On the management fourth-order Schr\"{o}dinger-Hartree equation

We consider the Cauchy problem associated to the fourth-order nonlinear Schr\"{o}dinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in $H^s$-spaces. We also analyze the scaling limit of fast dispersion management and the convergence to a model with averaged dispersions.

An additional point related to model (1.4), also motivated by models in nonlinear optics, corresponds to the case of dispersion managed α = α(t), β = β(t), modelling varying dispersion along the fiber, which permits to balance the effects of nonlinearity and dispersion in such a way that stable nonlinear pulses (solitary waves) are supported over long distances (cf. [1,2,20,23,24,26,31]). See also Carvajal, Panthee and Scialom [5], and some references therein, to the case of a third-order nonlinear Schrödinger equation with time-dependent coefficients.
The second situation is related to the posible interactions described by the potential V. An interesting interaction is given by the following coupled system i∂ t ψ(t, x) + ∆ψ(t, x) = V (t, x)ψ(t, x), x ∈ R n , t ∈ R, ∆V (t, x) = −|ψ(t, x)| 2 , x ∈ R n , t ∈ R, (1.5) where V (t, x) is a potential function. If n ≥ 3, the potential V can be explicitly written as a solution of the Poisson equation (1.5) 2 as V = C n (|x| −(n−2) * |ψ| 2 ), (1.6) where C n is a constant which only depends on n. Thus, substituting (1.6) into the Schrödinger equation (1.5) 1 we obtain the so called Schrödinger-Hartree equation i∂ t ψ(t, x) + ∆ψ(t, x) = (|x| −(n−2) * |ψ(t, x)| 2 )ψ(t, x), x ∈ R n , t ∈ R. (1.7) The nonlinearity in equation (1.7) has been generalized by considering the Hartree type nonlinearity (| · | −λ * |ψ| 2 )ψ, λ > 0, which is relevant to describing several physical phenomena, as for instance, the dynamics of the mean-field limits of many-body quantum systems such as coherent states and condensates, the quantum transport in semiconductors superlattices, the study of mesoscopic structures in Chemistry, among others (cf. [7,21,29]). From the mathematical point of view, some significative results on well-posedness in energy spaces has been obtained in [9,25,30] and references therein.
Based on the previous considerations, in this paper we study the Cauchy problem associated to the following fourth-order Schrödinger-Hartree equation with variable dispersion coefficients where the unknown u(x, t) is a complex-valued function in space-time R n × R, n ≥ 1, and u 0 denotes the initial data in t 0 ∈ R. The coefficients α, β are real-valued functions which represent the variable dispersion coefficients. The constant θ = 0 is a real coefficient which denotes the focusing or defocousing behavior (when diffraction and nonlinearity are working against or with each other). The nonlinearity coefficient λ > 0.
The general IVP (1.8) has not been considered in the literature. Thus, in this paper, we are interested in studying the well-posedness issues for the IVP (1.8) for given data based in the L 2 -Sobolev spaces and, α, β continuous or piecewise constant periodic functions. The novelty of our results is summarizes in the following aspects: For initial data u 0 ∈ H s (R n ), s ≥ {0, λ/2 − 2} and 0 < λ < n, we prove the existence of local in time solution u ∈ C([−T + t 0 , T + t 0 ]; H s (R n )). The proof is based on L p t L q x properties of the linear propagator, as well as the Hardy-Littlewood-Sobolev inequality which allow us to control the Hartree nonlinearity. For initial data in L 2 , by using the conserved quantity u(t) L 2 (R n ) = u 0 L 2 (R n ) we are able to extend the local solution globally. We also prove the existence of global solution in H 1 by combining the L 2 -conservative law, the local well-posedness in H 1 and argument of blow up alternative. If the nonlinearity is given by θ|u| 2 u, we also analyze the existence of global solution in H s , s ≥ 0. Finally, we will address the scaling limit to fast dispersion management, that is, for each ǫ > 0, we consider the ǫ-scaled fourth-order nonlinear Schrödinger equation by making β ǫ (t) = β( t ǫ ), α ǫ (t) = α( t ǫ ), and then, we analyze the scaling limit ǫ → 0 + of the solutions.
This article is organized as follows. In Section 2, we establish some linear estimates which are fundamental for obtaining our results of local and global mild solutions. In Section 3, we prove the existence of local solutions in H s for s ≥ λ/2. In Section 4, we analyze the existence of local solutions in H s for {0, λ/2 − 2} ≤ s < λ/2. In Section 5, we prove some results of global existence. Finally, in Section 6, we give a result about the scaling limit to fast dispersion management.

