Energy Solution to the Chern-Simons-Schrödinger Equations

Abstract

We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space 𝐻¹(ℝ²). The Coulomb gauge provides elliptic features for gauge fields 𝐴₀,𝐴_𝑗. The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.

1. Introduction

We study herein the initial value problem of the Chern-Simons-Schrödinger (CSS) equations 𝑖𝐷₀𝜙+𝐷_𝑗𝐷_𝑗𝜙=−𝜆𝜙²𝜕𝜙,₀𝐴₁−𝜕₁𝐴₀=−Im𝜙𝐷₂𝜙,𝜕₀𝐴₂−𝜕₂𝐴₀=Im𝜙𝐷₁𝜙,𝜕₁𝐴₂−𝜕₂𝐴₁1=−2𝜙², (1) where 𝑖 denotes the imaginary unit; 𝜕₀=𝜕/𝜕𝑡, 𝜕₁=𝜕/𝜕𝑥₁, and 𝜕₂=𝜕/𝜕𝑥₂ for (𝑡,𝑥₁,𝑥₂)∈ℝ¹⁺²; 𝜙∶ℝ¹⁺²→ℂ is the complex scalar field; 𝐴_𝜇∶ℝ¹⁺²→ℝ is the gauge field; 𝐷_𝜇=𝜕_𝜇+𝑖𝐴_𝜇 is the covariant derivative for 𝜇=0,1,2, and 𝜆>0 is a coupling constant representing the strength of interaction potential. The summation convention used involves summing over repeated indices and Latin indices are used to denote 1,2.

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 high-temperature 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 𝑀(𝑡)=_ℝ2𝜙(𝑡,𝑥)²𝑑𝑥=𝑀(0), (2) and the conservation of total energy 𝐸(𝑡)=_ℝ2𝐷_𝑗𝜙(𝑡,𝑥)²−𝜆2𝜙(𝑡,𝑥)⁴𝑑𝑥=𝐸(0). (3) Note that the terms |𝐹|²=(1/2)𝐹_𝜇𝜈𝐹^𝜇𝜈 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: 𝜙^𝑎(𝑡,𝑥)=𝑎𝜙𝑎²𝑡,𝑎𝑥,𝐴^𝑎_𝑗(𝑡,𝑥)=𝑎𝐴_𝑗𝑎²,𝐴𝑡,𝑎𝑥_𝑎0(𝑡,𝑥)=𝑎²𝐴₀𝑎²𝑡,𝑎𝑥. (4) Therefore, the scaled critical Sobolev exponent is 𝑠_𝑐=0 for 𝜙. 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: 𝜙→𝜙𝑒^𝑖𝜒,𝐴_𝜇→𝐴_𝜇−𝜕_𝜇𝜒, (5) where 𝜒∶ℝ²⁺¹→ℝ is a smooth function. Therefore, a solution to the CSS system is formed by a class of gauge equivalent pairs (𝜙,𝐴_𝜇). In this work, we fix the gauge by imposing the Coulomb gauge condition of 𝜕_𝑗𝐴_𝑗=0, under which the Cauchy problem of the CSS system may be reformulated as follows: 𝑖𝜕_𝑡𝜙−𝐴₀𝜙+Δ𝜙+2𝑖𝐴_𝑗𝜕_𝑗𝜙−𝐴²_𝑗𝜙=−𝜆𝜙²𝜕𝜙,₁𝐴₂−𝜕₂𝐴₁=−1/2𝜙²,𝜕₁𝐴₁+𝜕₂𝐴₂=0,Δ𝐴₀=lm𝑄₁₂𝜙,𝜙+𝜕₁𝐴₂𝜙²−𝜕₂𝐴₁𝜙², (6)(7)(8) where the initial data 𝜙(0,𝑥)=𝜙₀(𝑥). 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 𝐻²(ℝ²), and that there exists at least one global solution, 𝜙∈𝐿^∞(ℝ⁺;𝐻¹(ℝ²))∩𝐶_𝜔(ℝ⁺;𝐻¹(ℝ²)), provided that the initial data are made sufficiently small in 𝐿²(ℝ²) 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 pseudo-conformal 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 𝜙₀∈𝐻¹(ℝ²). 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 𝐻¹(ℝ²). We follow a rather direct means of constructing the 𝐻¹ 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 𝐴₀. In the MSM, 𝐴₀ can be written roughly as 𝑅_𝑗𝑅_𝑘(𝑢²)+|𝑢|², where 𝑅_𝑗=𝜕_𝑗(−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 𝐻^𝑠1(ℝ²) with 𝑠₁>1/2, and similarly, the uniqueness was proved in [14] for 𝐻^𝑠2(ℝ²) with 𝑠₂>3/4. To show the existence and uniqueness of the 𝐻¹ solution to the CSS system, the estimate of the gauge field, 𝐴₀, is important for situations in which special structures of nonlinear terms in the defining equation for 𝐴₀ are used. The following describes are our main results.

