GLOBAL SMOOTH SOLUTIONS FOR THE NONLINEAR SCHR¨ODINGER EQUATION WITH MAGNETIC EFFECT

. We consider the Cauchy problem of the nonlinear Schr¨odinger equation with magnetic eﬀect

1. Introduction. In this paper, we are concerned with the global existence of smooth solutions for the Cauchy problem of a set equations arising from plasma physics. The equations under study read with initial data E(0, x) = E 0 (x), (B(0, x), B t (0, x)) = (B 0 (x), B 1 (x)).
Here, E : R + × R 3 → C 3 is the slowly varying amplitude of the high-frequency electric field (E denotes the complex conjugate of E), and the function B : R + × R 3 → R 3 is the self-generated magnetic field. γ is the speed of electron, and the notation × means the cross product for R 3 or C 3 valued vectors. System (1) is a simplified model in plasma physics. It describes the nonlinear interaction between plasma-wave and particles [22], especially when the phase speed of plasma wave is much less than the speed of ions so that the fluctuation of the density satisfies a stationary equation It also exhibits that the self-generated magnetic field influences the high-frequency electric field directly by the coupled term iE × B, which in turn affects the density 1754 DAIWEN HUANG AND JINGJUN ZHANG of particles in a indirect way through the equation of n. Therefore, system (1) can also be regarded as a magnetic type Zakharov system with n satisfying (3).
Omitting the effect of the magnetic field B, the classical Zakharov system [31] plays an important role in plasmas, and there are a lot of mathematical results on the global existence of weak solutions or smooth solutions, local well-posedness and scattering theory for this system; cf. [1,2,3,4,7,12,13,14,15,21,28,30] and the references therein. In the presence of magnetic field, we refer to [22,26] for the background of physical importance and [8,16,17,20,25,27,32] for the mathematical theories of the magnetic type Zakharov systems.
Note that system (1) reduces to the cubic nonlinear Schrödinger equation if we ignore the effect of B in (1) or let γ → ∞ in the equation of B. Such equation has been studied by many researchers, see for example [6,18,19,24]. However, as far as we know, there are no results on the global existence of the solution for system (1). Hence, in this work, we are interested in the mathematical theories (especially, the global dynamics) for this system.
For s ∈ R, we denote H s (orḢ s ) the inhomogeneous (or homogeneous) Sobolev spaces, equipped with the norm u H s := (I − ∆) s/2 u L 2 = (1 + |ξ| 2 ) s/2 u L 2 , Then the Cauchy problem (1)-(2) has a unique global solution such that This paper is organized as follows. In Section 2, we write system (1) into an integral system by using the profiles, and introduce the work space and the linear decay estimates. Section 3 is devoted to dealing with the energy estimate. The weighted estimates for the magnetic field and the Schrödinger component are given in Section 4 and Section 5, respectively. In Section 6, we present the proof of our main result.

2.
Preliminaries. In order to prove Theorem 1.1, we rewrite system (1) into a first order system. To this end, we set then Cauchy problem (1)-(2) is reduced to a set of unknowns (E, M ) with initial data where we have used the identity Now we restate Theorem 1.1 in terms of (E, M ).
Theorem 2.1. Let N ≥ 100, γ > 0 and γ = 1. Then there exists a positive constant 0 1 such that if the initial data satisfies then the Cauchy problem (5)-(6) admits a unique global solution (E, M ) satisfying In view of (4), Theorem 1.1 follows immediately from the above theorem. Hence, from now on, we mainly focus on the proof of Theorem 2.1. To prove this result, we work on the framework of space-time resonance method [11,21,29]. Define the profiles from system (5), (f, g) satisfies Therefore, we have and where the phases ϕ, ψ ± and φ are The integral identities (12)- (13) are the main equations that we will discuss later. Inspired by the work [21,29], the important fact we observe is that there are several implicit relations for the phases The above identities show some null conditions of the phases, hence, we may regard the resonances for these four phases are null. Now for any T > 0, we define the norm associated to our work space Moreover, we set One of the basic ingredients in applying the method of space-time resonance is the liner dispersive estimate. For the Schrödinger operator e it∆ , it is known that for p ∈ [2, +∞], which can be found, for example, in [5]. Combining (16) and the inequalities we obtain the following linear decay estimates Similarly, we have The estimates (17)- (22) are important in our succeeding analysis.
The main aim of Sections 3-5 is to derive the a priori bound of the solution to (5) in the spaces X T and Y T , and obtain the following type estimate then Theorem 2.1 follows by a standard continuation argument. In the estimates for the energy norm, since the regularity of E and M is not at the same level and the nonlinear term contains two order derivatives, the usual energy method will introduce one order loss of derivative. To overcome this difficulty, we use the idea of [21] to exploit the following positive properties |ψ + | = |η||2ξ − (γ + 1)η| |η| 2 , |φ| = |ξ||(γ − 1)ξ + 2η| |ξ| 2 when |ξ| ∼ |η| |ξ − η|, then we apply normal form transformation to recover the loss of derivatives (see the next section). In the weighted estimates of the solution, we will mainly deal with the terms which contain the growth factor s (or s 2 ). For the magnetic field, thanks to the derivative nonlinear structure, we can use the null resonance relation (46) to integrate by parts in η and thus close the estimates. For the electric filed, we exploit the key implicit conditions for the phases (see (71) and (94)), then we perform integration by parts both in s and η to estimate the problematic terms. The weighted estimates are presented in Sections 4-5. Here, we remark that the growth bound (1 + t) 1/2 for |x| 2 f L 2 which comes from (111) and (112) seems to be sharp since the decay rate of E L 4 is optimal.

