# Energy Solution to the Chern-Simons-Schrödinger Equations

Authors: Hyungjin Huh
Journal: Abstract and Applied Analysis
 (2013)

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## 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 spaceH1(ℝ2). The Coulomb gauge provides elliptic features for gauge fields A0,Aj. The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.


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## Body

## 1. Introduction

We study herein the initial value problem of the Chern-Simons-Schrödinger (CSS) equations(1)iD0ϕ+DjDjϕ=-λ|ϕ|2ϕ,∂0A1-∂1A0=-Im(ϕ-D2ϕ),∂0A2-∂2A0=Im(ϕ-D1ϕ),∂1A2-∂2A1=-12|ϕ|2,
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; Aμ:ℝ1+2→ℝ is the gauge field; Dμ=∂μ+iAμ 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(2)M(t)=∫ℝ2‍|ϕ(t,x)|2dx=M(0),
and the conservation of total energy
(3)E(t)=∫ℝ2‍|Djϕ(t,x)|2-λ2|ϕ(t,x)|4dx=E(0).
Note that the terms |F|2=(1/2)FμνFμν 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:(4)ϕa(t,x)=aϕ(a2t,ax),Aja(t,x)=aAj(a2t,ax),A0a(t,x)=a2A0(a2t,ax).
Therefore, the scaled critical Sobolev exponent is sc=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:(5)ϕ→ϕeiχ,Aμ→Aμ-∂μχ,
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,Aμ). In this work, we fix the gauge by imposing the Coulomb gauge condition of ∂jAj=0, under which the Cauchy problem of the CSS system may be reformulated as follows:
(6)i∂tϕ-A0ϕ+Δϕ+2iAj∂jϕ-Aj2ϕ=-λ|ϕ|2ϕ,(7)∂1A2-∂2A1=-1/2|ϕ|2,∂1A1+∂2A2=0,(8)ΔA0=lm(Q12(ϕ-,ϕ))+∂1(A2|ϕ|2)-∂2(A1|ϕ|2),
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))∩Cω(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 RjRk(u2)+|u|2, where Rj=∂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 Hs1(ℝ2) with s1>1/2, and similarly, the uniqueness was proved in [14] for Hs2(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 dataϕ0 belong to H1(ℝ2). Then, there exists a local-in-time solution, ϕ, to (6)–(8) that satisfies
(9)ϕ∈L∞([0,T);H1(ℝ2))∩C([0,T);L2(ℝ2)),Jδϕ∈Lp(0,T;Lq(ℝ2)),
where 0<δ<1/2, 2<δq, 1/p+1/q=1/2 and J=(1-Δ)1/2.Theorem 2.
Letϕ 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
(10)ϕ,ψ∈C([0,T];L2(ℝ2)),∥ϕ∥LT∞H1≤M,∥ψ∥LT∞H1≤M,
for some constant M>0. One then has ∥(ϕ-ψ)(t,·)∥L2(ℝ2)=0 for 0≤t≤T.We present some preliminaries in Section2. 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 Ws,p, with the norm ∥f∥Ws,p=∥Jsf∥Lp and W˙s,p with the norm ∥f∥W˙s,p=∥|∇|sf∥Lp, where J=(1-Δ)1/2 and |∇|=(-Δ)1/2. The space Hs denotes Ws,2. We define the space time norm as ∥f∥LTpLq=(∫0T‍∥f(t,·)∥Lq(ℝ2)pdt)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 Theorems1 and 2. The following lemma is reminiscent of Wente's inequality (see [15, 16]).Lemma 3.
Letf and g be two functions in H1(ℝ2) and let u be the solution of
(11)Δu=∂1f∂2g-∂2f∂1ginℝ2,
where u is small at infinity. Then, u∈C(ℝ2)∩H˙1(ℝ2) and
(12)∥u∥L∞(ℝ2)+∥∇u∥L2(ℝ2)≤C∥∇f∥L2(ℝ2)∥∇g∥L2(ℝ2).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(13)i∂tu+Δu+2idiv(au)=F,
where a=(a1(t,x),a2(t,x)) and aj are real-valued functions. Then, for s≥0 there exists an absolute constant Cs>0 such that
(14)∥u(t,·)∥H˙s≤∥u(0,·)∥H˙s+Cs∫0t‍(∥∇a∥H˙s∥u∥L∞+∥∇a∥L∞∥u∥H˙s+∥F∥H˙s)ds,
wherein one means the homogeneous Sobolev space H˙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.
LetT≤1 and v be a solution to the equation
(15)i∂tv+Δv=F1+F2,(t,x)∈(0,T)×ℝ2.
Then, for δ∈R and ε>0, one has
(16)∥Jδv∥LTpLq≲∥v∥LT∞Hδ+1/2+ε+∥F1∥LT2Hδ-1/2+∥F2∥LT1Hδ,
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.
For2≤q<∞, one has
(17)∥u∥Lq(ℝ2)≤(4π)(2-q)/2q(q2)1/2∥u∥L2(ℝ2)2/q∥∇u∥L2(ℝ2)1-2/q.


