ON THE CAUCHY PROBLEM FOR THE XFEL SCHR¨ODINGER EQUATION

. In this paper, we consider the Cauchy problem for the nonlinear Schr¨odinger equation with a time-dependent electromagnetic ﬁeld and a Coulomb potential, which arises as an eﬀective single particle model in X-ray free electron lasers(XFEL). We ﬁrstly show the local and global well-posedness for the Cauchy problem under the assumption that the magnetic potential is unbounded and time-dependent, and then obtain the regularity by a ﬁxed point argument.

Under this assumption, the domain D((i∇ − A(t, x)) 2 ) = {u ∈ L 2 (R 3 ), (i∇ − A(t, x)) 2 u ∈ L 2 (R 3 )} does not depend on time t, see [15]. Therefore, for s ∈ N, the space H s A(t) := {u ∈ S (R 3 ), (1 + (i∇ − A(t, x)) 2 ) s/2 u ∈ L 2 (R 3 )} does not depend on time t, where S (R 3 ) denotes the space of tempered distribution on R 3 . Moreover, H s A(t) -norm can be defined by u H s A(t) = (i∇ − A(t)) s u L 2 + u L 2 . In particular, if A(t, x) is a bounded function and satisfies Assumption 1, then u H s A(t) = u H s . Because the space H A(t) is independent of time t, we often abbreviate H A(t) and · H s A(t) by H A and · H s A respectively. Equation (1) has important applications in physics. A solution u of this Schrödinger equation can be considered as the wave function of an electron beam, under the action of magnetic potential A(α(t), x), interacting with an atomic nucleus, located at the point β(t), interacting self-consistently through the Coulomb (Hartree) force with strength λ 2 , the local Fock approximation with strength λ 3 and exponent σ, see [13]. Since XFEL is more powerful by several orders of magnitude than more conventional lasers, the systematic investigation of many of the standard assumptions and approximations has attracted increased attention. For physical reasons we shall only consider the three-dimensional case here, that is the spatial variable x is assumed to be in R 3 .
When A(α(t), x) = 0, β(t) = 0, equation (1) is simplified to the following nonlinear Schrödinger equation It has received a great deal of attention from mathematicians, mainly with respect to the local and global properties; for instance, see [3,10,12,18] and the references therein. When the potential A depends only on spatial variable x and β(t) = 0, equation (1) is a class of nonlinear Schrödinger equations with a magnetic potential. Regarding the study of this kind of equations, the property of A plays an important role. When the magnetic field potential A is bounded, the spaces H 1 A and H 1 coincide and the Cauchy problem (1) can be solved in H 1 by using standard techniques. If the magnetic field is unbounded, it is impossible to solve the Cauchy problem in H 1 since the product u → Au is unbounded in L 2 . To solve this problem, Bouard [8], Nakamura and Shimomura [16] considered it in the weighted space Σ := {u ∈ H 1 and xu ∈ L 2 }. However, this space does not reflect the property of A. On the other hand, Cazenave and Esteban [4] investigated the special case where the magnetic field A is linear with respect to x. In one way, this paper is more satisfactory since they only require u 0 to belong to the energy space. Nevertheless, their results apply only to the linear magnetic field.
When α(t) = t and λ 1 = λ 2 = 0, Michel in [15] studied the local well-posedness of (1) by approximating the magnetic potential A(t, x) by potentials which are piecewise constant with respect to time. However, the terms 1 |x−β(t)| u and (| · | −1 * |u| 2 )u in (1) do not satisfy the assumption on nonlinearity in [15]. Recently, when the potential A depends only on time t and β(t) = 0, by using a simple change of coordinates and a phase shift, Antonelli et al. in [1] investigated the asymptotic behavior of solution of equation (1) with 0 < σ < 2 3 in the highly oscillating regime. |x−β(t)| brings some essential difficulties, see Remark 3.1 for a detailed analysis. Secondly, regarding the H 2 A regularity, in order to remove the smooth condition on the nonlinearity |u| 2σ u, we use the idea due to T.Kato [14] (see also [3]), based on the general idea for Schrödinger equation, that two space derivative cost the same as one time derivative. Because of the time-dependence of magnetic Laplacian (i∇ − A(α(t), x)) 2 , the compactness method can not be used to solve this problem, see section 4. This paper is organized as follows: in Section 2, we will collect some preliminaries such as Strichartz's estimates, the compactness results, etc. In section 3, we will establish the local and global well-posedness. In section 4, we will establish the H 2 A regularity for (1).