Linear propagator
Before studying the nonlinear Cauchy problem we give some properties of the linear problem associated to (1.8), which is given by For α and β being integrable functions, we define the cumulative dispersions A(t 0 , t) and B(t 0 , t) on the closed interval [t 0 , t] by We denote by U α,β (t, t 0 ) the linear propagator which describes the solution u(x, t) of (2.1). It holds that Then, for any t, r, l ∈ R, it holds We will use the notation U (t, t 0 ) := U α,β (t, t 0 ) and U (t) := U (t, 0). Then, (2.2)-(2.3) imply that For each s ∈ R, the propagator U (t, t 0 ) is an isometry on H s (R), that is, for any f ∈ H s (R) it holds unless α, β be constant functions. Thus, U (t, r) is not a group.
Next lemmas will be useful in order to estimate the nonlinearity in (1.8).
Following the arguments in Tomas [27], p.477, we get . (2.12) In order to conclude the proof it is enough to show that Note that for t, r, τ ∈ I m , we have |B(t, r) − B(τ, r)| = β ± |t − τ |, then by Fubini's Theorem, Lemma 2.5 and the Hardy-Littlewood-Sobolev inequality, we have

Now, from (2.12) and (2.13) we obtain
This finishes the proof of (2.10). Now we prove (2.11). By hypothesis, the points (p 2 , q 2 ) and (p 1 , q 1 ) are in the segment of the line connecting P = 1 2 , 0 with Q = if n ≥ 5. Therefore, without loss of generality we can assume that p 2 ∈ [2, p 1 ). This implies that .

From the last estimates and an interpolation argument we have
From the last inequality and an argument of duality, we obtain t t0 U (t, τ )g(·, τ )dτ which yields the result.

Linear propagator with continuous dispersion
In this section we analyze the propagator associated to the linear problem (2.1), where the dispersion functions α, β ∈ C([−T + t 0 , t 0 + T ]) such that β(t) = 0 for all t ∈ R. Below, we establish some Strichartz estimates related to the linear propagator U (t, r). From now on, we will use the notation Proposition 2.8 For each admisible pairs (q, p), (q 1 , p 1 ), (q 2 , p 2 ), it holds: (2.14) 2. There exists Proof: In order to proof (2.14) we again use a duality argument. Thus, it is enough to show that for From Fubini's Theorem, Cauchy-Schwarz inequality and Plancherel's identity, we arrived at Similarly to the proof of Proposition 2.7 we get In order to conclude the proof it is enough to show that Therefore, using the Fubini's Theorem, following Lemma 2.1 and the Hardy-Littlewood-Sobolev in- This rest of the proof of (2.14) is similar to Proposition 2.7. On the other hand, a duality argument analogous to the proof of (2.11) permits to prove (2.15).