Theorem 1. Let initial data 𝜙₀ belong to 𝐻¹(ℝ²). Then, there exists a local-in-time solution, 𝜙, to (6)–(8) that satisfies 𝜙∈𝐿^∞[0,𝑇);𝐻¹ℝ²∩𝐶[0,𝑇);𝐿²ℝ²,𝐽^𝛿𝜙∈𝐿^𝑝0,𝑇;𝐿^𝑞ℝ², (9) where 0<𝛿<1/2, 2<𝛿𝑞, 1/𝑝+1/𝑞=1/2 and 𝐽=(1−Δ)^1/2.

Theorem 2. Let 𝜙 and 𝜓 be solutions to (6)–(8) on (0,𝑇)×ℝ² in the distribution sense with the same initial data to that outlined vide supra. Moreover, one assumes that 𝜙,𝜓∈𝐶[0,𝑇];𝐿²ℝ²,𝜙_{𝐿∞𝑇𝐻1}≤𝑀,𝜓_{𝐿∞𝑇𝐻1}≤𝑀, (10) for some constant 𝑀>0. One then has ‖(𝜙−𝜓)(𝑡,⋅)‖_𝐿2(ℝ2)=0 for 0≤𝑡≤𝑇.

We present some preliminaries in Section 2. Theorems 1 and 2 are proved in Sections 3 and 4, respectively. We conclude the current section by providing a few notations. We denote space time derivatives by 𝜕=(𝜕₀,𝜕₁,𝜕₂) and ∇ is used for spacial derivatives. We use the standard Sobolev spaces 𝑊^𝑠,𝑝, with the norm ‖𝑓‖_{𝑊𝑠,𝑝}=‖𝐽^𝑠𝑓‖_𝐿𝑝 and 𝑊̇^𝑠,𝑝 with the norm ‖𝑓‖_{̇𝑊𝑠,𝑝}=‖|∇|^𝑠𝑓‖_𝐿𝑝, where 𝐽=(1−Δ)^1/2 and |∇|=(−Δ)^1/2. The space 𝐻^𝑠 denotes 𝑊^𝑠,2. We define the space time norm as ‖𝑓‖_𝐿^𝑝_𝑇𝐿𝑞=(∫^𝑇₀‖𝑓(𝑡,⋅)‖^𝑝_{𝐿𝑞(ℝ2)}𝑑𝑡)^1/𝑝. We use 𝑐,𝐶 to denote various constants. Because we are interested in local solutions, we may assume that 𝑇≤1. Thus, we replace the smooth function of 𝑇,𝐶(𝑇) with 𝐶. We also use the convention of writing 𝐴≲𝐵 as shorthand for 𝐴≤𝐶𝐵.

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 𝑓 and 𝑔 be two functions in 𝐻¹(ℝ²) and let 𝑢 be the solution of Δ𝑢=𝜕₁𝑓𝜕₂𝑔−𝜕₂𝑓𝜕₁𝑔inℝ², (11) where 𝑢 is small at infinity. Then, 𝑢∈𝐶(ℝ²)∩𝐻̇¹(ℝ²) and ‖𝑢‖_𝐿^∞_(ℝ²₎+‖∇𝑢‖_𝐿²_(ℝ²₎≤𝐶∇𝑓_𝐿2(ℝ2)∇𝑔_𝐿2(ℝ2). (12)

The following energy estimate in [17, 18] is used for estimating a solution to the magnetic Schrödinger equation.