Notation. Throughout this paper, we use the following notation. C > 0 will stand for a constant that may be different from line to line when it does not cause any confusion. Since we exclusively deal with R 3 , we often use the abbreviations L r = L r (R 3 ), H s = H s (R 3 ) in what follows. Given any interval I ⊂ R, the norm of mixed spaces L q (I, L r (R 3 )) is denoted by · L q t L r x (I) . We recall that a pair of exponents (q, r) is Schrödinger-admissible if 2 q = 3( 1 2 − 1 r ) and 2 ≤ r ≤ 6. Then, for any space-time slab I × R 3 , we can define the Strichartz norm where the supremum is taken over all admissible pairs of exponents (q, r). For simplicity, we always denote 2. Preliminaries. In this section, we will recall some known facts and give some elementary results which will be used and play important roles later. Firstly, we recall the following compactness lemma, see [3] for a detailed presentation.
Lemma 2.1. [3] Let X → Y be two Banach spaces, I be a bounded, open interval of R, and (u n ) n∈N be a bounded sequence in C(Ī, Y ). Assume that u n (t) ∈ X for all (n, t) ∈ N × I and that sup{ u n (t) X , (n, t) ∈ N × I} = K < ∞. Assume further that u n is uniformly equicontinuous in Y . If X is reflexive, then there exist a function u ∈ C(Ī, Y ) which is weakly continuousĪ → X and some subsequence (u n k ) k∈N such that for every t ∈Ī, u n k (t) u(t) in X as k → ∞.
From this lemma, we can deduce the following compactness lemma.
Lemma 2.2. Let I be a bounded interval of R, and (u n ) n∈N be a bounded sequence of L ∞ (I, Next, we recall some results established in [15,19]. Assume that A satisfies Assumption 1 and α ∈ C(−∞, ∞). It is established in [19] that one can define (1) For any admissible pair (q, r), there exists C q such that For all admissible pairs (γ, ρ) and (γ 1 , ρ 1 ), there exists C = C γ,γ1 independent of s such that , x (s, s + T ) and T > 0. Finally, we recall the following Gronwall-type estimate which will be used to prove the uniqueness of the weak solution of (1), see [11].
for all 0 < t < T , then 3. The local existence and global existence. In this section, we establish the local and global well-posedness for (1). Firstly, we consider the local well-posedness of (1).
In addition, the following properties hold: (i) There is the blow-up alternative, i.e., either The following equalities follow. and Remark 3.1. Because the magnetic field potential A(α(t), x) is unbounded with respect to x, we cannot apply a fixed point argument and Strichartz's estimates to prove the local well-posedness of (1). Indeed, according to When . Motivated by the ideas in [15,17], we approximate the magnetic potential A(α, x) and the Coulomb potential 1 |x−β(t)| by potentials which are piecewise constant with respect to time. Michel in [15] applied the boundedness of ∂ γ x A (see Assumption 1) to obtain a priori estimate and pass to limit. However, the Coulomb potential 1 |x−β(t)| is unbounded and does not satisfy this property. The methods of a priori estimate and passing to limit in [15] cannot work for our case. Therefore, it is interesting to introduce some other methods to solve the related problems. In order to establish the local well-posedness for (1), we firstly solve the Cauchy problem (1), where A(α(t), x) = A(x) and β(t) ≡ β 0 are time independent.
Lemma 3.2. Let A be time independent and satisfy Assumption 1, β(t) ≡ β 0 for some β 0 ∈ R 3 . Assume that M > 0, 0 < σ < 2. Then, there exist 0 < T max ≤ ∞ depending only on M such that for all where Proof. This lemma can be proved by a similar method as that of Proposition 2.6 in [15]. Let us notice that the nonlinearity |u| 2σ u satisfies Assumption 2 in [15], but the nonlinearities 1 |x| u and (| · | −1 * |u| 2 )u do not satisfy this assumption. Following [6], let It is easy to check that g 1,m − g 6,m satisfy the Lemmas 2.3 and 2.4 in [15]. With the above results at hand, one can prove this lemma along the lines of Proposition 2.6 in [15], so we omit it.
