g(x)e−ixξ dx,

gˆ(ξ) = F(g)(ξ) = Rd

and the inverse Fourier transform by F −1 (g)(x) = (2π)−d

g(ξ)eixξ dξ.

Rd

Notation 1.4 (Time-space norms). For p, q ∈ [1, ∞], we denote by Lpt , Lqx = Lp , Lq the space Lp (R, Lq (Rd )). For p, q ∈ [1, ∞], s ∈ R, we denote by Lpt , Wxs,q = Lp , W s,q the space Lp (R, W s,q (Rd )). In the case q = 2 we also write it Lp , H s or Lpt , Hxs . When p = q, we may write Lpt,x for Lpt , Lpx . For p, q ∈ [1, ∞], σ, σ ˜ ∈ R, we denote by Lpt , (Bqσ,˜σ )x = Lp , Bqσ,˜σ the space Lp (R, Bqσ,˜σ (Rd )).

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The proper definition of inhomogeneous Besov spaces is given in Section 2, Definition 2.3. Notation 1.5 (Probability-time-space norms). For p, q ∈ [1, ∞], we denote by L2ω , Lpt , Lqx = L2ω , Lp , Lq the space L2 (Ω, Lp (R, Lq (Rd ))). For p, q ∈ [1, ∞], s ∈ R, we denote by L2ω , Lpt , Wxs,q = L2ω , Lp , W s,q the space L2 (Ω, Lp (R, W s,q (Rd ))). In the case q = 2 we also write it L2ω , Lp , H s or L2ω , Lpt , Hxs . For p, q ∈ [1, ∞], σ, σ ˜ ∈ R, we denote by L2ω , Lpt , (Bqσ,˜σ )x = L2ω , Lp , Bqσ,˜σ the space L2 (Ω, Lp (R, Bqσ,˜σ (Rd ))). Notation 1.6 (Time-space-probability norms). For p, q ∈ [1, ∞], s ∈ R, we denote by Lpt , Wxs,q , L2ω = Lp , W s,q , L2ω the space (1 − x )−s/2 Lp (R, Lq (Rd , L2 (Ω))). In the case q = 2 we also write it Lp , H s , L2ω , and note that Lp , H s , L2ω = Lp L2ω H s . In the case s = 0, we also write it Lp , Lq , L2ω . For p, q ∈ [1, ∞], σ, σ ˜ ∈ R, we denote by Lpt , (Bqσ,˜σ )x , L2ω = Lp , Bqσ,˜σ , L2ω the space induced by the norm 1/2     2jσ  p . gLp ,Bqσ,˜σ ,L2 =  2 gj 2Lqx ,L2 + 22j σ˜ gj 2Lqx ,L2 L (R) ω

ω

ω

j<0

j≥0

2. Toolbox In this section, we present existing results in the literature, either classical or more recent ones such as dispersive estimates and matters related to Littlewood-Paley decomposition. Take η a smooth function with support included in the annulus {ξ ∈ Rd | |ξ| ∈ (1/2, 2)}, and define for  j ∈ Z, ηj (ξ) = η(2−j ξ). We assume that on Rd  {0}, j ηj = 1. For any tempered distribution f ∈ S  (R), we write fj = Δj f where Δj is the Fourier multiplier by ηj i.e. fˆj = ηj fˆ. Notation 2.1. We have f =

 j∈Z

fj and we call this the Littlewood-Paley decomposition of f .

Lemma 2.2. [Bernstein’s lemma] Let a ≥ b ≥ 1, there exists C such that for all j ∈ Z and all f ∈ S  such that fj ∈ Lb (Rd ), we have fj La ≤ C2jd( b − a ) fj Lb . 1

1

We now introduce inhomogeneous Besov spaces where the inhomogeneity comes from a different treatment of high and low frequencies. Definition 2.3. [[21]] Let σ, σ ˜ ∈ R and p ≥ 1, we define Bpσ,˜σ by the space induced by the norm f Bpσ,˜σ =

 j<0

22jσ fj 2Lp +

 j≥0

22j σ˜ fj 2Lp

1/2 .

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σ σ ˜ Remark 2.1. This corresponds to taking the homogeneous Besov norm B˙ p,2 for the low frequencies and B˙ p,2 for the high frequencies.

We state a few properties of these spaces. Proposition 2.4. Let σ1 ≤ σ2 and σ ˜1 ≥ σ ˜2 and p ≥ 1. We have that for all f ∈ Bpσ1 ,˜σ1 , f also belongs to Bpσ2 ,˜σ2 and we have f Bpσ2 ,˜σ2 ≤ f Bpσ1 ,˜σ1 . Theorem 2.5. [Littlewood-Paley theorem] Let s ≥ 0 and p ∈ (1, ∞). If p ≥ 2, there exists C such that for all f ∈ Bp0,s we have f W s,p ≤ Cf Bp0,s . If p ≤ 2 then there exists C such that for all f ∈ W s,p , we have f 0,s Bp ≤ f W s,p . We cite here bilinear estimates from [19]. Proposition 2.6. [Lemma 4.1 [19]] Let (σj )1≤j≤3 ∈ R3 , (˜ σj )1≤j≤3 ∈ R3 , (pj )1≤j≤3 ∈ [2, ∞[3 such that for all j ∈ {1, 2, 3}, max(0, σj , σ1 + σ2 + σ3 ) ≤ d

1  1 1 + + −1 ≤σ ˜1 + σ ˜2 + σ ˜3 , σ ˜j ≤ σ ˜1 + σ ˜2 + σ ˜3 and σj pj < d. p1 p2 p3 σ ,˜ σj

There exists C such that for all (fj )1≤j≤3 such that fj ∈ Bpjj

for all j ∈ {1, 2, 3},

3      fj B σj ,˜σj .  f1 f2 f3  ≤ C j=1

Rd

pj

We now state some dispersive estimates for the linear flow. Define S(t) = e−it(m− ) . Proposition 2.7. Take p, q ∈ [2, ∞] (note that d ≥ 3 so that we are excluding the endpoint) such that 2 d d + = . p q 2 There exists C such that for all u ∈ L2 , S(t)uLpt ,Lqx ≤ CuL2 . Exploiting Bernstein’s lemma and Littlewood-Paley theorem, we get the following. Corollary 2.8. Take p, q ∈ [2, ∞[ and σ, σ1 ≥ 0 such that 2 d d + = − σ1 , p q 2 there exists C, C  such that for all u ∈ B2σ1 ,σ1 +σ , S(t)uLpt ,W σ,q ≤ C  S(t)uLpt ,Bq0,σ ≤ CuB σ1 ,σ1 +σ ≤ CuH σ1 +σ . 2

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3. Set-up In this section, we reduce the problem to finding a correct functional setting for our fixed point problem. To lighten the notation, and f being fixed, we sometimes write Y instead of Yf . Writing X = Y + Z, the perturbation Z satisfies i∂t Z = (m − )Z + w ∗ (2ReE(Y Z) + E(|Z|2 ))(Y + Z). Let an initial perturbation Z0 ∈ L2ω , H s , we have that Z solves the Cauchy problem 

i∂t Z = (m − )Z + w ∗ (2ReE(Y Z) + E(|Z|2 ))(Y + Z) Z|t=0 = Z0

(7)

if and only if the couple perturbation/induced potential (Z, V ) solves the Cauchy problem ⎧ ⎪ ⎨ i∂t Z = (m − )Z + w ∗ V (Y + Z), V = 2Re(E(Y¯ Z)) + E(|Z|2 ), ⎪ ⎩ Z|t=0 = Z0 . The idea is to set up spaces for Z and V , ΘZ and ΘV such that ΘZ × ΘZ is embedded in ΘV in the sense that there exists a constant C such that for all u, v ∈ ΘZ , E(uv)ΘV ≤ CuΘZ vΘZ . Indeed, one key idea behind the proof is that even if V contains a linear term in Z, it behaves like a quadratic term on Z in terms of function spaces, which is better regarding the use of dispersive estimates, and this because of cancellations due to the randomness of the equation. We formulate the problem in Duhamel form: t Z(t) = S(t)Z0 − i

t S(t − s)(w ∗ V (s))Z(s)ds − i

0

S(t − s)(w ∗ V (s))Y (s)ds

(8)