Local well-posedness in
In this section we prove the local existence in H s (R n ) with s ≥ λ 2 . We assume that the variable dispersion α, β verifies either α, β ∈ C([−T + t 0 , T + t 0 ]) with β(t) = 0, for all t ∈ [−T + t 0 , T + t 0 ], or α, β are periodic piecewise constants. Results of local well-posedness in the case of the Schrödinger-Hartree equation with constant dispersion (α(t) =constant and β(t) = 0) were obtained in Miao et al [25]. The proof is obtained through the contraction mapping argument. For that, as usual, we consider the solution of (1.8) via the Duhamel's formula which is given by (3.1)

Local well-posedness with continuous dispersion
From (2.4), Lemmas 2.2, 2.3 and 2.4, and the Sobolev embedding, we obtain and CT 0 R 2 ≤ 1 2 , we have that Φ 1 maps X s T0,R to itself. Now, from the Hölder inequality, Lemma 2.2 and the Sobolev embedding, we get Thus, if we take T 0 ≤ T small enough, Φ 1 is a contraction. Consequently, Φ 1 has a unique fixed point at X s T0,R which is solution of (3.1). Finally, we will prove the time-continuity of the solution. For that, Then, taking the H s -norm of the difference between (3.1) and (3.5) we get Notice that .
Since A(t, t 0 ) and B(t, t 0 ) are continuous in the variable t, and u 0 ∈ H s , then the Lebesgue Dominated Convergence Theorem implies that lim t→t1 J 1 = 0. On the other hand we have From (2.2), (2.4) and taking into account that Thus we conclude the proof of Theorem 3.1.
In this section we prove the local existence in H s (R n ) with max{0, λ 2 − 2} ≤ s < λ 2 . As before, we assume that the variable dispersion α, β satisfies either α, β ∈ C([−T + t 0 , T + t 0 ]) with β(t) = 0, for all t ∈ [−T + t 0 , T + t 0 ] or α, β are periodic piecewise constants. The proof is obtained through the contraction mapping argument. However, it is not easy to obtain the solution by using the contraction mapping approach only in C([0, T ]; H s (R n )). As usual, we use the Strichartz estimates, obtained in Section 3, and the Hardy and Hardy-Littlewood-Sobolev inequalities in order to obtain the existence of local solutions in C([0, T ]; H s (R n )) ∩ L q T (H s p (R n )), for some admissible pair (q, p). Consider the mapping Φ 1 defined in (3.2), and let

Local well-posedness with continuous dispersion
3n−2s−2λ . In the same way, we also have Thus, if we choose R and T 0 ≤ T such that C u 0 H s ≤ R 2 and CT ρ 0 R 2 ≤ 1 2 , we have that Φ 1 maps Y s T0,R to itself. Now, let u, v ∈ Y s T,R . Then, from Proposition 2.8 we get Using Lemma 2.4, the Hardy and Hölder inequalities, we have In a similar way, Thus, if T 0 ≤ T is small enough, Φ 1 is a contraction. Consequently, Φ 1 has a unique fixed point at Y s T0,R which is solution of (3.1). The time-continuity of the solution follows in the same spirit of the end of the proof of Theorem 3.1.

Global existence
The aim of this section is to analyze the global well-posedness of (1.8). We prove that the local solution of the initial value problem (1.8), with initial data in L 2 and H 1 , can be extended to the real line R.

Global existence in L 2 (R n )
In this subsection, we analyze the global existence of solutions for the model (1.8) with α, β verifying either α, β ∈ C(R) with β(t) = 0, for all t ∈ R, or α, β are periodic piecewise constants. Taking into account the mass conservation u(t) L 2 = u(t 0 ) L 2 and the local theory in L 2 , we are able to extend the local solution obtained in Theorem 4.1 globally in time. This is the content of next theorem.
Theorem 5.1 Let u 0 ∈ L 2 (R n ) and 0 < λ < min{n, 4}. Then, the local solution to the initial value problem (1.8) obtained in Theorems 4.1, 4.2 can be extended to the real line R.
Proof: First we consider the case α, β are periodic piecewise constants. Note that in the proof of Theorem 4.2, the time existence of the solution u(t) depends only on u 0 L 2 . More exactly, T can be taken as T Since u(t) X 0 T 0 ,R ≤ C u 0 L 2 and u(t) L 2 = u 0 L 2 for all t on the time-interval of the existence, a standard continuity argument implies that, on each subinterval I 1 m and I 2 m , there exists a solution u ∈ L ∞ (I 1,2 m ; L 2 (R n )). Considering the union of sub-intervals I m , we infer the existence of a solution u ∈ L ∞ (R; L 2 (R n )). The continuity in time is obtained in a similar way as in Theorem 3.1, therefore there exists a solution u ∈ C(R; L 2 (R n )).
In the case α, β ∈ C(R) with β(t) = 0, for all t ∈ R, the L 2 -conservative law and the local theory in L 2 provided by Theorem 4.1, also give the global existence in L 2 .

