1.Introduction HyungjinHuh EnergySolutiontotheChern-Simons-SchrödingerEquations ResearchArticle

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Volume 2013, Article ID 590653,7pages http://dx.doi.org/10.1155/2013/590653

Research Article

Energy Solution to the Chern-Simons-Schrödinger Equations

Hyungjin Huh

Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea

Correspondence should be addressed to Hyungjin Huh; [email protected] Received 19 November 2012; Accepted 17 January 2013

Academic Editor: Graziano Crasta

Copyright © 2013 Hyungjin Huh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove that the Chern-Simons-Schr¨odinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space𝐻¹(R²). The Coulomb gauge provides elliptic features for gauge fields𝐴₀, 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-Schr¨odinger (CSS) equations

𝑖𝐷₀𝜙 + 𝐷_𝑗𝐷_𝑗𝜙 = −𝜆󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²𝜙,

𝜕₀𝐴₁− 𝜕₁𝐴₀= −Im(𝜙𝐷₂𝜙) ,

𝜕₀𝐴₂− 𝜕₂𝐴₀=Im(𝜙𝐷₁𝜙) ,

𝜕₁𝐴₂− 𝜕₂𝐴₁= −1 2 󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²,

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where𝑖denotes the imaginary unit;𝜕₀ = 𝜕/𝜕𝑡,𝜕₁ = 𝜕/𝜕𝑥₁, and𝜕₂ = 𝜕/𝜕𝑥₂ for(𝑡, 𝑥₁, 𝑥₂) ∈ R¹⁺²;𝜙 : R¹⁺² → Cis the complex scalar field;𝐴_𝜇 : R¹⁺² → Ris the gauge field;

𝐷_𝜇 = 𝜕_𝜇 + 𝑖𝐴_𝜇 is the covariant derivative for𝜇 = 0, 1, 2, and𝜆 > 0is a coupling constant representing the strength of interaction potential. The summation convention used involves summing over repeated indices and Latin indices are used to denote1, 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) = ∫

R²󵄨󵄨󵄨󵄨𝜙(𝑡,𝑥)󵄨󵄨󵄨󵄨²𝑑𝑥 = 𝑀 (0) , (2) and the conservation of total energy

𝐸 (𝑡) = ∫

R²󵄨󵄨󵄨󵄨󵄨𝐷^𝑗𝜙(𝑡, 𝑥)󵄨󵄨󵄨󵄨󵄨²−𝜆

2󵄨󵄨󵄨󵄨𝜙(𝑡,𝑥)󵄨󵄨󵄨󵄨⁴𝑑𝑥 = 𝐸 (0) . (3) Note that the terms |𝐹|² = (1/2)𝐹_𝜇]𝐹^𝜇] are missing in (3) when compared to the Maxwell-Schr¨odinger 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) = 𝑎𝜙 (𝑎²𝑡, 𝑎𝑥) , 𝐴^𝑎_𝑗(𝑡, 𝑥) = 𝑎𝐴_𝑗(𝑎²𝑡, 𝑎𝑥) , 𝐴^𝑎₀(𝑡, 𝑥) = 𝑎²𝐴₀(𝑎²𝑡, 𝑎𝑥) . (4) Therefore, the scaled critical Sobolev exponent is𝑠_𝑐= 0for𝜙.

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

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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 = −𝜆󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²𝜙, (6)

𝜕₁𝐴₂− 𝜕₂𝐴₁= −1/2󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨², 𝜕₁𝐴₁+ 𝜕₂𝐴₂= 0, (7) Δ𝐴₀=lm(𝑄₁₂(𝜙, 𝜙)) + 𝜕₁(𝐴₂󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) − 𝜕₂(𝐴₁󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) , (8) where the initial data𝜙(0, 𝑥) = 𝜙₀(𝑥). For the formulation of (6)–(8) we refer the reader toSection 3.

The initial value problem of the CSS system was investi- gated in [6,7]. It was shown in [6] that the Cauchy problem is locally well posed in𝐻²(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- Schr¨odinger 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 data𝜙₀ ∈ 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 Schr¨odinger map (MSM) was studied. In fact, the CSS and MSM systems have several similarities except for the defining equation for𝐴₀. In the MSM,𝐴₀can be written roughly as𝑅_𝑗𝑅_𝑘(𝑢²) + |𝑢|², where𝑅_𝑗 = 𝜕_𝑗(−Δ)^−1/2denotes the Riesz transform. The local existence of a solution to the MSM was proved in [12] for the initial data in𝐻^𝑠¹(R²)with 𝑠₁> 1/2, and similarly, the uniqueness was proved in [14] for 𝐻^𝑠²(R²)with𝑠₂> 3/4. To show the existence and uniqueness of the𝐻¹ 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 for𝐴₀ are used. The following describes are our main results.