0

and t V (t) = E(|Z| ) + 2ReE(Y¯ (t)S(t)Z0 ) − 2ReE(iY¯ (t)

S(t − s)(w ∗ V (s))Z(s)ds

2

0

t − 2ReE(iY¯ (t)

S(t − s)(w ∗ V (s))Y (s)ds) 0

with the Schrödinger group S(t) = e−it(m− ) . This representation is possible as we look for solutions in the function spaces described in Definition 3.2. The initial value problem for Z is indeed well posed in such spaces from a direct adaptation of Proposition 3.3 in [12]. We set for all V , and all Z, t WV (Z) = −i

S(t − s)(w ∗ V (s))Z(s)ds. 0

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80

In other terms, to solve the Cauchy problem, we have to solve the fixed point

Z V

= AZ 0

Z V



⎞ Z ⎟ V ⎟ =⎜ ⎝ 2 Z ⎠ AZ 0 V ⎛



1 ⎜ AZ 0

(9)

with A1Z0

Z V

= S(t)Z0 + WV (Y ) + WV (Z)

and A2Z0

Z V

= E(|Z|2 ) + 2ReE(Y¯ S(t)Z0 ) + 2ReE(Y¯ WV (Y )) + 2ReE(Y¯ (t)WV (Z)).

The map AZ0 has a constant part (or source term) CZ0 a linear part L and a quadratic part Q given by AZ 0

Z V

= CZ0 + L

Z V

+Q

Z V

(10)

where

S(t)Z0 CZ0 = , 2ReE(Y¯ S(t)Z0 )

WV (Y ) Z , L = V 2ReE(Y¯ WV (Y ))

(11) (12)

and Q

Z V

=

WV (Z) E(|Z|2 ) + 2ReE(Y¯ WV (Z))

.

(13)

The linear term can be written under the form L=

0 L2 0 L1

with L2 (V ) = WV (Y ) and L1 (V ) = 2ReE(Y¯ WV (Y )). Solving the fixed point equation is now a problem of finding the right function spaces and showing suitable continuity estimates. The properties we are going to show are summarized in the following proposition. Proposition 3.1. Let Θ = ΘZ × ΘV and Θ0 be two Banach spaces such that 1. 2. 3. 4. 5.

ΘZ × ΘZ is embedded in ΘV in the sense that for all u, v ∈ ΘZ , E(uv)ΘV  uΘZ vΘZ , there exists C such that for all Z0 ∈ Θ0 , CZ0 Θ ≤ CZ0 Θ0 , 1 − L is continuous, invertible with continuous inverse as a linear operator of Θ, there exists C1 such that for all (Z, V ) ∈ Θ, WV (Z)ΘZ ≤ C1 V ΘV ZΘZ , there exists C2 such that for all (Z, V ) ∈ Θ, 2ReE(Y¯ WV (Z))ΘV ≤ C2 V ΘV ZΘZ ,

then there exists ε > 0 such that for all Z0 ∈ Θ0 with Z0 Θ0 ≤ ε, the Cauchy problem (7) has a unique solution in ΘZ such that 2ReE(Y¯ Z) + E(|Z|2 ) ∈ ΘV and the flow thus defined is continuous in the initial datum.

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81

The proof of this proposition and of the fact that finding such spaces implies the theorem is presented in Section 8. One important feature is that the terms WV (Z), 2Re(E(Y¯ WV (Z))) are all bilinear, E(|Z|2 ) is solved by the embedding of ΘZ × ΘZ in ΘV and thus we can do a contraction argument directly from this setting. We end this section by defining the solution spaces ΘZ and ΘV . Definition 3.2. Let for d ≥ 3, 0, 1

ΘZ = C(R, H s (Rd , L2ω )) ∩ Lp (R, W s,p (Rd , L2ω )) ∩ Ld+2 (R, Ld+2 (Rd , L2ω )) ∩ L4 (R, Bq 4 (Rd , L2ω )) where s = Let

d 2

− 1 is the critical regularity for the cubic Schrödinger equation, p = 2 d+2 d , q = d+2

d+2

−1/2,0

ΘV = Lt 2 , Lx 2 + L2t , B2

4d d+1 .

.

(14)

In the next sections, we check that Θ = ΘZ × ΘV satisfies assumptions 1, 3, 4, in Proposition 3.1 for d ≥ 3 and assumption 5 only for d ≥ 4. 4. Embeddings and Strichartz estimates In this section, we check assumption 1 in Proposition 3.1, and dispersive estimates for the linear flow which induce assumption 4. In the whole section, d ≥ 3. Proposition 4.1. The space ΘZ × ΘZ is embedded in ΘV as in for all u, v ∈ ΘZ , E(uv)ΘV  uΘZ vΘZ . (d+2)/2

(d+2)/2 , Lx . −1/2,0 2 is embedded in L , B2 . The temporal part 0, 14 0, 1 −1/2,0 proving the embedding Bq × Bq 4 in B2 .

Proof. We have that Ld+2 , Ld+2 × Ld+2 , Ld+2 is embedded in Lt t t x x It remains to prove that

0, 1 L4 , Bq 4

0, 1 × L4 , Bq 4

of the norm

works by Hölder inequality. We are left with We use Lemma 4.1 in [19], that we mentioned in the toolbox, Proposition 2.6 with σ1 = σ2 = 0, σ3 = 12 , σ ˜1 = σ ˜2 = 14 , σ ˜3 = 0, and p1 = p2 = q, p3 = 2. We have for all j = 1, 2, 3, pj ≥ 2, σj < pdj , σ ˜j ≤ σ ˜1 + σ ˜2 + σ ˜3 . We also have max(0, σ1 , σ2 , σ3 , σ1 + σ2 + σ3 ) =

1 1 1 1 = d( + + − 1) = σ ˜1 + σ ˜2 + σ ˜3 . 2 p1 p2 p3

Indeed, d(

1 1 1 2 1 d+1 1 1 + + − 1) = d( − ) = d( − )= . p1 p2 p3 q 2 2d 2 2



Proposition 4.2. There exists C such that for all u ∈ L2ω , H s , S(t)uΘZ ≤ CuL2ω ,H s .

(15)

Proof. First of all, let us mention that since p, d + 2 and q are bigger than 2, we have, by Minkowski 0, 1 inequality for all f ∈ L2ω , C(R, H s ) ∩ Lp , W s,p ∩ Ld+2 , Ld+2 ∩ L4 , Bq 4 ), f ΘZ ≤ f 

0, 1 4

L2ω ,C(R,H s )∩L2ω ,Lp ,W s,p ∩L2ω ,Ld+2 ,Ld+2 ∩L2ω ,L4 ,Bq

.

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82

For Lp , W s,p , we have 2 d d+2 d + = = p p p 2 hence Strichartz estimates from Proposition 2.7 apply. For Ld+2 , Ld+2 , we have that 2 d d + =1= −s d+2 d+2 2 hence Strichartz inequalities from Corollary 2.8 apply. 0, 1 For L4 , Bq 4 , we have S(t)u

0, 1 L4t ,Bq 4

1/2      4. = S(t)uj 2Lq + 2j/2 S(t)uj 2Lq L t

j<0

j≥0

Since 4 ≥ 2, by convexity we have, S(t)u

Let q1 =

2d d−1

and s1 =

d−3 4 .

0, 1 L4t ,Bq 4

≤



S(t)uj 2L4 ,Lq +

j<0



2j/2 S(t)uj 2L4 ,Lq

1/2 .

j≥0

Since d ≥ 3 and s1 =

d q1

− dq (we recall q =

4d d+1 ),

we have by Bernstein lemma,

S(t)uj Lq ≤ C2js1 S(t)uj Lq1 and since 1 d d + = 2 q1 2 we get by Strichartz estimates, 2.7, S(t)uj L4t ,Lqx1 ≤ Cuj L2 . We deduce S(t)u We have s1 ≥ 0 and s1 +

1 4

=

d−1 4

≤

d−2 2

0, 1 4

L4t Bq

≤ CuB s1 ,s1 +1/4 . 2

= s, by 2.4, we get

S(t)u

0, 1 4

L4t ,Bq

≤ CuB20,s ≤ CuH s .



