Global solutions of two coupled Maxwell systems in the temporal gauge

In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.

1. Introduction. The Lagrangian density of the (3+1)-dimensional Maxwell-Klein-Gordon system and the (2+1)-dimensional Maxwell-Chern-Simons-Higgs system are given respectively by and where A α ∈ R is the gauge fields, φ is a complex scalar field, N is a real scalar field, F αβ = ∂ α A β − ∂ β A α is the curvature, D µ = ∂ µ − iA µ is the covariant derivative for the MKG system and D µ = ∂ µ − ieA µ is the covariant derivative for the MCSH system, e is the charge of the electron, κ > 0 is the Chern-Simons constant, v is a nonzero constant, ǫ µνρ is the totally skew-symmetric tensor with ǫ 012 = 1. For the MKG system, indices are raised and lowered with respect to the Minkowski metric g αβ = diag(−1, 1, 1, 1), while for the MCSH system, indices are raised and lowered with respect to the metric g αβ = diag(1, −1, −1). We use the convention that the Greek indices such as α, β run through {0, 1, 2, 3} for MKG system, while they run through {0, 1, 2} for MCSH system; the Latin indices such as j, k run through {1, 2, 3}, while they run through {1, 2} for MCSH system; and repeated indices are summed.
For the nontopological boundary condition, we introduce N satisfying N + ev 2 /k = N . Then we have (φ, N , A 1 , A 2 ) → 0 as |x| → ∞. In this case, U N in the system (7) changes to U N respectively. For the topological case, we will discuss a subcase of this case, we assume lim |x|→∞ φ = λ for a fixed complex scalar λ with |λ| = v, i.e., φ tends to be constant at the infinity, this assumption is very natural. We introduce ϕ satisfying ϕ+λ = φ. Then we also have (ϕ, N, A 1 , A 2 ) → 0 as |x| → ∞.
The system (3)-(4) are invariant under the gauge transformations TWO COUPLED MAXWELL SYSTEMS IN THE TEMPORAL GAUGE   3 The system (7) is also invariant under the gauge transformations Hence one may impose an additional gauge condition on A. Usually there are three gauge conditions to choose, Coulomb gauge The Maxwell-Klein-Gordon system is a classical system which has been studied extensively, see e.g. [2], [3]. For the temporal gauge case, in [2], the authors worked in the Coulomb gauge. In this gauge, by exploiting the null structure of the nonlinearity, they obtained the global existence of finite energy solutions of the system. Then, by choosing a suitable χ, they use the gauge transform (10) to transform the obtained global solution in the Coulomb gauge to satisfy the temporal gauge, so they also obtained the global finite energy solution in the temporal gauge. In this paper, we will work directly on the temporal gauge, and obtain the the global existence of finite energy solutions in this gauge. We state our results as follows: ) satisfying the Maxwell-Klein-Gordon system in the distributional sense.
Following the approach in [4] to investigate the Yang-Mills system in the temporal gauge, we will decompose the spatial gauge potentials into divergence free parts and curl free parts, and use the X s,b type spaces to obtain the local well-posedness of the Maxwell-Klein-Gordon system in the temporal gauge. Here, we see that the estimates for the Maxwell-Klein-Gordon system is similar to that for the Yang-Mills case, so we can directly use the estimates which have been already proved in [4]. To show the finite energy local solution of the system extends globally, as in [3], [5], 1 we see that when the initial data satisfies the coulomb gauge, then theirḢ 1 norms can be controlled by the energy of the system and L 2 norm of the solution, and also we see in the investigation of the local well-posedness of the system, we have transformed A to A ′ such that A ′ satisfy (A ′ ) cf = 0, so we have divA ′ = div(A ′ ) df = 0. By combining these two facts, we see that the local solution extends globally. Now we turn our attention to the Cauchy problem of (7). In [6], the authors show that the system is globally well-posed in the Lorenz gauge in H 2 × H 1 , and in [7], this was extended to H 1 × L 2 regularity . Recently, in [8], the author investigate the low regularity of the system in the Lorenz gauge. In [9], the authors show that the system is globally well-posed in the temporal gauge in H 2 ×H 1 . And in [10], the present author show that the system is locally well-posed in the energy regularity H 1 × L 2 and above.
In this paper, by using the approach just described to get the global finite energy solutions of the Maxwell-Klein-Gordon system in the temporal gauge, we can also get the global finite energy solutions of the Maxwell-Chern-Simons-Higgs system in the temporal gauge. Since the nontopological boundary condition is similar to the topological boundary condition case, we just state the results for the former.
) satisfying the Maxwell-Chern-Simons-Higgs system in the distributional sense.
with the same initial data for a sufficiently small T > 0, satisfying for i = 1, 2, |τ |=|ξ|,± (ST ) < +∞, for some b > 1 2 , and some sufficiently small α > 0 and δ > 0 such that Some notations: Sometimes in the paper we will abbreviate Maxwell-Klein-Gordon as MKG and Maxwell-Chern-Simons-Higgs as MCSH. H s (s ∈ R) are Sobolev spaces with respect to the norms f H s = ξ sf L 2 , wheref (ξ) = F f (ξ) is the Fourier transform of f (x) and we use the shorthand ξ = (1 + |ξ| 2 ) 1 2 . We use the shorthand X Y for X ≤ CY , where C >> 1 is a constant which may depend on the quantities which are considered fixed. X ∼ Y means X Y X. We use b+ to denote b + ǫ, for a sufficiently small positive ǫ, and ✷ := ∂ tt − ∆.
In section 2, we will consider the local well-posedness of MKG system in section 2.1, and then in section 2.2, we will show the global existence of finite energy solution of the system. In Section 3, we will show the global existence of finite energy solution of the MCSH system.
Now we decompose A into the divergence free parts A df and curl free parts A cf .
Recall that for a vector field functions X(x) : Since divcurl = 0 and curl∇ = 0, this expresses X as the sum of its divergence free and curl free parts. Let P denote the projection operator onto the divergence-free vector fields on R 3 , P := (−∆) −1 curlcurl. Then the system (12)- (14) becomes We do not expand out A i in (16)-(17), but remember that here Q ij (u, v) := ∂ i u∂ j v − ∂ j u∂ i v, 1 ≤ i, j ≤ 3 denote the null forms, and R i are the Riesz transformations defined by R i := |D| −1 ∂ i . We can also assume that A cf (0) = 0. This can be established by using the transform (10), and let χ = −∆ −1 divA(0).
Here we construct our solutions in the X s,b type spaces. We recall some definitions and some basic properties of X s,b and H s,b spaces.

