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

1. Introduction

In this paper, we are interested in the Cauchy problem of the Chern-Simons-Schrodinger equations coupled with a neutral field (CSSn) in [R.sup.1+2]:

i[D.sub.0][psi] + [D.sub.j][D.sub.j][psi] = [[absolute value of ([psi])].sup.2] [psi] + 2N[psi], (1)

[[partial derivative].sub.00]N - [DELTA]N + N = -2[[absolute value of ([psi])].sup.2], (2)

[[partial derivative].sub.0][A.sub.1] - [[partial derivative].sub.1][A.sub.0] = 2 Im ([bar.[psi]][D.sub.2][psi]), (3)

[[partial derivative].sub.0][A.sub.2] - [[partial derivative].sub.2][A.sub.0] = -2 Im ([bar.[psi]][D.sub.1][psi]), (4)

[[partial derivative].sub.1][A.sub.2] - [[partial derivative].sub.2][A.sub.1] = [[absolute value of ([psi])].sup.2]. (5)

Here, [psi](t, x) : [R.sup.1+2] [right arrow] C is the matter field, N(t, x) : [R.sup.1+2] [right arrow] r is the neutral field, and [A.sub.[mu]](t, x) : [R.sup.1+2] [right arrow] R is the gauge field. [D.sub.[mu]] = [[partial derivative].sub.[mu]] - [A.sub.[mu]] is the covariant derivative, i = [square root of (-1)], [mathematical expression not reproducible], and [DELTA] = [[partial derivative].sub.j][[partial derivative].sub.j]. We use notation A = ([A.sub.0], [A.sub.j]) = ([A.sub.0], [A.sub.1], [A.sub.2]). From now on, Latin indices are used to denote T, 2 and the summation convention will be used for summing over repeated indices.

The CSSn system exhibits both conservation of the charge,

[mathematical expression not reproducible], (6)

and conservation of the total energy

[mathematical expression not reproducible], (7)

The CSSn system is invariant under the following gauge transformations:

[psi] [right arrow] [psi][e.sup.i[chi]],

N [right arrow] N,

[A.sub.[mu]] [right arrow] [A.sub.[mu]] + [[partial derivative].sub.[mu]][chi], (8)

where [chi] : [R.sup.1+2] [right arrow] R is a smooth function. Therefore, a solution to the CSSn system is formed by a class of gauge equivalent pairs ([psi], N, A). In this paper, we fix the gauge by adopting the Coulomb gauge condition [[partial derivative].sub.j][A.sub.j] = 0, which provides elliptic features for gauge fields A. Under the Coulomb gauge condition, the Cauchy problem of the CSSn system is reformulated as follows:

i[[partial derivative].sub.t][psi] + [DELTA][psi] = -[A.sub.0][psi] + [A.sup.2.sub.j][psi] + 2i[A.sub.j][[partial derivative].sub.j][psi] + [[absolute value of ([psi])].sup.2] [psi] + 2N[psi], (19)

[[partial derivative].sub.tt]N - [DELTA]N + N = -2[[absolute value of ([psi])].sup.2], (10)

[DELTA][A.sub.0] = 2 Im ([[partial derivative].sub.2][bar.[psi]][[partial derivative].sub.1][psi] - [[partial derivative].sub.1][bar.[psi]][[partial derivative].sub.2][psi]) + 2[[partial derivative].sub.2] ([A.sub.1] [[absolute value of ([psi])].sup.2]) (11)

-2[[partial derivative].sub.2] ([A.sub.2] [[absolute value of ([psi])].sup.2]), [DELTA][A.sub.1] = -[[partial derivative].sub.2] ([[absolute value of ([psi])].sup.2]), (12)

[DELTA][A.sub.2] = [[partial derivative].sub.1] ([[absolute value of ([psi])].sup.2]), (13)

with the initial data [psi](0, x) = [[psi].sub.0](x), N(0, x) = [n.sub.0](x), [[partial derivative].sub.t]N(0, x) = [n.sub.1](x). Note that [psi], N are dynamical variables and A are determined by [psi] 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-Schrodinger (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 [psi] and the neutral field N. The CSS system reads as

[mathematical expression not reproducible] (14)

and has conservation of the total energy

[mathematical expression not reproducible] (14)

We remark that [mathematical expression not reproducible] 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 [H.sup.2], [H.sup.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 finite-time blow-up solution constructed in [2,4]. The CSSn system also has difficulty with nondefiniteness of N[[absolute value of ([psi])].sup.2] in the total energy, but we could obtain a global solution by controlling it with [H.sup.1]-norm.