Proposition 4.3. Let d ≥ 3. There exists C1 such that for all Z ∈ ΘZ and V ∈ ΘV , WV (Z)ΘZ ≤ C1 ZΘZ V L(d+2)/2 ≤ C1 ZΘZ V ΘV . t,x

Proof. Thanks to the previous proposition, Christ-Kiselev lemma and dual Strichartz estimates, we have WV (Z)ΘZ  (w ∗ V )uLp ,W s,p ,L2ω

C. Collot, A.-S. de Suzzoni / J. Math. Pures Appl. 137 (2020) 70–100

with p the conjugate of p that is p = 2 d+2 d+4 . We have

1 p

=

1 p

+

2 d+2

83

hence, thanks to Hölder’s inequality,

WV (u)ΘZ  w ∗ V W s,(d+2)/2 ZLp ,W s,p ,L2ω . The potential w allows us to lose s derivatives, that is, w ∗ V W s,(d+2)/2 ≤ wW s,1 V L(d+2)/2 , t,x

and we get WV (u)ΘZ  V ΘV ZΘZ .



5. Linear term We study in this section the linearized operator L defined in (12). We prove that 1 − L is continuous, invertible with continuous inverse on ΘZ × ΘV . We recall that 1−L=

1 −L2 0 1 − L1

,

(16)

hence it is sufficient to prove that 1 − L1 is continuous, invertible with continuous inverse from ΘV to ΘV and that L2 is continuous from ΘV to ΘZ . We start with the continuity of L2 . Proposition 5.1. The operator t L2 : V → WV (Y ) = −i

S(t − s) [(w ∗ V (s))Y (s)] ds, 0

is continuous from ΘV to ΘZ . Proposition 5.1 is a corollary of the following estimates. Proposition 5.2. Let σ, σ1 , p1 , q1 be such that σ, σ1 ≥ 0, σ ≤ s , p1 > 2 and 2 d d + = − σ1 . p1 q1 2 Assume moreover, that for all α ∈ N d with |α| ≤ 2 σ , |∂ α h|  ξ−2 . There exists C > 0 such that for all − 12 ,σ1 +σ

U ∈ L2t , B2

,  sup 

A

 S(t − s)Y (s)U (s)dsLp1 ,Bq0,σ ,L2 ≤ CU  t

A∈R

1

ω

− 1 ,σ1 +σ 2

L2t ,B2

0 σ − 12 ,σ1 +σ

and if σ + σ1 ≤ s , there exists C > 0 such that for all U ∈ L2t , B2 1  sup  A∈R

A

 S(t − s)Y (s)U (s)dsLp1 ,W σ,q1 ,L2 ≤ CU  t

0

ω

,

σ1 − 1 ,σ1 +σ 2

L2t ,B2

.

C. Collot, A.-S. de Suzzoni / J. Math. Pures Appl. 137 (2020) 70–100

84

Proof of Proposition 5.1. Take p1 > 2, σ1 ≥ 0 and q1 ≥ 2 such that 2 d d + = − σ1 . p1 q1 2 Assuming Proposition 5.2, we get by Christ-Kiselev lemma, since p1 > 2,  

t

 S(t − s)Y (s)U (s)dsLp1 ,Bq0,σ ,L2 ≤ CU  t

ω

1

− 1 ,σ1 +σ 2

L2t ,B2

.

0

We use it with p1 = 4, q1 = q, σ =  

t

1 4

and σ1 =

d−3 4

≤ s − σ, to get

 S(t − s)Y (s)U (s)dsL4 ,Bq0,1/4 ,L2 ≤ CU  t

ω

− 1 ,s 2

L2t ,B2

.

0

We also have  

t

 S(t − s)Y (s)U (s)dsLp1 ,W q1 ,σ ,L2 ≤ CU  t

ω

σ1 − 1 ,σ1 +σ 2

L2t ,B2

.

0

We apply it to (p1 , q1 , σ1 , σ) equal to either (p, p, 0, s), (d + 2, d + 2, s, 0) or (∞, 2, 0, s) to get WV (Y )ΘZ ≤ Cw ∗ V L2 ,B −1/2,s t

2

and thus, by putting the derivatives on high frequency on w and on low frequency on V , WV (Y )ΘZ ≤ CwW s,1 V L2 ,B −1/2,0 . t

2



It remains to prove Proposition 5.2. To do so, we first establish preliminary lemmas. 2 A Lemma 5.3. Let U ∈ L∞ t Lx and define L3 (U ) =

2 E(|LA 3 (U )| )

=

A 0

S(t − s)Y (s)U (s)ds. We have

 A 2  dη|f (η)|  Sη (t − s)U (s)ds , 2

(17)

0

where Sη (t) = e−it(− −2iη·) . 2 Proof. Let U ∈ L∞ t Lx . We write U (s)Y (s) as a Wiener integral:



2

f (η)eiη.x−is(m+|η| ) U (s)dW (η).

U (s)Y (s) = η∈Rd

(18)

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85

Above, we remark that thanks to Fubini and to the Wiener integration ⎞ ⎛     ⎜ ⎟ iη.x−is(m+|η|2 ) = E ⎝ f (η)e U (s, x)dW (η)⎠ dx   η∈Rd  = |U (s, x)|2 |f (η)|2 dηdx = U (s)2L2x f 2L2 .

U (s)Y (s)2L2ω ,L2x

η∈Rd

Therefore, almost everywhere U (s)Y (s, ω) ∈ L2x , making S(t − s)(U (s)Y (s)) well defined. From the com2 mutator relation S(t)(eiη.x U ) = eiη.x−it(m+|η| ) Sη (t)U we infer that: S(t − s)(U (s)Y (s)) =

f (η)eiη.x−is(m+|η|

2

)−i(t−s)(m+|η|2 )

Sη (t − s)(U (s))dW (η)

η∈Rd



2

f (η)eiη.x−it(m+|η| ) Sη (t − s)(U (s))dW (η).

=

(19)

η∈Rd

From the definition of the Wiener integral, taking the expectation one obtains: ⎛ 2 ⎞  A 

A   ⎟ 2 ⎜ E ⎝ S(t − s)(U (s)Y (s))ds ⎠ = E f (η)eiη.x−it(m+|η| ) Sη (t − s)(U (s))dW (η)ds   d 0

η∈R

0



A

×

  2 f¯(η  )e−iη .x+it(m+|η | ) Sη (t − s )(U (s ))dW (η  )ds



η  ∈Rd 0

A A

=

η∈Rd 0



|f (η)|2 Sη (t − s)(U (s))Sη (t − s )(U (s ))dsds dη

0

 A 2  |f (η)|  Sη (t − s)U (s)ds dη, 2

=

0

η∈Rd

which ends the proof of the identity (17).  We start the proof of Proposition 5.2 with the case σ = 0 for Hölder spaces and σ = σ1 = 0 for Besov spaces. Lemma 5.4. Let σ1 , p1 , q1 be such that σ1 ≥ 0, p1 > 2 and 2 d d + = − σ1 . p1 q1 2 σ − 12 ,σ1 +σ

There exists C > 0 such that for all U ∈ L2t , B2 1

,

p q sup LA 3 (U )Lt 1 ,Lx1 ,L2ω ≤ CU 

A∈R

σ1 − 1 ,σ1 2

L2t ,B2

.

