Semiclassical Analysis for Hartree equation

HAL Id: hal-00195618

https://hal.archives-ouvertes.fr/hal-00195618

Submitted on 11 Dec 2007

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Semiclassical Analysis for Hartree equation

Rémi Carles, Satoshi Masaki

To cite this version:

Rémi Carles, Satoshi Masaki. Semiclassical Analysis for Hartree equation. Asymptotic Analysis, IOS Press, 2008, 58 (4), pp.211-227. �10.3233/ASY-2008-0882�. �hal-00195618�

hal-00195618, version 1 - 11 Dec 2007

R ´EMI CARLES AND SATOSHI MASAKI

Abstract. We justify WKB analysis for Hartree equation in space di- mension at least three, in a r´egime which is supercritical as far as semi- classical analysis is concerned. The main technical remark is that the nonlinear Hartree term can be considered as a semilinear perturbation.

This is in contrast with the case of the nonlinear Schr¨odinger equation with a local nonlinearity, where quasilinear analysis is needed to treat the nonlinearity.

1. Introduction

We consider the semiclassical limit ε→0 for the Hartree equation (1.1) iε∂_tu^ε+ε²

2∆u^ε=λ(|x|^−γ∗ |u^ε|²)u^ε, γ >0, λ∈R, x∈Rⁿ, in space dimension n>3. We consider initial data of WKB type, (1.2) u^ε(0, x) =a^ε₀(x)e^iφ0(x)/ε,

where a^ε₀ typically has an asymptotic expansion as ε→0, a^ε₀ ∼

ε→0a₀+εa₁+ε²a₂+. . . , a_j independent of ε∈]0,1].

The approach that we follow is closely related to the pioneering works of P. G´erard [12], and E. Grenier [15], for the nonlinear Schr¨odinger equation with local nonlinearity:

(1.3) iε∂_tu^ε+ ε²

2∆u^ε=f |u^ε|² u^ε,

where the function f is smooth, and real-valued. In [15], the assumption f^′ > 0 is necessary for the arguments of the proof. More recently, this assumption was relaxed in [3], allowing to consider the case f(y) = +y^σ, σ ∈ N. Moreover, it is noticed in [3] that to carry out a WKB analysis in Sobolev spaces for (1.3), the assumption f^′ > 0 is essentially necessary.

Typically, in the case f^′ <0, working with analytic data is necessary, and sufficient as shown in [12, 21]. The reason is that the local nonlinearity is analyzed through quasilinear arguments in [15, 3], and f^′ determines the velocity of a wave equation: if f^′ >0, then the wave equation is hyperbolic, and f^′ <0, the underlying operator becomes elliptic.

The above discussion is altered for the Hartree type nonlinearity. Typi- cally, no assumption is made on the sign of λhere. As noticed in [1] in the special case of the Schr¨odinger–Poisson system, the nonlocal nonlinearity in (1.1) can be handled by semilinear arguments. However, a quasilinear analysis is needed to handle the convective coupling.

1

There are at least to motivations to study this question, besides the gen- eral picture of justifying approximations motivated by physics. As remarked in [6] in the case of a local nonlinearity, WKB analysis and a geometrical transform can help understand the behavior of a wave function near a focal point, in a supercritical r´egime. In [20], other informations were obtained thanks to a different approach, in the case of a Hartree type nonlinearity.

The approach of [20] and the results of the present paper will certainly be helpful to improve the understanding of the focusing phenomenon in semiclassical analysis. Another application of the WKB analysis for (1.1) concerns the Cauchy problem for the Hartree equation, that is (1.1) with ε= 1. Following the approach initiated in [4, 5, 9, 10, 18, 19], we can prove an ill-posedness result, together with a loss of regularity; see Corollary 1.9.

Assumption 1.1. Let n>3andmax(n/2−2,0)< γ6n−2. We suppose the following conditions with some s > n/2 + 1:

• The initial amplitude a^ε₀ ∈H^s(Rⁿ), uniformly forε∈[0,1]: there exists a constant C independent of ε such that ka^ε₀k_Hs 6C.

• The initial phaseφ₀ satisfies

|φ₀(x)|+|∇φ₀(x)| −→

|x|→∞0,

and ∇²φ₀ ∈ H^s(Rⁿ). Moreover, there exists q₀ ∈]n/(γ + 1), n[ such that

∇φ₀ ∈L^q0(Rⁿ).

Remark 1.2. We will see that the above assumption implies thatφ₀and∇φ₀ are bounded, and enjoy some extra integrability properties. See Remark 3.3.

Remark 1.3. By employing the geometrical reduction made in [1], we can relax the assumption|φ0(x)|+|∇φ0(x)| →0 as|x| → ∞in a sense. Indeed, we can replace the initial phase φ₀ withφ₀+φ_quad, where φ_quad ∈C^∞(Rⁿ) is a polynomial of degree at most two.