Lemma 4. Let u be a solution of 𝑖𝜕_𝑡𝑢+Δ𝑢+2𝑖div(𝑎𝑢)=𝐹, (13) where 𝑎=(𝑎₁(𝑡,𝑥),𝑎₂(𝑡,𝑥)) and 𝑎_𝑗 are real-valued functions. Then, for 𝑠≥0 there exists an absolute constant 𝐶_𝑠>0 such that ‖𝑢(𝑡,⋅)‖_̇𝐻^𝑠≤‖𝑢(0,⋅)‖_̇𝐻^𝑠+𝐶_𝑠^𝑡₀‖∇𝑎‖_̇𝐻^𝑠‖𝑢‖_𝐿^∞+‖∇𝑎‖_𝐿^∞‖𝑢‖_̇𝐻^𝑠+‖𝐹‖_̇𝐻^𝑠𝑑𝑠, (14) wherein one means the homogeneous Sobolev space 𝐻̇^𝑠 when 𝑠>0 and simply 𝐿² when 𝑠=0.

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𝜕_𝑡𝑣+Δ𝑣=𝐹₁+𝐹₂,(𝑡,𝑥)∈(0,𝑇)×ℝ². (15) Then, for 𝛿∈𝑅 and 𝜀>0, one has 𝐽^𝛿𝑣_{𝐿𝑝𝑇𝐿𝑞}≲‖𝑣‖_𝐿^∞_𝑇𝐻^{𝛿+1/2+𝜀}+𝐹_{1𝐿2𝑇𝐻𝛿−1/2}+𝐹_{2𝐿1𝑇𝐻𝛿}, (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‖_𝐿^𝑞_(ℝ²₎≤(4𝜋)^{(2−𝑞)/2𝑞}𝑞2^1/2‖𝑢‖_𝐿^2/𝑞_2(ℝ2)‖∇𝑢‖_𝐿^1−2/𝑞_2(ℝ2). (17)

3. The Proof of Theorem 1

Theorem 1 is proved in this section. Because the local well-posedness 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: 𝐷_𝛼𝜙𝐷_𝛽𝜙−𝐷_𝛽𝜙𝐷_𝛼𝜙=𝑄_𝛼𝛽𝐴𝜙,𝜙−𝑖_𝛼𝜕_𝛽𝜙²−𝐴_𝛽𝜕_𝛼𝜙²,𝐷_𝛼𝐷_𝛽𝜙−𝐷_𝛽𝐷_𝛼𝜙=𝑖𝐹_𝛼𝛽𝜙, (18) where 𝑄_𝛼𝛽(𝜙,𝜙)=𝜕_𝛼𝜙𝜕_𝛽𝜙−𝜕_𝛽𝜙𝜕_𝛼𝜙 and 𝐹_𝛼𝛽=𝜕_𝛼𝐴_𝛽−𝜕_𝛽𝐴_𝛼. Note that the second-order terms 𝜕_𝛼𝛽𝜙 are cancelled out. Combined with the above algebra, the equation for 𝐴₀ comes from the second and third equations in (1): Δ𝐴₀=𝜕₁Im𝜙𝐷₂𝜙−𝜕₂Im𝜙𝐷₁𝜙=Im𝑄₁₂𝜙,𝜙+𝜕₁𝐴₂𝜙²−𝜕₂𝐴₁𝜙². (19) We then have the formulation (6)–(8) in which 𝜙 is the only dynamical variable and 𝐴₁, 𝐴₂, and 𝐴₀ are determined through (7) and (8).

The constraint equation 𝜕₁𝐴₂−𝜕₂𝐴₁=−1/2|𝜙|² and the Coulomb gauge condition 𝜕₁𝐴₁+𝜕₂𝐴₂=0  provide an elliptic feature of 𝐴=(𝐴₁,𝐴₂); that is, the components 𝐴_𝑗 can be determined from 𝜙 by solving the elliptic equations Δ𝐴₁=𝜕₂12𝜙²,Δ𝐴₂=−𝜕₁12𝜙². (20) 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 𝐴=(𝐴₁,𝐴₂): 𝐴₁1=−𝑥4𝜋|𝑥|²²∗𝜙²,𝐴₂=1𝑥4𝜋|𝑥|¹²∗𝜙². (21)

3.1. Estimates for 𝐴 and 𝐴₀

We are now ready to estimate several quantities of 𝐴,𝐴₀. Making use of (20) and the representation (21), we obtain the following estimates for 𝐴.