Global existence in H 1 (R n )
If α and β are constants, the solution u of (1.8) satisfies the following energy conservation law Therefore, for some particular signs of α, β, θ, and by using the following generalized Gagliardo-Niremberg inequality Theorem 5.2 Let u 0 ∈ H 1 (R n ), 0 < λ < n and λ < 4. Assume that α, β ∈ C(R) with β(t) = 0, for all t ∈ R. Then, the local solution to the initial value problem (1.8) can be extended to R.
Proof: Recall that the L 2 -solutions u of (1.8)-(2.8) satisfies the mass conservation law Moreover, the following relation holds Then, if p ≥ 1 is such that 2 p + λ n = 1, from (5.2), Hardy-Littlewood-Sobolev and Hölder inequalities, we get Suppose that T max < ∞. Then, if 1 q1 + 1 q2 = 1 2 , from (5.3) and Gronwall inequality, for 0 ≤ t < T max we have that Without loss of generality we consider t 0 = 0; then, we use the equation in order to obtain an estimate of u L 2q 2 ([0,Tmax];L p (R n )) . Indeed, for 0 ≤ t ≤T 0 ≤ T max , we get At this point we need to consider that (2q 2 , p) is an admissible pair. This condition implies that q 2 = 4 λ and thus, we find the restriction λ ≤ 4. Therefore, from Hölder inequality and Lemma 2.3, we obtain .
By continuity there exists T * ≤T 0 such that Notice that (5.5) is also valid for T * instead ofT 0 . Replacing (5.6) in (5.5) we get that and this contradicts the choice ofT 0 . Therefore, considering T (5.7) If T 0 = T max we finish the proof. Suppose that T 0 < T max . Then, we repeat the above argument to obtain a priori estimate in the interval [0, 2T 0 ]. Indeed, from Duhamel's formula we have that For T 0 ≤ t ≤ T 0 +T 1 < T max , from the Hölder inequality and Lemma 2.3 we arrived at .
Again, taking 0 <T , we obtain (5.8) Therefore, we can choseT 1 = T 0 . From (5.7) and (5.8), we obtain Repeating this process a finite number of steps and using the value of T 0 we arrived at, Replacing (5.9) in (5.4) we get the estimate for any 0 ≤ t < T max , which is a contradiction to the blow-up alternative. Therefore, T max = ∞.

Remark 5.3
Combining the arguments in the proof of Theorem 3.3 with those in the proof of Theorem 5.2 we can prove that the local solution to the initial value problem (1.8) obtained in the case α, β piecewise constant, can be extended to R.

Remark 5.4
We could try to prove the global existence in H s combining the local existence in H k , k ∈ N, and an interpolation argument. We could use an induction argument on k to prove global existence for initial data in H k (R n ) with k ≥ 2 an integer. For that, we need an a priori estimate to show that the global existence of (1.8) in H k−1 (R n ) implies the global existence in H k (R n ). However, if we multiply the first equation in (1.8) by D 2α xū , where α is a multi-index with |α| ≤ k, k > 1, next, conjugate (1.8) and multiply it by D 2α x u, and then, we add the two obtained equations and use basic properties of the Laplacian and the operator ∆ 2 , we arrived at By Leibnitz's rule we have that Thus, from (5.10) we obtain Unfortunately, seems so difficult to control the right hand side of (5.11) in terms of the norms u H 1 and u H k−1 .