For s > n/2, we denote by X^s(Rⁿ) the Zhidkov space X^s(Rⁿ) ={u∈L^∞(Rⁿ)|∇u∈H^s−1(Rⁿ)}.

This space was introduced in [22] (see also [23]) in the case n = 1, and its study was generalized to the multidimensional case in [11]. We denote

kuk_Xs :=kuk_L^∞+k∇uk_Hs−1. We write H^s =H^s(Rⁿ) and X^s =X^s(Rⁿ).

Theorem 1.4. Let Assumption 1.1 be satisfied. There exists T > 0 inde- pendent ofεands >1+n/2, and a unique solutionu^ε∈C([0, T];H^s) to the equation (1.1)–(1.2). Moreover, it can be written in the form u^ε=a^εe^iφε/ε, where a^ε is complex-valued, φ^ε is real-valued, with

a^ε∈C([0, T];H^s), φ^ε∈C

[0, T];L^∞∩L

nq0 n−q0

, and ∇φ^ε∈C [0, T];X^s+1∩L^q0

.

Moreover, ifφ₀ ∈L^p0 for somep₀∈]n/γ, nq₀/(n−q₀)[thenφ^ε∈C([0, T];L^p0).

Note that obtaining a local existence time T which is independent ofε∈ ]0,1] is already a non-trivial information, at least for a focusing nonlinearity λ < 0. Using classical results on the Cauchy problem for Hartree equation [14], and a scaling argument, would yield an existence time that goes to zero with ε. Taking q₀ = 2, we immediately obtain the following corollary:

Corollary 1.5. Let Assumption 1.1 be satisfied. Let u^ε = a^εe^iφε/ε be the solution given in Theorem 1.4. If γ ∈]n/2−1, n−2]and∇φ0 ∈H^s+1, then

φ^ε∈C

[0, T];X^s+2∩Lⁿ²ⁿ⁻² .

With this local existence result, we can justify a WKB expansion, pro- vided that the initial data have a suitable expansion as ε→0.

Assumption 1.6. Let N be a positive integer. We suppose Assumption 1.1 with some s > n/2 + 2N+ 1. Moreover, the initial amplitude a^ε₀ writes

(1.4) a^ε₀=a₀+

N

X

j=1

ε^ja_j +ε^Nr_N^ε, where a_j ∈H^s (06j6N) and kr_N^εk_Hs →0 as ε→0.

Theorem 1.7. Let Assumption 1.6 be satisfied. Let u^ε = a^εe^iφε/ε be the unique solution given in Theorem 1.4. Then, there exist (b_j, ϕ_j)₀6j6N, with

b_j ∈C([0, T];H^s−2j), ϕ_j ∈C

[0, T];L^∞∩L

nq0 n−q0

and ∇ϕ_j ∈C [0, T];X^s−2j+1∩L^q0 , such that:

a^ε=b₀+

N

X

j=1

ε^jb_j+o ε^N

in C([0, T];H^s−2N),

φ^ε=ϕ₀+

N−1

X

j=1

ε^jϕ_j+o ε^N

in C([0, T];L^∞∩L

nq0 n−q0),

∇φ^ε=∇ϕ₀+

N

X

j=1

ε^j∇ϕ_j +o ε^N

in C [0, T];X^s−2N+1∩L^q0 . Moreover, for j > 1, ϕj ∈ L^p for all p > n/γ, and ∇ϕj ∈ L^q for all q > n/(γ + 1).

Corollary 1.8. Let Assumption 1.6 be satisfied. The solution u^ε given in Theorem 1.4 has the following asymptotic expansion, as ε→0:

u^ε=e^iϕ0/ε β₀+εβ₁+. . .+ε^N−1β_N−1+ε^N−1ρ^ε

, kρ^εk_Hs−2N+2−→

ε→00, where ϕ₀ is given by Theorem 1.7, and β_j ∈ C([0, T];H^s−2j) is a smooth function of (b_k, ϕ_k+1)_06k6j. For instance,

β₀ =b₀e^iϕ1 ; β₁ =b₁e^iϕ1+iϕ₂b₀e^iϕ1.

Corollary 1.9. Let n > 5, λ ∈ R, max(n/2−2,2) < γ 6 n−2 and 0< s < s_c=γ/2−1. There exists a sequence of initial data

(ψ^h₀)_0<h61, ψ₀^h∈ S(Rⁿ), kψ^h₀k_H^s−→

h→00, a sequence of times t^h →0, such that the solution to

i∂tψ^h+1

2∆ψ^h =λ

|x|^−γ∗ |ψ^h|²

ψ^h ; ψ_|t=0^h =ψ^h₀ satisfies

ψ^h(t^h) H^k−→

h→0+∞, ∀k > s

1 +s_c−s = s γ/2−s.