Proposition 7. Let 𝑠≥0 and 0<2/𝑞<𝛿<1. One also assumes that 2≤𝑝<∞ if 𝑠>0 or 2<𝑝<∞ if 𝑠=0. Then, one has ‖∇𝐴‖_̇𝐻^𝑠≲𝜙_𝐿∞𝜙_̇𝐻𝑠,‖∇𝐴‖_𝐿^∞≲𝜙_𝐿∞𝐽^𝛿𝜙_𝐿𝑞,‖𝐴‖_𝐿^∞≲𝜙_𝐿2𝐽^𝛿𝜙_𝐿𝑞,|∇|^𝑠𝐴_𝐿𝑝≲𝜙_𝐿𝑝𝜙_̇𝐻𝑠. (22)

Proof. The above can be checked by applying Calderon-Zygmund and Hardy-Littlewood-Sobolev inequalities. We refer to [2, Section 2] for the details.

To estimate 𝐴₀, the special algebraic structure 𝑄₁₂ and divergence form of the nonlinear terms in (19) are used.

Proposition 8. Let 𝐴₀ be the solution of (19). Then, one has 𝐴_0𝐿∞+∇𝐴_0𝐿2≲1+𝜙²_𝐿2∇𝜙²_𝐿2. (23)

Proof. Decompose 𝐴₀=𝐴^′₀+𝐴₀^′′ as follows: Δ𝐴^′₀=Im𝑄₁₂𝜙,𝜙,Δ𝐴₀^′′=𝜕₁𝐴₂𝜙²−𝜕₂𝐴₁𝜙². (24)(25) We first estimate the quantity ‖𝐴₀‖_{𝐿∞(ℝ2)}. Applying Lemma 3 to (24), we deduce that 𝐴^′_0𝐿∞≲∇𝜙²_𝐿2. (26) To estimate ‖𝐴₀^′′‖_{𝐿∞(ℝ2)} we use the Gagliardo-Nirenberg inequality with small 𝜖>0: ‖𝑢‖_𝐿^∞_(ℝ²₎≤𝐶_𝜖‖Δ𝑢‖^𝛼_{𝐿1+𝜖ℝ2}‖𝑢‖_𝐿^1−𝛼_4ℝ2,with𝛼=(1+𝜖)/(1+5𝜖). (27) Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce 𝐴₀^′′_𝐿4≲𝐴𝜙²_𝐿4/3≲‖𝐴‖_𝐿⁴𝜙²_𝐿4≲𝜙_𝐿2𝜙³_𝐿4≲𝜙_𝐿^5/2₂∇𝜙_𝐿^3/2₂, (28) where Proposition 7 and Lemma 6 are used. We can also derive the following from (25): Δ𝐴₀^′′_𝐿1+𝜖≲∇𝐴⋅𝜙²_{𝐿1+𝜖(i)}+𝐴∇𝜙²_{𝐿1+𝜖(ii)}. (29) The first term can be estimated as follows: (i)≲‖∇𝐴‖_𝐿²𝜙²_{𝐿(2+2𝜖)/(1−𝜖)}≲𝜙_𝐿^2/(1+𝜖)₂∇𝜙_𝐿^{(2+4𝜖)/(1+𝜖)}₂, (30) where ‖𝜙‖_𝐿^{(4+4𝜖)/(1−𝜖)}≲‖𝜙‖_𝐿^{(1−𝜖)/(2+2𝜖)}₂‖∇𝜙‖_𝐿^{(1+3𝜖)/(2+2𝜖)}₂ is used. The second term can be estimated as follows: (ii)≲‖𝐴‖_𝐿⁴𝜙_{𝐿(4+4𝜖)/(1−3𝜖)}∇𝜙_𝐿2≲𝜙_𝐿2𝜙_𝐿4𝜙_𝐿^{(1−3𝜖)/(2+2𝜖)}₂∇𝜙_𝐿^{(3+7𝜖)/(2+2𝜖)}₂≲𝜙_𝐿^2/(1+𝜖)₂∇𝜙_𝐿^{(2+4𝜖)/(1+𝜖)}₂, (31) where ‖𝜙‖_𝐿^{(4+4𝜖)/(1−3𝜖)}≲‖𝜙‖_𝐿^{(1−3𝜖)/(2+2𝜖)}₂‖∇𝜙‖_𝐿^{(1+5𝜖)/(2+2𝜖)}₂ is used. Therefore, we obtain with 𝜖=1/11, that is, 𝛼=3/4, 𝐴₀^′′_𝐿∞≲𝜙²_𝐿2∇𝜙²_𝐿2. (32) Therefore, we conclude that 𝐴_0𝐿∞≲∇𝜙²_𝐿21+𝜙²_𝐿2. (33) On the other hand, Lemma 3 shows that ∇𝐴^′_0𝐿2≲∇𝜙²_𝐿2. (34) We also have from (25) that ∇𝐴₀^′′_𝐿2≲𝐴|𝜙|²_𝐿2≲‖𝐴‖_𝐿⁴𝜙²_𝐿8≲𝜙_𝐿2𝜙_𝐿4𝜙²_𝐿8≲𝜙²_𝐿2∇𝜙²_𝐿2. (35) Therefore, we have ∇𝐴_0𝐿2≲∇𝜙²_𝐿21+𝜙²_𝐿2. (36)