Considering conservation of the energy (7), it is natural to study the Cauchy problem with the initial data [[psi].sub.0], [n.sub.0], [n.sub.1] [member of] [H.sup.1] x [H.sup.1] x [L.sup.2]. Our first result is concerned with a local solution in energy space.

Theorem 1. For the initial data ([[psi].sub.0], [n.sub.0], [n.sub.1]) [member of] [H.sup.1]([R.sup.2]) x [H.sup.1]([R.sup.2]) x [L.sup.2]([R.sup.2]), there are T > 0 and a unique local-in-time solution ([psi], N, A) to (9)-(13) such that

[mathematical expression not reproducible] (15)

where 2 < q < [infinity]. 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 ([[psi].sub.0], [n.sub.0], [n.sub.1]) [member of] [H.sup.1]([R.sup.2]) x [H.sup.1]([R.sup.2]) x [L.sup.2] ([R.sup.2]), there exists a unique global solution ([psi], N, A) to (9)-(13) such that

[mathematical expression not reproducible] (16)

where 2 < q < [infinity]. Moreover, the solution has continuous dependence on initial data.

Note that, considering (11)-(13), [A.sub.j] can be determined by [psi] as

[A.sub.j] = [(-1).sup.j+1]/2[pi] ([x.sub.j']/[[absolute value of (x)].sup.2] * [[absolute value of ([psi])].sup.2]), (17)

and then [A.sub.0] can be determined as

[mathematical expression not reproducible] (18)

where j' = 2 if j = 1, and j' = 1 if j = 2. We present estimates for A and refer to [3, 5] for proof.

Proposition 3. Let [psi] [member of] [H.sup.1]([R.sup.2]) and let A be the solution of (11)-(13). Then, we have, for 2 < q [infinity],

[mathematical expression not reproducible] (19)

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 [H.sup.s]([R.sup.2]) with the norm [mathematical expression not reproducible]. We will use c, C to denote various constants. When we are interested in local solutions, we may assume that T [less than or equal to] 1. Thus we shall replace smooth function of T, C(T) by C. We use A [??] B to denote an estimate of the form A [less than or equal to] CB.

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 [[absolute value of ([xi])].sup.2][psi] from the CSSn system, it is the same as the Klein-Gordon-Schodinger system with Yukawa coupling (KGS). There are many studies on the Cauchy problem of the KGS system in the Sobolev spaces [H.sup.s] [6-9]. Moreover, if we ignore the interaction with the neutral field N 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 [H.sup.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.

Theorem 4. Let (f, N, A) and ([??], [??], [??]) be solutions to (9)-(13) on (0, T) x [R.sup.2] in the distribution sense with the same initial data ([[psi].sub.0], [n.sub.0], [n.sub.1]) [member of] [H.sup.1]([R.sup.2]) x [H.sup.1]([R.sup.2]) x [L.sup.2] ([R.sup.2]) satisfying

[mathematical expression not reproducible] (20)

for some M > 0. Then, we have

[mathematical expression not reproducible] (21)

for 0 [less than or equal to] t [less than or equal to] T. 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 [L.sup.p] - [L.sup.p'] 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 f(t, x) : [R.sup.1+2] [right arrow] R be a solution to

[[partial derivative].sub.tt] f - [DELTA]f + f = F, (t, x) [member of] [R.sup.1+2], f (0, x) = 0, (22)

[[partial derivative].sub.t] f (0, x) = 0, (23)

and T(t) be the Klein-Gordon propagator. Then, we have

f (t, x) = [[integral].sup.t.sub.0] T (t - s) F(s) ds, (24)

and

[mathematical expression not reproducible] (25)

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 [I.sub.1] be the operator defined by

[mathematical expression not reproducible] (26)

If 1/q = 1/p - 1/2, 1 < p < 2, then we have

[mathematical expression not reproducible] (27)

The following Gagliardo-Nirenberg inequality with the explicit constant depending on q 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 Schrodinger equations.

Lemma 7. For 2 [less than or equal to] q < [infinity], we have

[mathematical expression not reproducible] (28)

We need the following Groonwall type inequality.

