Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field

Here,ψ(t, x) : R → C is the matter field,N(t, x) : R → R is the neutral field, and Aμ(t, x) : R → R is the gauge field.Dμ = ∂μ − iAμ is the covariant derivative, i = √−1, ∂0 = ∂t, ∂j = ∂xj , and Δ = ∂j∂j. We use notation A = (A0, Aj) = (A0, A1, A2). From now on, Latin indices are used to denote 1, 2 and the summation convention will be used for summing over repeated indices. The CSSn system exhibits both conservation of the charge, Q (t) fl 󵄩󵄩󵄩󵄩ψ (t, ⋅)󵄩󵄩󵄩󵄩L2 = Q (0) , (6) and conservation of the total energy

The CSSn system exhibits both conservation of the charge,

𝑄 (𝑡) fl
(, ⋅)     2 =  (0) , (6) and conservation of the total energy The CSSn system is invariant under the following gauge transformations: →   ,  → ,   →   +   , (8) where  : R 1+2 → R is a smooth function.Therefore, a solution to the CSSn system is formed by a class of gauge equivalent pairs (, , A).In this paper, we fix the gauge by adopting the Coulomb gauge condition     = 0, which provides elliptic features for gauge fields A. Under the 2
The CSSn system is derived from the nonrelativistic Maxwell-Chern-Simons model in [1] by regarding Maxwell term in the Lagrangian as zero.Compared with the Chern-Simons-Schrödinger (CSS) system which comes from the nonrelativistic Maxwell-Chern-Simons model by taking the Chern-Simons limit in [1], the CSSn system has the interaction between the matter field  and the neutral field .The CSS system reads as and has conservation of the total energy We remark that ‖(, ⋅)‖ 4  4 has opposite sign in (7) compared with (14).In fact, this difference causes different global behavior of solution.The local well-posedness of the CSS system in  2 ,  1 was shown in [2,3], respectively.We can prove the existence of a local solution of the CSSn system by applying similar argument.On the other hands, due to the nondefiniteness of total energy, the CSS system has a finitetime blow-up solution constructed in [2,4].The CSSn system also has difficulty with nondefiniteness of || 2 in the total energy, but we could obtain a global solution by controlling it with  1 -norm.
Our second result is concerned with a global solution in energy space.

Theorem 2. For the initial data
, there exists a unique global solution (, , A) to ( 9)-( 13) such that where 2 <  < ∞.Moreover, the solution has continuous dependence on initial data.
Note that, considering (11)- (13),   can be determined by  as and then  0 can be determined as where   = 2 if  = 1, and   = 1 if  = 2.We present estimates for A and refer to [3,5] for proof.
Proposition 3. Let  ∈  1 (R 2 ) and let A be the solution of ( 11)- (13).Then, we have, for ( We will prove Theorems 1 and 2 in Sections 2 and 3, respectively.We conclude this section by giving a few notations.We use the standard Sobolev spaces   (R 2 ) with the norm ‖‖   = ‖(1 − Δ) /2 ‖  2 .We will use ,  to denote various constants.When we are interested in local solutions, we may assume that  ≤ 1.Thus we shall replace smooth function of , () by .We use  ≲  to denote an estimate of the form  ≤ .

Proof of Theorem 1
In this section we address the local well-posedness of solution to ( 9)-( 13).We note that if we remove the gauge fields and the term || 2  from the CSSn system, it is the same as the Klein-Gordon-Schödinger system with Yukawa coupling (KGS).There are many studies on the Cauchy problem of the KGS system in the Sobolev spaces   [6][7][8][9].Moreover, if we ignore the interaction with the neutral field  which does not cause any difficulty in obtaining a local solution, a local solution for the CSSn system can be obtained in a similar way to the CSS system.We could obtain a local regular solution by referring to [2,8] and then construct a local energy solution by using the compactness argument introduced in [2, 3,5,6].In other words, a local  1 -solution is constructed by the limit of a sequence of more smooth solutions and it satisfies CSSn system in the distribution sense.For the proof, we follow the same argument as in [2].So we omit the detail of the local existence here.Since the compactness argument does not guarantee the uniqueness and the continuous dependence on initial data of a local solution, we would rather contribute this section to show the uniqueness and the continuous dependence on initial data of a local solution.
Before beginning the proof, we gather lemmas used for the proof of Theorem 4. We use the following   -   estimate proved in [10] which plays an important role to control the difference of solutions.It was used in [6] for the uniqueness of the KGS system.Lemma 5. Let (, ) : R 1+2 → R be a solution to and () be the Klein-Gordon propagator.Then, we have and The Hardy-Littlewood-Sobolev inequality is also used to control the difference of solutions.For the proof, we refer to Theorem 6.1.3 in [11].Lemma 6.Let  1 be the operator defined by The following Gagliardo-Nirenberg inequality with the explicit constant depending on  is used to show the uniqueness.It was proved in [12,13] and used in [3,5,12,13] to show the uniqueness of the nonlinear Schrödinger equations.Lemma 7.For 2 ≤  < ∞, we have ∇ We need the following Grönwall type inequality.
where ,  > 0 and  > 2. Then we have for  ∈ . (30) Proof.Define Then, the assumption (29) implies and the standard Grönwall' inequality gives Considering the definition of ℎ() in the above inequality, we have (30).
We also need the following inequality to show that the solution is continuously dependent on initial data.We refer to [14].

Proof of Theorem 2
In this section we study the existence of a global solution to (9)- (13).Firstly, we derive the conservation laws ( 6) and (7).
Multiplying (1) the left side of (63) becomes . (66) On the other hands, multiplying (3), ( 4) by    2 ,    1 , respectively, we have Adding the both sides, we have Replacing (iii) with this, integration by parts gives where (5) which leads to (7).Now we are ready to prove the existence of global solution.By the conservation laws ( 6) and ( 7), we have and