3.2. The Energy Solution to (CSS)

We now prove Theorem 1. Let us define 𝑋(𝑇)=𝜙_{𝐿∞𝑇𝐻1}+𝐽^𝛿𝜙_{𝐿𝑝𝑇𝐿𝑞}, (37) where 0<𝛿<1/2, 2<𝛿𝑞, and 1/𝑝+1/𝑞=1/2. We derive the following estimate: 𝑋≲𝜙_0𝐻1+𝑇^1/6𝜙_0𝐿21+𝜙₀²_𝐿2𝑋²+𝑋⁴, (38) from which Theorem 1 is proved by standard argument; see [2, Section 3].

To control ‖𝜙‖_𝐿^∞_𝑇𝐻¹, we apply Lemma 4 to the solution of (6)–(8).

Proposition 9. Let 𝜙 be a solution to (6)–(8). Then, one has 𝜙_{𝐿∞𝑇𝐿2}=𝜙_0𝐿2,𝜙_{𝐿∞𝑇̇𝐻1}≲𝜙_0̇𝐻1+1+𝜙₀²_𝐿2𝑇^{(𝑝−3)/𝑝}𝜙³_{𝐿∞𝑇̇𝐻1}+𝐽^𝛿𝜙³_{𝐿𝑝𝑇𝐿𝑞}, (39) where 2<𝛿𝑞 and 3<𝑝<∞.

Proof. From the conservation of mass, we derive the first estimate. We apply Lemma 4 to (6) with 𝐹=𝐴₀𝜙+𝐴²_𝑗𝜙−𝜆|𝜙|²𝜙 and 𝑠=1. Combined with Proposition 7, we have ‖∇𝐴‖_̇𝐻¹𝜙_𝐿∞≲𝜙²_{𝑊𝛿,𝑞}𝜙_̇𝐻1,‖∇𝐴‖_𝐿^∞𝜙_̇𝐻1≲𝜙²_{𝑊𝛿,𝑞}𝜙_̇𝐻1,𝐴²𝜙_̇𝐻1≲𝜙²_𝐿2𝜙³_{𝑊𝛿,𝑞}+𝜙³_̇𝐻1,𝜙²𝜙_̇𝐻1≲𝜙²_{𝑊𝛿,𝑞}𝜙_̇𝐻1, (40) where 2<𝛿𝑞. We are then left to estimate ‖𝐴₀𝜙‖_̇𝐻¹. By Proposition 8, we obtain 𝐴₀𝜙_̇𝐻1≲𝐴_0𝐿∞𝜙_̇𝐻1+𝐴_0̇𝐻1𝜙_𝐿∞≲1+𝜙²_𝐿2∇𝜙³_𝐿2+𝐽^𝛿𝜙³_𝐿𝑞. (41) Combining (40) and (41), we obtain 𝜙_{𝐿∞𝑇̇𝐻1}≲𝜙_0̇𝐻1+^𝑇₀1+𝜙₀²_𝐿2𝜙³_̇𝐻1+𝐽^𝛿𝜙³_𝐿𝑞≲𝜙_0̇𝐻1+1+𝜙₀²_𝐿2𝑇^{(𝑝−3)/𝑝}×𝜙³_{𝐿∞𝑇̇𝐻1}+𝐽^𝛿𝜙³_{𝐿𝑝𝑇𝐿𝑞}, (42) where 3<𝑝<∞ and 𝑇<1.

To estimate ‖𝐽^𝛿𝜙‖_𝐿^𝑝_𝑇𝐿𝑞, we apply Lemma 5 to the solution of (6)–(8).

Proposition 10. Let 𝜙 be a solution to (6)–(8). Then, one has 𝐽^𝛿𝜙_{𝐿𝑝𝑇𝐿𝑞}≲𝜙_{𝐿∞𝑇𝐻1}+𝑇^1/6𝜙_0𝐿21+𝜙₀²_𝐿2𝑋²+𝑋⁴, (43) where 2<𝛿𝑞, 3<𝑝<∞ and 1/𝑝+1/𝑞=1/2.