Lemma 8. Let f(t) be a continuous nonnegative function defined on I = [0 ,a) and has zero only at 0. Suppose that f satisfies

[mathematical expression not reproducible] (29)

where [alpah], [beta] > 0 and q > 2. Then we have

[[integral].sup.t.sub.0] f (s) ds [less than or equal to] [([alpha][e.sup.2[beta]t/q - [alpha]/[beta]).sup.q/2] for t [member of] I. (30)

Proof. Define

h (t) = q/2 [([[integral].sup.t.sub.0] f(s) ds).sup.2/q] + q[alpha]/s[beta]. (31)

Then, the assumption (29) implies

[mathematical expression not reproducible] (32)

and the standard Gronwall' inequality gives

h (t) [less than or equal to] h(0) [e.sup.2[beta]t/q] = q[alpha]/2[beta] [e.sup.2[beta]t/q]. (33)

Considering the definition of h(t) 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 q > 1 and a, b > 0. Let f : [0, [infinity]) [right arrow] [0, [infinity]) satisfy

f (t) [less than or equal to] a + b [[integral].sup.t.sub.0] [f.sup.1-1/q] (s)ds (34)

for all t [greater than or equal to] 0. Then, f(t) [less than or equal to] [([a.sup.1/q] + [bq.sup.-1]t).sup.q] for all t [greater than or equal to] 0.

Now we are ready to prove Theorem 4. The basic rationale is borrowed from [3, 5, 15]. Let ([psi], N, A) and ([??], [??], [??]) be solutions of (9)-(13) with the same initial data. If we set

[mathematical expression not reproducible] (35)

then the equations for u and v satisfy

[mathematical expression not reproducible] (36)

[[partial derivative].sub.tt]v - [DELTA]v + v = -2([[absolute value of ([psi])].sup.2] - [[absolute value of ([??])].sup.2]), (37)

where

u, v [member of] [L.sup.[infinity]] ([0, T); [H.sup.1] ([R.sup.2])) [intersection] C([0, T); [L.sup.2] ([R.sup.2])),

[[partial derivative].sub.t]v [member of] [L.sup.[infinity]] ([0, T); [L.sup.2] ([R.sup.2])). (38)

First of all, we will derive, for q > 2,

[mathematical expression not reproducible] (39)

where

[alpha] = [T.sup.2/q] [qM.sup.2] (1 + [M.sup.4/q] + [M.sup.2+4/q]) and [beta] = [M.sup.2]. (40)

Once we obtain (39), considering [mathematical expression not reproducible], Lemma 8 gives

[mathematical expression not reproducible] (41)

for 0 [less than or equal to] t [less than or equal to] T. We note that

[mathematical expression not reproducible] (42)

Let us take the time interval T' [less than or equal to] T satisfying (2 + [M.sup.2])(2[M.sup.2]T') < 1/2. Letting q [right arrow] [infinity] we have that [mathematical expression not reproducible] for 0 [less than or equal to] t [less than or equal to] T'. Using this argument repeatedly, we conclude that [mathematical expression not reproducible] for 0 [less than or equal to] t [less than or equal to] T.

To derive the estimate (39), multiplying (36) by u and integrating the imaginary part on [0, t] x [R.sup.2], we have

[mathematical expression not reproducible] (43)

Considering [[partial derivative].sub.j][[??].sub.j] = 0, we have (III) = 0. Except for the integral (VI), the right-hand side of (43) is bounded, by adopting the same manner described in [3, 5], as follows.

[mathematical expression not reproducible] (44)

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

[mathematical expression not reproducible] (45)

and

[mathematical expression not reproducible] (46)

where p is determined by 1/q = 1/p-1/2. For 1/r + 1/q = 1/2, the Holder's inequality and Gagliardo-Nirenberg inequality yield

[mathematical expression not reproducible] (47)

Thus, the Holder's inequality gives

[mathematical expression not reproducible] (48)

For the integral (IV), similar estimate shows

[mathematical expression not reproducible] (49)

which implies

[mathematical expression not reproducible] (50)

For the integral (VI), we first apply Lemma 5 to (37) which leads to

[mathematical expression not reproducible] (51)

Then, we have

[mathematical expression not reproducible] (52)

Collecting these bounds (44), (52), we obtain (39) which implies

[mathematical expression not reproducible] (53)

On the other hand, multiplying (37) by [[partial derivative].sub.t]v and integrating over [0, t] x [R.sup.2], we have

[mathematical expression not reproducible] (54)

for 0 [less than or equal to] t [less than or equal to] T. The Holder's inequality and Gagliardo-Nirenberg inequality give us

[mathematical expression not reproducible] (55)