Proof. We will approximate the magnetic potential A(α(t), x) and the Coulomb potential Step 1. Estimates on the sequence (u n ) n∈N .
According to Lemma 3.2, for all M > 0, there exists Next, we consider the following equation: (10) We deduce from uniqueness established in Lemma 3.2 that the solution of this equation given by where v 0,n is the solution of and for k ≥ 1, v k,n is the solution of the following equation In the following, we will prove that the functions v k,n , k = 0, . . . , n − 1 are welldefined. Since u 0 H 1 A ≤ M 4 , it follows from Lemma 3.2 that the Cauchy problem (12) is local well-posedness. Moreover, for any k ∈ {0, . . . , n − 1}, in order to prove that v k,n is well defined, it suffices to show that v k−1,n (t k n ) H 1 A ≤ M . Let k 1 ∈ {1, . . . , n − 1} be the greatest integer such that v k1−1,n (t k1 n ) H 1 A ≤ M . Then, the function u n given by (11) is well-defined for t ∈ [0, t k1+1 n ) and is continuous with Then, we deduce from Lemma 3.2 that for all k ∈ {1, . . . , This implies Taking the sum of (16) for k = 1, . . . , k 1 , we obtain On the other hand, since u n satisfies (10), there exists K(M ) independent of n ∈ N such that This yield that By some elementary calculations, we have
Therefore, note that the diamagnetic inequality

and the Hardy inequality
it follows from (20) and the mean-valued theorem that for all t ∈ [0, t k1+1 Since ∂ α A is bounded, the first term of the right hand of (17) is bounded by CtM 2 . By the Hardy and diamagnetic inequalities, the second term of the right hand of (17) is bounded by CtM 2 . Combining (14), (17)-(22), we obtain for any n > N and t ∈ [0, t k1+1 Taking T and δ sufficiently small, we have for any n > N and t ∈ [0, T ] Combining this estimate and the fact that u n is the solution of (10), we have Step 2. Passage to the limit. By applying (23), (24), and Lemma 2.2, we deduce that there exist u ∈ L ∞ ((0, T ), H 1 A ) and a subsequence, still denoted by (u n ) n∈N , such that, for a.e. t ∈ [0, T ], From the embedding W 1,∞ ((0, T ), Next, we note that for all z 1 , z 2 ∈ C, it holds It follows from (23), (26), (27), the embedding H 1 A → L r , Hölder's inequality that where r = 2σ + 2 and a = 1 − 3( 1 2 − 1 2σ+2 ). This implies (|u n | 2σ u n ) n∈N is a bounded sequence in C 0, a 2 ([0, T ], L r ). Therefore, we deduce from Lemma 2.1 that there exist a subsequence, still denoted by (|u n | 2σ u n ) n∈N , and Similarly, we can deduce that there exist a subsequence, still denoted by (( 1 |x| * |u n | 2 )u n ) n∈N and f 1 ∈ C 0, 1 2 ([0, T ], L 2 ) such that, for all t ∈ [0, T ], Next, in order to pass to limit for the term un |x−βn(t)| , we write and show that Indeed, applying the Hardy and diamagnetic inequalities, it follows from the dominated convergence theorem that for every ϕ ∈ C ∞ c (R 3 ) and η ∈ D(0, T ),

BINHUA FENG AND DUN ZHAO
For the second term on the right hand in (31), it follows from Hardy's inequality, (25) and the compact embedding In addition, for every t ∈ [0, T ], there exists k such that t ∈ [t k n , t k+1 n ), we consequently deduce from the definition of A(α n (t), x) and the mean-valued theorem that for every ϕ ∈ C ∞ c (R 3 ) This implies that for every t . On the other hand, since u n satisfies (10), it follows from that for every ϕ ∈ C ∞ c (R 3 ) and η ∈ D(0, T ), Applying (25), (29)-(36), and the dominated convergence theorem, we deduce easily that This implies that u satisfies In order to show that f 1 (t) = ( 1 |·| * |u(t)| 2 )u(t) and f 2 (t) = |u(t)| 2σ u(t), we need the following lemma. Proof. It suffices to show that for every bounded subset B of R 3 , To this end, we omit the time dependence and write Firstly, (30) implies a = 0. Next, it is obvious that b = 0. Finally, (25) implies that u n → u in L r (B). Therefore, c = 0, and f 1 , iu L r (B),L r (B) = 0. Similarly, we can prove that f 2 , iu L r (B),L r (B) = 0.