## 3. The Proof of Theorem1

Theorem1 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:
(18)Dαϕ¯Dβϕ-Dβϕ¯Dαϕ=Qαβ(ϕ-,ϕ)-i(Aα∂β(|ϕ|2)-Aβ∂α(|ϕ|2)),DαDβϕ-DβDαϕ=iFαβϕ,
where Qαβ(ϕ-,ϕ)=∂αϕ-∂βϕ-∂βϕ-∂αϕ and Fαβ=∂αAβ-∂βAα. 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):
(19)ΔA0=∂1Im(ϕ-D2ϕ)-∂2Im(ϕ-D1ϕ)=Im(Q12(ϕ-,ϕ))+∂1(A2|ϕ|2)-∂2(A1|ϕ|2).
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 Aj can be determined from ϕ by solving the elliptic equations
(20)ΔA1=∂2(12|ϕ|2),ΔA2=-∂1(12|ϕ|2).
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):
(21)A1=-14π(x2|x|2*|ϕ|2),A2=14π(x1|x|2*|ϕ|2).


### 3.1. Estimates forA and A0

We are now ready to estimate several quantities ofA,A0. Making use of (20) and the representation (21), we obtain the following estimates for A.Proposition 7.
Lets≥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
(22)∥∇A∥H˙s≲∥ϕ∥L∞∥ϕ∥H˙s,∥∇A∥L∞≲∥ϕ∥L∞∥Jδϕ∥Lq,∥A∥L∞≲∥ϕ∥L2∥Jδϕ∥Lq,∥|∇|sA∥Lp≲∥ϕ∥Lp∥ϕ∥H˙s.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 estimateA0, the special algebraic structure Q12 and divergence form of the nonlinear terms in (19) are used.Proposition 8.
LetA0 be the solution of (19). Then, one has
(23)∥A0∥L∞+∥∇A0∥L2≲(1+∥ϕ∥L22)∥∇ϕ∥L22.Proof.
DecomposeA0=A0′+A0′′ as follows:
(24)ΔA0′=Im(Q12(ϕ-,ϕ)),(25)ΔA0′′=∂1(A2|ϕ|2)-∂2(A1|ϕ|2).
We first estimate the quantity ∥A0∥L∞(ℝ2). Applying Lemma 3 to (24), we deduce that
(26)∥A0′∥L∞≲∥∇ϕ∥L22.
To estimate ∥A0′′∥L∞(ℝ2) we use the Gagliardo-Nirenberg inequality with small ϵ>0:
(27)∥u∥L∞(ℝ2)≤Cϵ∥Δu∥L1+ϵ(ℝ2)α∥u∥L4(ℝ2)1-α,withα=(1+ϵ)/(1+5ϵ).
Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce
(28)∥A0′′∥L4≲∥A|ϕ|2∥L4/3≲∥A∥L4∥ϕ∥L42≲∥ϕ∥L2∥ϕ∥L43≲∥ϕ∥L25/2∥∇ϕ∥L23/2,
where Proposition 7 and Lemma 6 are used. We can also derive the following from (25):
(29)∥ΔA0′′∥L1+ϵ≲∥∇A·|ϕ|2∥L1+ϵ︸(i)+∥A∇(|ϕ|2)∥L1+ϵ︸(ii).
The first term can be estimated as follows:
(30)(i)≲∥∇A∥L2∥ϕ2∥L(2+2ϵ)/(1-ϵ)≲∥ϕ∥L22/(1+ϵ)∥∇ϕ∥L2(2+4ϵ)/(1+ϵ),
where ∥ϕ∥L(4+4ϵ)/(1-ϵ)≲∥ϕ∥L2(1-ϵ)/(2+2ϵ)∥∇ϕ∥L2(1+3ϵ)/(2+2ϵ) is used. The second term can be estimated as follows:
(31)(ii)≲∥A∥L4∥ϕ∥L(4+4ϵ)/(1-3ϵ)∥∇ϕ∥L2≲∥ϕ∥L2∥ϕ∥L4∥ϕ∥L2(1-3ϵ)/(2+2ϵ)∥∇ϕ∥L2(3+7ϵ)/(2+2ϵ)≲∥ϕ∥L22/(1+ϵ)∥∇ϕ∥L2(2+4ϵ)/(1+ϵ),
where ∥ϕ∥L(4+4ϵ)/(1-3ϵ)≲∥ϕ∥L2(1-3ϵ)/(2+2ϵ)∥∇ϕ∥L2(1+5ϵ)/(2+2ϵ) is used. Therefore, we obtain with ϵ=1/11, that is, α=3/4,
(32)∥A0′′∥L∞≲∥ϕ∥L22∥∇ϕ∥L22.
Therefore, we conclude that
(33)∥A0∥L∞≲∥∇ϕ∥L22(1+∥ϕ∥L22).
On the other hand, Lemma 3 shows that
(34)∥∇A0′∥L2≲∥∇ϕ∥L22.
We also have from (25) that
(35)∥∇A0′′∥L2≲∥A|ϕ|2∥L2≲∥A∥L4∥ϕ∥L82≲∥ϕ∥L2∥ϕ∥L4∥ϕ∥L82≲∥ϕ∥L22∥∇ϕ∥L22.
Therefore, we have
(36)∥∇A0∥L2≲∥∇ϕ∥L22(1+∥ϕ∥L22).