Theorem 1. Let initial data𝜙₀belong to𝐻¹(R²). Then, there exists a local-in-time solution,𝜙, to(6)–(8)that satisfies

𝜙 ∈ 𝐿^∞([0, 𝑇) ; 𝐻¹(R²)) ∩ 𝐶 ([0, 𝑇) ; 𝐿²(R²)) , 𝐽^𝛿𝜙 ∈ 𝐿^𝑝(0, 𝑇; 𝐿^𝑞(R²)) , (9) where0 < 𝛿 < 1/2,2 < 𝛿𝑞,1/𝑝+1/𝑞 = 1/2and𝐽 = (1−Δ)^1/2.

Theorem 2. Let𝜙 and𝜓be solutions to(6)–(8)on(0, 𝑇) × R²in the distribution sense with the same initial data to that outlined vide supra. Moreover, one assumes that

𝜙, 𝜓 ∈ 𝐶 ([0, 𝑇] ; 𝐿²(R²)) ,

󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹≤ 𝑀, 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹ ≤ 𝑀, (10) for some constant𝑀 > 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 spaces𝑊^𝑠,𝑝, with the norm‖𝑓‖_𝑊𝑠,𝑝 = ‖𝐽^𝑠𝑓‖_𝐿𝑝 and𝑊̇^𝑠,𝑝with the norm‖𝑓‖_𝑊_̇^𝑠,𝑝 = ‖|∇|^𝑠𝑓‖_𝐿𝑝, where𝐽 = (1 − Δ)^1/2 and

|∇| = (−Δ)^1/2. The space 𝐻^𝑠 denotes𝑊^𝑠,2. We define the space time norm as‖𝑓‖_𝐿^𝑝

𝑇𝐿^𝑞 = (∫₀^𝑇‖𝑓(𝑡, ⋅)‖^𝑝_𝐿𝑞(R²)𝑑𝑡)^1/𝑝. We use𝑐, 𝐶to denote various constants. Because we are interested in local solutions, we may assume that 𝑇 ≤ 1. Thus, we replace the smooth function of𝑇, 𝐶(𝑇)with𝐶. We also use the convention of writing𝐴 ≲ 𝐵as shorthand for𝐴 ≤ 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. Let𝑓and𝑔be two functions in𝐻¹(R²)and let𝑢 be the solution of

Δ𝑢 = 𝜕₁𝑓𝜕₂𝑔 − 𝜕₂𝑓𝜕₁𝑔 inR², (11) where𝑢is small at infinity. Then,𝑢 ∈ 𝐶(R²) ∩ ̇𝐻¹(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 Schr¨odinger equation.

Lemma 4. Let u be a solution of

𝑖𝜕_𝑡𝑢 + Δ𝑢 + 2𝑖div(𝑎𝑢) = 𝐹, (13)

where𝑎 = (𝑎₁(𝑡, 𝑥), 𝑎₂(𝑡, 𝑥))and𝑎_𝑗are real-valued functions.

Then, for𝑠 ≥ 0there exists an absolute constant𝐶_𝑠 > 0such that

‖𝑢(𝑡, ⋅)‖ _̇𝐻^𝑠 ≤ ‖𝑢(0, ⋅)‖ _̇𝐻^𝑠 + 𝐶_𝑠∫^𝑡

0(‖∇𝑎‖ _̇𝐻^𝑠‖𝑢‖_𝐿^∞+‖∇𝑎‖_𝐿^∞‖𝑢‖ _̇𝐻^𝑠+‖𝐹‖ _̇𝐻^𝑠) 𝑑𝑠, (14) wherein one means the homogeneous Sobolev space ̇𝐻^𝑠when 𝑠 > 0and simply𝐿²when𝑠 = 0.

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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 Schr¨odinger equation.

Lemma 5. Let𝑇 ≤ 1andVbe a solution to the equation 𝑖𝜕_𝑡V+ ΔV= 𝐹₁+ 𝐹₂, (𝑡, 𝑥) ∈ (0, 𝑇) ×R². (15) Then, for𝛿 ∈ 𝑅and𝜀 > 0, one has

󵄩󵄩󵄩󵄩󵄩𝐽^𝛿V󵄩󵄩󵄩󵄩󵄩𝐿^𝑝_𝑇𝐿^𝑞≲ ‖V‖_𝐿^∞_𝑇_𝐻^{𝛿+1/2+𝜀} + 󵄩󵄩󵄩󵄩𝐹¹󵄩󵄩󵄩󵄩𝐿²_𝑇𝐻^𝛿−1/2+ 󵄩󵄩󵄩󵄩𝐹²󵄩󵄩󵄩󵄩𝐿¹_𝑇𝐻^𝛿, (16) where1/𝑝 + 1/𝑞 = 1/2and2 ≤ 𝑞 < ∞.

We use the following Gagliardo-Nirenberg inequality with the specific constant [21], especially for the proof of Theorem 2.

