Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

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Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

Laurent Lafleche

To cite this version:

Laurent Lafleche. Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial

Data. 2019. �hal-02046481v2�

VLASOV EQUATION FOR CONCENTRATED INITIAL DATA

LAURENT LAFLECHE^1,2

Abstract. We prove a quantitative andglobal in timesemiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimensiond≥3, including the case of a Coulomb singularity in dimensiond= 3. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an inter- mediate result, we also obtain quantitative bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects, uniform in the Planck constant~.

Table of Contents

1. Introduction 1

1.1. Main results 3

2. Free Transport 8

3. Propagation of moments 10

3.1. Classical case. 10

3.2. Boundedness of Eulerian moments 11

3.3. Application to the semiclassical limit 19

4. Acknowledgments 24

References 24

1. Introduction

The equation governing the dynamics of a large number of interacting particles of densityf =f(t, x, ξ) in the phase space is the Vlasov equation

(Vlasov) ∂tf +ξ· ∇xf+E· ∇ξf = 0,

Date: October 15, 2020.

2010Mathematics Subject Classification. 82C10, 35Q41, 35Q55 (82C05,35Q83).

Key words and phrases. Hartree equation, Nonlinear Schrödinger equation, Vlasov equation, Coulomb interaction, gravitational interaction, semiclassical limit.

1CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research Uni- versity, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16 France, [email protected].

2CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau cedex, France.

1

2 LAURENT LAFLECHE

where E =−∇V is the force field corresponding to the mean field potentialV = V(x) given by

V =K∗ρf = Z

R^d

K(x−y)ρf(y) dy, where we denote by ρf(x) := R

R^df(x, v) dv the spatial density and by K(x) the pair interaction potential between two particles at distance|x|.

The counterpart of the Vlasovequation in quantum mechanics is the Hartree equation

(Hartree) i~∂tρ= [H,ρ],

whereρ is a self-adjoint Hilbert-Schmidt operator called the density operator and the Hamiltonian is defined by

H =−h² 2 ∆ +V.

In this formula, the potential is defined byV =K∗ρwhere the spatial densityρis defined as the diagonal of the kernel%(x, y) of the operatorρ, i.e. ρ(x) =%(x, x).

In this paper, we study in a quantitative way the limit when~→0 of theHartree equation which is known to converge to theVlasovequation. The question of the derivation of this equation from the quantum mechanics is a very active topic of research. Non-constructive results in weak topologies have indeed already been proved, including the case of Coulomb interactions, starting from the work of Lions and Paul [36] and Markowich and Mauser [39]. See also [32,23,31,2,1].

Some more precise quantitative results have also more recently been proved for smooth forces which are always at least Lipschitz in [6,3,4,12,26]. In [27], Golse and Paul introduce a pseudo-distance on the model of the Wasserstein-(Monge- Kantorovitch) between classical phase space densities and quantum density oper- ators to get a rate of convergence for the semiclassical limit for Lipschitz forces.

This strategy has been used in the recent paper [35] of the present author to extend this result to more singular interactions, but only up to a fixed time in the case of potentials with a strong singularity such as the Coulomb interaction.

We also mention the work of Porta et al [48] and Saffirio [50] about the question of the mean-field limit for the Schrödinger equation to the Hartree equation for Fermions since this limit is coupled with a semiclassical limit. Results are proved for the Coulomb interaction under assumptions of propagation of regularity along the Hartree dynamics which is still an open problem. Other results about the mean-field limit can be found in [10,20,9] where non-quantitative results are established for the Coulomb potential, and more precise limits can be found in [49,47,26,40,27,29,30]

for Bosons and in [22,21,13,11,7,45,48,44] for Fermions.

Here, we extends the results of [35] by proving a global in time semiclassical limit under a smallness condition of space moments. We first prove a global in time bound on some modified space moments, from which we obtain the propagation of space and velocity moments. The same kind of results were already known for

~ = 1 (see Remark 1.4), and the main novelty is the fact that the bounds we prove are uniform in ~. The bound on the velocity is then sufficient to use the theory already used in the above mentioned paper to get a globalL^∞bound on the spatial density and the quantitative semiclassical limit in the quantum Wasserstein pseudo-distance.

The fact that the time decay due to the dispersion properties gives global esti- mates for theVlasovequation was already used in [8]. The modified space moments of order 2 are linked to a Lyapunov functional reminiscent of the conservation of energy, see [43,19]. The propagation of modified space moments was investigated further in [16,41,42].