### 3.2. The Energy Solution to (CSS)

We now prove Theorem1. Let us define
(37)X(T)=∥ϕ∥LT∞H1+∥Jδϕ∥LTpLq,
where 0<δ<1/2, 2<δq, and 1/p+1/q=1/2. We derive the following estimate:
(38)X≲∥ϕ0∥H1+T1/6∥ϕ0∥L2(1+∥ϕ0∥L22)(X2+X4),
from which Theorem 1 is proved by standard argument; see [2, Section 3].To control∥ϕ∥LT∞H1, we apply Lemma 4 to the solution of (6)–(8).Proposition 9.
Letϕ be a solution to (6)–(8). Then, one has
(39)∥ϕ∥LT∞L2=∥ϕ0∥L2,∥ϕ∥LT∞H˙1≲∥ϕ0∥H˙1+(1+∥ϕ0∥L22)T(p-3)/p(∥ϕ∥LT∞H˙13+∥Jδϕ∥LTpLq3),
where 2<δq and 3<p<∞.Proof.
From the conservation of mass, we derive the first estimate. We apply Lemma4 to (6) with F=A0ϕ+Aj2ϕ-λ|ϕ|2ϕ and s=1. Combined with Proposition 7, we have
(40)∥∇A∥H˙1∥ϕ∥L∞≲∥ϕ∥Wδ,q2∥ϕ∥H˙1,∥∇A∥L∞∥ϕ∥H˙1≲∥ϕ∥Wδ,q2∥ϕ∥H˙1,∥A2ϕ∥H˙1≲∥ϕ∥L22(∥ϕ∥Wδ,q3+∥ϕ∥H˙13),∥|ϕ|2ϕ∥H˙1≲∥ϕ∥Wδ,q2∥ϕ∥H˙1,
where 2<δq. We are then left to estimate ∥A0ϕ∥H˙1. By Proposition 8, we obtain
(41)∥A0ϕ∥H˙1≲∥A0∥L∞∥ϕ∥H˙1+∥A0∥H˙1∥ϕ∥L∞≲(1+∥ϕ∥L22)(∥∇ϕ∥L23+∥Jδϕ∥Lq3).
Combining (40) and (41), we obtain
(42)∥ϕ∥LT∞H˙1≲∥ϕ0∥H˙1+∫0T‍(1+∥ϕ0∥L22)(∥ϕ∥H˙13+∥Jδϕ∥Lq3)≲∥ϕ0∥H˙1+(1+∥ϕ0∥L22)T(p-3)/p×(∥ϕ∥LT∞H˙13+∥Jδϕ∥LTpLq3),
where 3<p<∞ and T<1.To estimate∥Jδϕ∥LTpLq, we apply Lemma 5 to the solution of (6)–(8).Proposition 10.
Letϕ be a solution to (6)–(8). Then, one has
(43)∥Jδϕ∥LTpLq≲∥ϕ∥LT∞H1+T1/6∥ϕ0∥L2(1+∥ϕ0∥L22)(X2+X4),
where 2<δq, 3<p<∞ and 1/p+1/q=1/2.Proof.
Applying Lemma5 with F1=A0ϕ-2iAj∂jϕ and F2=A2ϕ-λ|ϕ|2ϕ, we obtain
(44)∥Jδϕ∥LTpLq≲∥ϕ∥LT∞H1+∥A0ϕ∥LT2Hδ-1/2+∥A·∇ϕ∥LT2Hδ-1/2+∥A2ϕ∥LT1Hδ+∥|ϕ|2ϕ∥LT1Hδ,
where δ=1/2-ε, 3<p<∞ and 2<δq. Considering Proposition 8, we obtain
(45)∥A0ϕ∥LT2Hδ-1/2≲∥A0∥LT∞L∞∥ϕ∥LT2L2≲T1/2∥ϕ0∥L2(1+∥ϕ0∥L22)∥∇ϕ∥L22.
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
(46)∥A·∇ϕ∥LT2Hδ-1/2≲∥A∥LT2L∞∥∇ϕ∥LT∞L2≲∥ϕ0∥L2T(p-2)/2p∥Jδϕ∥LTpLq∥ϕ∥LT∞H1,(47)∥A2ϕ∥LT1Hδ≲∥A2∥LT1L4∥Jδϕ∥LT∞L4+∥A2∥LT2Wδ,2+ε∥ϕ∥LT2L(4+2ε)/ε≲T∥ϕ0∥L23/2∥ϕ∥LT∞H15/2+T1/4∥ϕ0∥L2∥ϕ∥LT∞H12∥Jδϕ∥LTpLq2,(48)∥ϕ3∥LT1Hδ≲∥Jδϕ∥LT∞L2∥ϕ∥LT2L∞2≲T(p-2)/p∥ϕ∥LT∞H1∥Jδϕ∥LTpLq2.
Plugging estimates (45)–(48) into (44) with p>3, we obtain
(49)∥Jδϕ∥LTpLq≲∥ϕ∥LT∞H1+T1/6∥ϕ0∥L2(1+∥ϕ0∥L22)(X2+X4).We finally obtain the estimate (38) by combining Propositions 9 and 10, which proves Theorem 1.