Note that unlike in the case of Schr¨odinger equations with local nonlin- earity, considering a large space dimension is necessary to observe this phe- nomenon: in low space dimensions, Hartree equations are locally well-posed in Sobolev spaces of positive regularity (see e.g. [8, 14]).

Using Sobolev embedding, one could infer a loss of regularity at the level of the energy space (consider s > 1 and _γ/2−s^s = 1, hence γ = 4s > 1), in the spirit of [19] (see also [2, 21] for Schr¨odinger equations), provided that the space dimension is n>7.

The rest of this paper is organized as follows. In the next paragraph, we present the general strategy adopted in this paper. In §3, we collect some technical estimates. Theorem 1.4 is proved in§4, and Theorem 1.7 is proved in §5, as well as Corollary 1.8. Finally, Corollary 1.9 is inferred in§6.

2. General strategy

To prove Theorem 1.4 and Theorem 1.7, we follow the same strategy as in [15]. Seek a solution u^ε to (1.1)–(1.2) represented as

(2.1) u^ε(t, x) =a^ε(t, x)e^{iφε(t,x)/ε},

with a complex-valued space-time function a^ε and a real-valued space-time functionφ^ε. Note thata^εis expected to be complex-valued, even if its initial value a^ε₀ is real-valued. We remark that the phase functionφ^ε also depends on the parameter ε. Substituting the form (2.1) into (1.1), we obtain

iε

∂_ta^ε+a^εi∂_tφ^ε ε

+ε²

2

∆a^ε+ 2

∇a^ε·i∇φ^ε ε

−a^ε|∇φ^ε|²

ε² +a^εi∆φ^ε ε

=λ(|x|^−γ∗ |a^ε|²)a^ε. To obtain a solution of the above equation (hence, of (1.1)), we choose to consider the following system:







∂_ta^ε+∇a^ε· ∇φ^ε+1

2a^ε∆φ^ε=iε 2∆a^ε

∂_tφ^ε+1

2|∇φ^ε|²+λ(|x|^−γ∗ |a^ε|²) = 0.

(2.2)

This choice is essentially the same as the one introduced by E. Grenier [15].

We consider this with the initial data

a^ε_|t=0 =a^ε₀, φ^ε_|t=0 =φ₀. (2.3)

From now on, we work only on (2.2)–(2.3). We first prove that it admits a unique solution with suitable regularity (see Theorem 1.4), hence providing a solution to (1.1)–(1.2). The asymptotic expansion (Theorem 1.7) then follows by the similar arguments.

To conclude this paragraph, we remark that uniqueness for (1.1)–(1.2) in the class C([0, T];H^s),s > n/2 + 1, is a straightforward consequence of [14]

(see also [8]) in space dimension 3 6 n 6 5, since the parameter ε ∈]0,1]

can be considered fixed. Indeed, in that case, one hasx7→ |x|^−γ ∈L^p+L^∞ for some p > 1 and p > n/4, since γ 6n−2 < 4. Uniqueness is actually obtained in the weaker class of finite energy solutions. Since we work at a higher degree of regularity, we can simply notice that the nonlinear potential

|x|^−γ∗ |u^ε|² is bounded in L^∞([0, T]×Rⁿ): for χ ∈ C₀^∞(Rⁿ; [0,1]), χ = 1 near x= 0, we have from Young’s inequality,

|x|^−γ∗ |u^ε|²

L^∞(Rⁿ)6

χ|x|^−γ

L¹(Rⁿ)ku^εk²_L^∞_(Rn)

+

(1−χ)|x|^−γ

L^∞(Rⁿ)ku^εk²_L2(Rⁿ).

Since we work with anH^s regularity,s > n/2 + 1, the above right hand side is bounded, and uniqueness follows from standard energy estimates in L².

3. Preliminary estimates

We first recall a consequence of the Hardy-Littlewood-Sobolev inequality, which can be found in [16, Th. 4.5.9] or [13, Lemma 7]:

Lemma 3.1. If ϕ∈ D^′(Rⁿ) is such that ∇ϕ∈ L^p(Rⁿ) for some p ∈]0, n[, then there exists a constantγ such thatϕ−γ ∈L^q(Rⁿ), with1/p= 1/q+1/n.

Remark 3.2. The limiting case γ =n−2 corresponds to the Schr¨odinger–

Poisson system considered in [1], with suitable conditions at infinity to in- tegrate the Poisson equation.

Remark 3.3. By Lemma 3.1 and Sobolev inequality, Assumption 1.1 implies φ₀ ∈ L

nq0

n−q0 ∩L^∞, and ∇φ₀ ∈ L^q0 ∩X^s+1. Note that 2n/(n−2) < n if n >5. Therefore, in this case, we can always find q₀ in ]n/(γ+ 1), n[ such that ∇φ₀ ∈L^q0.