(20)

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86

Proof. We start by taking U in the Schwartz class to allow the following computations and we conclude by density. We have from (17) and Minkowski’s inequality: 2 q1 LA 1 2 3 (U )Lp t ,Lx ,Lω



 =

 A 2   dη|f (η)|  Sη (t − s)U (s)ds Lp1 /2 ,Lq1 /2 x 2

t

0

Rd





2

≤

dη|f (η)|

A

2 Sη (t − s)U (s)dsLp1 ,Lqx1 . t

0

Rd

By Strichartz inequality and Bernstein lemma, Corollary 2.8, we get 2 q1 LA 1 2 3 (U )Lp t ,Lx ,Lω



2

≤

dη|f (η)|

A

2 Sη (−s)U (s)dsB σ1 ,σ1 . 2

0

Rd

ˆ1 (ξ) = |ξ|σ1 U ˆ (ξ), we have We introduce U1 defined by U  

A



2 Sη (−s)U (s)dsB σ1 ,σ1 =

A dξ

dt1

2

0

A

0

Rd

dt2 ei(t1 −t2 )(ξ

2

−2ξ·η)

ˆ1 (ξ, t1 )U ˆ1 (ξ, t2 ). U

0

We do the change of variables t = t2 − t1 , we get  

A

2 Sη (−s)U (s)dsB σ1 ,σ1 =







dξ

dt1 e−it(ξ

dt

2

0

−2ξ·η)

ˆ1 (ξ, t1 )U ˆ1 (ξ, t + t1 ), U

t1 ∈Dt

R

Rd

2

where Dt = [−t, A − t] ∪ [0, A]. We integrate over η reminding that h is the inverse Fourier transform of |f |2 , we get 2 LA 3 (U )Lp1 ,Lqx1 ,L2  t

dξ

ω

t1 ∈Dt

R

Rd

2 ˆ1 (ξ, t1 )U ˆ1 (ξ, t + t1 ). dt1 e−itξ h(−2tξ)U

dt

We use Cauchy-Schwarz inequality over t1 to get A 2 ˆ1 (ξ, t1 )2 2 L3 (U )Lp1 ,Lqx1 ,L2  dξ dt|h(−2tξ)|U Lt t

ω

1

R

Rd

We have |h(−2tξ)| ≤ t|ξ|−2 from the hypotheses of Theorem 1.1. We then do the change of variable τ = t|ξ| and get 2 q1 LA 1 2 3 (U )Lp t ,Lx ,Lω



dξ

ˆ1 (ξ, t1 )2 2 dτ τ −2 |ξ|−1 U Lt

1

Rd

R

and since τ −2 is integrable, we have 2 q1 LA 1 2 3 (U )Lp t ,Lx ,Lω



σ −1/2,σ1 −1/2

Lt ,B2 1

Rd

which is the desired result. 

ˆ1 (ξ, ·)2 2 dξ = U 2 |ξ|−1/2 U 2 Lt

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87

Lemma 5.5. Let p1 , q1 be such that p1 > 2 and d d 2 + = . p1 q1 2 − 12 ,0

There exists C > 0 such that for all U ∈ L2t , B2

,

0,0 sup LA 1 2 ≤ CU  3 (U )Lp t ,Bq ,L ω

1

A∈R

− 1 ,0 2

L2t ,B2

.

(21)

Proof. By Minkowski’s inequality, we have A 0,0 LA 1 2 ≤ L3 (U )L2 Lp1 ,B 0,0 . 3 (U )Lp q t ,Bq ,L t ω

1

ω

1

By Strichartz inequality we get  

0,0 LA 1 2 3 (U )Lp t ,Bq1 ,Lω

A S(−s)Y (s)U (s)dsL2ω ,L2x . 0

We use the formula (17) for LA 3 (U ) when t = 0 and obtain 2 LA 3 (U )Lp1 ,Bq0,0 ,L2

ω

1





2

|f (η)|

A Sη (−s)U (s)dsL2ω ,L2 . 0

Following exactly the same steps as in the proof of Lemma 5.4, we obtain: 2 LA 3 (U )Lp1 ,Bq0,0 ,L2  U  ω

1

 U 

− 1 ,− 1 2 2

B2

− 1 ,0 2

B2

,

which ends the proof of the lemma.  We can now end the proof of Proposition 5.2 thanks to the three lemmas above. Proof of Proposition 5.2. For vectors α = (α1 , . . . , αd ) ∈ N d and η = (η1 , . . . , ηd ) ∈ Rd we write |α| = d d αj α j=1 αj , ∂ = j=1 ∂j and Cαβ =

d 

Cαβjj and η α =

j=1



α

ηj j

j

β

where Cαjj is a binomial coefficient. We have for all α ∈ N d ,

∂ α LA 3 (U ) =

 β+γ=α

A S(t − s)∂ β Y (s)∂ γ U (s)ds.

Cαβ 0

Indeed, Y is almost everywhere differentiable and there holds, for |β| ≤ s : ∂ β Y (s) = Rd

i|β| η β f (η)eiη.x−is(m+|η| ) dW (η), 2

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88

meaning that replacing Y by ∂ β Y consists in replacing f (η) by i|β| η β f (η). Therefore, we have for σ1 ≥ 0, p1 , q1 > 2 such that p21 + qd1 = d2 − σ1 , and σ ∈ N ∩ [0, s ], for any A ∈ R: p LA 3 (U )Lt 1 ,W σ,q1 ,L2ω ≤ CU L2 ,B σ1 −1/2,σ1 +σ , t

2

thanks to Lemma 5.4. Above, the constant C depends on supα∈N d , |α|≤2σ ξ2 ∂ α hL∞ , because the Fourier transform of |η α |2 |f |2 is ∂ 2α h up to multiplication by a constant. By interpolation, p LA 3 (U )Lt 1 ,W σ,q1 ,L2ω ≤ CU L2 ,B σ1 −1/2,σ1 +σ t

for all σ ≥ 0. And for σ + σ1 ∈ N and

2 p1

+

d q1

=

d 2

2

− σ1 , we have by Bernstein lemma

A LA 3 (U )Lp1 ,Bq0,σ ,L2  L3 (U )Lp1 ,Bq0,σ+σ1 ,L2 ω

1

with

2 p1

+

d q2

2

ω

= d2 . Therefore, applying Lemma 21 with σ + σ1 ≤ s : 0,σ LA −1/2,σ1 +σ . 1 2 ≤ CU  2 3 (U )Lp L ,B t ,Bq ,L 1

ω

t

2

We get the result for σ ≥ 0, σ1 ≥ 0, σ + σ1 ≤ s by interpolation.  Having established the desired estimates for L2 , we now deal with the other part of the linear term, which involves L1 . We start by computing an explicit formula for this operator, and then show some continuity properties. Lemma 5.6. The operator L1 is a Fourier multiplier of symbol wm ˆ f (in both space and time) given by for d all t ∈ R and all ξ ∈ R , t Fx (L1 (V )) (t, ξ) = −2w(ξ) ˆ

sin(|ξ|2 (t − s))h(2ξ(t − s))Fx V (s, ξ)ds, 0

where h is the inverse Fourier transform of |f |2 , or, put another way: Ft,x (L1 (V )) (τ, ξ) = w(ξ)m ˆ f (τ, ξ)Ft,x V (τ, ξ), where 

mf (τ, ξ) = −2Ft sin(|ξ| t)h(2ξt)1t≥0 2



+∞ (τ ) = −2 e−iτ t sin(|ξ|2 t)h(2ξt)dt.

(22)

0 2 Proof of Lemma 5.6. Let V ∈ L∞ t Lx so that one can perform the computations below. Using the formula (19) we write WV (Y ) as a Wiener integral:

t WV (Y ) = −i

2

f (η)eiη.x−it(m+|η| ) Sη (t − s)(w ∗ V (s))dsdW (η).

Rd 0

By the property of Wiener integration:

C. Collot, A.-S. de Suzzoni / J. Math. Pures Appl. 137 (2020) 70–100

  E WV (Y )Y¯ = −iE

t

89

2

f (η)eiη.x−it(m+|η| ) Sη (t − s)(w ∗ V (s))dsdW (η)

Rd 0



   2 ¯ (η  ) f¯(η  )e−iη .x+it(m+|η | ) dW

× Rd

t = −i

|f (η)|2 Sη (t − s)(w ∗ V (s))dsdη.