5.3
Global well-posedness in H s (R n ) with s > 0 and nonlinearity |u| 2 u Taking into account the Remark 5.4, throughout this section we consider the model In this case, from Duhamel's formula we have, The proof of the next theorem is similar to that one of Theorem 4.1.
. Then there exists T 0 = T 0 ( u 0 H s ) ≤ T and a unique solution u of the integral equation (5.13) in the class C([−T 0 + t 0 , T 0 + t 0 ]; H s (R n )) ∩ L q T0 (H s p (R n )). Proof: Consider the mapping Since U (t, t 0 ) is unitary in H s , using Propositions 2.8, Lemma 2.4, Hölder inequality and Sobolev embeddings, we obtain 6−n+2s and b = 6n n−2s . The rest of the proof es very similar to that one in Theorem 4.1.
Next, we will analyze the global well-posedness in H s (R n ) with s ≥ 0. For that, next lemma will be useful.
Lemma 5.6 [3] Let f, g ∈ H r (R n ), with r > 1 2 and h ∈ H s (R n ), with 0 ≤ s ≤ r. Then Theorem 5.7 Let u 0 ∈ H s (R n ), with s ≥ 0, n < 4. Assume that α, β ∈ C(R) with β(t) = 0, for all t ∈ R. Then the local solution to the initial value problem (5.12) can be extended to R.
Proof: We already to known that the L 2 -solutions u of (5.12) also satisfies the mass conservation law Moreover, the following relation holds Let u 0 ∈ H 1 (R n ) and T max be the maximal existence time of the solution to (5.12). Suppose that T max < ∞. Then, for 0 < t < T max , from the (5.14) and the Hölder inequality we arrive at Without loss of generality we consider t 0 = 0; then, we use the equation in order to obtain an estimate of u L 2q 2 ([0,Tmax];L ∞ (R n )) . Indeed, At this point we need to use the inequality (2.14), which implies that q 2 = 4 n and n < 4. Hence, .
By continuity there exists T * ≤T 0 such that Notice that (5.16) is also valid for T * instead ofT 0 . Replacing (5.18) in (5.16) we get that and this contradicts the choice ofT 0 . Therefore, considering T 4−n 4 If T 0 = T max we finish the proof. Suppose that T 0 < T max . Then, we repeat the above argument to obtain a priori estimate in the interval [0, 2T 0 ]. Indeed, from Duhamel's formula we have that For T 0 ≤ t ≤ T 0 +T 1 < T max , from (2.14), we arrived at Repeating this process a finite number of steps and using the value of T 0 we arrived at, Replacing (5.21) in (5.15) we get the a priori estimate, for any 0 ≤ t < T max , which is a contradiction to the blow-up alternative. Therefore, T max = ∞.
Next, we use an induction argument on k to prove global well-posedness for initial data in H k (R n ) with k ≥ 2 an integer. For this we use an a priori estimate to show that the global well-posedness of (5.12) in H k−1 (R n ) implies the global well-posedness in H k−1 (R n ).
First, multiply equation (5.12) by D 2α xū , where α is a multi-index with |α| ≤ k, next conjugate (5.12) and multiply by D 2α x u, add the two equations obtained and from the properties of the Laplacian and the operator ∆ 2 , we arrived at By Leibnitz's rule we have that For 0 < β < α, using Proposition 5.6 with s = 0 we get For 0 < β < α, using Proposition 5.6 with s = 0 we get By Gromwall's inequality we get In order to obtain global well-posedness in the fractional Sobolev space H s (R n ), with s > 0 not an integer, a straightforward argument of nonlinear interpolation theory can be used, which finishes the proof of the theorem.
Remark 5.8 Combining the arguments in the proof of Theorem 3.3 with those in the proof of Theorem 5.7 we can prove that the local solution to the initial value problem (1.8) obtained in the case α, β piecewise constant (Theorem 3.3), can be extended to R.