Step 3. Conclusion. By using the classical argument based on Strichartz's estimates, we can obtain the following uniqueness of the weak solution u of (1).
Proof. It follows from Duhamel's formulation of (1) that In the following, we set ρ 1 = 2σ + 2, taking γ 1 such that (γ 1 , ρ 1 ) is an admissible pair. Applying Lemma 2.3 to (43), we deduce from Hölder's inequality that for every t ∈ (0, T ] x (0,t) where On the other hand, by (17), (25), the weak lower semicontinuity of the magnetic Sobolev norm (i∇ − A(α(s))) L 2 , it follows that E(u(t)) ≤E n (t, u n (t)) Finally, t > 0 being fixed, we deduce from the uniqueness that v n (s) = u n (t − s) is the solution of the following equation with initial data v n (0) = u n (t). Then we perform the same computations as above to get the converse inequality to (45) and hence (4) is proved. This completes the proof.
Next, we consider the global existence of (1).
Then the solution u of (1) exists globally.
Proof. In order to show this theorem, it suffices to show for every T > 0.
Using the diamagnetic and Hardy's inequalities, we have and When λ 3 > 0, using Young's inequality with ε, from (4), (47), (48), we have Hence, taking ε < 1, applying Gronwall's inequality, (46) follows. When λ 3 < 0 and 0 < σ < 2 3 , by using the same argument as above and the following Gagliardo-Nirenberg's inequality follows. This completes the proof of this theorem. 4. The H 2 A regularity. In this section, we investigate the H 2 A regularity for (1). The H s regularity for nonlinear Schrödinger equations is well-known if the nonlinearity is sufficiently smooth, see [3]. The smooth condition on the nonlinearity can be improved (removed, if s ≤ 2) by estimating time derivatives of the equation instead of space derivatives, see [3,14]. There are two possible techniques for studying the H 2 A regularity of equation (1). One is the compactness method, see Theorem 7.4.1 in [5]. However, a problem arises when we try to mimic proof of Theorem 7.4.1 in [5]. Indeed, the first step is to obtain a generalization of Theorem 7.2.1 in [5] in the time-dependent framework. In the time-dependent framework, the existence of smooth solution is not easy to prove. Indeed, the key point in [5] is that for any g ∈ C([0, T ], X) being Lipschitz continuous with respect to time, the function v(t) = t 0 U 1 (t, s)g(s)ds is also Lipschitz continuous with respect to time, where U 1 (t) := e it is the free Schrödinger propagator. This result is easily proved in autonomous case by using the identity U 1 (t + h, s) = U 1 (t, s − h). This fails to be true in the non-autonomous case. Moreover, we cannot mimic the argument in [15] by approximating the magnetic potential A(t, x) by potentials which are piecewise constant with respect to time. Indeed, since the nonlinearities we consider are not sufficiently smooth, we obtain some estimates by differentiating the approximating equations with respect to time variable. However, this magnetic potential is not differentiable with respect to time variable.
The other is Kato's method, based on a fixed point argument and Strichartz's estimates. When we apply this method to show the H 2 A regularity of equation (1), a difficulty occurs if A(α(t), x) is unbounded with respect to x. Indeed, we infer from (6) that there is no chance to control ∇(|u| 2σ ) by (i∇ − A(α(t)))(|u| 2σ ). Therefore, we consider the H 2 A regularity under the assumption that A(α(t), x) is bounded. How to obtain the H 2 A regularity for (1) when A is a time-dependent unbounded potential is an interesting open problem.
The magnetic field A(α(t), x) is a bounded function and satisfies Assumption 1.

A(t)
= u H 1 . It is thus easy to see that Φ maps E to itself and is a contraction in the S(0, l) norm, provided l is chosen sufficiently small. The contraction mapping theorem then implies the existence of a unique H 2 A solution to (1) on [0, l]. By a similar argument as Section 5.3 in [3], we have u ∈ C([0, T ], H 2 A )∩C 1 ([0, T ], L 2 ). This completes the proof.