Lemma 6. For2 ≤ 𝑞 < ∞, one has

‖𝑢‖_𝐿^𝑞_(R²₎≤ (4𝜋)^{(2−𝑞)/2𝑞} (𝑞

2)^1/2‖𝑢‖^2/𝑞_𝐿2(R²)‖∇𝑢‖^1−2/𝑞_𝐿2(R²). (17)

3. The Proof of Theorem 1

Theorem 1is proved in this section. Because the local well- posedness for smooth data is already known in [6], we simply present an a prioriestimate 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_𝛼𝛽(𝜙, 𝜙)

− 𝑖 (𝐴_𝛼𝜕_𝛽(󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) − 𝐴_𝛽𝜕_𝛼(󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²)) , 𝐷_𝛼𝐷_𝛽𝜙 − 𝐷_𝛽𝐷_𝛼𝜙 = 𝑖𝐹_𝛼𝛽𝜙,

(18) where𝑄_𝛼𝛽(𝜙, 𝜙) = 𝜕_𝛼𝜙𝜕_𝛽𝜙 − 𝜕_𝛽𝜙𝜕_𝛼𝜙and𝐹_𝛼𝛽= 𝜕_𝛼𝐴_𝛽− 𝜕_𝛽𝐴_𝛼. Note that the second-order terms 𝜕_𝛼𝛽𝜙 are cancelled out.

Combined with the above algebra, the equation for𝐴₀comes from the second and third equations in (1):

Δ𝐴₀= 𝜕₁Im(𝜙𝐷₂𝜙) − 𝜕₂Im(𝜙𝐷₁𝜙)

=Im(𝑄₁₂(𝜙, 𝜙)) + 𝜕₁(𝐴₂󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) − 𝜕₂(𝐴₁󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) . (19) We then have the formulation (6)–(8) in which 𝜙 is the only dynamical variable and𝐴₁,𝐴₂, and𝐴₀are determined through (7) and (8).

The constraint equation𝜕₁𝐴₂− 𝜕₂𝐴₁= −1/2|𝜙|²and the Coulomb gauge condition𝜕₁𝐴₁+𝜕₂𝐴₂= 0provide an elliptic feature of𝐴 = (𝐴₁, 𝐴₂); that is, the components𝐴_𝑗 can be determined from𝜙by solving the elliptic equations

Δ𝐴₁= 𝜕₂(1

2󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) , Δ𝐴₂= −𝜕₁(1

2󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) . (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 of𝐴 = (𝐴₁, 𝐴₂):

𝐴₁= − 1 4𝜋( 𝑥₂

|𝑥|² ∗ 󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) , 𝐴₂= 1 4𝜋( 𝑥₁

|𝑥|² ∗ 󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) . (21) 3.1. Estimates for𝐴and𝐴₀. We are now ready to estimate several quantities of 𝐴, 𝐴₀. Making use of (20) and the representation (21), we obtain the following estimates for𝐴.

Proposition 7. Let𝑠 ≥ 0and0 < 2/𝑞 < 𝛿 < 1. One also assumes that2 ≤ 𝑝 < ∞if𝑠 > 0or2 < 𝑝 < ∞if𝑠 = 0. Then, one has

‖∇𝐴‖ _̇𝐻^𝑠 ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ^̇𝐻^𝑠,

‖∇𝐴‖_𝐿^∞ ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑞,

‖𝐴‖_𝐿^∞ ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑞,

󵄩󵄩󵄩󵄩|∇|^𝑠𝐴󵄩󵄩󵄩󵄩𝐿^𝑝 ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩^𝐿^𝑝󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ^̇𝐻^𝑠.

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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 structure𝑄₁₂ and divergence form of the nonlinear terms in (19) are used.

Proposition 8. Let𝐴₀be the solution of(19). Then, one has

󵄩󵄩󵄩󵄩𝐴⁰󵄩󵄩󵄩󵄩^𝐿^∞+ 󵄩󵄩󵄩󵄩∇𝐴⁰󵄩󵄩󵄩󵄩^𝐿²≲ (1 + 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²^𝐿²) 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²^𝐿². (23) Proof. Decompose𝐴₀= 𝐴^󸀠₀+ 𝐴^󸀠󸀠₀ as follows:

Δ𝐴^󸀠₀=Im(𝑄₁₂(𝜙, 𝜙)) , (24)

Δ𝐴^󸀠󸀠₀ = 𝜕₁(𝐴₂󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) − 𝜕₂(𝐴₁󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) . (25) We first estimate the quantity‖𝐴₀‖_𝐿^∞_(R²₎. ApplyingLemma 3 to (24), we deduce that

󵄩󵄩󵄩󵄩󵄩𝐴^󸀠⁰󵄩󵄩󵄩󵄩󵄩𝐿^∞≲ 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿². (26) To estimate ‖𝐴^󸀠󸀠₀‖_𝐿∞(R²) we use the Gagliardo-Nirenberg inequality with small𝜖 > 0:

