Energy Solution to the Chern-Simons-Schrödinger Equations

and Applied Analysis 3 The following type of Strichartz estimate was used in [19, 20] for the study of the Benjamin-Ono equation. We refer to [12] for the counterpart to the Schrödinger equation. Lemma 5. Let T ≤ 1 and V be a solution to the equation i∂ t V + ΔV = F 1 + F 2 , (t, x) ∈ (0, T) ×R 2 . (15) Then, for δ ∈ R and ε > 0, one has 󵄩󵄩󵄩󵄩 J δ V 󵄩󵄩󵄩󵄩L p T L q ≲ ‖V‖ L ∞ T H δ+1/2+ε + 󵄩󵄩󵄩󵄩F1 󵄩󵄩󵄩󵄩L2 T H δ−1/2 + 󵄩󵄩󵄩󵄩F2 󵄩󵄩󵄩󵄩L1 T H δ , (16) where 1/p + 1/q = 1/2 and 2 ≤ q < ∞. We use the following Gagliardo-Nirenberg inequality with the specific constant [21], especially for the proof of Theorem 2. Lemma 6. For 2 ≤ q < ∞, one has ‖u‖Lq(R2) ≤ (4π) (2−q)/2q ( q

The CSS system of equations was proposed in [1,2] to deal with the electromagnetic phenomena in planar domains, such as the fractional quantum Hall effect or hightemperature superconductivity. We refer the reader to [3,4] for more information on the physical nature of these phenomena.
The CSS system exhibits conservation of mass and the conservation of total energy Note that the terms | | 2 = (1/2) ] ] are missing in (3) when compared to the Maxwell-Schrödinger equations studied in [5].
To figure out the optimal regularity for the CSS system, we observe that the CSS system is invariant under scaling: Therefore, the scaled critical Sobolev exponent is = 0 for . In view of (2) we may say that the initial value problem of the CSS system is mass critical.
The CSS system is invariant under the following gauge transformations: where : R 2+1 → R is a smooth function. Therefore, a solution to the CSS system is formed by a class of gauge 2 Abstract and Applied Analysis equivalent pairs ( , ). In this work, we fix the gauge by imposing the Coulomb gauge condition of = 0, under which the Cauchy problem of the CSS system may be reformulated as follows: where the initial data (0, ) = 0 ( ). For the formulation of (6)-(8) we refer the reader to Section 3.
The initial value problem of the CSS system was investigated in [6,7]. It was shown in [6] that the Cauchy problem is locally well posed in 2 (R 2 ), and that there exists at least one global solution, ∈ ∞ (R + ; 1 (R 2 )) ∩ (R + ; 1 (R 2 )), provided that the initial data are made sufficiently small in 2 (R 2 ) by finding regularized equations. They also showed, by deriving a virial identity, that solutions blow up in finite time under certain conditions. Explicit blow-up solutions were constructed in [8] through the use of a pseudoconformal transformation. The existence of a standing wave solution to the CSS system has also been proved in [9,10].
The adiabatic approximation of the Chern-Simons-Schrödinger system with a topological boundary condition was studied in [11], which provides a rigorous description of slow vortex dynamics in the near self-dual limit.
Taking the conservation of energy (3) into account, it seems natural to consider the Cauchy problem of the CSS system with initial data 0 ∈ 1 (R 2 ). Our purpose here is to supplement the original result of [6] by showing that there is a unique local-in-time solution in the energy space 1 (R 2 ). We follow a rather direct means of constructing the 1 solution and prove the uniqueness. We adapt the idea discussed in [12,13] where a low regularity solution of the modified Schrödinger map (MSM) was studied. In fact, the CSS and MSM systems have several similarities except for the defining equation for 0 . In the MSM, 0 can be written roughly as ( 2 ) + | | 2 , where = (−Δ) −1/2 denotes the Riesz transform. The local existence of a solution to the MSM was proved in [12] for the initial data in 1 (R 2 ) with Theorem 1. Let initial data 0 belong to 1 (R 2 ). Then, there exists a local-in-time solution, , to (6)-(8) that satisfies ∈ ∞ ([0, ) ; 1 (R 2 )) ∩ ([0, ) ; 2 (R 2 )) , (6)-(8) on (0, ) × R 2 in the distribution sense with the same initial data to that outlined vide supra. Moreover, one assumes that , ∈ ([0, ] ; 2 (R 2 )) ,

Preliminaries
We collect here a few lemmas used for the proof of Theorems 1 and 2. The following lemma is reminiscent of Wente's inequality (see [15,16]).

Lemma 3.
Let and be two functions in 1 (R 2 ) and let be the solution of where is small at infinity. Then, ∈ (R 2 ) ∩̇1(R 2 ) and The following energy estimate in [17,18] is used for estimating a solution to the magnetic Schrödinger equation.
Abstract and Applied Analysis 3 The following type of Strichartz estimate was used in [19,20] for the study of the Benjamin-Ono equation. We refer to [12] for the counterpart to the Schrödinger equation.
We use the following Gagliardo-Nirenberg inequality with the specific constant [21], especially for the proof of Theorem 2.

The Proof of Theorem 1
Theorem 1 is proved in this section. Because the local wellposedness for smooth data is already known in [6], we simply present an a priori estimate for the solution to (6)- (8). Let us first explain (8). To derive it, note the following identities: where ( , ) = − and = − . Note that the second-order terms are cancelled out. Combined with the above algebra, the equation for 0 comes from the second and third equations in (1): We then have the formulation (6)- (8) in which is the only dynamical variable and 1 , 2 , and 0 are determined through (7) and (8).
The constraint equation 1 2 − 2 1 = −1/2| | 2 and the Coulomb gauge condition 1 1 + 2 2 = 0 provide an elliptic feature of = ( 1 , 2 ); that is, the components can be determined from by solving the elliptic equations Taking into account that the Coulomb gauge condition in Maxwell dynamics deduces a wave equation, the previous observation was used in [6]. Using (20), we have the following representation of = ( 1 , 2 ): 3.1. Estimates for and 0 . We are now ready to estimate several quantities of , 0 . Making use of (20) and the representation (21), we obtain the following estimates for .
To estimate 0 , the special algebraic structure 12 and divergence form of the nonlinear terms in (19) are used.

Proposition 8.
Let 0 be the solution of (19). Then, one has Proof. Decompose 0 = 0 + 0 as follows: We first estimate the quantity ‖ 0 ‖ ∞ (R 2 ) . Applying Lemma 3 to (24), we deduce that To estimate ‖ 0 ‖ ∞ (R 2 ) we use the Gagliardo-Nirenberg inequality with small > 0: Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce where Proposition 7 and Lemma 6 are used. We can also derive the following from (25): . (29) The first term can be estimated as follows: where is used. The second term can be estimated as follows: where is used. Therefore, we obtain with = 1/11, that is, = 3/4, (32) Therefore, we conclude that On the other hand, Lemma 3 shows that We also have from (25) that Therefore, we have

The Proof of Theorem 2
In this section, we prove the uniqueness of the solution to (6). The basic rationale is borrowed from [12,22].