The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge

Abstract

We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition and the temporal gauge condition .

1. Introduction

In the current article, we study the following space-time monopole equations in one space dimension:Here , where is a Lie algebra of a matrix Lie group such as SO(), SU() with Lie bracket . We denote space-time derivatives by , .

The space-time monopole equations in can be written as follows:Equation (1) is obtained by the dimensional reduction of system (2). More precisely, we consider (2) independent of the coordinate and renaming as to get (1). The space-time monopole system (2) is a nonabelian gauge field theory and can be derived by dimensional reduction from anti-self-dual Yang-Mills equations; see [1], for instance. The system is an example of a completely integrable system and has an equivalent formulation as a Lax pair. It was first introduced by Ward in [2] as a hyperbolic analog of Bogomol’nyi equations and discussed from the point of view of twistors.

System (1) is invariant under the rescaling from which we deduce a scale invariant Lebesgue space and Sobolev space . Another important property of system (1) is an invariance under the gauge transformation where is smooth and compactly supported map into Lie group .

A broad survey on the space-time monopole equations is given in [1]. In particular, using the inverse scattering transform, they have shown global existence and uniqueness up to a gauge transformation for small initial data in . The survey [1] also contained a number of other interesting results related to the space-time monopole equations. Czubak showed in [3] that the space-time monopole system in Coulomb gauge is locally well-posed for small initial data in with . The Cauchy problem of the space-time monopole equations in under the Lorenz gauge condition was discussed in [4, 5]. In particular almost critical local well-posedness has been proved in [5]. The existence of global solution in , under Lorenz gauge condition , was proved in [6] for Lebesgue space with . In the current article, we will consider two gauge conditions and .

First we impose the spatial gauge condition . Then system (1) becomesWith the notations of and , system (5) can be rewritten asSystem (6)–(8) has the conservation of chargeFrom (8), we have the following representation, with a boundary condition ,We have showed that the initial value problem of (6)–(8) reduces to the study of the following systemwhere is defined by (10).

Theorem 1. For initial data   , the initial value problem for (11) has a unique, global in time solution which belongs to Moreover we get the following upper bound of norm

We will derive some asymptotic behaviors of solutions to (11) with . We refer to Remark 7 in Section 2.

Next we impose the temporal gauge condition . Then we haveWith the notations and , system (14) can be rewritten as

Theorem 2. For initial data , the initial value problem for (15)–(17) has a unique, global in time solution which belongs to Moreover we get the following upper bound of norm:

We obtain the following result for the critical case .

Theorem 3. For any , the initial value problem for (15)–(17) has a unique, global in time solution which belongs to

Now we summarize the algebraic definition and properties used for the proof of Theorems 1–3. Let be a Lie algebra of a matrix Lie group such as , . We define the following norm for : Let and denote inner product induced by the trace norm and Lie bracket, respectively. Then we have, for matrices ,

Theorem 1 is proved in Section 2. We show Theorems 2 and 3 in Sections 3 and 4, respectively. We conclude this section by giving a few notations. We use the standard Sobolev space with the norm , where and denotes the Fourier transform of . The space denotes . We define the space-time norm . We 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

To show the existence of local solution to (6)–(8), we introduce a linear estimate (see [7]).

Lemma 4. Let be the solution to the inhomogeneous equationwhere and for . Then for we have

We will prove the existence of local solution to (11) with . The case of can be treated similarly. Proof follows by standard arguments from a priori estimates of the following Propositions.

Proposition 5. Let be the solution of (11) in a strip with Define Then there exist constants and , depending only on , such that if then .

Proposition 6. Let and be two solutions of (11) verifying the hypothesis of Proposition 5 in a strip and let as in a Proposition 5 and be the corresponding quantity for the primed solution. Define Then there exist constants and , depending only on and , such that if then .

Proposition 6 follows by the similar argument to Proposition 5. We will only prove Proposition 5. We define It is easily shown that by applying Lemma 4. We will derive the inequality . Then the bootstrap argument completes the proof of Proposition 5. From representation (10), we have the following bound:Applying (29), the integrals in can be treated as follows: Then we get the relation which completes the proof of Proposition 5 by the bootstrap argument. Global existence for initial data can be proved taking charge conservation (9) into account.

We will prove the global existence of solutions to the initial problem (11) for initial data . Differentiating both sides of (11), we can obtain Considering (10), we obtain Integrating on and using Gagliardo-Nirenberg inequality we have Then we obtain Therefore we have an upper bound of norm to extend a local solution globally.