DAIWEN HUANG AND JINGJUN ZHANG
where In the following contents of the paper, for a > 0 we denote by P ≤ a the frequency projection operator defined by where θ is a radial, smooth function satisfying 0 ≤ θ ≤ 1, θ(x) = 1 for |x| ≤ 5/4 and suppθ ⊂ B 8/5 (0). Also, we define P >a := 1 − P ≤a . The main estimate of this section is stated in the following proposition.
where the constant C is independent of T .
Proof. Recall the definition (10), we know Using the fact E × E is purely imaginary, we see that the L 2 norm of E is conserved for all time as long as the solution exists. According to (23)- (24) and the conservation of the L 2 norm of E, in order to prove the bound (25), it suffices to show for all t ∈ [0, T ). By (17), the term F 1 can be directly estimated as To estimate F 2 , we split it into We first estimate theḢ N norm of F 2 . On the support of 1−χ 1 , there is |η| |ξ−η|, so the N derivatives can be put on E = e is∆ f , then by (20), For the term F 2 , we may assume |η| ≥ 1 since the case |η| ≤ 1 can be easily treated by using Hölder's inequality and the fact P ≤1 u H s u L 2 (s ≥ 0). As the frequency of g is higher than f , so if all the derivatives fall directly on the function M = e isγ∆ g, it will appear a loss of derivative. To recover the loss of derivatives, we should exploit the nonvanishing property of the phase ψ + (ξ, η). Note that on the support of χ 1 , there hold |ξ| ∼ |η| and This bound allows us to estimate F (2) 2 through integrating by parts in s. Define The lower bound (30) yields | χ 1 | 1. Moreover, by the property of χ 1 , we can see the symbol χ 1 satisfies Coifman-Meyer bound. Using the classical Coifman-Meyer multiplier theorem, we obtain Similarly, the term (32) can be estimated as

DAIWEN HUANG AND JINGJUN ZHANG
From (11), we have Therefore, there holds Following similar argument as F 2 , we can get Thus, the desired bound (26) follows from (28), (29), (35) and (36). It remains to show the bound (27). Since the arguments for G 1 and G 2 are similar, we only consider in detail the estimate for G 1 . Due to the structure of derivative nonlinearity, the part ofḢ −1 norm can be estimated directly as Also, the part of L 2 norm can be easily treated as Now we estimateḢ N −1 norm of G 1 . As the nonlinear term contains two order derivatives, it will again produce the loss of derivatives. To close our argument, we introduce a smooth cut-off function χ 2 (ξ, η) satisfying On the support of 1 − χ 2 (ξ, η) − χ 2 (ξ, ξ − η), we have |ξ − η| ∼ |η|. Hence the term G 1 is estimated by where we have used the following bounds in the last step: Note that the terms G correspond to |ξ − η| |η| ∼ |ξ| and |η| |ξ − η| ∼ |ξ|, respectively, so by symmetry, it is sufficient to estimate G 1 . On the support of χ 2 (ξ, η), we have Integrating by parts in time gives With similar arguments as the terms (31)-(34), we can obtain Therefore, combining (37), (38), (39) and (45) yields (27) as desired. This ends the proof of Proposition 1.

4.
Weighted estimate for the magnetic field. This section is concerned on the weighted estimate for the magnetic component. Note that Thanks to the derivative structure of the nonlinear term, we can use the relation (46) to exclude the phenomenon of time-space resonances.
where C is independent of T .
Proof. In order to prove (47), we know from (23) and (24) that it suffices to show for any t ∈ [0, T ) and j = 1, 2. As the essential structure of G 1 and G 2 is the same, for simplicity, we only concentrate on the estimate for G 1 in this proof. Recall Without loss of generality, we may assume |ξ − η| |η| in (49), otherwise we can make a change of variable ξ − η →η. Applying ∇ ξ to G 1 gives For (50), we put the derivatives on the function e −is∆ f . Using the bounds For (51), it is easy to see Before estimating (52), note that when N ≥ 100 and 0 < δ ≤ 1/30, This shows that the low order L 2 norm of the high frequency part can produce a good decay factor, so when estimating (52), one can reduce our argument to the case |ξ − η|, |η|, |ξ| (1 + s) δ .
Namely, due to the good decay rate and high order energy, the low order derivative can be converted into a small growth factor. Now we use (46) to get is∇ ξ φe isφ |ξ| 2 = is(2(γ − 1)ξ + 2η)e isφ ( ξ 2|ξ| · ∇ η φ)|ξ| = ((γ − 1)ξ + η)(ξ · ∇ η e isφ ), then integrating by parts in η gives Estimate for (56) is the same as (51). Using the assumption (55), we have From (53), (54) and (59), we thus get xG 1 L 2 A 2 T . Applying ∆ ξ to (49), we can get after a direct computation The term (60) is estimated similarly as (50). Applying the L 2 × L ∞ estimate, the treatment for (61) is trivial. The terms (63) and (64) can be estimated in the same way as (52). The estimate for (65) is essentially the same as (57). For (66), using the identity (46) to integrate by parts in η, we can obtain analogous terms as (60), (65) plus the term From the estimate (19), one gets Here, we point out that the growth factor (1 + t) 1/3 can also be replaced by (1 + t) α with α = 1/4+. However, for simplicity, we will not take this general bound in our arguments. Now, we are left to deal with the term (62). To eliminate the factor s 2 , we use (46) to integrate by parts in η twice and obtain the following contributions where m 1 , · · · , m 4 are homogenous symbols of order no more than 2. All the above terms can be bounded by applying previous arguments. Therefore, we get The proof of Proposition 2 is completed.