‖𝑢‖_𝐿^∞_(R²₎≤ 𝐶_𝜖‖Δ𝑢‖^𝛼_𝐿1+𝜖(^R²)‖𝑢‖^1−𝛼^𝐿⁴(^R²),

with𝛼 = (1 + 𝜖) / (1 + 5𝜖) . (27) Applying Hardy-Littlewood-Sobolev’s inequality to (25) we deduce

󵄩󵄩󵄩󵄩󵄩𝐴^󸀠󸀠⁰󵄩󵄩󵄩󵄩󵄩𝐿⁴ ≲ 󵄩󵄩󵄩󵄩󵄩𝐴󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²󵄩󵄩󵄩󵄩󵄩𝐿^4/3 ≲ ‖𝐴‖_𝐿⁴󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²^𝐿⁴

≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩³𝐿⁴≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩^5/2𝐿²󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩^3/2𝐿² , (28)

(4)

where Proposition 7 and Lemma 6 are used. We can also derive the following from (25):

󵄩󵄩󵄩󵄩󵄩Δ𝐴^󸀠󸀠⁰󵄩󵄩󵄩󵄩󵄩𝐿^1+𝜖≲ 󵄩󵄩󵄩󵄩󵄩∇𝐴 ⋅⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²󵄩󵄩󵄩󵄩󵄩𝐿^1+𝜖 (i)

+ 󵄩󵄩󵄩󵄩󵄩𝐴∇(󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟²)󵄩󵄩󵄩󵄩󵄩𝐿^1+𝜖 (ii)

. (29)

The first term can be estimated as follows:

(i) ≲ ‖∇𝐴‖_𝐿²󵄩󵄩󵄩󵄩󵄩𝜙²󵄩󵄩󵄩󵄩󵄩𝐿(2+2𝜖)/(1−𝜖) ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩^2/(1+𝜖)𝐿² 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩(2+4𝜖)/(1+𝜖)

𝐿² , (30)

where ‖𝜙‖_𝐿(4+4𝜖)/(1−𝜖) ≲ ‖𝜙‖(1−𝜖)/(2+2𝜖)

𝐿² ‖∇𝜙‖(1+3𝜖)/(2+2𝜖)

𝐿² is used.

The second term can be estimated as follows:

(ii) ≲ ‖𝐴‖_𝐿⁴󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿(4+4𝜖)/(1−3𝜖)󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩𝐿²

≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿⁴󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩(1−3𝜖)/(2+2𝜖)

𝐿² 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩(3+7𝜖)/(2+2𝜖) 𝐿²

≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩^2/(1+𝜖)𝐿² 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩(2+4𝜖)/(1+𝜖)

𝐿² ,

(31)

where‖𝜙‖_𝐿(4+4𝜖)/(1−3𝜖) ≲ ‖𝜙‖(1−3𝜖)/(2+2𝜖)

𝐿² ‖∇𝜙‖(1+5𝜖)/(2+2𝜖) 𝐿² is used.

Therefore, we obtain with𝜖 = 1/11, that is,𝛼 = 3/4,

󵄩󵄩󵄩󵄩󵄩𝐴^󸀠󸀠⁰󵄩󵄩󵄩󵄩󵄩𝐿^∞ ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿². (32) Therefore, we conclude that

󵄩󵄩󵄩󵄩𝐴⁰󵄩󵄩󵄩󵄩𝐿^∞ ≲ 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿²(1 + 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²) . (33) On the other hand,Lemma 3shows that

󵄩󵄩󵄩󵄩󵄩∇𝐴^󸀠⁰󵄩󵄩󵄩󵄩󵄩𝐿²≲ 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿². (34) We also have from (25) that

󵄩󵄩󵄩󵄩󵄩∇𝐴^󸀠󸀠⁰󵄩󵄩󵄩󵄩󵄩𝐿² ≲ 󵄩󵄩󵄩󵄩󵄩𝐴|𝜙|²󵄩󵄩󵄩󵄩󵄩𝐿²≲ ‖𝐴‖_𝐿⁴󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿⁸

≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿⁴󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿⁸≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿².

(35)

Therefore, we have

󵄩󵄩󵄩󵄩∇𝐴⁰󵄩󵄩󵄩󵄩𝐿² ≲ 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿²(1 + 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²) . (36)

3.2. The Energy Solution to (CSS). We now proveTheorem 1.

Let us define

𝑋 (𝑇) = 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹+ 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑝_𝑇𝐿^𝑞, (37) where0 < 𝛿 < 1/2,2 < 𝛿𝑞, and1/𝑝 + 1/𝑞 = 1/2. We derive the following estimate:

𝑋 ≲ 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐻¹+ 𝑇^1/6󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿²(1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) (𝑋²+ 𝑋⁴) , (38) from whichTheorem 1is proved by standard argument; see [2, Section 3].