Proof. Applying Lemma 5 with 𝐹₁=𝐴₀𝜙−2𝑖𝐴_𝑗𝜕_𝑗𝜙 and 𝐹₂=𝐴²𝜙−𝜆|𝜙|²𝜙, we obtain 𝐽^𝛿𝜙_{𝐿𝑝𝑇𝐿𝑞}≲𝜙_{𝐿∞𝑇𝐻1}+𝐴₀𝜙_{𝐿2𝑇𝐻𝛿−1/2}+𝐴⋅∇𝜙_{𝐿2𝑇𝐻𝛿−1/2}+𝐴²𝜙_{𝐿1𝑇𝐻𝛿}+𝜙²𝜙_{𝐿1𝑇𝐻𝛿}, (44) where 𝛿=1/2−𝜀, 3<𝑝<∞ and 2<𝛿𝑞. Considering Proposition 8, we obtain 𝐴₀𝜙_{𝐿2𝑇𝐻𝛿−1/2}≲𝐴_{0𝐿∞𝑇𝐿∞}𝜙_{𝐿2𝑇𝐿2}≲𝑇^1/2𝜙_0𝐿21+𝜙₀²_𝐿2∇𝜙²_𝐿2. (45) The other terms can be treated, as mentioned in Section 1, by similar arguments to those in [2, Section 3]. Applying Proposition 7, we have 𝐴⋅∇𝜙_{𝐿2𝑇𝐻𝛿−1/2}≲‖𝐴‖_𝐿²_𝑇𝐿^∞∇𝜙_{𝐿∞𝑇𝐿2}≲𝜙_0𝐿2𝑇^{(𝑝−2)/2𝑝}𝐽^𝛿𝜙_{𝐿𝑝𝑇𝐿𝑞}𝜙_{𝐿∞𝑇𝐻1},𝐴²𝜙_{𝐿1𝑇𝐻𝛿}≲𝐴²_{𝐿1𝑇𝐿4}𝐽^𝛿𝜙_{𝐿∞𝑇𝐿4}+𝐴²_{𝐿2𝑇𝑊𝛿,2+𝜀}𝜙_{𝐿2𝑇𝐿(4+2𝜀)/𝜀}≲𝑇𝜙_0𝐿^3/2₂𝜙_𝐿^5/2_∞𝑇𝐻1+𝑇^1/4𝜙_0𝐿2𝜙²_{𝐿∞𝑇𝐻1}𝐽^𝛿𝜙²_{𝐿𝑝𝑇𝐿𝑞},𝜙³_{𝐿1𝑇𝐻𝛿}≲𝐽^𝛿𝜙_{𝐿∞𝑇𝐿2}𝜙²_{𝐿2𝑇𝐿∞}≲𝑇^{(𝑝−2)/𝑝}𝜙_{𝐿∞𝑇𝐻1}𝐽^𝛿𝜙²_{𝐿𝑝𝑇𝐿𝑞}. (46)(47)(48) Plugging estimates (45)–(48) into (44) with 𝑝>3, we obtain 𝐽^𝛿𝜙_{𝐿𝑝𝑇𝐿𝑞}≲𝜙_{𝐿∞𝑇𝐻1}+𝑇^1/6𝜙_0𝐿21+𝜙₀²_𝐿2𝑋²+𝑋⁴. (49)

We finally obtain the estimate (38) by combining Propositions 9 and 10, which proves Theorem 1.

4. 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].