1.1. Main results. We adopt the same notational conventions as in [35]. In par- ticular,L^p,∞ denotes the weak Lebesgue spaces of functions onR^d and we define the quantum version of the phase space Lebesgue and weighted Lebesgue spaces as

L^p:=n

ρ∈B(L²),kρk_Lp:=h^−d/p0Tr(|ρ|^p)^1p <∞o L^p₊:={ρ∈ L^p,ρ=ρ^∗≥0}

L^p(m) :={ρ∈ L^p,ρm∈ L^p},

whereB(L²) denotes the set of bounded linear operators onL²and Tr denotes the trace. We also define the quantum probability measures by

P:=

ρ∈ L¹₊,Tr(ρ) = 1 . Moreover, in order to ensure well-posedness

We will denote by p:=−i~∇ the quantum impulsion, which is an unbounded operator on L², and by M₀ the common total mass of the densities in both the quantum and the classical setting

M0:= Tr(ρ) = Z Z

R^2d

f(t, x, v) dxdv.

Our first result states that if the spatial density is concentrated enough, then theEulerian momentsTr(|x−tp|ⁿρ) are bounded globally in time.

Theorem 1. Let d≥3,n∈2N\ {0},r∈[1,∞] and define b_n:= ^nr_n+1^0+d. Assume (1) ∇K∈L^b,∞ with b∈ max ^d₃,b4,bn

,^d₂ , and letρ be a solution of theHartreeequation with initial condition

ρⁱⁿ∈ L^r∩ L¹₊(1 +|x|ⁿ+|p|ⁿ).

Then there exists an explicit constantC>0depending onM₀,Tr(|p|ⁿρⁱⁿ),k∇Kk_Lb,∞, ρⁱⁿ

L^r and not on ~, such that if

(2) Tr(|x|ⁿρⁱⁿ)<C,

then

Tr(|x−tp|ⁿρ)∈L^∞(R+), uniformly in ~.

Remark 1.1. The theorem applies in particular in the case of interaction kernels K with a singularity like the Coulomb interaction. For example for any ε >0

K(x) =±1

|x|1|x|≤1+ ±1

|x|^1+ε1|x|>1.

4 LAURENT LAFLECHE

An other interesting case of application is the case of the Yukawa potential that is commonly used as an approximation in the case when there are particles with positive and negative charge, and which is of the form

K(x) =e^−|x|/λD

|x| ,

where λD > 0 is the Debye length, which represents the characteristic size of the interaction.

Remark 1.2. An other good example of potentials verifying the assumptions of the theorem are potentials of the form

(3) K(x) = ±1

|x|^a,

with a= ^d_b −1 ∈(1,⁸₇) when d= 3. In dimension d= 4,d= 5 and d≥6, one can even better take respectively a∈(1,³₂),a ∈(1,¹⁶₉) and a∈(1,2). Of course, regular potentials also enter the scope of this first theorem as long as they decay sufficiently at infinity, so that we can take for example

K(x) = ±1 1 +|x|^a, for any a >1.

Remark 1.3. Since ρ∈ L¹₊, it is an Hilbert-Schmidt operator that can be written as a integral operator of kernel%(x, y)and it can also be diagonalized by the spectral theorem. Hence, we can write for anyϕ∈L²

(4) ρϕ(x) =

Z

R^d

%(x, y)ϕ(y) dy=X

j∈J

λ_j|ψjihψj|ϕi,

where (ψ_j)_j∈J ∈(L²)^J with J ⊂N is an orthogonal basis. The space density can then be written

ρ(x) :=%(x, x) =X

j∈J

λ_j|ψ_j(x)|², and the space moments

Tr(|x|ⁿρ) = Z

R^d

ρ(x)|x|ⁿdx.

For even integers n∈2N, the velocity moments can be written Tr(|p|ⁿρ) =~ⁿ

X

j∈J

λj

Z

R^d

∇ⁿ²ψj

2. (5)

Remark 1.4. Notice that the existence theory for both Hartreeand Vlasov equa- tions is already quite well understood, see for example[24,25,33,14,34,15]for the Hartreeequation and[37,51,46]for theVlasovequation. For more singular poten- tials than the Coulomb potential, remark that by the real interpolation definition of Lorentz spaces, our hypothesis (1)on ∇K implies

∇K∈L^p+L^q for any(p, q)such that0< p <b< q <∞,

where one can takep= ^2d+8_d+8 sinceb>b4. Moreover, by Sobolev embeddings, since b>^d₃,

K∈L^r0+L^∞ with r0>^d₂,

and also with r0 > ^d+4₄ when d ≥ 3, since b > b4. Therefore, our assumptions implies hypotheses (90) and (91) in [36] and so the existence of solutions for both equations. Remark that, as in our previous paper [35], we are not trying to prove here the propagation of regularity forψ. In particular the global in time propagation for ~ = 1 of the multi-Sobolev norms defined by Formula (5) is proved in [15, Appendix]in the case of the Coulomb potential, where they are denoted bykΨk_Hn(λ)

forΨ = (ψ_j)_j∈J. The same analysis can be performed for our class of potentials.