Remark 7. Let us consider the initial value problem of (6)–(8) for initial data with compact , . Considering (22), we obtain from (11) which implies Then we havewhich implies for , . Therefore we obtain the following system:with initial data . Hence we get the following solution to (39) for : We can check a decay of the local norm in spite of conservation (9) where is any finite interval.

3. Proof of Theorem 2

We introduce the following bilinear estimates for the proof of existence of local solution to (15)–(17).

Lemma 8. Let , be the solution to inhomogeneous equationswhere and , for . Then the solutions , to (42) satisfy the following estimates:for any .

Proof. We refer to [7] for the proof of (43). We will show estimate (44). We have solution representations of , Applying change of variables and Fubini’s theorem, we can derive the following four estimates: Combining the above estimates, we obtain the desired result (44).

We will prove the existence of local solution to (15)–(17) with , . Let us consider the following iteration system corresponding to (15)–(17):with the initial data , , , and . We will denote and show the following boundedness of iteration:for sufficiently small . It is obvious for .

Applying Lemma 4, it follows from Hölder inequality that for sufficiently small . The estimate for is similar. We also have from the third equation of (47) for sufficiently small . Then we obtain (48) for iteration.

We apply Lemma 8 to obtain (49) for sufficiently small . Similarly one may obtain the estimate for small . Applying Lemma 8, we also have for sufficiently small

For the difference estimate we denote We will show that for sufficiently small which implies the existence of a local solution to (15)–(17).

Let us show how to estimate and . Other terms can be bounded in a similar way. Applying Lemma 4, we can derive from the first equation (48) thatFor , we estimate from Lemma 8We can derive Applying Lemma 8, we have We can take care of other terms of in the same manner as (56) and (57) to obtain We choose sufficiently small to obtain (55). Then we can prove the local existence, uniqueness, and Lipschitz continuity of solution by applying standard contraction mapping argument. Note that the existence time depends only on the size of norms of initial data.

Now we focus on global existence for initial data. Differentiating both sides of (15)–(17), we can obtainIntegrating (61) on , we obtainIntegrating (62) on , we obtainCombining (63) and (64), we arrive at We estimate bound from (15), (16) Then Gronwall’s inequality leads us to

4. Proof of Theorem 3

We consider the critical space . We will show the local existence of solution for small data and construct the local solution for the initial data of arbitrary size applying the finite speed of propagation of solution. Applying the nonconcentration property in the Lebesgue space which was introduced originally in [8], we will extend the local solution globally.

First we prove the existence of local solution for a small data. We will show the following boundedness by induction with respect to .for sufficiently small . It is obvious for .

Applying Lemma 4, we have for sufficiently small . The estimate for is similar. We also have for sufficiently small .

Applying Lemma 8, we have for sufficiently small . The estimate for is similar. Applying Lemma 8, we also have, for sufficiently small ,

For the difference estimate we denote We will show that for sufficiently small which implies the existence of a local solution to (15)–(17).

Let us show how to estimate , , , and . Other terms can be bounded in a similar way. Applying Lemma 4, we can deriveFor , we estimate from Lemma 8We can deriveFor , we estimate from Lemma 8We can take care of other terms of in the same manner as (75)–(78) to obtain

Next we make use of the finite speed of propagation of solution to construct the local solution for the initial data of arbitrary size. Choose satisfyingWith this , let us split in two ways for each . We also denote that Applying the finite speed of propagation and considering the smallness condition (80), we can construct the corresponding solution , , in , for each . Let us consider the following regions: Then we get a solution , , on . Note that this solution is not influenced by changing the initial data on the complement of . Using the uniqueness of solution we can gather these solutions to obtain a solution on Then we know that the solution , , belongs to .

To complete Theorem 3, we will show the nonconcentration property of the solution to extend the local solution globally. Let be the maximal time of the existence such that the solution to (15)–(17) with initial data . Now we assume that . We will show that there exists such thatfor any . If condition (84) holds, then there exist and satisfying Taking a new initial data , we can extend the solution over which gives a contradiction. Therefore we must have and then prove the existence of global solution.

Let be the smooth solution to (15)–(17). From (15) and (16), we obtain where (22) is taken into account. Then we derivefrom which we have

From (17), we derive where (87) is used. Integrating on and applying change of variables , and Fubini’s theorem, we have Then we obtainWe can verify the nonconcentration of solution (84) for taking sufficiently small in (88) and (91).

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this article and regarding the funding that they have received.

Acknowledgments

Hyungjin Huh was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014R1A1A2053747).