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

Abstract

We study the Cauchy problem of the Chern-Simons-Schrödinger equations with a neutral field, under the Coulomb gauge condition, in energy space . We prove the uniqueness of a solution by using the Gagliardo-Nirenberg inequality with the specific constant. To obtain a global solution, we show the conservation of total energy and find a bound for the nondefinite term.

1. Introduction

In this paper, we are interested in the Cauchy problem of the Chern-Simons-Schrödinger equations coupled with a neutral field (CSSn) in :Here, is the matter field, is the neutral field, and is the gauge field. is the covariant derivative, , , , and . We use notation . From now on, Latin indices are used to denote and the summation convention will be used for summing over repeated indices.

The CSSn system exhibits both conservation of the charge,and conservation of the total energy

The CSSn system is invariant under the following gauge transformations: where is a smooth function. Therefore, a solution to the CSSn system is formed by a class of gauge equivalent pairs . In this paper, we fix the gauge by adopting the Coulomb gauge condition , which provides elliptic features for gauge fields . Under the Coulomb gauge condition, the Cauchy problem of the CSSn system is reformulated as follows:with the initial data ,  ,  . Note that ,   are dynamical variables and are determined by through (11)–(13).

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 asand has conservation of the total energyWe remark that 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 , 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 finite-time blow-up solution constructed in [2, 4]. The CSSn system also has difficulty with nondefiniteness of in the total energy, but we could obtain a global solution by controlling it with -norm.

Considering conservation of the energy (7), it is natural to study the Cauchy problem with the initial data . Our first result is concerned with a local solution in energy space.

Theorem 1. For the initial data , there are and a unique local-in-time solution to (9)–(13) such that where . Moreover, the solution has continuous dependence on initial data.

Our second result is concerned with a global solution in energy space.

Theorem 2. For the initial data , there exists a unique global solution to (9)–(13) such that where . Moreover, the solution has continuous dependence on initial data.

Note that, considering (11)–(13), can be determined by asand then can be determined aswhere if , and if . We present estimates for and refer to [3, 5] for proof.

Proposition 3. Let and let 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 with the norm . We will use to denote various constants. When we are interested in local solutions, we may assume that . Thus we shall replace smooth function of by . We use to denote an estimate of the form .

2. 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 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 [6–9]. 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 -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.

Theorem 4. Let and be solutions to (9)–(13) on in the distribution sense with the same initial data satisfyingfor some . Then, we have for . Moreover, the solution depends on initial data continuously.

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 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 in [11].

Lemma 6. Let be the operator defined by If , , then we have

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 , we have

We need the following Grönwall type inequality.

Lemma 8. Let be a continuous nonnegative function defined on and has zero only at 0. Suppose that satisfieswhere , and . Then we have

Proof. Define Then, the assumption (29) implies and the standard Grönwall’ inequality gives Considering the definition of 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].

Lemma 9. Let and . Let satisfy for all . Then, for all .

Now we are ready to prove Theorem 4. The basic rationale is borrowed from [3, 5, 15]. Let and be solutions of (9)–(13) with the same initial data. If we set then the equations for and satisfywhere

First of all, we will derive, for ,where Once we obtain (39), considering , Lemma 8 gives for . We note that Let us take the time interval satisfying . Letting we have that for . Using this argument repeatedly, we conclude that for

To derive the estimate (39), multiplying (36) by and integrating the imaginary part on , we haveConsidering , we have . Except for the integral (VI), the right-hand side of (43) is bounded, by adopting the same manner described in [3, 5], as follows.We will provide, for instance, the bound for (II) and (IV). The rest can be proved in a similar way. Due to (17), Lemma 6 and Lemma 7 lead to and where is determined by . For , the Hölder’s inequality and Gagliardo-Nirenberg inequality yield Thus, the Hölder’s inequality gives For the integral (IV), similar estimate shows which implies For the integral (VI), we first apply Lemma 5 to (37) which leads to Then, we haveCollecting these bounds (44), (52), we obtain (39) which implies

On the other hand, multiplying (37) by and integrating over , we have for . The Hölder’s inequality and Gagliardo-Nirenberg inequality give us

Finally, continuous dependence on initial data follows from the same estimates above and the same argument in [14]. Let and be solutions of (9)–(13) with the initial data and , respectively. If we set and , the above estimates showApplying Lemma 9 to (56), we have and this implies that the solution depends on initial data continuously in locally uniformly in time.

3. 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) by and taking its conjugate, we have Subtracting (59) from (58), we obtain Then, integration by parts on gives which implies (6).

Multiplying (1) by and taking its conjugate, we have Summing the both sides and integrating by parts on , we obtainConsidering and the left side of (63) becomesOn the other hands, multiplying (3), (4) by , , respectively, we have Adding the both sides, we have Replacing (iii) with this, integration by parts giveswhere (5) is used. Inserting (66) and (69) into (63), we haveNow, multiplying (2) by and integrating on provideAdding (70) and (71), we finally obtain which leads to (7).

Now we are ready to prove the existence of global solution. By the conservation laws (6) and (7), we haveandBecause we do not know the sign of the last term , the energy conservation (74) does not imply a global energy solution directly. Therefore we would find a bound for the last term and then a uniform bound for -norm of solution which leads to global existence. We refer to [8].

Using the Hölder’s inequality, Lemma 7, and Young’s inequality, we haveFrom (75) and (74), it follows that which implies Referring to Proposition 3, the Hölder’s inequality and Young’s inequality give which yields

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Hyungjin Huh was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03028308).