To control‖𝜙‖_𝐿^∞

𝑇𝐻¹, we applyLemma 4to the solution of (6)–(8).

Proposition 9. Let𝜙be a solution to(6)–(8). Then, one has

󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐿²= 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿²,

󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇 ̇𝐻¹≲ 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩 ̇𝐻¹

+ (1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) 𝑇^{(𝑝−3)/𝑝}(󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩³𝐿^∞_𝑇 ̇𝐻¹+ 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩³𝐿^𝑝_𝑇𝐿^𝑞) , (39) where2 < 𝛿𝑞and3 < 𝑝 < ∞.

Proof. From the conservation of mass, we derive the first estimate. We applyLemma 4to (6) with𝐹 = 𝐴₀𝜙 + 𝐴²_𝑗𝜙 − 𝜆|𝜙|²𝜙and𝑠 = 1. Combined withProposition 7, we have

‖∇𝐴‖ _̇𝐻¹󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞ ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝑊^𝛿,𝑞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ̇𝐻¹,

‖∇𝐴‖_𝐿^∞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ̇𝐻¹≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝑊^𝛿,𝑞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ̇𝐻¹,

󵄩󵄩󵄩󵄩󵄩𝐴²𝜙󵄩󵄩󵄩󵄩󵄩 ̇𝐻¹ ≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²(󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩³𝑊^𝛿,𝑞+ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩³̇𝐻¹) ,

󵄩󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²𝜙󵄩󵄩󵄩󵄩󵄩 ̇𝐻¹≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝑊^𝛿,𝑞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ̇𝐻¹,

(40)

where 2 < 𝛿𝑞. We are then left to estimate ‖𝐴₀𝜙‖ _̇𝐻¹. By Proposition 8, we obtain

󵄩󵄩󵄩󵄩𝐴⁰𝜙󵄩󵄩󵄩󵄩 ̇𝐻¹≲ 󵄩󵄩󵄩󵄩𝐴⁰󵄩󵄩󵄩󵄩𝐿^∞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩 ̇𝐻¹+ 󵄩󵄩󵄩󵄩𝐴⁰󵄩󵄩󵄩󵄩 ̇𝐻¹󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞

≲ (1 + 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²) (󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩³𝐿²+ 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩³𝐿^𝑞) . (41) Combining (40) and (41), we obtain

󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇 ̇𝐻¹ ≲ 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩 ̇𝐻¹+ ∫^𝑇

0 (1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) (󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩³̇𝐻¹+ 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩³𝐿^𝑞)

≲ 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩 ̇𝐻¹+ (1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) 𝑇^{(𝑝−3)/𝑝}

× (󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩³𝐿^∞_𝑇 ̇𝐻¹+ 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩³𝐿^𝑝_𝑇𝐿^𝑞) ,

(42) where3 < 𝑝 < ∞and𝑇 < 1.

To estimate‖𝐽^𝛿𝜙‖_𝐿^𝑝_𝑇_𝐿^𝑞, we applyLemma 5to the solution of (6)–(8).

Proposition 10. Let𝜙be a solution to(6)–(8). Then, one has

󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑝_𝑇𝐿^𝑞≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹+ 𝑇^1/6󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿²(1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) (𝑋²+ 𝑋⁴) , (43) where2 < 𝛿𝑞,3 < 𝑝 < ∞and1/𝑝 + 1/𝑞 = 1/2.

Proof. ApplyingLemma 5with𝐹₁= 𝐴₀𝜙−2𝑖𝐴_𝑗𝜕_𝑗𝜙and𝐹₂= 𝐴²𝜙 − 𝜆|𝜙|²𝜙, we obtain

󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑝_𝑇𝐿^𝑞≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹+ 󵄩󵄩󵄩󵄩𝐴⁰𝜙󵄩󵄩󵄩󵄩𝐿²_𝑇𝐻^𝛿−1/2

+ 󵄩󵄩󵄩󵄩𝐴 ⋅ ∇𝜙󵄩󵄩󵄩󵄩𝐿²_𝑇𝐻^𝛿−1/2+ 󵄩󵄩󵄩󵄩󵄩𝐴²𝜙󵄩󵄩󵄩󵄩󵄩𝐿¹_𝑇𝐻^𝛿+ 󵄩󵄩󵄩󵄩󵄩󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²𝜙󵄩󵄩󵄩󵄩󵄩𝐿¹_𝑇𝐻^𝛿, (44)

(5)

where𝛿 = 1/2 − 𝜀,3 < 𝑝 < ∞and2 < 𝛿𝑞. Considering Proposition 8, we obtain

󵄩󵄩󵄩󵄩𝐴⁰𝜙󵄩󵄩󵄩󵄩𝐿²_𝑇𝐻^𝛿−1/2

≲ 󵄩󵄩󵄩󵄩𝐴⁰󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐿^∞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²_𝑇𝐿²≲ 𝑇^1/2󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿²(1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) 󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩²𝐿². (45) The other terms can be treated, as mentioned inSection 1, by similar arguments to those in [2, Section 3]. Applying Proposition 7, we have