## 3.1. Estimates forA and A0

We are now ready to estimate several quantities ofA,A0. Making use of (20) and the representation (21), we obtain the following estimates for A.Proposition 7.
Lets≥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
(22)∥∇A∥H˙s≲∥ϕ∥L∞∥ϕ∥H˙s,∥∇A∥L∞≲∥ϕ∥L∞∥Jδϕ∥Lq,∥A∥L∞≲∥ϕ∥L2∥Jδϕ∥Lq,∥|∇|sA∥Lp≲∥ϕ∥Lp∥ϕ∥H˙s.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 estimateA0, the special algebraic structure Q12 and divergence form of the nonlinear terms in (19) are used.Proposition 8.
LetA0 be the solution of (19). Then, one has
(23)∥A0∥L∞+∥∇A0∥L2≲(1+∥ϕ∥L22)∥∇ϕ∥L22.Proof.
DecomposeA0=A0′+A0′′ as follows:
(24)ΔA0′=Im(Q12(ϕ-,ϕ)),(25)ΔA0′′=∂1(A2|ϕ|2)-∂2(A1|ϕ|2).
We first estimate the quantity ∥A0∥L∞(ℝ2). Applying Lemma 3 to (24), we deduce that
(26)∥A0′∥L∞≲∥∇ϕ∥L22.
To estimate ∥A0′′∥L∞(ℝ2) we use the Gagliardo-Nirenberg inequality with small ϵ>0:
(27)∥u∥L∞(ℝ2)≤Cϵ∥Δu∥L1+ϵ(ℝ2)α∥u∥L4(ℝ2)1-α,withα=(1+ϵ)/(1+5ϵ).
Applying Hardy-Littlewood-Sobolev's inequality to (25) we deduce
(28)∥A0′′∥L4≲∥A|ϕ|2∥L4/3≲∥A∥L4∥ϕ∥L42≲∥ϕ∥L2∥ϕ∥L43≲∥ϕ∥L25/2∥∇ϕ∥L23/2,
where Proposition 7 and Lemma 6 are used. We can also derive the following from (25):
(29)∥ΔA0′′∥L1+ϵ≲∥∇A·|ϕ|2∥L1+ϵ︸(i)+∥A∇(|ϕ|2)∥L1+ϵ︸(ii).
The first term can be estimated as follows:
(30)(i)≲∥∇A∥L2∥ϕ2∥L(2+2ϵ)/(1-ϵ)≲∥ϕ∥L22/(1+ϵ)∥∇ϕ∥L2(2+4ϵ)/(1+ϵ),
where ∥ϕ∥L(4+4ϵ)/(1-ϵ)≲∥ϕ∥L2(1-ϵ)/(2+2ϵ)∥∇ϕ∥L2(1+3ϵ)/(2+2ϵ) is used. The second term can be estimated as follows:
(31)(ii)≲∥A∥L4∥ϕ∥L(4+4ϵ)/(1-3ϵ)∥∇ϕ∥L2≲∥ϕ∥L2∥ϕ∥L4∥ϕ∥L2(1-3ϵ)/(2+2ϵ)∥∇ϕ∥L2(3+7ϵ)/(2+2ϵ)≲∥ϕ∥L22/(1+ϵ)∥∇ϕ∥L2(2+4ϵ)/(1+ϵ),
where ∥ϕ∥L(4+4ϵ)/(1-3ϵ)≲∥ϕ∥L2(1-3ϵ)/(2+2ϵ)∥∇ϕ∥L2(1+5ϵ)/(2+2ϵ) is used. Therefore, we obtain with ϵ=1/11, that is, α=3/4,
(32)∥A0′′∥L∞≲∥ϕ∥L22∥∇ϕ∥L22.
Therefore, we conclude that
(33)∥A0∥L∞≲∥∇ϕ∥L22(1+∥ϕ∥L22).
On the other hand, Lemma 3 shows that
(34)∥∇A0′∥L2≲∥∇ϕ∥L22.
We also have from (25) that
(35)∥∇A0′′∥L2≲∥A|ϕ|2∥L2≲∥A∥L4∥ϕ∥L82≲∥ϕ∥L2∥ϕ∥L4∥ϕ∥L82≲∥ϕ∥L22∥∇ϕ∥L22.
Therefore, we have
(36)∥∇A0∥L2≲∥∇ϕ∥L22(1+∥ϕ∥L22).