Thus, since we assume initially bounded velocity moments of order n ≥ 2, this implies that our solutions will always satisfy Ψ ∈ L^∞_loc((0, T), Hⁿ(λ)) but with a bound a priori not uniform in ~. Hence, the difficulty lies in the fact to obtain ~ independent bounds, which prevent for example to estimate separately each part of the commutator appearing in Hartreeequation.

We can state the analogue of this theorem for solutions of theVlasovequation Proposition 1.1. Let d≥ 3, n ∈ 2N, r ∈ [1,∞] and assume ∇K verifies Con- dition (1). Let f be a solution of the Vlasov equation with nonnegative initial condition

fⁱⁿ∈L^r_x,ξ∩L¹(1 +|x|ⁿ+|ξ|ⁿ).

Then there exists an explicit constant C>0 such that if Z Z

R^2d

fⁱⁿ(x, ξ)|x|ⁿdxdξ <C, then

Z Z

R^2d

f(·, x, ξ)|x−tξ|ⁿdxdξ∈L^∞(R+).

Remark 1.5. For theVlasovequation, contrarily to theHartreeequation, we gen- erally not have strong solutions in our setting (the velocity moments are not deriva- tives in the classical case). However, we still have global existence of renormalized solutions (see[17,18]).

We can use the first theorem to obtain good estimates on the space and velocity moments and on the spatial density that do not depend on~.

Theorem 2. Let d≥3,r∈[d⁰,∞],n∈(2N)\{0,2} and assume (6) ∇K∈L^b,∞ with b∈ max b4,^d₃

,^d₂ ,

and letρ be a solution of theHartreeequation with initial condition

ρⁱⁿ∈ L^r∩ L¹₊(1 +|x|⁴+|p|ⁿ),

for a given even integer n≥4. Then there exists a constant C depending on M₀, Tr

|p|⁴ρⁱⁿ

,k∇Kk_Lb,∞, and ρⁱⁿ

_Lr such that if Tr(|x|⁴ρⁱⁿ)≤ C,

6 LAURENT LAFLECHE

then there existscn=cd,n,r ≥0andC >0depending on the initial conditions such that

Tr(|p|ⁿρ)≤Chti^cn (7)

Tr(|x|ⁿρ)≤Chti^n(cn+1) (8)

kρk_Lp≤Chti^−d/p0, (9)

wherep⁰ =r⁰+^d₄ andhti=√

1 +t². Moreover, if Tr |x|ⁿρⁱⁿ

is sufficiently small, then we can get more precise estimates

Tr(|p|ⁿρ)≤C Tr(|x|ⁿρ)≤Chtiⁿ

kρk_Lpn ≤Chti^−d/p0n, wherep⁰_n=r⁰+_n^d.

We can once more state the analogue result for theVlasovequation.

Proposition 1.2. Let d ≥ 3, r ∈ [d⁰,∞], n ∈ (2N)\{0,2} and assume K veri- fies (6). Letf be a solution of the Vlasovequation with nonnegative initial condi- tion

fⁱⁿ∈L^r_x,ξ∩L¹_x,ξ(1 +|x|⁴+|ξ|ⁿ),

for a given even integer n≥4. Then there exists C>0 such that if Z Z

R^2d

fⁱⁿ(x, ξ)|x|⁴dxdξ≤ C,

then there existscn=cd,n,r >0andC >0depending on the initial conditions such that

Z Z

R^2d

f(t, x, ξ)|ξ|ⁿdxdξ≤Chti^cn Z Z

R^2d

f(t, x, ξ)|x|ⁿdxdξ≤Chti^n(cn+1) kρfk_Lp≤Chti^−d/p0,

wherep⁰ =r⁰+^d₄. As in the previous theorem, one can take cn= 0 andp⁰=r⁰+^d_n if the initial space moments of ordernare small.

Before stating the result about the semiclassical limit, we recall the definition of the semiclassical Wasserstein-(Monge-Kantorovitch) distance introduced by Golse and Paul in [27]. We say thatγ ∈L¹(R^2d,P) is a semiclassical coupling between a classical kinetic density f ∈L¹∩ P(R^2d) and a density operatorρ∈P and we writeγ∈ C_~(f,ρ) when

Tr(γ(z)) =f(z) Z

R^2d

γ(z) dz=ρ.