󵄩󵄩󵄩󵄩𝐴 ⋅ ∇𝜙󵄩󵄩󵄩󵄩𝐿²_𝑇𝐻^𝛿−1/2 ≲ ‖𝐴‖_𝐿²_𝑇_𝐿^∞󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐿²

≲ 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿² 𝑇^{(𝑝−2)/2𝑝} 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑝_𝑇𝐿^𝑞󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹, (46)

󵄩󵄩󵄩󵄩󵄩𝐴²𝜙󵄩󵄩󵄩󵄩󵄩𝐿¹_𝑇𝐻^𝛿≲ 󵄩󵄩󵄩󵄩󵄩𝐴²󵄩󵄩󵄩󵄩󵄩𝐿¹_𝑇𝐿⁴󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐿⁴

+ 󵄩󵄩󵄩󵄩󵄩𝐴²󵄩󵄩󵄩󵄩󵄩𝐿²_𝑇𝑊^𝛿,2+𝜀󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²_𝑇𝐿^{(4+2𝜀)/𝜀}

≲ 𝑇 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩^3/2𝐿²󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩^5/2𝐿^∞_𝑇𝐻¹

+ 𝑇^1/4󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿²󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿^∞_𝑇𝐻¹󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩²𝐿^𝑝_𝑇𝐿^𝑞,

(47)

󵄩󵄩󵄩󵄩󵄩𝜙³󵄩󵄩󵄩󵄩󵄩𝐿¹_𝑇𝐻^𝛿≲ 󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐿²󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿²_𝑇𝐿^∞

≲ 𝑇^{(𝑝−2)/𝑝}󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩²𝐿^𝑝_𝑇𝐿^𝑞.

(48)

Plugging estimates (45)–(48) into (44) with𝑝 > 3, we obtain

󵄩󵄩󵄩󵄩󵄩𝐽^𝛿𝜙󵄩󵄩󵄩󵄩󵄩𝐿^𝑝_𝑇𝐿^𝑞≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿^∞_𝑇𝐻¹+ 𝑇^1/6󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩𝐿²(1 + 󵄩󵄩󵄩󵄩𝜙⁰󵄩󵄩󵄩󵄩²𝐿²) (𝑋²+ 𝑋⁴) . (49)

We finally obtain the estimate (38) by combining Propo- sitions9and10, which provesTheorem 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 set𝜔 = 𝜙 − 𝜓, then the equation for𝜔is

𝑖𝜕_𝑡𝜔+Δ𝜔 = 𝐴₀𝜔+(𝐴₀− 𝐵₀) 𝜓 − 2𝑖𝐴 ⋅ ∇𝜔−2𝑖 (𝐴 − 𝐵) ⋅ ∇𝜓 + 𝐴²𝜔+(𝐴²− 𝐵²) 𝜓 −𝜆󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²𝜔−𝜆 (󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²− 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²) 𝜓.

(50) We will derive

𝜕_𝑡‖𝜔‖²_𝐿² ≲ 𝑞^1/2𝑀²‖𝜔‖^2−4/𝑞_𝐿2 + 𝑞𝑀^2+4/𝑞(1 + 𝑀²) ‖𝜔‖^2−4/𝑞_𝐿2 , (51)

where𝑀is a constant inTheorem 2and𝑞 > 2. Then we have

𝜕_𝑡‖𝜔‖^4/𝑞_𝐿2 ≲1

𝑞(𝑞^1/2𝑀²+ 𝑞𝑀^2+4/𝑞(1 + 𝑀²)) . (52) Considering‖𝜔(0, ⋅)‖_𝐿²= 0and2 < 𝑞, we obtain

‖𝜔‖_𝐿²≲ (𝑇(𝑀²+ 𝑀^4+4/𝑞))^𝑞/4. (53) Letting𝑞 → ∞, for the time interval satisfying 𝑇(𝑀² + 𝑀^4+4/𝑞) ≤ 1/2, we conclude that‖𝜔(𝑡, ⋅)‖_𝐿² = 0for0 ≤ 𝑡 ≤ 𝑇, which thus provesTheorem 2.

In the remainder of this section, we derive inequality (51). Multiplying𝜔to both sides of (50) and integrating the imaginary part ofR², we have

𝜕_𝑡‖𝜔‖²_𝐿2= ∫ 2 (𝐴⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟₀− 𝐵₀)Im(𝜓𝜔)

(I)

− 2𝐴⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟_𝑗𝜕_𝑗|𝜔|²

(II)

− 4(𝐴⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟_𝑗− 𝐵_𝑗)Re(𝜕_𝑗𝜓𝜔)

(III)

𝑑𝑥

+ ∫ 2 (𝐴⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟²− 𝐵²)Im(𝜓𝜔)

(IV)

− 2𝜆(󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟²− 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²)Im(𝜓𝜔)

(V)

𝑑𝑥.

