Nonlocal Nonlinear Schrödinger Equations in R3

This paper studies a class of nonlocal nonlinear Schrödinger equations in R3, which occurs in the infinite ion acoustic speed limit of the Zakharov system with magnetic fields in a cold plasma. The magnetic fields induce some nonlocal effects in these nonlinear Schrödinger systems, and the main goal of this paper is to understand these effects. The key is to establish some a priori estimates on the nonlocal terms generated by the magnetic field, through which we obtain various conclusions including finite time blow-ups, sharp threshold of global existence and instability of standing waves for these equations.


Introduction
This paper concerns the equations that arise in the infinite ion acoustic speed limit (c 0 → ∞) of the magnetic Zakharov system in a cold plasma in R 3 : where E(t, x) is a complex vector-valued function from R + ×R 3 into C 3 , n(t, x) is a real function from R + × R 3 into R, B(t, x) is a real vector-valued function from R + × R 3 into R 3 , η and δ are two constants with η > 0 and δ 0, ∧ denotes the exterior product of vector-valued functions and E denotes the complex conjugate of E.
Here, the Zakharov system (ZSM) describes the spontaneous generation of a magnetic field in a cold plasma. E denotes a slowly varying complex amplitude of the high-frequency electric field, B the self-generated magnetic field, n the fluctuation of the electron density from its equilibrium, c 0 is a parameter which tends to +∞ in the subsonic limit [6,11,27,28]. In this limit, the Zakharov system (ZSM) formally reduces to the following vector nonlinear Schrödinger equations with magnetic field: (SB-1) Using Fourier transforms, we can solve the second equation in (SB-1): for E ∈ H 1 (R 3 ), B(E) ∈ L 2 (R 3 ) and where F and F −1 denote the Fourier transform and the Fourier inverse transform, respectively (see [13,[16][17][18]). Due to rotational invariance of (SB-1), we let E = (E 1 , E 2 , 0) and ξ = (ξ 1 , ξ 2 , 0) for simplicity. Then (SB-1) becomes the following nonlinear Schrödinger system with nonlocal terms: along with the initial data Our main interest is to study the influence of magnetic fields on the solutions of (1.1)- (1.2). For this purpose, we first consider the case in which E 1 (t, x), E 2 (t, x) have the same frequency ω, with ω > 0. Let E 1 (t, x) = e iwt u(x), E 2 (t, x) = e iwt v(x). Here (u(x), v(x)) is a pair of complex valued functions which satisfies the following equations: It is easy to check that the nonlocal operator F −1 η|ξ | 2 |ξ | 2 −δ F(E 1 E 2 − E 1 E 2 ) in (1.1)-(1.2) has the symbol σ (ξ) = η|ξ | 2 |ξ | 2 −δ (see [8,20,21]). In [7], a simplified version was considered.
If one neglects the magnetic field, the Equations (SB-1) become the cubic focusing vector nonlinear Schrödinger equation: which is a classical nonlinear model in quantum mechanics. Ginibre and Velo [10] established the local existence of the Cauchy problem for Equation (S) in the energy class H 1 (R N ) when E(t, x) is a scalar-valued function from [0, +∞) × R N into C. Glassey [9] and Ogawa-Tsutsumi [22,23] proved that the solutions to the Cauchy problem for Equation (S) blow up in a finite time for some initial data, especially for a class of sufficiently large data. When E(t, x) is a two-or threedimensional vector-valued function, Mckinstrie in [15] obtained a similar finite time blow-up result for the solution to the Cauchy problem.
Returning to the nonlocal nonlinear Schrödinger equations (1.1)-(1.2), to our best knowledge, there is no rigorous proof of existence of singular solutions for (1.1)-(1.2), although physical reasonings suggest solutions may collapse in a finite time [11]. Moreover, it would be interesting to consider the sharp threshold for the global existence of a Cauchy problem (see [29,30]) for such nonlinear Schrödinger equations with nonlocal effects, as with equations (1.1)-(1.2). As in many previous works [1,2,7,14,22,[25][26][27], we obtain finite time blow-ups, a sharp threshold of global existence and orbital instability of standing waves for (1.1)-(1.2) in R 3 . Due to a hiding symmetry, the nonlocal (magnetic field) effect preserves the underlying variational structure. This is a very useful fact that leads to conservation laws of energy and mass of the system. On the other hand, the nonlocal term leads to various complications in applications of virial identities (which have played a fundamental role in many of classical papers on the subject). We have introduced several new techniques to improve some of the earlier ideas. In particular, the proof of existence of ground states (minimal energy standing waves) contains some very general methods that may be useful for other related problems, as well. Our analysis also provides some preliminary understandings of the effect of a self-generated magnetic field in a cold plasma.
For simplicity, we shall denote various positive constants by C throughout this paper.