Rd 0

 Set T (t, ξ) the Fourier transform in space of E WV (Y )Y . From (18) and Fubini we infer: t T (t, ξ) = −i

|f (η)|2 e−i(t−s)(|ξ|

2

+2ξ.η)

w(ξ) ˆ Vˆ (s, ξ)dsdη

Rd 0

t = −i

h(2ξ(t − s))e−i(t−s)|ξ| w(ξ) ˆ Vˆ (s, ξ)dsdη. 2

0

The space Fourier transform of 2ReE(WV1 (Y )Y ) is T (t, ξ) + T (t, −ξ). Given that w ∗ V is real and |f |2 is even and real, and thus h is real and even, we have t sin(ξ 2 (t − s))h(2ξ(t − s))w(ξ) ˆ Vˆ (s, ξ)ds

T (t, ξ) + T (t, −ξ) = −2 0

which gives the result.  Since L1 is a Fourier multiplier with symbol wm ˆ f given by (22), the continuity and the invertibility of 1 − L1 on L2 based space-time function spaces is equivalent to the boundedness and the non-vanishing of 1 − wm ˆ f . This issue was studied in [22] where the author showed the following: Proposition 5.7 (Lewin Sabin Corollary 1 [22]). Let f be a radial map in L2 ∩L∞ (Rd ). Consider the Fourier multiplier mf defined by (22). Assume:  • Rd |ξ|1−d |f f | < ∞, • writing r = |ξ| the radial variable, ∂r |f |2 < 0 for r > 0, • |ξ|2−d h ∈ L1 (Rd ). If w ∈ L1 (Rd ) is an even function such that ⎛ (w) ˆ −  L∞ ⎝



Rd

⎞ |h| dx⎠ < 2|Sd−1 |, |x|d−2

and such that g w(0) ˆ + < 1, where g := lim inf Remf (τ, ξ), (τ,ξ)→(0,0)

then there holds

90

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min

(τ,ξ)∈R×Rd

|w(ξ)||m ˆ f (τ, ξ) − 1| > 0 − 12 ,0

and the operator 1 − L1 is invertible on L2t,x and L2t B2

(23)

.

√ Remark 5.1. The f in the paper by Lewin and Sabin corresponds to g(r) = |f ( re)|2 for any unitary vector of Rd , e. The assumptions they make on g are implied by the ones we make on f . Proof. For the proof, we refer to [22]. There the authors show that under the assumptions of the proposition, d −1 (τ, ξ) → w(ξ)m ˆ ˆ f (τ, ξ) is uniformly bounded on R × R , and that (23) holds. Therefore, (1 − w(ξ)m f (τ, ξ)) − 12 ,0

is a bounded function. This implies the continuity of 1 − L1 both on L2t,x and L2t B2

.



The continuity of 1 − L1 on ΘV is a consequence of the above continuity on L2t,x and of other mild assumptions on f and w. Proposition 5.8 (Lewin-Sabin Proposition 3 [22]). Assume that the hypothesis of Proposition 5.7 hold and  that moreover (|h| + |∇h|)dr < +∞ (that is h and h in |ξ|d−1 L1 (Rd )) and (1 + |ξ|)w ˆ ∈ L2(d+2)/(d−2) , d+2

then 1 − L1 and (1 − L1 )−1 are continuous from ΘV into Lt,x2 . Proof. A first direction to prove Proposition 5.8 is to show that the Fourier multiplier w(ξ)mf (τ, ξ) and its inverse are continuous operators on Lebesgue spaces using standard harmonic analysis tools. This is done in [22] where the authors show that this multiplier satisfies suitable conditions ensuring the application of Stein’s and Marcinkiewicz’s theorems. One can check using the computations there that its inverse also satisfies the same suitable conditions. We give here another proof avoiding the use of advanced harmonic analysis. We claim that d+2

2 1+d w(ξ)m ˆ ). f (τ, ξ) ∈ L d−2 (R

(24)

Assuming the above bound, then one computes the following, using the continuity of the Fourier transform  from Lp into Lp for 1 ≤ p < +∞ and Hölder inequality. We have by definition L1 V 

d+2

Lt,x2

 w(ξ)m ˆ f (τ, ξ)Ft,x V 

L

d+2 d

(R1+d )

.

We use Hölder inequality to get L1 V 

d+2

Lt,x2

 w(ξ)m ˆ f (τ, ξ)

L

2 d+2 d−2

(R1+d )

Because of the bound (24), and that Ft,x V L2 (R1+d ) ≤ V  L1 V 

 V 

d+2

Lt,x2

Ft,x V L2 (R1+d ) 1

L2t ,B − 2 ,0

, we have

1

L2t ,B − 2 ,0 d+2

which is in turn less than V ΘV . This shows the continuity of L1 from ΘV into Lt,x2 . Similarly, using −1 the fact that (1 − w(ξ)m ˆ is uniformly bounded from Proposition 5.7, we get through algebraic f (τ, ξ)) considerations (1 − L1 )−1 V 

d+2

Lt,x2

 V 

d+2

Lt,x2 

+ (1 − (1 − L1 )−1 )V 

d+2

Lt,x2

By definition, and using that for p ≥ 2, and χ ∈ Lp , χ ˆ Lp ≤ χLp , we have

.

C. Collot, A.-S. de Suzzoni / J. Math. Pures Appl. 137 (2020) 70–100

(1 − (1 − L1 )−1 )V 

d+2 Lt,x2

≤

91

w(ξ)m ˆ f (τ, ξ) Ft,x V  d+2 1+d . L d (R ) 1 − w(ξ)m ˆ f (τ, ξ)

Using that 1 − w(ξ)m ˆ f (τ, ξ) is bounded from below we get (1 − (1 − L1 )−1 )V 

d+2

Lt,x2

 w(ξ)m ˆ f (τ, ξ)

L

2 d+2 d−2

(R1+d )

Ft,x V L2 (R1+d ) d+2

and we conclude as previously. This shows the continuity of (1 − L1 )−1 from ΘV into Lt,x2 . Hence it remains to prove (24). We compute for |ξ| > 1 and τ = 0, doing the change of variables t ← t|ξ|: +∞ +∞

τ 2 ξ −i |ξ| t −iτ t 2 mf (τ, ξ) = −2 e sin(|ξ| t)h(2ξt)dt = − e sin(|ξ|t)h 2 t dt. |ξ| |ξ| 0

We write

τ t 1 −i |ξ| |ξ| e

0

= iτ −1 ∂t e−i |ξ| t to get τ

2i mf (τ, ξ) = − τ

+∞



 τ ξ −i |ξ| t ∂t e sin(|ξ|t)h 2 t dt |ξ| 0

and we integrate by parts to get ⎛

⎞ +∞ +∞



τ τ 2i ⎝ ξ ξ ξ mf (τ, ξ) = |ξ| e−i |ξ| t cos(|ξ|t)h 2 t dt + 2 e−i |ξ| t sin(|ξ|t) .∇h 2 t dt⎠ τ |ξ| |ξ| |ξ| 0

0

Finally, we use the fact that h is radially symmetric and the integrability assumptions to get mf (τ, ξ) 

1 + |ξ| |τ |

Since mf is uniformly bounded from Proposition 5.7, one deduces that |mf (τ, ξ)|  (1 + |ξ|)(1 + |τ |)−1 . Therefore one concludes that: d+2 d+2 d+2 1 2 d−2 |w(ξ)m ˆ (τ, ξ)| dτ dξ  (1 + |ξ|)2 d−2 |w| ˆ 2 d−2 (ξ) f d+2 dξdτ 2 d−2 (1 + |τ |) 1+d 1+d R

and since

R

1 (1+|τ |)

2 d+2 d−2

is integrable, we have

d+2

2 |w(ξ)m ˆ ˆ f (τ, ξ)| d−2  (1 + |ξ|)w

L

2 d+2 d−2

(Rd )

< +∞.



R1+d

6. The remaining quadratic term We have already dealt with the quadratic terms E(|Z|2 ) and WV (Z) in Propositions 4.1 and 4.3 respectively, it remains to prove assumption 5 in Proposition 3.1. In what follows, we assume d ≥ 4, it is the only part of the paper that does not work in dimension 3. Proposition 6.1. There exists C > 0 such that for all (Z, V ) ∈ Θ, 2ReE(Y¯ WV (Z))ΘV ≤ CV ΘV ZΘZ .