(54)

The integrals (II)–(V), that is, those not containing𝐴₀, 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|²𝑑𝑥 = 0, (III) , (IV) , (V) ≲ 𝑞𝑀^2+4/𝑞(1 + 𝑀²) ‖𝜔‖^2−4/𝑞_𝐿2

(55)

for which we omit the proof.

We simply present how to control integral (I), for which we have

󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫(𝐴⁰− 𝐵₀)Im(𝜓𝜔) 𝑑𝑥󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≲󵄩󵄩󵄩󵄩𝐴⁰− 𝐵₀󵄩󵄩󵄩󵄩𝐿^𝑎󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿^𝑏‖𝜔‖_𝐿^𝑐, (56) where1/𝑎 + 1/𝑏 + 1/𝑐 = 1, 2 ≤ 𝑎, 𝑏, 𝑐. ApplyingLemma 6, we obtain

󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿^𝑏≲ 𝑏^1/2󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩^2/𝑏𝐿² 󵄩󵄩󵄩󵄩∇𝜓󵄩󵄩󵄩󵄩^1−2/𝑏𝐿² ≲ 𝑏^1/2𝑀^1−2/𝑏,

‖𝜔‖_𝐿^𝑐 ≲ 𝑐^1/2‖𝜔‖^2/𝑐_𝐿2‖∇𝜔‖^1−2/𝑐_𝐿2 ≲ 𝑐^1/2‖𝜔‖^2/𝑐_𝐿2𝑀^1−2/𝑐. (57) To control‖𝐴₀− 𝐵₀‖_𝐿^𝑎, we consider the equation for𝐴₀− 𝐵₀ Δ (𝐴₀− 𝐵₀) = 𝜕₁Im(𝜙𝜕₂𝜙) − 𝜕₂Im(𝜙𝜕₁𝜙) − 𝜕₁Im(𝜓𝜕₂𝜓) + 𝜕₂Im(𝜓𝜕₁𝜓) + 𝜕₁(𝐴₂󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) − 𝜕₂(𝐴₁󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²)

− 𝜕₁(𝐵₂󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²) + 𝜕₂(𝐵₁󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²) .

(58)

(6)

Decomposing𝐴₀and𝐵₀as (24) and (25), we have Δ (𝐴^󸀠₀− 𝐵^󸀠₀) = 𝜕₁Im(𝜙𝜕₂𝜔) − 𝜕₂Im(𝜓𝜕₁𝜔)

+ 𝜕₁Im(𝜔𝜕₂𝜓) − 𝜕₂Im(𝜔𝜕₁𝜙) , (59) Δ (𝐴^󸀠󸀠₀− 𝐵^󸀠󸀠₀) = 𝜕₁(𝐴₂(󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²− 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²)) − 𝜕₂(𝐴₁(󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²− 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²))

+ 𝜕₁((𝐴₂− 𝐵₂) 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²) − 𝜕₂((𝐴₁− 𝐵₁) 󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²) . (60) Taking into account

𝜕₁Im(𝜙𝜕₂𝜔) = 𝜕₁(𝜕₂Im(𝜙𝜔) −Im(𝜔𝜕₂𝜙)) ,

𝜕₂Im(𝜓𝜕₁𝜔) = 𝜕₂(𝜕₁Im(𝜓𝜔) −Im(𝜔𝜕₁𝜓)) , (61) we can rewrite the equation for𝐴^󸀠₀− 𝐵^󸀠₀as follows:

Δ (𝐴^󸀠₀− 𝐵^󸀠₀) = 𝜕₁(Im(𝜔𝜕₂𝜓) −Im(𝜔𝜕₂𝜙))

+ 𝜕₂(Im(𝜔𝜕₁𝜓) −Im(𝜔𝜕₂𝜙)) , (62) where𝜕₁𝜕₂Im(𝜙𝜔) − 𝜕₂𝜕₁Im(𝜓𝜔) = 𝜕₁𝜕₂Im(𝜔𝜔) = 0should be noted. Using the Hardy-Littlewood-Sobolev inequality, we have 󵄩󵄩󵄩󵄩󵄩𝐴^󸀠⁰− 𝐵^󸀠₀󵄩󵄩󵄩󵄩󵄩𝐿^𝑎 ≲ 󵄩󵄩󵄩󵄩󵄩|𝑥|⁻¹∗ (𝜔∇𝜓)󵄩󵄩󵄩󵄩󵄩𝐿^𝑎

