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

Equation (1) is obtained by the dimensional reduction of system (2). More precisely, we consider (2) independent of the y coordinate and renamingA2 asN to get (1). The spacetime monopole system (2) is a nonabelian gauge field theory and can be derived by dimensional reduction from anti-selfdual 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


Introduction
In the current article, we study the following space-time monopole equations in one space dimension: Here , ,  = ( 0 ,  1 ) : R 1+1 → g, where g is a Lie algebra of a matrix Lie group such as SO(), SU() with Lie bracket [⋅, ⋅].We denote space-time derivatives by  0 =   ,  1 =   .
The space-time monopole equations in R 2+1 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  2 as  to get (1).The spacetime monopole system (2) is a nonabelian gauge field theory and can be derived by dimensional reduction from anti-selfdual 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.
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  2,1 (R 2 ).
The survey [1] also contained a number of other interesting 2 Advances in Mathematical Physics 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   (R 2 ) with  > 1/4.The Cauchy problem of the spacetime monopole equations in R 2+1 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 R 1+1 , under Lorenz gauge condition  0  0 −  1  1 = 0, was proved in [6] for Lebesgue space   (R) with 1 ≤  ≤ ∞.In the current article, we will consider two gauge conditions  1 = 0 and  0 = 0.
First we impose the spatial gauge condition  1 = 0. Then system (1) becomes With the notations of  +  =  and  −  = V, system (5) can be rewritten as System ( 6)-( 8) has the conservation of charge From ( 8), we have the following representation, with a boundary condition  0 (−∞) = 0, We have showed that the initial value problem of ( 6)-( 8) reduces to the study of the following system where  0 is defined by (10).
Theorem 1.For initial data  0 , V 0 ∈   (R) ( = 0, 1), the initial value problem for (11) has a unique, global in time solution which belongs to Moreover we get the following upper bound of  1 norm We will derive some asymptotic behaviors of solutions to (11) with  0 , V 0 ∈   (R).We refer to Remark 7 in Section 2.
Next we impose the temporal gauge condition  0 = 0. Then we have With the notations  +  =  and  −  = V, system (14) can be rewritten as Theorem 2. For initial data  0 , V 0 ,  0 ∈   (R) ( = 0, 1), 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  1 norm: We obtain the following result for the critical case  1 .
Theorem 3.For any  0 , V 0 ,  0 ∈ Now we summarize the algebraic definition and properties used for the proof of Theorems 1-3.Let g be a Lie algebra of a matrix Lie group such as SO(), SU().We define the following norm for  ∈ g: Let ⟨⋅, ⋅⟩ and [⋅, ⋅] denote inner product induced by the trace norm and Lie bracket, respectively.Then we have, for matrices , ,  ∈ g, 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  , (R) with the norm ‖‖  , (R) = ‖Λ  ‖   (R) , where ( Λ )() = (1 + || 2 ) /2 f() and f denotes the Fourier transform of .The space   (R) denotes  ,2 (R).We define the space-time norm ‖‖      = (∫  0 ‖(, ⋅)‖    (R) ) 1/ .We 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
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 equation where  ±0 ∈   and  ± ∈ We will prove the existence of local solution to (11) with  0 , V 0 ∈  2 (R).The case of , V ∈  1 (R) can be treated similarly.Proof follows by standard arguments from a priori estimates of the following Propositions.Proposition 5. Let , V be the solution of (11 Define Then there exist constants  > 0 and  * > 0, depending only on (0), such that if  <  * then () ≤ (0).Proposition 6.Let , V and   , V  be two solutions of (11) verifying the hypothesis of Proposition 5 in a strip R×[0, ] and let () as in a Proposition 5 and   () be the corresponding quantity for the primed solution.Define Then there exist constants  > 0 and  * > 0, depending only on (0) and   (0), such that if  <  * then () ≤ (0).
Proposition 6 follows by the similar argument to Proposition 5. We will only prove Proposition 5. We define It is easily shown that () ≲ (0) + () by applying Lemma 4. We will derive the inequality () ≲ ((0) + ()) 3 .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: (30) Then we get the relation () ≲ ((0) + ()) 3 which completes the proof of Proposition 5 by the bootstrap argument.Global existence for  2 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  0 , V 0 ∈  1 (R).Differentiating both sides of (11), we can obtain Considering (10), we obtain Integrating on R and using Gagliardo-Nirenberg inequality we have Then we obtain Therefore we have an upper bound of  1 norm to extend a local solution globally.

Proof of Theorem 2
We introduce the following bilinear estimates for the proof of existence of local solution to (15)-(17).
Lemma 8. Let  ± ,  1 be the solution to inhomogeneous equations where Then the solutions  ± ,  1 to (42) satisfy the following estimates: for any 0 <  < ∞.
Proof.We refer to [7] for the proof of (43).We will show estimate (44).We have solution representations of  ± ,  1 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).

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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  > 0 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  1 (R) initial data.Differentiating both sides of (15)-( 17), we can obtain Integrating (61) on R, we obtain Integrating (62) on R, we obtain Combining ( 63) and (64), we arrive at We estimate  ∞ bound from (15), ( 16) Then Gronwall's inequality leads us to

Proof of Theorem 3
We consider the critical space  1 .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 .
Applying Lemma 4, we have for sufficiently small  > 0. The estimate for ‖V (+1) ‖  ∞   1 is similar.We also have for sufficiently small  > 0.
Advances in Mathematical Physics 7 Applying Lemma 8, we have for sufficiently small  > 0. The estimate for ‖ (+1) 1 is similar.Applying Lemma 8, we also have, for sufficiently small  > 0,       (+1) V (+1)      (72) For the difference estimate we denote We will show that for sufficiently small  > 0 which implies the existence of a local solution to (15)-( 17).

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Then we get a solution  , , V , ,  , 1 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 , V,  1 on ⋃ =1,2 ⋃ ∈ Ω   .Then we know that the solution , V,  1 belongs to  1   1 (R).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 , V,  1 ∈  1    1 (R) to ( 15)-( 17) with initial data  0 , V 0 ,  1 .Now we assume that  < ∞.We will show that there exists  = (, ‖ 0 ‖  1 , ‖V 0 ‖  1 , ‖ We can verify the nonconcentration of solution (84) for taking  sufficiently small in (88) and (91).