The next two lemmas can be found in [17]:

Lemma 3.4 (Commutator estimate). Let s > 0 and 1 < p < ∞. Set Λ = (1−∆)^1/2. Then, it holds that

kΛ^s(f g)−fΛ^sgk_Lp 6c(k∇fk_L^∞

Λ^s−1g

L^p+kΛ^sfk_Lpkgk_L^∞).

Lemma 3.5. Let s >0 and1< p <∞. There existsC >0 such that kΛ^s(f g)k_Lp 6C(kΛ^sfk_Lpkgk_L^∞+kfk_L^∞kΛ^sgk_Lp), ∀f, g∈W^s,p∩L^∞.

The following lemma is crucial for our analysis:

Lemma 3.6. Let n > 3, k > 0, and s₁, s₂ ∈ R. Let γ > 0 satisfying n/2−k < γ6n−k−s₁+s₂. Then, there exists C_s such that

|∇|^k(|x|^−γ∗f)

H^s1 6C_s(kfk_Hs2 +kfk_L1), ∀f ∈L¹∩H^s2.

Proof. Since F|x|^−γ =C|ξ|^−n+γ, it holds that

|∇|^k(|x|^γ∗f)

H^s1 =C

hξi^s1|ξ|^−n+γ+kFf L².

The high frequency part (|ξ|>1) is bounded by Ckfk_Hs2 if−n+γ+k+ s₁−s₂60. On the other hand, the low frequency part (|ξ|61) is bounded by

CkFfk_L^∞ Z

|ξ|61

|ξ|^2(−n+γ+k)dξ6Ckfk_L1

if 2(−n+γ +k)>−n, that is, ifγ > n/2−k.

4. Existence result: proof of Theorem 1.4

Operating ∇ to the equation for φ^ε in (2.2) and putting v^ε := ∇φ^ε, we obtain the following system:

(4.1)







∂_ta^ε+v^ε· ∇a^ε+1

2a^ε∇ ·v^ε=iε

2∆a^ε, a^ε_|t=0 =a^ε₀,

∂_tv^ε+v^ε· ∇v^ε+λ∇ |x|^−γ∗ |a^ε|²

= 0, v_|t=0^ε =∇φ₀. We first construct the solution (a^ε, v^ε) to the system (4.1).

Proposition 4.1. Let Assumption 1.1 be satisfied. There exists T > 0 independent of ε and s, such that for all ε ∈ [0,1], (4.1) has a unique solution

(a^ε, v^ε)∈C

[0, T];H^s×

X^s+1∩L

nq0 n−q0

. Moreover, the norm of (a^ε, v^ε) is bounded uniformly forε∈]0,1].

4.1. Regularized system. We shall prove the existence of the solution to the system (4.1) by taking the limit of the solutions to the corresponding regularized system. We take ϕ∈C₀^∞(Rⁿ), withR

Rⁿϕ(x)dx= 1 and ϕ>0 and set

(4.2) J_δf =ϕ_δ∗f

where ϕ_δ =δ⁻ⁿϕ(x/δ). We first treat the following regularized system:

(4.3)







∂_ta^ε_δ+J_δ(v_δ^ε· ∇J_δa^ε_δ) + 1

2a^ε_δ∇ ·J_δv_δ^ε=iε

2∆J_δ²a^ε_δ ; a^ε_δ|t=0 =a^ε₀.

∂_tv^ε_δ+J_δ(v^ε_δ· ∇J_δv_δ^ε) +λ∇J_δ(|x|^−γ∗ |a^ε_δ|²) = 0 ; v_δ|t=0^ε =∇φ₀. The point is that the regularized equations (4.3) have been chosen so that the Cauchy problem can be solved as in the standard framework of Sobolev and Zhidkov spaces:

Lemma 4.2. Let Assumption 1.1 be satisfied. For allε∈[0,1]andδ ∈]0,1], there existsT_δ^ε>0such that the Cauchy problem (4.3)has a unique solution (a^ε_δ, v_δ^ε)∈C¹([0, T_δ^ε], H^s+1×X^s+2∩Lⁿ²⁻²ⁿ ).

Proof. The proof is based on the usual theorem for ordinary differential equations. We use the following estimates

kJ_δ(v^ε_δ· ∇J_δa^ε_δ)k_Hs+1 6Ckv^ε_δk_Hs+1k∇J_δa^ε_δk_Hs+1

6Cδ⁻¹kv^ε_δk_Hs+2ka^ε_δk_Hs+1,

and

ka^ε_δ∇ ·J_δv_δ^εk_Hs+1 6Cδ⁻¹ka^ε_δk_Hs+1kv_δ^εk_Xs+2, ∆J_δ²a^ε_δ

H^s+1 6Cδ⁻²ka^ε_δk_Hs+1, kJ_δ(v_δ^ε· ∇J_δv^ε_δ)k_Xs+2 6Cδ⁻¹kv_δ^εk²_Xs+2,

∇J_δ(|x|^−γ∗ |a^ε_δ|²)

X^s+2 6C

∆(|x|^−γ∗ |a^ε_δ|²) H^s+1

6Cka^ε_δk²_Hs+1.