Let (𝜙,𝐴₀,𝐴) and (𝜓,𝐵₀,𝐵) be solutions of (6)–(8) with the same initial data. If we set 𝜔=𝜙−𝜓, then the equation for 𝜔 is 𝑖𝜕_𝑡𝜔+Δ𝜔=𝐴₀𝜔+𝐴₀−𝐵₀𝜓−2𝑖𝐴⋅∇𝜔−2𝑖(𝐴−𝐵)⋅∇𝜓+𝐴²𝜔+𝐴²−𝐵²𝜓−𝜆𝜙²𝜔−𝜆𝜙²−𝜓²𝜓. (50) We will derive 𝜕_𝑡‖𝜔‖²_𝐿2≲𝑞^1/2𝑀²‖𝜔‖_𝐿^2−4/𝑞₂+𝑞𝑀^2+4/𝑞1+𝑀²‖𝜔‖_𝐿^2−4/𝑞₂, (51) where 𝑀 is a constant in Theorem 2 and 𝑞>2. Then we have 𝜕_𝑡‖𝜔‖_𝐿^4/𝑞₂≲1𝑞𝑞^1/2𝑀²+𝑞𝑀^2+4/𝑞1+𝑀². (52) Considering ‖𝜔(0,⋅)‖_𝐿²=0 and 2<𝑞, we obtain ‖𝜔‖_𝐿²≲𝑇(𝑀²+𝑀^4+4/𝑞)^𝑞/4. (53) Letting 𝑞→∞, for the time interval satisfying 𝑇(𝑀²+𝑀^4+4/𝑞)≤1/2, we conclude that ‖𝜔(𝑡,⋅)‖_𝐿²=0 for 0≤𝑡≤𝑇, which thus proves Theorem 2.

In the remainder of this section, we derive inequality (51). Multiplying 𝜔 to both sides of (50) and integrating the imaginary part of ℝ², we have 𝜕_𝑡‖𝜔‖²_𝐿2=2𝐴₀−𝐵₀Im𝜓𝜔_(I)−2𝐴_𝑗𝜕_𝑗|𝜔|²_(II)−4(𝐴_𝑗−𝐵_𝑗)Re(𝜕_𝑗𝜓𝜔)_(III)𝐴𝑑𝑥+2²−𝐵²Im𝜓𝜔_(IV)−2𝜆(𝜙²−𝜓²)Im(𝜓𝜔)_(V)𝑑𝑥. (54) The integrals (II)–(V), that is, those not containing 𝐴₀, can be controlled by applying similar arguments to those described in [2, Section 4]. Integral (II) can be estimated, considering 𝜕_𝑗𝐴_𝑗=0, by −𝐴_𝑗𝜕_𝑗|𝜔|²𝑑𝑥=𝜕_𝑗𝐴_𝑗|𝜔|²(),(),(V)𝑑𝑥=0,IIIIV≲𝑞𝑀^2+4/𝑞1+𝑀²‖𝜔‖_𝐿^2−4/𝑞₂ (55) for which we omit the proof.

We simply present how to control integral (I), for which we have 𝐴₀−𝐵₀Im𝜓𝜔𝑑𝑥≲𝐴₀−𝐵_0𝐿𝑎𝜓_𝐿𝑏‖𝜔‖_𝐿^𝑐, (56) where 1/𝑎+1/𝑏+1/𝑐=1, 2≤𝑎,𝑏,𝑐. Applying Lemma 6, we obtain 𝜓_𝐿𝑏≲𝑏^1/2𝜓_𝐿^2/𝑏₂∇𝜓_𝐿^1−2/𝑏₂≲𝑏^1/2𝑀^1−2/𝑏,‖𝜔‖_𝐿^𝑐≲𝑐^1/2‖𝜔‖_𝐿^2/𝑐₂‖∇𝜔‖_𝐿^1−2/𝑐₂≲𝑐^1/2‖𝜔‖_𝐿^2/𝑐₂𝑀^1−2/𝑐. (57) To control ‖𝐴₀−𝐵₀‖_𝐿^𝑎, we consider the equation for 𝐴₀−𝐵₀Δ𝐴₀−𝐵₀=𝜕₁Im𝜙𝜕₂𝜙−𝜕₂Im𝜙𝜕₁𝜙−𝜕₁Im𝜓𝜕₂𝜓+𝜕₂Im𝜓𝜕₁𝜓+𝜕₁𝐴₂𝜙²−𝜕₂𝐴₁𝜙²−𝜕₁𝐵₂𝜓²+𝜕₂𝐵₁𝜓². (58) Decomposing 𝐴₀ and 𝐵₀ as (24) and (25), we have Δ𝐴^′₀−𝐵^′₀=𝜕₁Im𝜙𝜕₂𝜔−𝜕₂Im𝜓𝜕₁𝜔+𝜕₁Im𝜔𝜕₂𝜓−𝜕₂Im𝜔𝜕₁𝜙,Δ𝐴₀^′′−𝐵₀^′′=𝜕₁𝐴₂𝜙²−𝜓²−𝜕₂𝐴₁𝜙²−𝜓²+𝜕₁𝐴₂−𝐵₂𝜓²−𝜕₂𝐴₁−𝐵₁𝜙². (59)(60) Taking into account 𝜕₁Im𝜙𝜕₂𝜔=𝜕₁𝜕₂Im𝜙𝜔−Im𝜔𝜕₂𝜙,𝜕₂Im𝜓𝜕₁𝜔=𝜕₂𝜕₁Im𝜓𝜔−Im𝜔𝜕₁𝜓, (61) we can rewrite the equation for 𝐴^′₀−𝐵^′₀ as follows: Δ𝐴^′₀−𝐵^′₀=𝜕₁Im𝜔𝜕₂𝜓−Im𝜔𝜕₂𝜙+𝜕₂Im𝜔𝜕₁𝜓−Im𝜔𝜕₂𝜙, (62) where 𝜕₁𝜕₂Im(𝜙𝜔)−𝜕₂𝜕₁Im(𝜓𝜔)=𝜕₁𝜕₂Im(𝜔𝜔)=0 should be noted. Using the Hardy-Littlewood-Sobolev inequality, we have 𝐴^′₀−𝐵^′_0𝐿𝑎≲|𝑥|⁻¹∗𝜔∇𝜓_𝐿𝑎≲𝜔∇𝜓_𝐿𝑟≲‖𝜔‖_𝐿^𝑠∇𝜓_𝐿2, (63) where 1/𝑎=1/𝑟−1/2 and 1/𝑟=1/𝑠+1/2, from which we deduce 𝑎=𝑠. Then, we have ‖𝜔‖_𝐿^𝑎∇𝜓_𝐿2≲𝑎^1/2‖𝜔‖_𝐿^2/𝑎₂‖∇𝜔‖_𝐿^1−2/𝑎₂𝑀≲𝑎^1/2𝑀^2−2/𝑎‖𝜔‖_𝐿^2/𝑎₂. (64)