Then we define the semiclassical Wasserstein-(Monge-Kantorovich) pseudo-distance in the following way

(10) W2,~(f,ρ) :=

inf

γ∈C~(f,ρ)

Z

R^2d

Tr(c_~(z)γ(z)) dz ¹₂

,

where c_~(z)ϕ(y) = |x−y|²+|ξ−p|²

ϕ(y), z = (x, ξ) and p = −i~∇y. This is not a distance but it is comparable to the classical Wasserstein distance W2

between the Wigner transform of the quantum density operator and the normal kinetic density, in the sense of the following theorem.

Theorem 3(Golse & Paul [27]). Letρ∈P andf ∈ P(R^2d)be such that Z

R^2d

f(x, ξ)

|x|²+|ξ|²

dxdξ <∞.

Then one hasW2,~(f,ρ)²≥d~and for the Husimi transform f˜_~ ofρ, it holds (11) W₂(f,f˜_~)²≤W_2,~(f,ρ)²+d~.

See [28] for more results about this pseudo-distance and the definition of the Husimi transform.

Our last theorem uses these results to obtain the semiclassical limit. We also recall the following theorem which will gives us our assumptions on the classical solution of theVlasovequation.

Theorem 4(Lions & Perthame [37], Loeper [38]). Assumefⁱⁿ∈L^∞_x,ξ(R⁶)verify (12)

Z Z

R⁶

fⁱⁿ|ξ|ⁿ⁰dxdξ < C for a given n0>6, and for all R >0,

(13) sup ess

(y,w)∈R⁶

{fⁱⁿ(y+tξ, w),|x−y| ≤Rt²,|ξ−w| ≤Rt} ∈L^∞_loc(R+, L^∞_x L¹_ξ).

Then there exists a unique solution to the Vlasov equation with initial condition ft=0=fⁱⁿ. Moreover, in this case, the spatial density verifies ρf ∈L^∞_loc(R+, L^∞).

Theorem 5. Let d≥3 and assume

∇K∈B_1,∞¹ ∩L^b withb∈ max b4,^d₃ ,^d₂

.

Let ρbe a solution of theHartreeequation with initial conditionρⁱⁿ∈P verifying ρⁱⁿ∈ L^∞∩ L¹₊(1 +|p|ⁿ¹)

∀i∈[[1, d]],pⁿ_i⁰ρⁱⁿ∈ L^∞,

wherep_i:=−i~∂i,[[1, d]] = [1, d]∩N, and(n0, n)∈(2N)² is such that n₀> d

n≥d+b(n0−1) b−1 .

Let f is a solution of the Vlasov equation with initial condition fⁱⁿ verifying the hypotheses of Theorem4and of mass M₀= 1. Then there exists a constant C>0 depending onTr(|p|⁴ρⁱⁿ),k∇Kk_Lb,∞ and

ρⁱⁿ

L^∞ such that if

Tr(|x|⁴ρⁱⁿ)≤ C,

then there exists a constantC >0depending on the initial conditions such that kρ(t)k_L∞ ≤Chtiⁿ⁰(¹⁺_b^c0)

(14)

8 LAURENT LAFLECHE

wherec=cn is given by (7). Again, if additionallyTr |x|ⁿρⁱⁿ

is also sufficiently small, then one can take c = 0. Moreover, the following semiclassical estimate holds

W_2,~(f(t),ρ(t))≤W_2,~(fⁱⁿ,ρⁱⁿ)^eθλ(t)e^eλ(t), with

θ=sign(lnW2,~(fⁱⁿ,ρⁱⁿ)) 2

λ(t) =Cⁱⁿhti¹⁺ⁿ⁰(¹⁺_b^c0) Cⁱⁿ= 1 +Ck∇Kk_B1

1,∞sup

t

kρf(t)k_L∞+kρ(t)k_L∞

hti^n0(1+c/b0) , for some constant C >0 independent from the initial conditions.

Remark 1.6. Again, the additional assumption ∇K ∈ B_1,∞¹ is compatible with a kernel with a Coulomb singularity in dimension d= 3 such as the one given in Remark1.1. However, higher local singularities such that the one from Equation(3) are not admissible for this result. This seems natural since even the uniqueness of solutions for theVlasov equation is not known for such singular potentials.

Remark 1.7. This theorem implies a result of convergence in the classical Wasser- stein distance at a rate Ct~^Ct as soon as the quantity W2,~(fⁱⁿ,ρⁱⁿ) is initially smaller than some power of ~. In particular, this implies weak convergence of the Wigner transform of the solution of the Hartree equation to the solution of the Vlasovequation. Remark that by[27, Theorem 2.4],W_2,~(fⁱⁿ,ρⁱⁿ)is always larger than √

d~. The fact that W2,~ can be controlled by the classical Wasserstein dis- tance up to an error term√

d~can be proved for example when the initial states are superposition of coherent states and this leads to results that can be written uniquely in term of the classical Wasserstein distance as in [35, Section 7].