We have applied Lemma 3.6 with k= 2 ands₁ =s₂ =s+ 1. We note that the space X^s+2∩Lⁿ²ⁿ⁻² with normk·k_Xs+2 is complete.

4.2. Uniform bound. We shall establish an upper bound for theH^snorm and X^s+1norm of a^ε_δand v^ε_δ fors > n/2 + 1, respectively. We first estimate the H^s norm of a^ε_δ. We use the following convention for the scalar product in L²:

hϕ, ψi:=

Z

Rⁿ

ϕ(x)ψ(x)dx.

Set Λ = (I−∆)^1/2. We shall estimate d

dtka^ε_δk²_Hs = 2 Reh∂tΛ^sa^ε_δ,Λ^sa^ε_δi.

Since [Λ^s,∇] = 0 and [Λ^s, J_δ] = 0, by commuting Λ^s with the equation for a^ε_δ, we find:

(4.4) ∂_tΛ^sa^ε_δ+J_δΛ^s(v_δ^ε· ∇J_δa^ε_δ) +1

2Λ^s(a^ε_δ∇J_δ·v_δ^ε)−iε

2J_δ∆J_δΛ^sa^ε_δ= 0.

The coupling of the second term and Λ^sa^ε_δ is written as hΛ^s(v^ε_δ· ∇J_δa_δ^ε), J_δΛ^sa^ε_δi=hv^ε_δ· ∇J_δΛ^sa_δ^ε, J_δΛ^sa^ε_δi

+h[Λ^s, v^ε_δ]· ∇J_δa^ε_δ, J_δΛ^sa^ε_δi,

where we have use the fact that hJ_δf, gi=hf, J_δgi for anyf andg. We see from the integration by parts that

(4.5) |Rehv_δ^ε· ∇J_δΛ^sa^ε_δ, J_δΛ^sa^ε_δi |6 1

2k∇v^ε_δk_L^∞kJ_δΛ^sa^ε_δk²_L2. Moreover, the commutator estimate shows that

(4.6) |Reh[Λ^s, v_δ^ε]· ∇J_δa^ε_δ, J_δΛ^sa^ε_δi |

6C(k∇v^ε_δk_Hs−1kJ_δ∇a^ε_δk_L^∞+k∇v_δ^εk_L^∞kJ_δ∇a^ε_δk_Hs−1)kJ_δΛ^sa^ε_δk_L2

We estimate the third term of (4.4) by the Kato–Ponce inequality as (4.7) |RehΛ^s(a^ε_δ∇J_δ·v^ε_δ),Λ^sa^ε_δi |

6C(ka^ε_δk_L^∞kJ_δ∇v^ε_δk_Hs +ka^ε_δk_HskJ_δ∇v^ε_δk_L^∞)kΛ^sa^ε_δk_L² and the last term vanishes since

(4.8) Reh−i∆J_δΛ^sa^ε_δ, J_δΛ^sa_δ^εi= Reik∇J_δΛ^sa^ε_δk²_L2 = 0.

Therefore, summarizing (4.4)–(4.8), we end up with d

dtka^ε_δk²_Hs 6C(ka_δ^εk_W1,∞+k∇v_δ^εk_L^∞)(ka^ε_δk_Hs+kv_δ^εk_Xs+1)ka^ε_δk_Hs,

hence

(4.9) d

dtka^ε_δk²_Hs 6C(ka^ε_δk_W1,∞+k∇v_δ^εk_L^∞)(ka^ε_δk²_Hs+kv_δ^εk²_Xs+1).

Let us proceed to the estimate of v_δ^ε. We denote the operator Λ^s∇ by Q.

From the equation for v_δ^ε, we have

(4.10) ∂_tQv_δ^ε+J_δQ(v^ε_δ· ∇J_δv_δ^ε) +Q∇Λ^sJ_δ(|x|^−γ∗ |a^ε_δ|²) = 0

We consider the coupling of this equation and Qv^ε_δ. The second term can be written as

hQ(v^ε_δ· ∇J_δv_δ^ε), J_δQv^ε_δi=hv^ε_δ· ∇J_δQv_δ^ε, J_δQv_δ^εi

+h[Q, v^ε_δ]· ∇J_δv^ε_δ, J_δQ∇v^ε_δi. As the previous case, integration by parts shows

(4.11) |Rehv_δ^ε· ∇J_δQv^ε_δ, J_δQv_δ^εi |6 1

2k∇ ·v^ε_δk_L^∞kJ_δQv^ε_δk_L², and the commutator estimate also shows

(4.12) |Reh[Q, v_δ^ε]· ∇J_δv^ε_δ, J_δQv^ε_δi |

6C(k∇v^ε_δk_HskJ_δ∇v^ε_δk_L^∞+k∇v^ε_δk_L^∞kJ_δ∇v^ε_δk_Hs)kJ_δQv_δ^εk_L2. For the estimate of the Hartree nonlinearity, we use Lemma 3.6 with k= 2 and s₁ =s₂=s, to obtain

λJ_δΛ^s∇²(|x|^−γ∗ |a^ε_δ|²)

L² 6C(

|a^ε_δ|² H^s +

|a^ε_δ|² L¹) 6C(ka^ε_δk_Hska^ε_δk_L^∞+ka^ε_δk²_L²).

