GLOBAL EXISTENCE AND SCATTERING OF EQUIVARIANT DEFOCUSING CHERN-SIMONS-SCHR¨ODINGER SYSTEM

. In this paper, we consider the following equivariant defocusing where φ ( t,x 1 ,x 2 ) : R 1+2 → R is a complex scalar ﬁeld, A µ ( t,x 1 ,x 2 ) : R 1+2 → R is the gauge ﬁeld for µ = 0 , 1 , 2, A r = x 1 and p > 4. When p > 4, the system is in the mass supercritical and energy subcrtical range. By using the conservation law of the system and the concentration compactness method introduced in [17], we show that the solution of the system exists globally and scatters.

When p > 4, the system is in the mass supercritical and energy subcrtical range. By using the conservation law of the system and the concentration compactness method introduced in [17], we show that the solution of the system exists globally and scatters.

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and In [20], the authors introduce which are easily seen to satisfy Using these transformations, we can express A 1 , A 2 , ∂ 1 , ∂ 2 in terms of A r , A θ , ∂ r , ∂ θ . In particular, Now, the equation (1.10) has the following representation which can be written in more compact form (1.14) Write the curvature F = dA in terms of variables t, r, θ, with By the formula (1.12) for A 1 and A 2 , we can assume for some v(t, r), where r = |x|.

JIANJUN YUAN
By calculating, we have A r = 0 and A θ is a radial function. By the equivariance of φ, the system (1.1)-(1.4) transforms further to In [20], the authors consider the p = 4 case, they prove that the solution of the system scatters when λ ≤ 0 in the equivariant case, and also scatters in the focusing case with the mass below the threshold when λ < 1 1 .
When p > 4, λ > 0, in [3], the authors show there are standing wave solutions of the system, so scattering does not always occur in the focusing case.
The m = 0 case corresponds to the radial case, when λ < 0, by using the concentration compactness method, in [28], we show that the solution of the system scatters. So we will focus on m = 0 case in this paper. Let If the term λ|φ| p−2 φ does not exist in (1.13), the equation is invariant under the transform φ(t, x) → µφ(µ 2 t, µx) for µ > 0, and µφ(µ 2 t, µx) L 2 = φ(t, x) L 2 , so it corresponds to the L 2 -critical case, and λ|φ| p−2 φ for p > 4 can be seen as a perturbation of the L 2 -critical case. This is similar to the combined nonlinearities with power exponents studied in [5], The term |u| q−1 u can also seen as a perturbation of the L 2 -critical equation. In [5], the authors use the concentration compactness method to prove the solution scatters below the ground state when d ≤ 4 in the radial case. We also refer the reader to [5] for a review of combined nonlinearities.
In section 2, we will make some preparations which will be used in section 3. In this section, we will give the well-posedness result of (1.19)-(1.23), and use the conservation laws of the system to show the global existence of the solution. In section 3, we will establish the short-time and long-time perturbation theory. The perturbation terms need to be carefully analysed, here we put them in L part of the Theorem 1.1. If Theorem 1.1 fails, firstly, by using the concentrationcompactness method and inducting on E(φ) + M (φ), we show that there exists a nonzero critical element with some infinite Lebesgue space-time norms which is compact in H 1 (R 2 ) on its trajectories in t, then following [20], we establish the localized viral identity to give a contradiction.
Notation. Throughout the article we will use the letter C to denote various constants whose exact value may change from line to line. We will use the notation X Y whenever there exists some positive constant C so that X ≤ CY .

Preliminaries.
We assume that all spatial L p spaces are based on the 2-dimensional Lebesgue measure. Define the operators [r −n ∂ r ] −1 and [r∂ r ] −1 by Then by the one dimensional Hardy's inequality, we have Since φ(t, x) = e imθ u(t, |x|), |∇φ| 2 = u 2 r + m 2 r 2 u 2 . In particular, when m = 0, . Let (q, r) and (q,r)be admissible pairs of exponents. Then where the prime denotes the dual exponent, i.e., 1 For φ defined on an interval I, we define Proof. We assume |T | ≤ 1. We define the space Y (I) and the map T by We can verify that Y (I) becomes a metric space with the metric ρ(φ 1 , φ 2 ) := φ 1 − φ 2 S I . By using Lemma 2.2, we have By combining (2.25) and (2.26), we have Similarly, We have shown that T is contraction in (Y (I), ρ), so we get the solution φ on I by the contraction mapping principle.
Theorem 2.5. (Global existence). The solution φ in Theorem 2.4 can be extended globally, i.e., φ ∈ C((−∞, +∞), H 1 (R 2 )). The solution φ(t) conserves mass and energy, for t ∈ R, we have Proof. Mass and energy conservation is standard. We show the global existence. By Theorem 2.3, it suffices to show the boundedness of φ(t) H 1 on its existence time interval. By (1.12), we have From the energy conservation, we have So ∇φ(t) L 2 is bounded on its existence time interval, which combines the conservation of mass, shows φ(t) H 1 is bounded on its existence time interval, and prove the theorem. Now, we give a sufficient condition to guarantee the scattering of the global solution φ(t).
Now, we define The construction of φ − is similar.
Proof. By using the conservation laws of the system, ∇φ(t) L ∞ t∈R L 2 x ≤ C. By using Lemma 2.3, we have By continuity of S 1 norm, Similarly as Theorem 2.5, we can define φ ± , and prove (2.39).
Remark 2.8. If φ 0 H 1 is sufficiently small, then ∇ e it∆ φ 0 L 4 t,x φ 0 H 1 is also sufficiently small, and we can use Theorem 2.7 to deduce that φ 0 H 1 and the solution scatters.
GLOBAL EXISTENCE AND SCATTERING OF EQUIVARIANT...
Suppose we also have the energy bound for some constant A 1 > 0. Let t 0 ∈ I and let u(t 0 ) ∈ H 1 be close to w(t 0 ), for some A 2 > 0. Moreover, assume the smallness conditions Then, there exists a solution u to (1.19)-(1.23) on I × R 2 with the initial data u(t 0 ) at time t = t 0 that satisfies where 0 < α < 2 p−1 . (3.8) Proof. By the wellposedness theory, the solution u exists on I. By time symmetry, we may assume t 0 = inf I.
with v(t 0 ) = u(t 0 ) − w(t 0 ). For the term 2m For the term 1