≲ 󵄩󵄩󵄩󵄩𝜔∇𝜓󵄩󵄩󵄩󵄩𝐿^𝑟 ≲ ‖𝜔‖𝐿^𝑠󵄩󵄩󵄩󵄩∇𝜓󵄩󵄩󵄩󵄩𝐿², (63) where1/𝑎 = 1/𝑟 − 1/2and1/𝑟 = 1/𝑠 + 1/2, from which we deduce𝑎 = 𝑠. Then, we have

‖𝜔‖_𝐿^𝑎󵄩󵄩󵄩󵄩∇𝜓󵄩󵄩󵄩󵄩𝐿²≲ 𝑎^1/2‖𝜔‖^2/𝑎_𝐿2 ‖∇𝜔‖^1−2/𝑎_𝐿2 𝑀 ≲ 𝑎^1/2𝑀^2−2/𝑎‖𝜔‖^2/𝑎_𝐿2 . (64) The term𝐴^󸀠󸀠₀− 𝐵^󸀠󸀠₀ can be bounded as follows:

󵄩󵄩󵄩󵄩󵄩𝐴^󸀠󸀠⁰ − 𝐵^󸀠󸀠₀󵄩󵄩󵄩󵄩󵄩𝐿^𝑎≲ 󵄩󵄩󵄩󵄩󵄩|𝑥|⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⁻¹∗ (|𝐴| 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²− 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²󵄨󵄨󵄨󵄨󵄨)󵄩󵄩󵄩󵄩󵄩𝐿^𝑎 (1)

+ 󵄩󵄩󵄩󵄩󵄩|𝑥|⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⁻¹∗ (|𝐴 − 𝐵| (󵄨󵄨󵄨󵄨𝜙󵄨󵄨󵄨󵄨²+ 󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨²))󵄩󵄩󵄩󵄩󵄩𝐿^𝑎 (2)

. (65)

Since||𝜙|²− |𝜓|²| ≤ (|𝜙| + |𝜓|)|𝜔|, we have (1) ≲ 󵄩󵄩󵄩󵄩|𝐴|(|𝜙| + |𝜓|)󵄩󵄩󵄩󵄩𝐿²‖𝜔‖_𝐿^𝑎

≲ ‖𝐴‖_𝐿⁶(󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿³+ 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿³) ‖𝜔‖_𝐿^𝑎 (66)

≲ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿³(󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿³+ 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿³) ‖𝜔‖_𝐿^𝑎

≲ 𝑎^1/2𝑀^2−2/𝑎‖𝜔‖^2/𝑎_𝐿2 . (67) Since|𝐴_𝑗− 𝐵_𝑗| ≲ |𝑥|⁻¹∗ ((|𝜙| + |𝜓|)|𝜔|), we may check

(2) ≲ 󵄩󵄩󵄩󵄩󵄩𝐴^𝑗− 𝐵_𝑗󵄩󵄩󵄩󵄩󵄩𝐿^𝑎(󵄩󵄩󵄩󵄩󵄩𝜙²󵄩󵄩󵄩󵄩󵄩𝐿²+ 󵄩󵄩󵄩󵄩󵄩𝜓²󵄩󵄩󵄩󵄩󵄩𝐿²)

≲ (󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩𝐿²+ 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿²) ‖𝜔‖_𝐿^𝑎(󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩²𝐿⁴+ 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩²𝐿⁴)

≲ 𝑎^1/2‖𝜔‖^2/𝑎_𝐿2‖∇𝜔‖^1−2/𝑎_𝐿2 (󵄩󵄩󵄩󵄩∇𝜙󵄩󵄩󵄩󵄩𝐿²+ 󵄩󵄩󵄩󵄩∇𝜓󵄩󵄩󵄩󵄩𝐿²)

≲ 𝑎^1/2𝑀^2−2/𝑎‖𝜔‖^2/𝑎_𝐿2 .

(68)

Then, we have

󵄩󵄩󵄩󵄩𝐴⁰− 𝐵₀󵄩󵄩󵄩󵄩𝐿^𝑎 ≲ 𝑎^1/2𝑀^2−2/𝑎‖𝜔‖^2/𝑎_𝐿2 . (69) Combining estimates (57) and (69), and denoting𝑏 = 𝑞/2, we obtain

󵄩󵄩󵄩󵄩𝐴⁰− 𝐵₀󵄩󵄩󵄩󵄩𝐿^𝑎󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩𝐿^𝑏‖𝜔‖_𝐿^𝑐 ≲ (𝑎𝑞𝑐)^1/2𝑀²‖𝜔‖^2−4/𝑞_𝐿2 , (70) where1/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.

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1.Introduction HyungjinHuh EnergySolutiontotheChern-Simons-SchrödingerEquations ResearchArticle

Research Article

Energy Solution to the Chern-Simons-Schrödinger Equations

Hyungjin Huh

1. Introduction

2. Preliminaries

3. The Proof of Theorem 1

4. The Proof of Theorem 2

Acknowledgments

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Mathematics

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