(4.13)

Summarizing (4.10)–(4.13), we deduce that d

dtk∇v_δ^εk²_Hs 6C(ka_δ^εk_L2+ka^ε_δk_L^∞+k∇v_δ^εk_L^∞)(ka^ε_δk_Hs+k∇v_δ^εk_Hs)k∇v^ε_δk_Hs, and so that

(4.14) d

dtk∇v^ε_δk²_Hs 6C(ka_δ^εk_L2+ka^ε_δk_L^∞+k∇v^ε_δk_L^∞)(ka^ε_δk_H²s+k∇v^ε_δk²_Hs).

Using Lemma 3.1, we see that the above estimate yields an L^2n/(n−2) esti- mate forv^ε_δ. Interpolating with a suitable ˙H^knorm shows that theL^∞norm of v^ε_δ is estimated as above. Alternatively, integrating the second equation of (4.3) with respect to time, Sobolev embedding directly yields a similar estimate for v^ε_δ inL^∞(Rⁿ).

Now, puttingM_δ^ε(t) :=ka^ε_δ(t)k²_Hs+k∇v^ε_δ(t)k²_Hs+kv_δ^ε(t)k²_L∞, we conclude from (4.9) and (4.14) that

(4.15) M_δ^ε 6C+C Z _t

0

(ka^ε_δk_L2 +ka^ε_δk_W1,∞+k∇v^ε_δk_L^∞)M_δ^εdτ.

We obtain the following Lemma.

Lemma 4.3. Let Assumption 1.1 be satisfied with s > n/2 + 1. There exists T independent of δ and εsuch that the solution (a^ε_δ, v^ε_δ) is bounded in C([0, T];H^s×X^s+1) uniformly in δ∈]0,1].

Proof. We only estimate the above M_δ^ε(t). Note that v^ε_δ vanishes at spatial infinity. It implies thatkv_δ^εk_L∞ is bounded byk∇v_δ^εk_Hs with some constant, since n>3. ands > n/2 + 1. Sobolev embedding and (4.15) yield

(4.16) M_δ^ε(t)6C+C Z t

0

(M_δ^ε(τ))^3/2dτ.

Therefore, there exists T_δ^ε>0 depending only on M_δ^ε(0) such thatM_δ^ε(t) is bounded by constant times M_δ^ε(0) uniformly in t ∈[0, T_δ^ε]. SinceM_δ^ε(0) is bounded independent ofδandεby assumption,T_δ^εcan be taken independent of δ andε, as well as the upper bound ofM_δ^ε(t).

4.3. Existence of the solution to the nonlinear hyperbolic system.

Next we prove the existence of the solution to (4.1). From Lemma 4.3, we see that the sequences {a^ε_δ}_δ and {v_δ^ε}_δ are uniformly bounded in C([0, T];H^s) and C([0, T];X^s+1 ∩Lⁿ²ⁿ⁻²), respectively. Therefore, from Ascoli–Arzela’s theorem, for a subsequence δ^′ of δ,

a^ε_δ^′ ⇀ a^ε weakly inC([0, T];H^s), v_δ^ε′ ⇀ v^ε weakly inC([0, T];Lⁿ²ⁿ⁻²),

∇v^ε_δ^′ ⇀∇v^ε weakly inC([0, T];H^s)

as δ^′ →0. Moreover, we have (a^ε, v^ε)∈C_w([0, T];H^s×X^s+1∩Lⁿ²ⁿ⁻²).

We shall show that (a^ε, v^ε) satisfies (4.1) inD^′(]0, T]×Rⁿ). We fix somet.

We choose some s^′ so that s > s^′ > n/2 + 1. Then, the above convergences imply (a^ε_δ^′,∇v^ε_δ^′) → (a^ε,∇v^ε) strongly in C([0, T];H_loc^s′ ×H_loc^s′ ) as δ^′ → 0 . Then, we deduce that a^ε_δ^′, ∇a^ε_δ^′, v_δ^ε′, and ∇v_δ^ε′ converge uniformly in any compact subset of Rⁿ, since s > n/2 + 1 and

v^ε_δ^′−v^ε

L^∞ is bounded by

∇v_δ^ε′ − ∇v^ε

H^s′ with some constant. Then, we can pass to the limit in all the terms in (4.1), except possibly the Hartree term. Since (f ∗g)∗h = f ∗(g∗h) and hf∗g, hi = hf,ˇg∗hi with ˇg(x) = ¯g(−x), the Hartree term can be rewritten as

λ∇J_δ^′(|x|^−γ∗ |a^ε_δ^′|²), ϕ

=−λ

J_δ^′|a^ε_δ^′|²,|x|^−γ∗ ∇ϕ .

