Well-posedness of a higher-order Schrödinger–Poisson–Slater system

In this paper, we show the global well-posedness of a higher-order nonlinear Schrödinger equation. Specifically, we consider a system of infinitely many coupled higher-order Schrödinger–Poisson–Slater equations with a self-consistent Coulomb potential. We prove the existence and uniqueness global in time of solutions in L2(R3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}( \mathbb{R}^{3})$\end{document} and in the energy space.


Introduction
This paper is devoted to the mathematical analysis of the following higher-order Schrödinger-Poisson-Slater mixed system (SPS in short) in three space dimensions: System (1) belongs to the family of higher-order Schrödinger equations that have been developed in, e.g., [7,8]. These are Schrödinger-type equations involving higher-order Schrödinger operator converging, in principle, to the semi-relativistic bound-state equation called the spinless Salpeter equation; see, e.g., [9][10][11][12]. The semi-relativistic spinless Salpeter equation is of high importance in particle physics and provides a particularly well defined approximation to the Bethe-Salpeter formalism designed for the Lorentzcovariant description of bound states within relativistic quantum field theory. The spinless Salpeter equation can be formally obtained using a three-dimensional reduction and suitable justified physical hypotheses. More importantly, it can be regarded as a generalization of the classical Schrödinger equation with the inclusion of the relativistically correct freeparticle kinetic energy. We refer the reader to [6] for a more detailed discussion of the physical background and application of the higher-order Schrödinger equations (also see [7] for an application to the case of alpha particles and see [13,14] for alternative applications).
In the sequel, we assume that the sequence of nonnegative real numbers {γ j , j ∈ N} ⊂ 1 (N) are fixed. That is, they do not depend on time as in the time dependent multiconfiguration models (see e.g. [15] ). Next, we recall the definition of some functional spaces, originally introduced in [16,17]. We denote γ := (γ 1 , γ 2 , . . .) and we define, for all 1 ≤ p ≤ q ≤ +∞, the L p (γ ) space as follows: For a given time interval I, we denote L q,p I (γ ) := L q (I; L p (γ )). We shall use the notation L q,p T (γ ) when I = [0, T]. Also, we shall use the shorthand notation L q,p loc (γ ) and L q,p (γ ) for L q loc ([0, +∞); L p (γ )) and L q ([0, +∞); L p (γ )), respectively. Now, we introduce Eventually, we define the set The main result of this paper is the following.
Theorem 1.1 Let J ≥ 2 and φ = (φ i ) i∈N be a set of initial data in L 2 (γ ) and (q, p) ∈ J such that 3 < p ≤ 6. Then the SPS system (1) has a unique solution ψ(t, x) satisfying This theorem shows the well-posedness of system (1) with initial data in L 2 instead of the energy space. Also, it is worth mentioning that Theorem 1.1 can be extended as follows.
There is a very rich mathematical literature dedicated to the analysis of the Hartree and Hartree-Fock equations. The well-posedness in the energy space H 1 (R 3 ) of the Hartree-Fock equations (δ = 1, k = 1, . . . , N ) was obtained in [18] (see also [19]). An L 2 theory was established, independently, in [16] and [17]. The authors established Strichartz' type estimates for the free Schrödinger semigroup in some weighted spaces, and used a contraction argument to conclude. In [20], the authors showed the well-posedness of the classical SPS system in L 2 (R 3 ), and in the energy space H 1 (R 3 ). In the context of higher-order Hartree-Fock equations, the existence and uniqueness of solutions in L 2 (R 3 ), H 2J (R 3 ) and H J (R 3 ) is proved for J ≥ 2 in [6]. Theorem 1.1 extends this result to the case of the SPS system.

Proof of Theorem 1.1
In this section we prove Theorem 1.1 by proceeding in two steps. First, we collect a few technical lemmas providing necessary Lipschitz bounds to set a contraction argument and obtain the existence and uniqueness of solutions in L q,p (γ ). The second step consists in the proof of the existence and uniqueness of solutions in H 2J (γ ). From this point onward, κ will denote variant universal constants that may change from line to line of an inequality, and we will emphasize it when it depends on important parameters. Also, we shall use the notation ζ (0) instead of ζ (t = 0) for all functions, and Ψ := ψϕ and ρ Ψ := ρ ψρ ϕ when there is no confusion.
For all J ∈ N , we denote by U J (t) the propagator of the free higher-order Schrödinger operator. More precisely, In particular, U J (t) is an unitary operator on every H s for all s ∈ R (see Ref. [6], Proposition 3.4). In order to prove the existence of solutions, we seek a fixed point of the following functional: in a well-chosen closed ball. To ease the notation, we shall use the following: for all time intervals I, so that we can write

L q,p (γ ) solutions
In this subsection, we prove the first part of Theorem 1.1. For that purpose, we need the following local Strichartz estimates for the propagator U J (t).