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92

−1/2,0

Proof. The proof relies on a duality argument. We first prove the L2 B2 We write U = U1 + U2 with U1 =



Uj and U2 =

j<0



1/2,0

estimate. Let U ∈ L2 , B2

.

Uj

j≥0

where we use the Littlewood-Paley decomposition of U . We have, integrating by parts:

∞  ¯ U1 , E(Y WV (Z)) = E  S(s − t)U1 (t)Y (t)dt, (w ∗ V )Z . s

This yields  U1 , E(Y¯ WV (Z)) ≤ 

∞

 Sη (s − t)U1 (t)Y (t)dtLq1 ,L2 (w ∗ V )Z t,x

ω

q

1 ,L2 Lt,x ω

,

s

with q1 = d + 2. We apply Lemma 5.4 with σ1 = s =  

∞

d 2

− 1:

 S(s − t)U1 (t)Y (t)dtLq1 ,L2 ≤ U1  t,x

ω

σ1 − 1 ,σ1 2

L2t ,B2

s

and since U1 contains only the low frequencies and s1 −  

∞

1 2

≥ 12 , we get

 S(s − t)U1 (t)Y (t)dtLq1 ,L2 ≤ U  t,x

ω

1 ,0

L2t ,B22

.

s

By Hölder inequality, since q1 = d + 2, we get (w ∗ V )Z

q

1 ,L2 Lt,x ω

≤ w ∗ V L2t ,L2x ZLpt ,Lpx ,L2ω .

The estimates above imply for the low frequency part:   U1 , E(Y¯ WV (Z))  U 

1/2,0

B2

V ΘV ZΘZ .

(25)

We now deal with high frequencies. We have

∞    U2 , E(Y¯ WV (Z)) = E  S(s − t)U2 (t)Y (t)dt, (w ∗ V )Z . s

We get by Hölder inequality    U2 , E(Y¯ WV (Z)) ≤ 

∞

 S(s − t)U2 (t)Y (t)dtLp ,Lpx ,L2 V L(d+2)/2 ,L(d+2)/2 ZLpt ,Lpx ,L2ω . x t

ω

t

s

From the above estimate we get the control for the high frequency part, by applying Lemma 5.4 with σ1 = 0,   U2 , E(Y¯ WV (Z))  U2 

−1/2,0

B2

V ΘV ZLpt ,Lpx ,L2ω  U B 1/2,0 V ΘV ZLpt ,Lpx ,L2ω 2

(26)

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93

where we used the fact that U2 contains only high frequencies. Combining (25) and (26) gives E(Y¯ WV (Z))L2 ,B −1/2,0 ≤ C(f, w)V ΘV ZΘZ . t

(27)

2

(d+2)/2 (d+2)/2 It remains to prove that E(Y¯ WV (Z)) belongs to Lt , Lx . Note that E(Y¯ WV (Z)) belongs to p2 q2 Lt , Lx if p2 > 2, q2 ≥ 2 and

d 2 d d d − s2 := + ∈ [ − s, ]. 2 p2 q2 2 2 2 This is due to the fact that Y ∈ L∞ x , Lω and by Strichartz inequalities

WV (Z)Lpt 2 ,Lqx2 ,L2ω  V L(d+2)/2 ZLpt ,W s2 ,p ,L2ω  V ΘV ZΘZ t,x

and that s2 ∈ [0, s]. We recall that by definition

d 2

− s = 1. We have

4 2d d + = 2 ∈ [1, ], d+2 d+2 2 (d+2)/2

thus E(Y¯ WV (Z)) belongs to Lt

(d+2)/2

, Lx

with:

E(Y¯ WV (Z))

d+2

Lt,x2

 V ΘV ZΘZ .

(28)

Gathering (27) and (28) one obtains the desired continuity estimate E(Y¯ WV (Z))ΘV ≤ C(f, w)V ΘV ZΘZ .



7. A space for the initial datum One has the following compatibility result between the space for the perturbation at initial time, and the leading order term for the solution and the potential as given in (11). Here, we prove assumption 2 in Proposition 3.1 for dimension higher than 4 but the proof can be adapted to dimension 3. 2d/(d+2)

Lemma 7.1. There exists a universal constant C > 0 such that for all Z0 ∈ L2ω , H d/2−1 ∩ Lx has CZ0 ∈ ΘZ × ΘV with

, L2ω , one

CZ0 ΘZ ×ΘV ≤ C Z0 

d

L2ω ,H 2 −1

+ Z0 

2d

Lxd+2 ,L2ω

.

(29)

Proof. We have that S(t)Z0 belongs to ΘZ because of Strichartz estimates and that Z0 ∈ L2ω , H s . Recall the definition of ΘV , (14). The control of the space-time Lebesgue norm of 2ReE(Y¯ S(t)Z0 ) uses standard Strichartz estimates, while the control on its Besov-type norm involves some extra dispersion in the interaction with Y . 2 We start with the space-time Lebesgue norm. Since Y belongs to L∞ x , Lω we get by Cauchy-Schwarz: E(Y¯ S(t)Z0 ) We recall that

d+2

Lt,x2

 S(t)Z0 

d+2

Lt,x2 ,L2ω

.

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94

2d d 4 + = 2 ∈ [1, ], d+2 d+2 2 hence by Strichartz estimates, we obtain the first continuity estimate: E(Y¯ S(t)Z0 )

 Z0 L2ω ,Hxs .

d+2

Lt,x2

(30)

For the Besov norm, by duality one has:

E(Y¯ S(t)Z0 )

− 1 ,0 2

L2t ,B2

      ¯ U )Z0  .  = sup S(−t)( Y   1= U 2 1/2,0   L ,B t

t,x,ω

2

We write U = U1 + U2 where U1 = P|ξ|≤1 U . For the low frequency part we apply Lemma 5.4 with p1 = +∞, q1 = 2d/(d − 2) and σ1 = 1: S(−t)(Y¯ U1 )

 t

2d

Lxd−2 ,L2ω

 U1 

1 ,1

L2t ,B22

 U 

1 ,0

L2t ,B22

.

For the high frequency part, we apply Lemma 5.4 with p1 = +∞, q1 = 2 and σ1 = 0: S(−t)(Y¯ U2 )L2x ,L2ω  U2 



− 1 ,0 2

L2t ,B2

 U 

1 ,0

L2t ,B22

.

t

Therefore, one has that of Banach spaces) with:

 t

2d/(d−2)

S(−t)(Y¯ U ) ∈ Lx

, L2ω + L2x , L2ω (endowed with the canonical norm for sums

S(−t)(Y¯ U )L2d/(d−2)  U  ,L2 +L2 ,L2 x



ω

x

ω

1 ,0

L2t ,B22

.

t

Hence, by duality:    

  ¯   S(−t)(Y U )Z0   U  2 12 ,0 Z0  d+2 + Z0 L2x ,L2ω 2d  Lt ,B2 Lx ,L2ω   t,x,ω

Therefore, by duality one obtains the second estimate: E(Y¯ S(t)Z0 )

− 1 ,0 2

L2t ,B2

 Z0 

2d

Lxd+2 ,L2ω

+ Z0 L2x ,L2ω .

The bounds (30) and (31) then imply the continuity estimate: E(Y¯ S(t)Z0 )ΘV  Z0 

d

H 2 −1 ,L2ω

+ Z0 

2d

Lxd+2 ,L2ω

.

The identity (11), the above bound and (15) yield the desired result (29). 