The function |x|^−γ∗ ∇ϕis not compactly supported, but an ε/3-argument shows that the right hand side tends to −λ

|a^ε|²,|x|^−γ∗ ∇ϕ

. Thus, we obtain the solution (a^ε, v^ε)∈C_w([0, T];H^s×X^s+1∩Lⁿ²ⁿ⁻²). We now claim that this solution is strongly continuous in time. To prove this, we only have to show that the solution is norm continuous, that is, the function M^ε(t) :=ka^ε(t)k²_Hs+k∇v^ε(t)k²_Hs is continuous in time. In the same way as (4.15), we have

(4.17) d

dtM^ε6C(ka^εk_L2+ka^εk_W1,∞+k∇v^εk_L^∞)M^ε.

Since the right hand side is bounded, M^ε is upper semi-continuous. Weak continuities of a^ε andv^εimply the lower semi-continuity ofM^ε. Hence,M^ε is continuous.

Lemma 4.4. Let Assumption 1.1 be satisfied. Supposes > n/2+1. LetT be given in Lemma 4.3. For all ε∈[0,1], there exists (a^ε, v^ε)∈C([0, T];H^s× X^s+1∩Lⁿ²ⁿ⁻²) which solves (4.1) in D^′.

4.4. Uniqueness. We next prove the uniqueness of the solution (a^ε, v^ε) by showing that if (a^ε₁, v^ε₁) and (a^ε₂, v^ε₂) are solutions to (4.1), in the class C([0, T];H^s×X^s+1 ∩L^2n/(n−2)) for some s > n/2 + 1, then the distance (a^ε₁ −a^ε₂, v₁^ε −v₂^ε) is equal to zero in L^∞([0, T];L² ×H˙¹) sense. Denote (d^ε_a, d^ε_v) := (a^ε₁ −a^ε₂, v^ε₁−v^ε₂). Then, from (4.1), the system for (d^ε_a, d^ε_v) is rewritten as

∂_td^ε_a+d^ε_v· ∇a^ε₁+v₂^ε· ∇d^ε_a+1

2d^ε_a· ∇v₁^ε+1

2a^ε₂· ∇d^ε_v =iε 2∆d^ε_a, (4.18)

∂_td^ε_v+d_v^ε· ∇v₁^ε+v₂^ε· ∇d_v^ε+λ∇(|x|^−γ∗(d^ε_aa^ε₁+a^ε₂d^ε_a)) = 0.

(4.19)

Now estimate the L² norm ofd^ε_a. From the equation in (4.18), it holds that d

dtkd^ε_ak²_L2 = 2 Reh∂_td^ε_a, d^ε_ai

6C|Rehd^ε_v· ∇a^ε₁, d^ε_ai |+C|Rehv₂^ε· ∇d^ε_a, d^ε_ai |+C|Rehd^ε_a· ∇v^ε₁, d^ε_ai | +C|Reha^ε₂· ∇d^ε_v, d_a^εi |+|Rehi∆d^ε_a, d^ε_ai |.

Now, H¨older’s inequality and integration by parts show that

|Reha^ε₂· ∇d^ε_v, d^ε_ai |6ka^ε₂k_L^∞k∇d^ε_vk_L²kd^ε_ak_L²,

|Rehv₂^ε· ∇d^ε_a, d^ε_ai |+|Rehd^ε_a· ∇v₁^ε, d_a^εi |6(k∇v₁^εk_L^∞+k∇v₂^εk_L^∞)kd_ak²_L2, Rehi∆d^ε_a, d^ε_ai= 0.

Another use of H¨older’s and Sobolev inequalities shows

| hd^ε_v· ∇a^ε₁, d^ε_ai |6kd^ε_vk

L

2n

n−2 k∇a^ε₁k_Lnkd^ε_ak_L2

6Ck∇a^ε₁k

Hⁿ²⁻¹k∇d^ε_vk_L2kd^ε_ak_L2

Thus, we end up with the estimate d

dtkd^ε_ak²_L2 6C

kd^ε_ak²_L2 +k∇d^ε_vk²_L2

, (4.20)

where the constantCdepends onka^ε₁k_Hn/2,ka^ε₂k_L^∞, andk∇v_k^εk_L2 (k= 1,2).