2.
If I contains the origin, there exists κ := κ(p 1 , p 2 ) such that This lemma is an extension of the results of [21]. The adaptation of the classical Strichartz' estimates to our weighted spaces is straightforward, and we refer the reader to [17] for a proof. Thanks to Lemma 2.1, we have whenever the right hand side is finite. Next, we recall the following mass-conservation property of the SPS system.

Lemma 2.2
Let T > 0 and φ = (φ i ) i∈N a set of initial data in L 2 (γ ). If there exists a weak solution ψ(t) = (ψ i (t)) i∈N to the SPS system on [0, T], then (ψ i (t)) i∈N is such that Proof On the one side, we multiply the SPS system (1) byψ l and integrate with respect to space. On the opposite side, we replace k by l in (1), take the complex conjugate of the system, multiply by φ l , integrate with respect to space, and integrate by parts the term Eventually, summing up the two equalities leads to the desired the result. Now, we estimate the linear and nonlinear parts in B q,p T (γ ) for all (q, p) ∈ J and T > 0. First, we have Lemma 2.3 Let T > 0, (q, p) ∈ J such that 3 < p ≤ 6 and ψ, ϕ ∈ B q,p T (γ ). Then there exists κ := κ(p) > 0 such that Proof The proof is based on a cut-off of the Coulomb potential 1 |x| . Indeed, for all function χ ∈ C ∞ such that χ(r) = 1 for 0 ≤ r ≤ 1 and χ(r) = 0 for r ≥ 2, we have clearly On the one side, we set b = +∞ and obtain On the opposite side, using Hölder's inequality in time and space, combined with the fact that (q, p) ∈ J, we obtain The right hand side of this estimate is finite for all 3 < p ≤ 6, which ensures that p p-2 ∈ [ 3 2 , 3[. Summing up the two inequalities we get (4). Now, let us make clear the relation between the norms of the density ρ ψ and the wave- Also, using the reverse Minkowski inequality for all p ≥ 2, we obtain ρ 1 2 ψ L p ≤ ψ L p (γ ) . We shall use this property tacitly in the sequel. Now, the Poisson part of the system satisfies the following.
Proof The proof is the same as in Ref. [17], we sketch it here for the reader's convenience. We have .
We shall use the cut-off of the Coulomb potential given in the proof of Lemma 2.3. On the one hand, recalling that ρ ψ = i γ i |ψ i | 2 and using Young's and Hölder's inequalities, we get Equivalently, we have Gathering these estimates, we obtain (5). Observe that the condition 3 < p ≤ 6 is needed to bound the short-range part of the potential as in Lemma 2.3. Now, we provide estimates for the Slater part of the SPS system.
Proof We follow the argument of [20] using different L q,p (λ) spaces. First, using a firstorder Taylor expansion, we can write for all θ ∈ [0, 1] Indeed, the Fréchet derivative of ρ 1 3 ζ ζ is given by where the symbol denotes the real part of complex numbers, and h, ψ 2 (γ ) denotes the Euclidean scalar product in 2 (γ ), h, ψ 2 (γ ) := j γ j h jψj . Our aim is to estimate T 1 and The proof of this fact is a straightforward application of Minkowski's inequality. Indeed, let (q, p) ∈ J, then using Hölder's inequality in space and time we get In order to pass from the second to the third line of the estimate above, we used the fact that ρ 1 2 ζ L q,p ≤ ζ L q,p (γ ) for all p ≥ 2 and ζ ∈ L q,p (γ ). Observe that Jp Jp-1 is the conjugate of Jp and 6p 3p+4 is the conjugate of 6p 3p-4 , and that the pair (Jp, 6p 3p-4 ) ∈ J. Eventually, we have obviously 1 -1 2J > 0 for all J ≥ 2. Now, we make an estimate of T 1 : In the first line, we used the Cauchy-Schwartz inequality, | h, ψ 2 (γ ) | 2 ≤ ρ ψ ρ h and the notation |ζ | = (|ζ j |) j∈N . In the second line, we used the fact that ζ L q,p (γ ) ≤ ρ 1 2 ζ L q,p for all p ≤ 2 and ζ ∈ L q,p (γ ). Clearly, we have 6p 3p+4 ≤ 2. Eventually, the proof of the lemma is achieved by setting (r, s) = (Jp, 6p 3p-4 ). Now, we are able to set up a contraction argument to prove the first part of Theorem 1.1. Let 0 < T ≤ 1 to be fixed later on, φ = (φ i ) i∈N be a set of orthonormal initial data in L 2 (γ ), and ψ = (ψ i ) i∈N , ϕ = (ϕ i ) i∈N in the following closed ball: Using Lemmas 2.1-2.5, we can write for all (q, p) ∈ J such that 3 < p ≤ 6 and J ≥ 2, We have Therefore, we get This shows that Λ maps B R into itself. Equivalently, Therefore, if we now let then there exists a constant 0 <η < 1 such that We infer that Λ is a strict contraction on B R . Therefore, the SPS system has a solution in L ∞ ([0, T]; L 2 (γ )). Moreover, the solution is unique in L ∞ ([0, T]; L 2 (γ )) ∩ L q,p T for all (q, p) ∈ J such that 3 < p ≤ 6. The fact that the solution is in C 0 ([0, T]; L 2 (γ )) follows using standard arguments. The last point to make clear is the global existence of the solution, which is a consequence of Lemma 2.2. Indeed, the time T > 0 fixed above depends only on the L 2 norm of the initial data, which is conserved by the dynamics of the SPS system. Thus, one can reiterate the contraction argument to cover the whole real line.