Conservation Laws of Mass and Energy
Using the inner structure of Equations (1.1)-(1.2), we get the conservation laws of total mass and total energy.
then total mass and total energy are conserved: we conclude (2.1). Multiplying (1.1) by 2∂ t E 1 and (1.2) by 2∂ t E 2 , taking the real parts, and then integrating with respect to space variable x on R 3 , we obtain we conclude (2.2). This completes the proof of Lemma 2.1.

Variational Structures
), a pair of complex-valued functions, we define the functionals S(u, v) and R(u, v) by Furthermore, we define two sets M and M 1 by (2.6) In view of (2.3), (2.4), (2.5) and (2.6), we define two constrained variational problems ), η > 0, δ 0, the Sobolev's embedding theorem and the properties of Fourier transforms, it follows that functionals S(u, v) and R(u, v) are both well defined. Remark 2.1. We note that for all θ 1, We also note that if (u, v) is a critical point of (2.3), and hence a solution of (1.4) and (1.5) then (E 1 , E 2 ) = (e iωt u, e iωt v) is a standing wave solution of (1.1)-(1.2).
For the constrained variational problem (2.8), we can obtain the following result: (2.9) Noting that On the other hand, in view of from R(u, v) = 0 and the Gagliardo-Nirenberg inequality v 4 it follows that , which implies that Thus, by Young's inequality, we have Combining (2.9), (2.10), (2.11) and (2.12), we conclude that Therefore, we get d 1 > 0 from (2.8).

Main Blow-Up Results
Let The main results of this section can be stated as follows: Theorem 3.1. Let η > 0 and δ 0. Assume that and one of the following three conditions holds: Then there exists 0 < T < +∞ such that , then the following finite time blow-up result holds.

Key Ingredients
The following Lemma 3.1 and Proposition 3.1 will play key roles in the proof of the main results of this section, Theorem 3.1 and Theorem 3.2.

Lemma 3.1. [26] Let f be a scalar-valued function. If |x| f and ∇ f belong to
In addition, one can check that Differentiating (3.5) again with respect to t, after a careful computation and proper groupings, we get These nonlocal expressions have to be handled with extra care. We verify them one by one, as follows: Using integration by parts, the Parseval identity, and properties of Fourier transforms, we derive that (C-1)-(C-5) imply the conclusion of Proposition 3.1.

Proofs of Theorem 3.1 and Theorem 3.2
With the preparations in the previous subsection, we can now prove Theorems 3.1 and 3.2.
Proof of Theorem 3.1. We show this theorem by contradiction. Assume that the maximal existence time T of the solution to the Cauchy problem (1.
which together with η > 0, δ 0 and Proposition 3.1 implies where J (t) is defined by (3.2). Integrating (3.8) twice with respect to t, we have Under hypothesis (i), (ii) or (iii), (3.3) and (3.9), we conclude that if On the other hand, by (2.1) and Lemma 3.1, one has , which together with (3.10) yields The latter implies that the maximal existence time T max T * , and hence is a contradiction.

Existence of Standing Waves
In this section, we discuss the existence of minimal energy standing waves of the system (1.1)-(1.2) for η > 0 and δ 0. The main result can be stated as: and it follows from Theorem 4.1 and virial identity that In order to prove Theorem 4.1, we need a series of basic facts. First of all, one has the following well-known result: Lemma 4.1. (Strauss [25]) For 1 < σ < 6, the embedding is a function of |x| alone}.
Next, we have the following proposition.
On the other hand, multiplying (1.4) by x∇u and (1.5) by x∇v, then integrating with respect to x on R 3 and taking the real parts for the resulting equations, we obtain By a direct calculation, we find Putting (S-1)-(S-5) into (4.5), we obtain (4.7) Since for θ = 1, 2, where η > 0 and δ 0, one obtains The latter, together with (4.7) and ω > 0, implies on M that This completes the proof of Proposition 4.2.

Sharp Threshold of Global Existence in R 3
In this section, we discuss the sharp threshold of global existence to the Cauchy problem (1.1)-(1.3). As in [1,2], our arguments are based on the local well-posedness (Proposition 2.1) established in Section 2. The main result of this section can be described by the following. , We start with two propositions which are key to the proof of Theorem 5.1.