2. Free Transport

We want to use the time decay properties of the kinetic free transport equation which writes forf =f(t, x, ξ)

∂tf+ξ· ∇xf = 0.

In quantum mechanics, free transport is given by the free Schrödinger equation i~∂tψ=H0ψ,

where ~= _2π^h and H0 =−^~2₂^∆ which can be written H0 = ^|p|₂² with the notation p=−i~∇. The solution corresponding to the initial conditionψⁱⁿ can be written T_tψⁱⁿwhere the semigroup T_tis given by

(15) Ttψ=e^−itH0/~ψ= e^{−iπ|x|2/(ht)} (iht)^d/2 ∗ψ.

The corresponding equation for density operatorsρ∈P is

(16) i~∂_tρ= [H₀,ρ],

whose solution isS_tρⁱⁿ where the semigroup S_tis defined by

(17) S_tρ:=T_tρT_−t.

As it can be easily noticed, it holdsT^∗_t =T⁻¹_t =T−t and for any (ρ₁,ρ₂)∈P², S(ρρ₂) =S(ρ)S(ρ₂). Moreover, a straightforward computation shows that

S_tp=p Stx=x−tp.

(18)

By the spectral theory, it implies thatSt(f(x)) =f(x−tp) for any nice functionf. By analogy, we can define the operator of translation of the impulsionpby

T˜tψ(x) :=e^−π|x|

2t

ih ψ(x) =Gt(x)ψ(x)

˜S_tρ:= ˜T_tρT˜_−t, (19)

which verifies the equation

i~∂_t(˜S_tρ) =

"

− |x|² 2 ,˜S_tρ

# , and the two following relations

˜S_tx=x S˜tp=p−tx.

(20)

We recall the quantum kinetic interpolation inequality that was already used in [35, Theorem 6]. Fork∈2Nwe define

ρ_k :=X

j∈J

λ_j|p^k2ψ_j|²= diag

p^k2ρ·p^k2 ,

and forr≥1 and 0≤k≤n, we define the exponentpn,k by its Hölder conjugate

(21) p⁰_n,k=n

k 0

p⁰_n withp⁰_n=

r⁰+d n

. Then the following inequality holds

Proposition 2.1. Let(k, n)∈(2N)² be such thatk≤nandρ∈ L¹(|ρ|ⁿ)∩ L^r₊for a givenr∈[1,∞]. Then there existsC=Cd,r,n,k>0 such that

kρ_kk_Lpn,k ≤CTr(|p|ⁿρ)^1−θkρk^θ_Lr, (22)

whereθ=_p^r0⁰ n,k

.

From this result, we can get an inequality with an additional time decay if we replace the velocity moments by the Eulerian moments Tr(|x−tp|ⁿρ).

Corollary 2.1. Letn∈2N,r∈[1,∞],p⁰ :=r⁰+^d_n andθ= ^r_p⁰0. Then kρk_Lp≤ 1

t^d/p0 Tr(|x−tp|ⁿρ)^1−θkρk^θ_Lr.

Proof of Corollary 2.1. We just remark that by Formula (20), we get t⁻ⁿTr(|x−tp|ⁿρ) = Tr(|p−x/t|ⁿρ)

= Tr ˜S_1/t(|p|ⁿ)ρ

= Tr |p|ⁿ˜S_−1/t(ρ) . Moreover, since ˜T is a unitary transformation, the following identities hold

diag ˜Stρ

=Gt(x)%(x, x)G_−t(x) =ρ(x)

˜S_−1/tρ

L^r =kρk_Lr.

10 LAURENT LAFLECHE

Then by the interpolation Inequality (22) we get kρk_Lp=

diag ˜S_−1/tρ _Lp

≤Tr |p|ⁿ˜S_−1/t(ρ)^1−θ

S˜_−1/tρ

θ L^r

≤t^−n(1−θ)Tr(|x−tp|ⁿρ)^1−θkρk^θ_Lr.

Finally, we remark thatn(1−θ) = ^n(r0_r^+d/n−r0+d/n ⁰⁾ =_p^d0 to get the result.

3. Propagation of moments

3.1. Classical case. In this section, we define the classical Eulerian, velocity and space moments by

Ln:=

Z Z

R^2d

f(t, x, ξ)|x−tξ|ⁿdxdξ Mn:=

Z Z

R^2d

f(t, x, ξ)|ξ|ⁿdxdξ Nn:=

Z Z

R^2d

f(t, x, ξ)|x|ⁿdxdξ.