H 2J (γ ) solutions
In this section we prove the second part of Theorem 1.1. For that purpose, we follow the method of Kato (see Refs. [22,23]) by setting a fixed point argument. From this point onward, we shall use the notation C Proof The first assertion follows immediately from the expression of U J (t) in Fourier variables. Now, let ζ be an initial data to the equation Then the second assertion is a consequence of Strichartz' estimates given in Lemma 2.1 and the fact that This finishes the proof.
This lemma provides an estimate which is equivalent to the first estimate in (3). The equivalent of the second estimate in (3) is a direct consequence of the following. and Proof The proof is based on Strichartz' estimates given in Lemma 2.1. Indeed, the first assertion is nothing but the second estimate in Lemma 2.1 with the pair (1, 2) as the conjugate of (∞, 2) ∈ J. To prove the second assertion, we observe that

This formula is justified by the fact that
In particular ζ (0) makes sense, actually it belongs to L 2 (γ ). Thanks to Lemma 2.1, we have In order to prove the last assertion, we write This formulation is justified by the same argument as above, thus we have (-) J Λ [0,T] (ζ ) ∈ L ∞,2 T (γ ), and the desired estimate follows immediately using the second assertion. Now, we have C(ψ) + P(ψ) -S(ψ) : H 2J (γ ) → L 2 . Thus, it is rather easy to see that there exists 0 ≤ s < 2J such that C(ψ) + P(ψ) -S(ψ) ∈ C 0 (H s (γ ), L 2 (γ )). For instance, if we fix That is, C(ψ)+P(ψ)-S(ψ) ∈ L ∞,2 T (γ ) for all ψ ∈ L ∞ T H s (γ ). Also, there exists a continuous function η : Let us write C = C 1 + C 2 , and equivalently for P with respect to the short and long ranges of 1 |x| . Then we have the following.
Let 0 ≤ s < 2J such that the three estimates (8) hold true, and (q, p) ∈ J such that p = max{p,p} of Lemma 2.8. Also, let T, K > 0 be fixed hereafter, and introduce the space Notice that L(γ ) = ∅ for all γ > 0 since ψ(t) = ϕ ∈ L and that (L, d) is a complete metric space when endowed with the distance d(ψ, ϕ) = ψϕ B q,p T . Thanks to (8), in particular (9), we see that Λ [0,T] (C(ψ) + P(ψ) + S(ψ))(t, x) belongs to C 0 T L 2 (γ ), therefore it is well defined. Now, thanks to Hölder's inequality, if ψ ∈ L(γ ), then ψ ∈ W 1,q T Lp(γ ). Indeed, With Lemma 2.8, one deduces the following estimates: Next, using (9) with the first estimate of Lemma 2.7, we obtain the existence of a constant κ independent of K and T such that The other estimates of Lemma 2.7 show that Λ [0,T] (C(ψ) + P(ψ) + S(ψ)) ∈ L ∞ T H 2J (γ ) ∩ W 1,l T L m (γ ) for all ψ ∈ L(γ ) and all pairs (l, m) ∈ J. Also, using Lemma 2.7 and estimates (9)-(10), we obtain (by possibly choosing a largerκ) and Combining these estimates with Lemma 2.6 and (10), we obtain (by possibly choosing largerκ) Also, Next, we choose K in L(γ ) such that Hence, using (12) we obtain Eventually, using the interpolation inequality ψ H s (γ ) ≤ ψ 2-s 2 H 2J (γ ) , and the estimates (11) and (13), and possibly choosing T smaller, we obtain In particular, the functional Λ maps L(γ ) into itself. A slight modification of the estimates of the nonlinear terms of the SPS