## 3.2. The Energy Solution to (CSS)

We now prove Theorem1. Let us define
(37)X(T)=∥ϕ∥LT∞H1+∥Jδϕ∥LTpLq,
where 0<δ<1/2, 2<δq, and 1/p+1/q=1/2. We derive the following estimate:
(38)X≲∥ϕ0∥H1+T1/6∥ϕ0∥L2(1+∥ϕ0∥L22)(X2+X4),
from which Theorem 1 is proved by standard argument; see [2, Section 3].To control∥ϕ∥LT∞H1, we apply Lemma 4 to the solution of (6)–(8).Proposition 9.
Letϕ be a solution to (6)–(8). Then, one has
(39)∥ϕ∥LT∞L2=∥ϕ0∥L2,∥ϕ∥LT∞H˙1≲∥ϕ0∥H˙1+(1+∥ϕ0∥L22)T(p-3)/p(∥ϕ∥LT∞H˙13+∥Jδϕ∥LTpLq3),
where 2<δq and 3<p<∞.Proof.
From the conservation of mass, we derive the first estimate. We apply Lemma4 to (6) with F=A0ϕ+Aj2ϕ-λ|ϕ|2ϕ and s=1. Combined with Proposition 7, we have
(40)∥∇A∥H˙1∥ϕ∥L∞≲∥ϕ∥Wδ,q2∥ϕ∥H˙1,∥∇A∥L∞∥ϕ∥H˙1≲∥ϕ∥Wδ,q2∥ϕ∥H˙1,∥A2ϕ∥H˙1≲∥ϕ∥L22(∥ϕ∥Wδ,q3+∥ϕ∥H˙13),∥|ϕ|2ϕ∥H˙1≲∥ϕ∥Wδ,q2∥ϕ∥H˙1,
where 2<δq. We are then left to estimate ∥A0ϕ∥H˙1. By Proposition 8, we obtain
(41)∥A0ϕ∥H˙1≲∥A0∥L∞∥ϕ∥H˙1+∥A0∥H˙1∥ϕ∥L∞≲(1+∥ϕ∥L22)(∥∇ϕ∥L23+∥Jδϕ∥Lq3).
Combining (40) and (41), we obtain
(42)∥ϕ∥LT∞H˙1≲∥ϕ0∥H˙1+∫0T‍(1+∥ϕ0∥L22)(∥ϕ∥H˙13+∥Jδϕ∥Lq3)≲∥ϕ0∥H˙1+(1+∥ϕ0∥L22)T(p-3)/p×(∥ϕ∥LT∞H˙13+∥Jδϕ∥LTpLq3),
where 3<p<∞ and T<1.To estimate∥Jδϕ∥LTpLq, we apply Lemma 5 to the solution of (6)–(8).Proposition 10.
Letϕ be a solution to (6)–(8). Then, one has
(43)∥Jδϕ∥LTpLq≲∥ϕ∥LT∞H1+T1/6∥ϕ0∥L2(1+∥ϕ0∥L22)(X2+X4),
where 2<δq, 3<p<∞ and 1/p+1/q=1/2.Proof.
Applying Lemma5 with F1=A0ϕ-2iAj∂jϕ and F2=A2ϕ-λ|ϕ|2ϕ, we obtain
(44)∥Jδϕ∥LTpLq≲∥ϕ∥LT∞H1+∥A0ϕ∥LT2Hδ-1/2+∥A·∇ϕ∥LT2Hδ-1/2+∥A2ϕ∥LT1Hδ+∥|ϕ|2ϕ∥LT1Hδ,
where δ=1/2-ε, 3<p<∞ and 2<δq. Considering Proposition 8, we obtain
(45)∥A0ϕ∥LT2Hδ-1/2≲∥A0∥LT∞L∞∥ϕ∥LT2L2≲T1/2∥ϕ0∥L2(1+∥ϕ0∥L22)∥∇ϕ∥L22.
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
(46)∥A·∇ϕ∥LT2Hδ-1/2≲∥A∥LT2L∞∥∇ϕ∥LT∞L2≲∥ϕ0∥L2T(p-2)/2p∥Jδϕ∥LTpLq∥ϕ∥LT∞H1,(47)∥A2ϕ∥LT1Hδ≲∥A2∥LT1L4∥Jδϕ∥LT∞L4+∥A2∥LT2Wδ,2+ε∥ϕ∥LT2L(4+2ε)/ε≲T∥ϕ0∥L23/2∥ϕ∥LT∞H15/2+T1/4∥ϕ0∥L2∥ϕ∥LT∞H12∥Jδϕ∥LTpLq2,(48)∥ϕ3∥LT1Hδ≲∥Jδϕ∥LT∞L2∥ϕ∥LT2L∞2≲T(p-2)/p∥ϕ∥LT∞H1∥Jδϕ∥LTpLq2.
Plugging estimates (45)–(48) into (44) with p>3, we obtain
(49)∥Jδϕ∥LTpLq≲∥ϕ∥LT∞H1+T1/6∥ϕ0∥L2(1+∥ϕ0∥L22)(X2+X4).We finally obtain the estimate (38) by combining Propositions 9 and 10, which proves Theorem 1.