(31)

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8. Proof of Theorem 1.1 We recall that the proof of Theorem 1.1 relies on finding a solution to the fixed point equation (9) for the perturbation Z and the induced potential V . According to (10), the fixed point equation can be written in the form:

Z Z −1 = (Id − L) CZ0 + Q . V V This is now a standard routine to solve the above equation thanks to the various estimates derived previously. To solve the above equation, one defines Φ[Z0 ](Z, V ) as the mapping Φ[Z0 ] : ΘZ ×

ΘV Z V

→ ΘZ × ΘV

Z −1 → (Id − L) CZ0 + Q . V

2d/(d+2)

Let us denote by Θ0 = L2ω H d/2−1 ∩ Lx L2ω the space for the initial datum with associated norm  · Θ0 . We claim that for Z0 small enough, the mapping Φ[Z0 ] is a contraction on B(0, CZ0 Θ0 ). Indeed, the identity (16) and the continuity results of Propositions 5.1, 5.8 and 5.7 give that: −1

(Id − L)

=

1 L2 (1 − L1 )−1 0 (1 − L1 )−1

∈ L(ΘZ × ΘV ).

Hence, for the leading order part, from this and the bound (29) for CZ0 one obtains: −1

 (Id − L)

(CZ0 ) ΘZ ×ΘV  Z0 Θ0 .

For the quadratic part, recall (13). In particular, one has Q(Z, V ) =

0 E(|Z|2 )

+ Q1 (Z, V )

where Q1 is bilinear. From Propositions 4.1, 4.3 and 6.1, if (Z, V ) ∈ B(0, CZ0 Θ0 ) then: Q(Z, V )ΘZ ×ΘV ≤ C(Z, V )2ΘZ ×ΘV ≤ CZ0 2Θ0 and, due to bilinearity:

 Q(Z, V ) − Q(Z  , V  )ΘZ ×ΘV ≤ C (Z, V )ΘZ ×ΘV + (Z  , V  )ΘZ ×ΘV (Z − Z  , V − V  )ΘZ ×ΘV ≤ CZ0 Θ0 (Z − Z  , V − V  )ΘZ ×ΘV . From the above estimates, one gets that Φ[Z0 ] is indeed a contraction on B(0, CZ0 Θ0 ) for some universal C if Z0 Θ0 is small enough. Applying Banach’s fixed point theorem yields the existence and uniqueness of a solution to (9) in B(0, CZ0 Θ0 ). To prove the scattering result, one rewrites (8) as: ⎛

⎞ +∞ +∞ Z(t) = S(t) ⎝Z0 − i S(−s)(w ∗ V (s))Z(s)ds − i S(−s)(w ∗ V (s))Y (s)ds⎠ 0

+i

0

+∞ +∞ S(t − s)(w ∗ V (s))Z(s)ds + i S(−s)(w ∗ V (s))Y (s)ds. t

t

C. Collot, A.-S. de Suzzoni / J. Math. Pures Appl. 137 (2020) 70–100

96

Applying Proposition 4.3, one obtains that

 +∞ 0

S(−s)(w ∗ V (s))Z(s)ds ∈ L2ω H d/2−1 and that:

+∞  S(t − s)(w ∗ V (s))Z(s)dsL2ω H d/2−1  Z(s)1s≥t ΘZ V (s)1s≥t ΘV → 0 t

as t → +∞. Similarly, from Proposition 5.1 one has that



 +∞ 0

S(−s)(w ∗ V (s))Y (s)ds ∈ H d/2−1 and that

+∞ S(−s)(w ∗ V (s))Y (s)dsL2ω H d/2−1  ΘZ V (s)1s≥t ΘV → 0 t

as t → +∞. Therefore, there exists indeed Z∞ ∈ H d/2−1 L2ω such that, as t → +∞: Z(t) = S(t)Z+∞ + oH d/2−1 L2ω (1). This ends the proof of Theorem 1.1. 9. Example of instability for a rough momentum distribution function We study here the linearization of the dynamics near the superposition of two waves which are orthogonal in probability and propagate in opposite directions. This corresponds to an equilibrium of the form (4) with a rough momentum distribution function f . Even in the defocussing case w ˆ ≥ 0, the form of f is involved to ensure linear stability. Indeed, we will prove linear instability for the present example.  Consider a potential w satisfying, without loss of generality if the equation is defocussing, Rd w = 1, a mass m ≥ 0 and a frequency ξ ∈ Rd . Let two functions in probability gi : Ω → C for i = 1, 2, with   |g |2 (ω)dω = 1/2 and Ω g¯1 (ω)g2 (ω)dω = 0. The following function is a solution of (1): Ω i Y [m, k](ω, t, x) :=

√

me−i(|ξ|

2

+m)t



 g1 (ω)eiξ.x + g2 (ω)e−iξ.x ,

which is not stationary, but is at equilibrium. We study a perturbation under the form X = Y + Z and decompose: Z(ω, t, x) := g1 (ω)ei(ξ.x−(|ξ|

2

+m)t)

ε1 (x, t) + g2 (ω)ei(−ξ.x−(|ξ

2

|+m)t)

ε2 (x, t) + ε3 (ω, t, x),

 where for almost all t, x ∈ I × Rd , one has Ω ε3 (ω, t, x)gi (ω)dω = 0 for i = 1, 2. At the linear level, ε1 and ε2 do not interact with ε3 and their evolution equation is:

∂t ε1 ∂t ε2

=

iΔε1 − 2ξ.∇ε1 − im (w ∗ Re(ε1 ) + w ∗ Re(ε2 )) iΔε2 + 2ξ.∇ε2 − im (w ∗ Re(ε1 ) + w ∗ Re(ε2 ))

+

−ie−i(ξ.x−(|ξ| +m)t) E (¯ g1 N L) 2 g2 N L) −ie(ξ.x+(|ξ| +m)t) E (¯ 2

,

where the nonlinear term is N L := 2(w∗Re(E(Y¯ Z)))Z +(w∗E(|Z|2 ))(Y +Z). We now focus on the linearized operator for (ε1 , ε2 ). We decompose between real and imaginary parts, writing u1 = Reε1 , u2 = Im ε1 , u3 = Reε2 and u4 = Im ε2 . One has the identity

iΔε1 − 2ξ.∇ε1 − im (w ∗ Re(ε1 ) + w ∗ Re(ε2 )) iΔε2 + 2ξ.∇ε2 − im (w ∗ Re(ε1 ) + w ∗ Re(ε2 ))

−Δu2 − 2ξ.∇u1 + i(Δu1 − 2ξ.∇u2 − mw ∗ u1 − mw ∗ u3 ) = . −Δu4 + 2ξ.∇u3 + i(Δu3 + 2ξ.∇u4 − mw ∗ u1 − mw ∗ u3 )

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Consequently the linear coupled dynamics for ε1 and ε2 can be written as the following system: ⎛

⎞ ⎛ ⎞ ⎛ ⎞ u1 u1 −2ξ.∇ −Δ 0 0 −mw∗ 0 ⎟ ⎜u ⎟ ⎜u ⎟ ⎜ Δ − mw∗ −2ξ.∇ ∂t ⎝ 2 ⎠ = A ⎝ 2 ⎠ , A := ⎝ . u3 u3 0 0 2ξ.∇ −Δ ⎠ u4 u4 −mw∗ 0 Δ − mw∗ 2ξ.∇

(32)

Note that A is a matrix of Fourier multipliers. We now study its spectrum. In the particular case m = 0, we retrieve for A the block diagonal form ⎛

−2ξ.∇ −Δ 0 −2ξ.∇ 0 ⎜ Δ A := ⎝ 0 0 2ξ.∇ 0 0 Δ

⎞ 0 0 ⎟ . −Δ ⎠ 2ξ.∇

Each of the two matrix operators only has imaginary spectrum, and corresponds to a linear Schrödinger equation in a moving frame. In the particular case ξ = 0, A and its symbol are given by: ⎛