5.
Weighted estimate for the Shcrödinger component. In this section, we want to prove the following proposition.
The bound (68) follows immediately from (23), (24), (70) and (90). Hence, in the remaining parts of this section, it is sufficient for us to show Lemmas 5.1 and 5.4 below.

5.2.
Estimate for the quadratic term. The aim of this subsection is to prove Lemma 5.4 below. To this aim, we need the following two lemmas.
From this identity and (11), one can compute Moreover, (86) also gives Hence, (84) follows from the above two equalities. Then by (17), (18) and (20), T . Similar to (40), we can convert the derivative ∇ into (1 + s) 1/12 due to the a priori energy bound on E, so T . Thus (85) follows. The proof of Lemma 5.2 is finished.
where the expressions of F 2 and F 3 are given by (24).
With a direct L 2 × L ∞ estimate, the bound for (92) is trivial. To estimate (93), the key observation is Using this identity, we have For (95), we integrate by parts in time and obtain In virtue ofḢ −1 norm of M , we get

DAIWEN HUANG AND JINGJUN ZHANG
The term (98) can be estimated similarly. For (99) and (100), we use (11) to obtain Therefore, the desired bound for (95) is obtained. For (96), we integrate by parts in η and get From the above estimates, we see (91) holds. Next, we prove that for all t ∈ [0, T ), Notice that The term (102) can be estimated easily by using a direct L 2 × L ∞ estimate. Using the relation (94) to integrate by parts in s and η, we rewrite (103) as The estimates for (105) and (106) are similar, and we take (105) as an example. Indeed, we use the fact g Ḣ−1/2 g Ḣ−1 + g L 2 A T and (18) to get For (108), note that by (11), The argument for (109) is essentially the same as (102). For the term (110), we use (19) and (22) to obtain It remains to estimate (104). By (94), one sees Using this identity to integrate by parts in time and frequency, we obtain In virtue of (19) and (22), the term (111) is estimated as (111), we can deal with the term (112). Using (11), we have The term (115) is analogous to (103), so we can apply the same strategy used in (103) to obtain the desired bound. For the last term (116), we again use the relation (94) to get In order to estimate (117)-(119), note first that |∇| −1 e isγ∆ (xg) L 3 e isγ∆ (xg) Ḣ−1/2 xg L 3/2 xg 1/2 L 2 |x| 2 g 1/2 L 2 (1 + s) 1/6 A T , then by (17), (18), (20) and (11), we have Therefore, combining the above estimates, we obtain (101) as desired. Finally, we should prove for all t ∈ [0, T ), there holds xF 3 (t) L 2 + (1 + t) −1/2 |x| 2 F 3 (t) L 2 A 2 T , where F 3 is given in (24). Note that the phase ψ − has similar null structure as ψ + (see (14)-(15)), one can apply analogous arguments as (91) and (101) to obtain the above bound. Since the proof is similar, we omit further details. This finishes the proof of Lemma 5.4. 6. Proof of Theorem 2.1. Now, by using the a priori estimates obtained in Propositions 1, 2 and 3, we now present the proof of Theorem 2.1.
By the condition (7)-(8) and the continuity of the solution, there exists a time T > 0 such that A T ≤ 4 0 . Let T be the supremun of T satisfying A T ≤ 4 0 , then from Proposition 1, Proposition 2 and Proposition 3, we have A T ≤ 2 0 + CA 2 T , where C is independent of T . Then we can show T = T * provided that 0 ≤ (16C) −1 . Indeed, if T < T * , the above inequality gives 4 0 ≤ 2 0 + 16C 2 0 , which is a contradiction for sufficiently small 0 . Therefore, we conclude that if 0 ≤ (16C) −1 , then A T * ≤ 4 0 , which in turn implies T * = +∞. Then the decay estimates in (9) follow easily from (17) and (20). This completes the proof of Theorem 2.1.