## 4. The Proof of Theorem2

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
(50)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.
We will derive
(51)∂t∥ω∥L22≲q1/2M2∥ω∥L22-4/q+qM2+4/q(1+M2)∥ω∥L22-4/q,
where M is a constant in Theorem 2 and q>2. Then we have
(52)∂t∥ω∥L24/q≲1q(q1/2M2+qM2+4/q(1+M2)).
Considering ∥ω(0,·)∥L2=0 and 2<q, we obtain
(53)∥ω∥L2≲(T(M2+M4+4/q))q/4.
Letting q→∞, for the time interval satisfying T(M2+M4+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
(54)∂t∥ω∥L22=∫‍2(A0-B0)Im(ψω-)︸(I)-2Aj∂j|ω|2︸(II)-4(Aj-Bj)Re(∂jψω-)︸(III)dx+∫‍2(A2-B2)Im(ψω-)︸(IV)-2λ(|ϕ|2-|ψ|2)Im(ψω-)︸(V)dx.
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 ∂jAj=0, by
(55)∫-Aj∂j|ω|2dx=∫∂jAj|ω|2dx=0,(III),(IV),(V)≲qM2+4/q(1+M2)∥ω∥L22-4/q
for which we omit the proof.We simply present how to control integral (I), for which we have(56)|∫(A0-B0)Im(ψω-)dx|≲∥A0-B0∥La∥ψ∥Lb∥ω∥Lc,
where 1/a+1/b+1/c=1, 2≤a,b,c. Applying Lemma 6, we obtain
(57)∥ψ∥Lb≲b1/2∥ψ∥L22/b∥∇ψ∥L21-2/b≲b1/2M1-2/b,∥ω∥Lc≲c1/2∥ω∥L22/c∥∇ω∥L21-2/c≲c1/2∥ω∥L22/cM1-2/c.
To control ∥A0-B0∥La, we consider the equation for A0-B0(58)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).
Decomposing A0 and B0 as (24) and (25), we have
(59)Δ(A0′-B0′)=∂1Im(ϕ-∂2ω)-∂2Im(ψ-∂1ω)+∂1Im(ω-∂2ψ)-∂2Im(ω-∂1ϕ),(60)Δ(A0′′-B0′′)=∂1(A2(|ϕ|2-|ψ|2))-∂2(A1(|ϕ|2-|ψ|2))+∂1((A2-B2)|ψ|2)-∂2((A1-B1)|ϕ|2).