0 −Δ 0 ⎜ Δ − mw∗ A := ⎝ 0 0 −mw∗ 0

⎞ ⎛ 0 0 0 −mw∗ 0 ⎟ ˆ ⎜ −|k|2 − mw(k) , mA (k) = ⎝ 0 −Δ ⎠ 0 Δ − mw∗ 0 −mw(k) ˆ

|k|2 0 0 0

0 −mw(k) ˆ 0 −|k|2 − mw(k) ˆ

⎞ 0 0 ⎟ . |k|2 ⎠ 0

 The eigenvalues of mA are by a direct check λ±,± = ± −|k|4 − (1 ± 1)m|k|2 w(k) ˆ ∈ iR in the defocussing case w ˆ ≥ 0. This linear operator is similar to the one arising in the linearization of the Gross-Pitaevskii equation near the trivial state, and has thus in the defocussing case only imaginary spectrum. The situation is more involved when ξ = 0 and m = 0. In the case of a Dirac potential one has the following instability result, whose proof can be extended to include other additional potentials. Lemma 9.1. Let w ˆ = 1, m > 0 and ξ = 0. Then the Fourier multiplier of the differential  matrix A given by (32) possesses positive and negative eigenvalues in the vicinity of the frequency k = ξ 4 − min(2, m/|ξ|2 ). Proof. We compute the characteristic polynomial of A, using the notations a = 2ξ.∇, b = −Δ and c = mw∗ to ease computations: ⎛⎛

⎞⎞ X + 2ξ.∇ Δ 0 0 mw∗ 0 ⎜⎜ −Δ + mw∗ X + 2ξ.∇ ⎟⎟ PA (X ) := det(X Id − A) = det ⎝⎝ ⎠⎠ 0 0 X − 2ξ.∇ Δ mw∗ 0 −Δ + mw∗ X − 2ξ.∇ = X 4 + 2((b + c)b − a2 )X 2 + ((b + c)b + a2 )2 − b2 c2 . We set Y = X 2 and compute the discriminant:     4D2 = (2(b + c)b − a2 )2 − 4 ((b + c)b + a2 )2 − b2 c2 = 4b bc2 − 4(b + c)a2 . Therefore the roots in Y of the above polynomial are: Y± := a2 − (b + c)b ± D = 4(ξ.∇)2 + (−Δ + mw∗)Δ ±

 −Δ (−Δ(mw∗)2 − 16(−Δ + mw∗)(ξ.∇)2 )

and the roots in X are:  X±,± = ± Y± .

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98

If Y+ or Y− has some positive spectrum, then at least one of the X±,± has some positive and negative spectrum which signals a linear instability. For r > 0, the symbol associated to Y+ evaluated at rξ is:  −4r2 |ξ|4 − r4 |ξ|4 − m|ξ|2 r2 + r2 |ξ|2 (r2 |ξ|2 m2 + 16(r2 |ξ|2 + m)|ξ|4 r2 )

  = |ξ|2 r2 −4|ξ|2 − r2 |ξ|2 − m + m2 + 16r2 |ξ|4 + 16m|ξ|2 . The above quantity is positive when the following polynomial

    m 4|ξ|2 + r2 |ξ|2 + m − m2 + 16r2 |ξ|4 + 16m|ξ|2 = |ξ|4 (r2 − 4) r2 − 4 + 2 2 |ξ| is negative. Therefore, at the frequency k = ξ and a negative eigenvalue. 



4 − min(2, m/|ξ|2 ), the Fourier multiplier of A has a positive

Acknowledgements C. Collot is supported by the ERC-2014-CoG 646650 SingWave. A-S de Suzzoni is supported by ESSED ANR-18-CE40-0028. Part of this work was done when C. Collot was visiting IHÉS and he thanks the institute. Appendix A. Proof of Corollary 1.2 We give here a proof of Corollary 1.2, which provides a scattering result for the operator associated to the solution X in the density matrices framework. Proof of Corollary 1.2. We first prove the continuity of the mapping from L2ω H d/2−1 into Hd/2−1 : Z → E(|ZZ|) + E(|ZYf (t = 0)|) + E(|Yf (t = 0)Z|)

(A.1)

which to Z : Rd × Ω associates a density matrix (operator). The integral kernel of the first term is of the form E(Z1 (x)Z¯2 (y)) so that from Cauchy-Schwarz and Fubini: E(|Z1 Z2 |)2 d −1 H2

= ∇

d 2 −1

E(|Z1 Z2 |)∇

d 2 −1

2 d −1 H2

|E(∇ 2 −1 Z1 (x)∇ 2 −1 Z¯2 (y))|2 dxdy d

=

d

Rd ×Rd



|E∇ 2 −1 Z1 (x)|2 |E∇ 2 −1 Z¯2 (y)|2 dxdy ≤ Z1  d

≤

d

d

L2ω H 2 −1

Z2 

d

L2ω H 2 −1

.

Rd ×Rd

For the second term, as a preliminary step we get from assumption (iii) of Theorem 1.1: 2

f (R)R

d−2

≤ C(d)f (R) 2

|ξ|≤R

C(d) ξd−2 dξ = Rd Rd

2

d−2

f (R)ξ |ξ|≤R

C(d) dξ ≤ Rd

f 2 (ξ)ξd−2 dξ |ξ|≤R

for any R ≥ 1. As from assumption (i), f 2 (ξ)ξd−2 ∈ L1 , and as f is bounded near the origin, one obtains from the above inequality that ξd/2−1 f L∞ ≤ C(f ) is finite. This implies that given u ∈ L2 (Rd ), the expression ∇d/2−1 Y¯f (t = 0), uL2 is a Gaussian random variable with:

C. Collot, A.-S. de Suzzoni / J. Math. Pures Appl. 137 (2020) 70–100

99

⎛ ⎛ 2 ⎞ 2 ⎞      ⎟  ⎟ d d ⎜ ⎜ u(ξ)dW (ξ) ⎠ E ⎝ ∇ 2 −1 Y¯f (t = 0, y)u(y)dy  ⎠ = E ⎝ ξ 2 −1 f¯(ξ)ˆ   d   d R

R



ξd−2 |f (ξ)|2 |ˆ u(ξ)|2 dξ ≤ C(f )u2L2 (R2 ) .

= Rd

With the above inequality we estimate the second term in (A.1) by duality, using Cauchy-Schwarz: E(|ZYf (t = 0)|)2 d −1 H2 d d = |E(∇ 2 −1 Z(x)∇ 2 −1 Y¯f (t = 0, y))|2 dxdy Rd ×Rd

      d d   = sup E(∇ 2 −1 Z(x)∇ 2 −1 Y¯f (t = 0, y))u(x, y)dxdy    

u L2 (R2d ) =1 Rd ×Rd  ⎛ ⎛

≤

sup

(E(|∇

d 2 −1

u L2 (R2d ) =1 Rd

Z(x)|2 )) ⎝E ⎝|

sup

⎛ d 1 (E(|∇ 2 −1 Z(x)|2 )) 2 ⎝

u L2 (R2d ) =1 Rd

≤C(f )Z

d

L2ω H 2 −1

∇

d 2 −1

Y¯f (t = 0, y)u(x, y)dy|2 ⎠⎠ dx

Rd

≤C(f )

⎞⎞ 12



1 2

⎞ 12



|u(x, y)|2 dy ⎠ dx

Rd

.

Similarly for the last term: E(|Yf (t = 0)Z|)2Hd/2−1 ≤ C(f )ZL2ω H d/2−1 . From (6) one gets the identity: e−itΔ (γ − γf )eitΔ − γ± = e−it(Δ−m) (E(|XX|) − E(|Yf Yf |))eit(Δ−m) − γ± = e−it(Δ−m) (E(|ZZ|) + E(|ZYf |) + E(|Yf Z|))eit(Δ−m) − γ± = E(|e−it(Δ−m) Ze−it(Δ−m) Z|) + E(|e−it(Δ−m) ZYf (t = 0)|) + E(|Yf (t = 0)e−it(Δ−m) Z|) − γ± = E(|e−it(Δ−m) Z − Z± e−it(Δ−m) Z|) + E(|Z± e−it(Δ−m) Z − Z± |) + E(|e−it(Δ−m) Z − Z± Yf (t = 0)|) + E(|Yf (t = 0)e−it(Δ−m) Z − Z± |). Corollary 1.2 is then a consequence of the above identity, the previous estimates, and the result (5) of the main Theorem: e−itΔ (γ − γf )eitΔ − γ± 

d

H 2 −1

 (1 + Z

d

L2ω H 2 −1

+ Z± 

d

L2ω H 2 −1

)Z − Z± 

d

−→ 0.

L2ω H 2 −1 t→±∞



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