Finally, continuous dependence on initial data follows from the same estimates above and the same argument in [14]. Let ([psi], N, A) and ([??], [??], [??]) be solutions of (9)-(13) with the initial data ([[psi].sub.0], [n.sub.0], [n.sub.1]) and ([[??].sub.0], [[??].sub.0], [[??].sub.1]), respectively. If we set u = [psi] - [??] and [u.sub.0] = [[psi].sub.0] - [[??].sub.0], the above estimates show

[mathematical expression not reproducible] (56)

Applying Lemma 9 to (56), we have

[mathematical expression not reproducible] (57)

and this implies that the solution depends on initial data continuously in [L.sup.2] 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 [bar.[psi]] and taking its conjugate, we have

[mathematical expression not reproducible] (58)

[mathematical expression not reproducible] (59)

Subtracting (59) from (58), we obtain

i[[partial derivative].sub.t] [[absolute value of ([psi])].sup.2] + [DELTA][psi][bar.[psi]] - [DELTA][bar.[psi]][psi] = 2i[[partial derivative].sub.j] ([A.sub.j] [[absolute value of ([psi]).sup.2]). (60)

Then, integration by parts on [R.sup.2] gives

[mathematical expression not reproducible] (61)

which implies (6).

Multiplying (1) by [[partial derivative].sub.t][bar.[psi]] and taking its conjugate, we have

[mathematical expression not reproducible] (62)

Summing the both sides and integrating by parts on [R.sup.2], we obtain

[mathematical expression not reproducible] (63)

Considering

[mathematical expression not reproducible] (64)

and

[mathematical expression not reproducible] (65)

the left side of (63) becomes

[mathematical expression not reproducible] (66)

On the other hands, multiplying (3), (4) by [[partial derivative].sub.t][A.sub.2], [[partial derivative].sub.t][A.sub.1], respectively, we have

[mathematical expression not reproducible] (67)

Adding the both sides, we have

[mathematical expression not reproducible] (68)

Replacing (iii) with this, integration by parts gives

[mathematical expression not reproducible] (69)

where (5) is used. Inserting (66) and (69) into (63), we have

[mathematical expression not reproducible] (70)

Now, multiplying (2) by [[partial derivative].sub.t]N and integrating on [R.sup.2] provide

[mathematical expression not reproducible] (71)

Adding (70) and (71), we finally obtain

d/dt E(t) = 0, (72)

which leads to (7).

Now we are ready to prove the existence of global solution. By the conservation laws (6) and (7), we have

[mathematical expression not reproducible] (73)

and

[mathematical expression not reproducible] (74)

Because we do not know the sign of the last term N[[absolute value of ([psi])].sup.2], 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 [H.sup.1-norm of solution which leads to global existence. We refer to [8].

Using the Holder's inequality, Lemma 7, and Young's inequality, we have

[mathematical expression not reproducible] (75)

From (75) and (74), it follows that

[mathematical expression not reproducible] (76)

which implies

[mathematical expression not reproducible] (77)

Referring to Proposition 3, the Holder's inequality and Young's inequality give

[mathematical expression not reproducible] (78)

which yields

[mathematical expression not reproducible] (79)

https://doi.org/10.1155/2018/3962062

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).

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[7] J.-B. Baillon and J. M. Chadam, "The Cauchy problem for the coupled Schroedinger-Klein-Gordon equations," in Contemporary Developments in Continuum Mechanics and Partial Differential Equations, vol. 30 of North-Holland Mathematics Studies, pp. 37-44, North-Holland, Amsterdam, the Netherlands, 1978.

[8] I. Fukuda and M. Tsutsumi, "On coupled Klein-Gordon-Schrodinger equations, II," Journal of Mathematical Analysis and Applications, vol. 66, no. 2, pp. 358-378, 1978.

[9] N. Hayashi and W. V. Wahl, "On the global strong solutions of coupled Klein-Gordon-Schrodinger equations," Journal of the Mathematical Society of Japan, vol. 39, no. 3, pp. 489-497, 1987.

[10] B. Marshall, W. Strauss, and S. Wainger, "Lp-Lq estimates for the Klein-Gordon equation," Journal de Mathematiques Pures et Appliquees, vol. 59, no. 4, pp. 417-440, 1980.

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Hyungjin Huh (iD) and Bora Moon (iD)

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

Correspondence should be addressed to Bora Moon; boramoon@cau.ac.kr

Received 26 March 2018; Accepted 8 May 2018; Published 4 June 2018

Academic Editor: Stephen C. Anco
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