JIANJUN YUAN
The restriction space X s,b |τ |=|ξ| (S T ) is defined analogously. Now we consider the following linear Cauchy problem Then for F ∈ X s,b−1+δ |τ |=|ξ|,± (S T ), u 0 ∈ H s , the Cauchy problem has a unique solution u ∈ X s,b |τ |=|ξ|,± (S T ), satisfying the first equation in the sense of D ′ (S T ). Moreover, For the linear Cauchy problem we have the following lemma where C only depends on b. For S T = (0, T ) × R n , the restriction space u X s,b τ =0 (ST ) is defined similar to X s,b |τ |=|ξ|,± (S T ) and X s,b |τ |=|ξ| (S T ). Now we consider the following linear Cauchy problem ∂ t u = F, u| t=0 = u 0 . (26) (S T ), u 0 ∈ H s , the Cauchy problem has a unique solution u ∈ X s,b τ =0 (S T ), satisfying the first equation in the sense of D ′ (S T ). Moreover, We have the following local existence results: , satisfying the constraint: then, there exists a time T > 0, which is a decreasing and continuous function of the data norm We will take the contraction spaces as .
And by using the Lemma 2.3 and Lemma 2.5, it is standard that the proof of Lemma 2.6 is reduced to the following estimates. (28) (30) These estimates have appeared in [4], please see the estimates (15)-(19) in [4], we refer the reader to the proof of these estimates in this paper, and we complete the proof of Lemma 2.6.

2.2.
Global existence of MKG system. Now we turn to the global existence part of Theorem 1.1. We will work on the finite energy space of (A, φ), i.e. we will work on H 1 × L 2 regularity for them. Suppose (A, φ) are the solutions of the system (12)-(14) on [0, T ], then we have on [0, T ], d dt Also, we have d dt (39) By combining (37) and (39), we have Now we turn to estimate theḢ 1 norms of A and φ, and we recall a Lemma from [3].