system shows that one can choose T possibly smaller in order to show that the mapping Λ is a strict contraction on the complete metric (L(γ ), d). Therefore, we can deduce the existence of a fixed point ψ ∈ L(γ ) solving the system (1) with initial data φ ∈ H 2J (γ ). Lemma 2.6 and Lemma 2.7 allow one to deduce that ψ(t, x) ∈ C 0 T H 2J (γ ) ∩ C 1 T L 2 (γ ). For this purpose, the technical point is to show that C(ψ) + P(ψ) -S(ψ) ∈ C 0 T L 2 (γ ). This can be achieved by observing that ψ ∈ C 0 T L 2 (γ ) and using (13) to obtain ψ ∈ L ∞ T H 2J (γ ). Eventually, the interpolation inequality used before leads to ψ ∈ C 0 T H s (γ ), hence C(ψ) + P(ψ) -S(ψ) ∈ C 0 T L 2 (γ ) since C(ψ) + P(ψ) -S(ψ) ∈ C(H s (γ ), L 2 (γ )). Observe that ψ ∈ W 1,l T L m (γ ) for all pairs (l, m) ∈ J, and the uniqueness of the solution readily follows using a difference argument (the proof is in the same spirit of the one showing that the mapping Λ is a strict contraction). The blow-up alternative and the continuous dependence on the initial data can be shown using standard arguments, and we refer to, e.g., [23]. Eventually, notice that the time T depends on J, p,p,p and K , and that K depends only on the initial data, therefore one can reiterate the argument (with initial data ψ(T), ψ(2T), . . .) and cover the whole real line, which finishes the proof of Theorem 1.1.

H J (γ ) solutions
The proof of Corollary 1.2 can be achieved using a limiting procedure based on Theorem 1.1. Indeed, we construct a solution in H J (γ ) as a limit of a sequence of solutions in H 2J (γ ). The proof is standard, we skip it and refer to [23] for similar arguments (see also [20]). The key point is to show the boundedness of ψ in H J (γ ), which is a consequence of the following. Lemma 2.9 Let T > 0 and φ = (φ i ) i∈N a set initial data in H 2J (γ ) and ψ(t) = (ψ i (t)) i∈N the associated solution to the SPS system. Then, for all t ∈ Proof The proof is standard. The higher-order operator and the Slater terms do not bring any new difficulties, and we refer the reader to [18] for an explicit proof in the case of Hartree and Hartree-Fock, and to [15,24] for a proof in a more general setting (see also [25] for the pseudo-relativistic case, that is, when the kinetic operator is given by -α 2 x i + α -4 where α > 0 denotes Sommerfeld's fine structure constant).
To show that there exists a constant such that ψ L ∞ T H J (γ ) ≤ C for all T > 0, we need to control the kinetic part of the energy which obviously depends on the parity of J. For this purpose, we need the following Gagliardo-Nirenberg inequality: Therefore, it is enough to focus on the leading term in the kinetic energy part, namely (-1) J+1 α(J) 2J m 2J-1 c 2J-2 J 2 ψ L 2 (γ ) . Let us assume that J is odd. Then, using (14) with the mass conservation and the estimate  , we infer that the leading term in the kinetic part dominates the Slater correction since J > 3 8 . Thus ψ ∈ L ∞ T H J (γ ). Now, assume J is even, then again using (14) with the mass conservation and the estimates , we infer that the leading term of the kinetic part dominates the potential part. Therefore ψ ∈ L ∞ T H J (γ ).