Proposition 3.1 (Classical large time estimate). Let (r,b) ∈ [1,∞]×[bn,∞],

∇K∈L^b,∞andf ∈L^∞(R+, L^r_x,ξ∩L¹_x,ξ)be a nonnegative solution ofVlasovequa- tion. Then for any n∈2N, there exists a constant C=Cd,r,n>0such that

dLn

dt

≤Ck∇Kk_Lb,∞M₀^Θ0 fⁱⁿ

r0 b

L^r_x,ξ

L¹⁺

a

n n(t) t^a , wherea= ^d_b−1 andΘ0= 1−^a_n−^r_b⁰.

Proof. We writef =f(t, x, ξ),E=E(x) and we compute dL_n

dt = Z Z

R^2d

|x−tξ|ⁿ(−ξ· ∇xf−E· ∇ξf)− |x−tξ|ⁿ⁻²(x−tξ)·ξfdxdξ

=−t Z Z

R^2d

f|x−tξ|ⁿ⁻²(x−tξ)·Edξdx.

By Hölder’s inequality, we deduce fort≥0 and anyp∈(1,∞)

dLn

dt

≤t Z

R^d

f|x−tξ|ⁿ⁻¹dξ _Lp

x

kEk_Lp0

≤tⁿ Z

R^d

f ^x_t −ξ

n−1dξ _Lp

x

k∇K∗ρ_fk_Lp0

≤CKtⁿ Z

R^d

f(t, x, ξ+^x_t)|ξ|ⁿ⁻¹dξ _Lp

x

Z

R^d

fdξ _Lq

,

where we used Hardy-Littlewood-Sobolev’s inequality withCK =k∇Kk_Lb,∞ and p∈(1,∞) such that

(23) 1

p⁰ + 1 q⁰ =1

b.

Then we want to use the classical kinetic interpolation inequality which tells that forp⁰_n,k = ⁿ_k⁰

r⁰+_n^d

andg=g(x, ξ)≥0, it holds (24)

Z

R^d

g(x, ξ)|ξ|^kdξ _Lpn,k

≤Cd,r,n

Z Z

R^2d

g(x, ξ)|ξ|ⁿdξdx

1−r⁰/p⁰_n,k

kgk^r

0/p⁰_n,k L^r_x,ξ . Since

1

p⁰_n,n−1 + 1

p⁰_n = 1 r⁰+d/n

1−n−1 n + 1

= n+ 1 nr⁰+d = 1

bn

≥ 1 b,

we can choose p ≤p_n,n−1 and q ≤p_n,0 verifying (23). Take p:= p_n,n−1. Then 1< q≤p_n and by interpolation

Z

R^d

fdξ _Lq

≤M¹⁻

p0 n q0

0

Z

R^d

fdξ

p⁰_n/q⁰

L^pn

.

Using the above inequality and then the interpolation inequality (24) fork= 0 and k=n−1 yields

dLn

dt

≤Cd,r,nCKtⁿM¹⁻

p0 n q0

0

Z

R^d

f(t, x, ξ+^x_t)|ξ|ⁿ⁻¹dξ _Lp

x

Z

R^d

fdξ

p⁰_n/q⁰

L^pn

≤Cd,r,nCKtⁿM¹⁻

r0

b+_n¹(^1−d_b)

0 kfk^r

0 b

L^r_x,ξ

Z Z

R^2d

f(t, x, ξ+^x_t)|ξ|ⁿdξ

¹⁺_n¹(^d_b⁻¹) . Witha=^d_b −1 and by a change of variable, we get

dL_n dt

≤C_d,r,nC_KtⁿM^1−r

0 b−^a_n 0 kfk^r

0 b

L^r_x,ξ

Z Z

R^2d

f(t, x, ξ) ξ−^x_t

ndξ 1+^a_n

≤Cd,r,nCKt^−aM¹⁻

r0 b−_n^a 0 kfk^r

0 b

L^r_x,ξ

Z Z

R^2d

f(t, x, ξ)|x−tξ|ⁿdξ ^1+a_n

,

which is the expected inequality.

3.2. Boundedness of Eulerian moments. We define ˜ρ := ˜S_−1/t(ρ) and for n∈2N

˜

ρn:= diag

p^n/2ρ˜·p^n/2 l_n:=tⁿρ˜_n.

We also introduce the following notations for the Eulerian, velocity and space mo- ments

Ln:= Tr(|x−tp|ⁿρ) Mn:= Tr(|p|ⁿρ)

Nn:= Tr(|x|ⁿρ),

as well as the corresponding moments ˜Mn and ˜Nn for ˜ρ. In particular, since we have

Ln = Tr(|x−tp|ⁿρ) =tⁿTr(|p−x/t|ⁿρ)

=tⁿTr(|p|ⁿ˜ρ) =tⁿTr(p^n/2ρ˜·p^n/2), we obtain with these notationsL_n=R

R^dl_n,M_n=R

R^dρ_nandN_n=R

R^dρ(x)|x|ⁿdx.