Taking into account
(61)∂1Im(ϕ-∂2ω)=∂1(∂2Im(ϕ-ω)-Im(ω∂2ϕ-)),∂2Im(ψ-∂1ω)=∂2(∂1Im(ψ-ω)-Im(ω∂1ψ-)),
we can rewrite the equation for A0′-B0′ as follows:
(62)Δ(A0′-B0′)=∂1(Im(ω-∂2ψ)-Im(ω∂2ϕ-))+∂2(Im(ω∂1ψ-)-Im(ω-∂2ϕ)),
where ∂1∂2Im(ϕ-ω)-∂2∂1Im(ψ-ω)=∂1∂2Im(ω-ω)=0 should be noted. Using the Hardy-Littlewood-Sobolev inequality, we have
(63)∥A0′-B0′∥La≲∥|x|-1*(ω∇ψ)∥La≲∥ω∇ψ∥Lr≲∥ω∥Ls∥∇ψ∥L2,
where 1/a=1/r-1/2 and 1/r=1/s+1/2, from which we deduce a=s. Then, we have
(64)∥ω∥La∥∇ψ∥L2≲a1/2∥ω∥L22/a∥∇ω∥L21-2/aM≲a1/2M2-2/a∥ω∥L22/a.The termA0′′-B0′′ can be bounded as follows:
(65)∥A0′′-B0′′∥La≲∥|x|-1*(|A|||ϕ|2-|ψ|2|)∥La︸(1)+∥|x|-1*(|A-B|(|ϕ|2+|ψ|2))∥La︸(2).
Since ||ϕ|2-|ψ|2|≤(|ϕ|+|ψ|)|ω|, we have
(66)(1)≲∥|A|(|ϕ|+|ψ|)∥L2∥ω∥La≲∥A∥L6(∥ϕ∥L3+∥ψ∥L3)∥ω∥La(67)≲∥ϕ∥L32(∥ϕ∥L3+∥ψ∥L3)∥ω∥La≲a1/2M2-2/a∥ω∥L22/a.
Since |Aj-Bj|≲|x|-1*((|ϕ|+|ψ|)|ω|), we may check
(68)(2)≲∥Aj-Bj∥La(∥ϕ2∥L2+∥ψ2∥L2)≲(∥ϕ∥L2+∥ψ∥L2)∥ω∥La(∥ϕ∥L42+∥ψ∥L42)≲a1/2∥ω∥L22/a∥∇ω∥L21-2/a(∥∇ϕ∥L2+∥∇ψ∥L2)≲a1/2M2-2/a∥ω∥L22/a.
Then, we have
(69)∥A0-B0∥La≲a1/2M2-2/a∥ω∥L22/a.
Combining estimates (57) and (69), and denoting b=q/2, we obtain
(70)∥A0-B0∥La∥ψ∥Lb∥ω∥Lc≲(aqc)1/2M2∥ω∥L22-4/q,
where 1/a+2/q+1/c=1. We then obtain (51) by combining (55) and (70).



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