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for some function e. Assume e = g + h, 2 i=1 ∂ i e = g 1 + h 1 for some functions g, h, g 1 and h 1 .
Suppose we also have the energy bound for some constant A 1 > 0, and for some constant B > 0.
Let t 0 ∈ I and let u(t 0 ) ∈ H 1 be close to w(t 0 ),
Assume also the smallness conditions Then, there exists a solution u to (1.19)-(1.23) on I × R 2 with the initial data u(t 0 ) at time t = t 0 which satisfies where 0 < α < 2 p−1 . By (3.39), we can divide I into into finite subintervals I k , such that then apply Theorem 3.1 on I k sucessively to deduce the Theorem, see e.g., [13].
Since the procedure is standard, we omit the details.

4.
Scattering. In this section, we will argue by contradiction to prove the scattering part of Theorem 1.  [22], and adapt it to the equivariant case. This type profile decomposition originates in [1] for the wave equation, for the Schrödinger equation it appeared e.g. in [22], [26].  such that for any k ∈ N, there exists w k n ∈ V m , The remainder w k n satisfies lim where (q, r) is L 2 -admissible and 2 < q < ∞.
Moreover, we have the following decoupling properties: ∀k ∈ N, To this end, for m > 0, let Applying Theorem 4.1 to {φ n (0, x)} and obtaining some subsequence of {φ n (0, x)}(still denoted by the same symbol), then there exists ϕ j ∈ H 1 (R 2 ) and (θ j n , t j n ) n≥1 of sequences in R/2πZ × R, with such that for any k ∈ N, there exists w k n ∈ H 1 (R 2 ), where (q, r) is L 2 -admissible, and 2 < q < ∞.
Moreover, we have the following decoupling properties: ∀k ∈ N,  Now, we construct the nonlinear profile. We define the nonlinear profile u j ∈ C (−∞, +∞), H 1 (R 2 ) to be the solution of (1.19)-(1.23) such that Let u j n = e iθ j n u j (t − t j n ). For the linear profile decomposition, we can give the corresponding nonlinear profile decomposition We will show u k n is a good approximation for u n provided that each nonlinear profile has a finite global Strichartz norm, which is the key to show the existence of the critical element.
Lemma 4.2. There exists j 0 ∈ N, such that for j > j 0 , Proof. By (4.13), we have which shows ∞ j=1 ϕ j 2 H 1 < ∞, and therefore ϕ j H 1 → 0, as j → ∞. By Remark 2.7, we have when j is large enough, u j S 1 (R) ϕ j H 1 , and prove the Lemma.

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and there exists B > 0 such that we have (4.25) In the above, u j n u m n L p−2 By using (4.13), and boundedness of φ n (0) L 2 x , we have lim sup  Proof. Now, u k n satisfies where e k n : = k j=1 2m By Lemma 4.3, there is a large constant A 1 , B (independent of k) with the property that for any k, there exists n 0 = n 0 (k) such that for n > n 0 , Also, by (4.13), there is a large constant A 2 (independent of k) with the property that for any k, there exists n 1 = n 1 (k) such that for n > n 1 , Moreover, we verify that for each k and δ > 0 there exists n 2 = n 2 (k, δ) such that for n > n 2 , e k n = f k n + g k n , Note that since u k n (0) − u n (0) = w k n , there exists k 1 = k 1 (δ) sufficiently large, such that for each k > k 1 there exists n 2 = n 2 (k) such that n > n 2 ∇ e it∆ u k n (0) − u n (0) L 4 t∈(−∞,+∞),x∈R 2 ≤ δ. 2m (4.37) Since |t j1 n − t j2 n | → 0 as n → ∞, we have u j1 n (t − (t j1 n − t j2 n ), ·)u j2 n (t, ·) L 2 t,x → 0 as n → ∞, so, m r 2 r 0 u j1 n u j2 n sdsu j n L (4.40) Since |t j1 n − t j n | → 0 as n → ∞, we have 2m → 0 as n → ∞. 2m Similarly, we also have x .
x . (4.50) x . (4.52) If j 1 = j 2 = j 3 = j 4 = j, ∞ r s 0 u j1 n u j2 n tdt s u j3 n u j4 n dsu j n L x . (4.55) ∞ r s 0 u j1 n u j2 n tdt s u j3 n u j4 n dsu j n . For i = 1, 2, x . (4.61) r 0 u j1 n u j2 n sds r |u k n | 2 x i r u j n . (4.62) If j 1 = j 2 , r 0 u j1 n u j2 n sds r |u k n | 2 x i r u j n L 4 3 t,x u j1 n (t − (t j1 n − t j2 n ), ·)u j2 n (t, ·) L 2 t L 2 x u j n L 4 t L 4 x u k n 2 L ∞ t L 4 x . (4.63) If j 1 = j 2 = j, r 0 u j1 n u j2 n sds r |u k n | 2 x i r u j n L