12 LAURENT LAFLECHE

3.2.1. Long time estimate. To obtain a differential inequality which will give us the long time behavior of the solution, we first need the following time dependent interpolation inequalities.

Proposition 3.2. Let 0 ≤k ≤n and p⁰_n,k := ⁿ_k0

p⁰_n andp∈[1, pn,k]. Then for any α≤k, there exists a constant C=Cd,r,n,k>0 such that

kρkk_Lp≤C Mα^1−θn,k,αM^θn,k,α−

r0 p0

n kρk

r0 p0

L^r

(25)

klkk_Lp≤C t^−d/p0L^1−θα ^n,k,αL^θn,k,α−

r0 p0

n kρk

r0 p0

L^r, (26)

where

θn,k,α=p⁰_n,α

p⁰ + k−α n−α. Proof. By the kinetic interpolation inequality (22),

kρkk_Lpn,k ≤C_d,r,n,kM

1− ^r0

p0

n n,kkρk

r0 p0

n,k

L^r .

Therefore, sincep≤pn,k, by interpolation betweenL^p spaces, we get kρkk_Lp≤ kρkk^1−θ_L1 kρkk^θ_Lpn,k

≤CM_k^1−θM^θ−

r0 p0

n kρk

r0 p0

L^r, whereθ=θn,k,k= ^p

0 n,k

p⁰ and we used the fact thatkρkk_L1=Mk. It already proves Inequality (25) fork=α. Sincek∈[α, n], we can also bound Mk in the following way

Mk≤M¹⁻

k−α n−α

α M

k−α n−α

n ,

which yields Inequality (25). To get (26), we follow the proof of Corollary 2.1.

Since ˜Spreserves the Schatten norms, we can write k˜ρk_Lr =kρk_Lr.

Hence, by replacing ρby ˜ρ in the kinetic interpolation inequality (22) and multi- plying byt^k, we obtain

kl_kk_Lpn,k =t^kkρ˜_kk_Lpn,k ≤Ct^k Tr |p|ⁿS˜_−1/tρ¹⁻

r0 p0

n,k

S˜_−1/tρ

r0 p0

n,k

L^r

≤Ct^k(Tr(|p−x/t|ⁿρ))¹⁻

r0 p0

n,kkρk

r0 p0

n,k

L^r

≤Ct^k−n+

nr0 p0

n,kL

1− ^r0

p0

n n,k kρk

r0 p0

n,k

L^r . Next we remark that

k−n+ nr⁰

p⁰_n,k =k−n+

1−k n

nr⁰

r⁰+d/n =−(n−k)

d/n r⁰+d/n

=− d p⁰_n,k, and we deduce Inequality (26) again by interpolation ofL^p betweenL¹ andL^pn,k

and by interpolation ofL_k betweenL_α andL_n.

Proposition 3.3(Large time estimate). Let (r,b)∈[1,∞]×[bn,∞],∇K∈L^b,∞

and ρ ∈ L^∞(R+,L^r∩ L¹₊) be a solution of the Hartree equation. Then for any n∈2N, there exists a constantC=Cd,r,n>0 such that

dL_n dt

≤Ck∇Kk_Lb,∞M₀^Θ0 ρⁱⁿ

r0 b

L^r

L¹⁺n ^an(t) t^a , wherea= ^d_b−1 andΘ0= 1−^a_n−^r_b⁰.

Proof. We first remark that by Formula (18) and spectral theory, we deduce

|x−tp|ⁿ =St(|x|ⁿ). Therefore by definingH0:=^|p|₂², by definition of St

i~∂t(St(|x|ⁿ)) = [H0,St(|x|ⁿ)] = [H0,|x−tp|ⁿ]. Hence, by differentiatingLn with respect to time, we obtain

i~∂_tL_n= Tr([H₀,|x−tp|ⁿ]ρ+|x−tp|ⁿ[H₀+V,ρ])

= Tr([H₀,|x−tp|ⁿ]ρ+ [|x−tp|ⁿ, H₀+V]ρ)

= Tr([|x−tp|ⁿ, V]ρ).