The term 𝐴₀^′′−𝐵₀^′′ can be bounded as follows: 𝐴₀^′′−𝐵₀^′′_𝐿𝑎≲|𝑥|⁻¹∗|𝐴|𝜙²−𝜓²_𝐿𝑎(1)+|𝑥|⁻¹∗|𝐴−𝐵|𝜙²+𝜓²_𝐿𝑎(2). (65) Since ||𝜙|²−|𝜓|²|≤(|𝜙|+|𝜓|)|𝜔|, we have (1)≲|𝐴|(|𝜙|+|𝜓|)_𝐿2‖𝜔‖_𝐿^𝑎≲‖𝐴‖_𝐿⁶𝜙_𝐿3+𝜓_𝐿3‖𝜔‖_𝐿^𝑎≲𝜙²_𝐿3𝜙_𝐿3+𝜓_𝐿3‖𝜔‖_𝐿^𝑎≲𝑎^1/2𝑀^2−2/𝑎‖𝜔‖_𝐿^2/𝑎₂. (66)(67) Since |𝐴_𝑗−𝐵_𝑗|≲|𝑥|⁻¹∗((|𝜙|+|𝜓|)|𝜔|), we may check (2)≲𝐴_𝑗−𝐵_𝑗𝐿𝑎𝜙²_𝐿2+𝜓²_𝐿2≲𝜙_𝐿2+𝜓_𝐿2‖𝜔‖_𝐿^𝑎𝜙²_𝐿4+𝜓²_𝐿4≲𝑎^1/2‖𝜔‖_𝐿^2/𝑎₂‖∇𝜔‖_𝐿^1−2/𝑎₂∇𝜙_𝐿2+∇𝜓_𝐿2≲𝑎^1/2𝑀^2−2/𝑎‖𝜔‖_𝐿^2/𝑎₂. (68) Then, we have 𝐴₀−𝐵_0𝐿𝑎≲𝑎^1/2𝑀^2−2/𝑎‖𝜔‖_𝐿^2/𝑎₂. (69) Combining estimates (57) and (69), and denoting 𝑏=𝑞/2, we obtain 𝐴₀−𝐵_0𝐿𝑎𝜓_𝐿𝑏‖𝜔‖_𝐿^𝑐≲𝑎𝑞𝑐^1/2𝑀²‖𝜔‖_𝐿^2−4/𝑞₂, (70) where 1/𝑎+2/𝑞+1/𝑐=1. We then obtain (51) by combining (55) and (70).

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2011-0015866), and was also partially supported by the TJ Park Junior Faculty Fellowship.