Also for vector field functions
So, by using the Lemma 2.8 for U = D i φ for i = 1, 2, 3 and (42), when A cf (0) = 0, we have Under the temporal gauge Now for the initial data (A i (0), φ(0)) of the system (12)-(14), firstly we make a gauge transform of the form (6) with χ = −∆ −1 divA(0) to let the transformed initial data A ′ satisfy (A ′ ) cf (0) = 0. For the transformed initial data (A ′ (0), φ ′ (0)), since A ′ (0) = (A ′ (0)) df + (A ′ (0)) cf = (A ′ (0)) df , and note that the energy E(t) is invariant under the transform (10), so by Lemma 2.7, (42)-(44), the correspondinġ H 1 norm of (A ′ (0), φ ′ (0)), ∂ t A i L 2 , and ∂ t φ L 2 can be controlled by the L 2 norm of (A(0), φ(0)) and E(0). Also for the L 2 norms of A ′ (0) and φ ′ (0), we have and Now use the Lemma 2.6 for the transformed initial data (A ′ (0), φ ′ (0)), we can get a local solution on [0, T 1 ] with T 1 depending on the A(0) L 2 , φ(0) L 2 and E(0), then we use the inverse transformation of the form (6) with χ = ∆ −1 divA(0) to transform back to get the solution for the original initial data, and thus we obtain the solution of the system on [0, T 1 ]. Then, we begin with (A(T 1 ), φ(T 1 )) as the initial data for the system (12)-(14), for the transformed initial data ( 3. Maxwell-Chern-Simons-Higgs system. Under the temporal gauge and nontopological boundary condition, we rewrite the Euler-Lagrange equations (7) in terms of (A, φ, N ) as follows with given initial data We do not expand out N in the right hand side of system (47), but remember that N = N + ev 2 k . We decompose A = (A 1 , A 2 ) into divergence free part A df = (A df 1 , A df 2 ) and curl free part A cf = (A cf 1 , A cf 2 ), such that A = A df + A cf , divA df = 0, and curlA cf = 0. Remember that for u = (u 1 (x, y), u 2 (x, y)), divu(x, y) = u 1,x (x, y) + u 2,y (x, y), curlu(x, y) = u 2,x (x, y) − u 1,y (x, y). And by calculating, we have We can also assume This can be established by using the transformation (1.11), and let χ = −∆ −1 divA(0). In [10], by setting and consider the equations satisfied by A cf i , A df i,± , φ ± , N ± , we obtain the local wellposedness result of the MCSH in the temporal gauge, see Theorem 3.2 and Theorem 3.1 in [10]. From which, we can deduce the following local well-posedness results for the system (49) under the conditions (48) and (50).
for some b > 1 2 , and some sufficiently small α > 0 and δ > 0 such that α ≤ 1 − b − δ, where S T = [−T, T ] × R 2 . The solution is unique in this regularity class. Now we turn to the global existence part of Theorem 1.1. We will work on the finite energy space of (A, φ, N ), i.e. we will work on H 1 × L 2 regularity for them. Similar to the Maxwell-Klein-Gordon system, we can control the L 2 norm of A(t) and φ(t). For the L 2 norms of N (t), we see by the expression of the energy E(t), we have so we have By combining these, we have Now for vector field functions (A 1 (x), A 2 (x)), if ∂ 1 A 2 −∂ 2 A 1 = B, ∂ 1 A 1 +∂ 2 A 2 = 0, then we have ∇A L 2 ≤ B L 2 . So by using this inequality for A df , we have Also, for i = 1, 2, since So, ∇φ(0) L 2 ≤ C + CE(0) + C φ(0) 2 L 2 A df 2 L 2 .
By combining (51), (53) and (55), we see that the H 1 norms of (A, φ, N ) can be controlled by E(0) and their L 2 norms, so begin with the initial data, we can proceed as in the Maxwell-Klein-Gordon case to get a global finite energy solution of the Maxwell-Chern-Simons-Higgs system. The uniqueness part of Theorem 1.2 is obvious, and we complete the proof of Theorem 1.2.