Then we use the operator ˜Stof translation in thexdirection defined in (19). By for- mulas (20) and spectral theory, we deduce that for anyt∈R, ˜S_tV =V. Therefore, we deduce

i~∂_tL_n=tⁿTr([(|p−x/t|ⁿ), V]ρ)

=tⁿTr S˜_1/t(|p|ⁿ), V ρ

=tⁿTr S˜_1/t(|p|ⁿ),S˜_1/t(V) ρ

=tⁿTr ˜S_1/t([|p|ⁿ, V])ρ

=tⁿTr([|p|ⁿ, V] ˜ρ).

As it has been proved in [35, Equation (38)], this expression can be bounded in the following way

|Tr([|p|ⁿ, V] ˜ρ)| ≤C_K~M˜n¹² sup

|a+b+c|=n/2−1

ρ˜_2|a|

1 2

L^α

ρ˜_2|b|

1 2

L^β

ρ˜_2|c|

1 2

L^γ, where (a, b, c)∈(N^d)³are multi-indices with|a|=a1+...+ad and

2 b= 1

α⁰ + 1 β⁰ + 1

γ⁰ (27)

C_K=C_d,nk∇Kk_Lb,∞.

As in [35, Proof of Theorem 3, Step 2], we remark that for the exponents p_n,k defined in (21) and multi-indices such that|a+b+c|=n/2−1, we have

1

p⁰_n,2|a|+ 1

p⁰_n,2|b|+ 1 p⁰_n,2|b| = 1

p⁰_n

3−2|a|+ 2|b|+ 2|c|

n

=2 (n+ 1) nr⁰+d = 2

bn

. Therefore, since b≥b_n, we can find (α, β, γ)∈[1, p_n,2|a|]×[1, p_n,2|b|]×[1, p_n,2|b|] verifying (27) and use the interpolation inequality (26) forα= 0. By the definition

14 LAURENT LAFLECHE

oflk and the fact thatL0=M0, we deduce

∂tLn≤CKtn−n/2−(|a|+|b|+|c|)L

1

n2 sup

|a+b+c|=n/2−1

l_2|a|

1 2

L^α

l_2|b|

1 2

L^β

l_2|c|

1 2

L^γ

≤CKt L

1

n2 sup

|a+b+c|=n/2−1

l_2|a|

1 2

L^α

l_2|b|

1 2

L^β

l_2|c|

1 2

L^γ

≤

C_d,r,nC_KM₀^Θ0kρk^Θ_Lr¹

t^−aL^Θ_n, where

a= d 2

1 α⁰ + 1

β⁰ + 1 γ⁰

−1 = d b−1 Θ₀= 1

2

3−p⁰_n 1

α⁰ + 1 β⁰ + 1

γ⁰

−2|a|+ 2|b|+ 2|c|

n

= 1 + 1 n−p⁰_n

b = 1−a n−r⁰

b Θ1= r⁰

2 1

α⁰ + 1 β⁰ + 1

γ⁰

= r⁰ b Θ = 1

2

1 +p⁰_n 1

α⁰ + 1 β⁰ + 1

γ⁰

+2|a|+ 2|b|+ 2|c|

n −r⁰

1 α⁰ + 1

β⁰ + 1 γ⁰

= 1 + 1 n

d b−1

= 1 + a n. We conclude by recalling thatkρk_Lr =

ρⁱⁿ

_Lrsince theHartreeequation preserves

the Schatten norm.

3.2.2. Short time estimate. To prove the short time estimate, we will use the bound- edness ofMn andNnfor short times to get the boundedness ofLn. To achieve this, we first need some lemmas to bound traces expressions with products ofxand p byMn andNn.

Lemma 3.1 (Interpolation for weighted traces). Let 0 ≤ k ≤n and A = |p| or A=|x|. Then for any operator ρ≥0 we have the following inequalities

Tr A^kρ

≤Tr(Aⁿρ)^knTr(ρ)^1−kn. (28)

Proof. By our definition of the absolute value, for two positive operators we have

|AB|² = BA²B. Therefore, the lemma follows from Hölder’s inequality for the trace and Araki-Lieb-Thirring inequality [5] since

Tr A^kρ

= A^k2ρ¹²

2 2≤

A^k2ρ^2nk

2

2n k

ρ¹²(^1−kn)

2

2n 2n−k

≤ Aⁿ²ρ¹²

2k n

2 Tr(ρ)^1−kn, and the right-hand side here is exactly the right-hand side of Inequality (28).

In the more general case of mixed product ofxandp, for any i∈[[1, d]]ⁿ we can define the set

Z_kⁿ(i) ={zi₁. . . zi_n,∀j∈[[1, n]], zi_j =xi_j or zi_j =p_ij,

{j, zj=p_j} =k}.

of operatorszconsisting of a product ofkpartial derivatives andn−kmultiplica- tions by a coordinate ofx, and look at the following quantities

Tr(zρ) = Tr(z_i1. . . z_inρ).