Similarly, for all 16i, j6n, we have the estimates for ∂_id^ε_v,j:

((∂_id^ε_v)· ∇)v^ε_1,j, ∂_id^ε_v,j 6C

∇v^ε_1,j

L^∞k∂_id^ε_vk²_L2,

(d^ε_v· ∇)∂iv^ε_1,j, ∂id^ε_v,j 6C

∇∂iv^ε_2,j

Lⁿkd^ε_vk

Lⁿ²ⁿ⁻²

∂id^ε_v,j

L²,

((∂_iv^ε₂)· ∇)d^ε_v,j, ∂_id^ε_v,j

6Ck∂_iv₂^εk_L^∞

∇d^ε_v,j

2 L²,

(v₂^ε· ∇)∂_id^ε_v,j, ∂_id^ε_v,j

6Ck∇v₁^εk_L^∞

∂_id^ε_v,j

2 L²,

∂_i∂_j(|x|^−γ∗(d_a^εa^ε₁)), ∂_id^ε_v,j

6C(ka^ε₁k_L^∞+ka₁^εk_L2)kd^ε_ak_L2

∂_id^ε_v,j L²,

∂_i∂_j(|x|^−γ∗(a₂^εd^ε_a)), ∂_id^ε_v,j

6C(ka^ε₂k_L^∞+ka₂^εk_L2)kd^ε_ak_L2

∂_id^ε_v,j L², where v_1,j^ε and d^ε_v,j denote the j-th components of v^ε₁ and d^ε_v, respectively.

Summing up over iand j, we obtain d

dtk∇d^ε_vk²_L2 6C

kd^ε_ak²_L2 +k∇d^ε_vk²_L2

. (4.21)

Denote D(t) :=kd^ε_a(t)k²_L² +k∇d^ε_v(t)k²_L²: since D(0) = 0, Gronwall lemma shows that D(t) = 0 for all t∈[0, T].

Lemma 4.5. The solution (a^ε, v^ε) ∈ C([0, T];H^s×X^s+1 ∩Lⁿ²ⁿ⁻²) to the system (4.1) given in Lemma 4.4 is unique.

4.5. Completion of the proof of Proposition 4.1. Now we complete the proof of existence result.

Proof of Proposition 4.1. We have already shown that the system (4.1) has a unique solution (a^ε, v^ε)∈C([0, T];H^s×X^s+1∩Lⁿ²⁻²ⁿ ).

We show that the existence time T is independent of s, thanks to tame estimates. In the above proof, the existence timeT depends ons. However, once we show the existence of the solution in [0, T_s0] for some s₀> n/2 + 1, then, for any s₁> n/2 + 1, we deduce from (4.17) that

M_s^ε₁(t)6M_s^ε₁(0) exp

Ct sup

06τ6t

(ka^ε(τ)k_L² +ka^ε(τ)k_W^1,∞+k∇v^ε(τ)k_L^∞)

6M_s^ε₁(0) exp(CtsupM_s^ε₀),

and so that M_s^ε₁(t) < ∞ holds for t ∈ [0, T_s0]. It means that the solution (a^ε, v^ε) extends to timeTs0 as a H^s1×X^s1+1∩Lⁿ²⁻²ⁿ -valued function, that is, T_s1 >T_s0. The same argument also shows T_s0 >T_s1. Therefore, T does

not depend on s.

4.6. Construction ofφ^ε. We finally constructφ^εfromv^εdefined in Propo- sition 4.1. Since v^ε is known, in view of (2.2), it is natural to define φ^ε as

(4.22) φ^ε(t, x) =φ₀(x)− Z _t

0

1

2|v^ε(τ, x)|²+λ |x|^−γ∗ |a^ε|² (τ, x)

dτ.

By Assumption 1.1 and Lemma 3.1,φ₀∈L^∞∩L

nq0

n−q0. Proposition 4.1 shows that

|v^ε|² ∈C

[0, T];X^s+1∩L

nq0 n−q0

. Lemma 3.6 with k= 2 and s₁=s₂ =sshows that

∇² |x|^−γ∗ |a^ε|²

∈C([0, T];H^s).

By the Hardy–Littlewood–Sobolev inequality, for all n/(γ+ 1)< q <∞, it holds that

∇(|x|^−γ∗ |a^ε|²)

L^q 6C

(|x|^−γ−1∗ |a^ε|²)

L^q 6Cka^εk²_Hs, and ∇ |x|^−γ∗ |a^ε|²

∈ C [0, T];H^s+1

. Moreover, the Sobolev inequality shows |x|^−γ ∗ |a^ε|² ∈ L^∞. Therefore, φ^ε has the regularity announced in Theorem 1.4. To conclude, we simply notice the identity

∂_t(∇φ^ε−v^ε) =∇∂_tφ^ε−∂_tv^ε = 0,

so that v^ε = ∇φ^ε, and (4.22) yields the second equation in (2.2). This completes the proof of Theorem 1.4.