Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge

We consider the Maxwell-Chern-Simons-Higgs system in Lorenz gauge and use a null condition to show local well-psoedness for low regularity data. This improves a recent result of J. Yuan.

We now further lower down the regularity of the data to (φ 0 , φ 1 ) ∈ H s ×H s−1 , Moreover it is easy to see that as a consequence of these results unconditional uniqueness holds for the solutions of Yuan, i.e. , for φ, A µ , N ∈ C 0 ([0, T ], H s ) ∩ C 1 ([0, T ], H s−1 ) and s > 3 4 , especially global well-posedness for finite energy solutions (s=1).
Whereas Chae-Chae only used standard energy type estimates Yuan applied bilinear Strichartz type estimates which was given in the paper of d'Ancona, Foschi and Selberg [AFS]. We also use this type of estimates but additionally take advantage of a crucial null condition of the term A µ ∂ µ φ in the wave eqution for φ. This was detected by Klainerman-Machedon [KM] and Selberg-Tesfahun [ST] for the Maxwell-Klein-Gordon equations and also by Selberg-Tesfahun [ST1] for the corresponding problem for the Chern-Simons-Higgs equations. When combined with the bilinear Strichartz type estimates this leads to the improved lower bound for the regularity for the problem at hand.
We denote the Fourier transform F with respect to space as well as to space and time by . The operator ∇ α is defined by F Define a+ := a + ǫ for ǫ > 0 sufficiently small, so that a < a+ < a + + and similarly a − − < a− < a. Besides the Sobolev spaces H k,p we use the spaces X s,b ± of Bougain-Klainerman-Machedon type defined as the completion of S(R 3 ) with respect to the norm We now formulate our main results. One easily checks that a solution of (1),(2),(3) (with (9)) under the Lorenz condition also fulfills the following system Here we replaced by + 1 by adding a linear terms on both sides of the equations in oder to avoid the operator (−∆) − 1 2 , which is unpleasent especially in two dimensions. Defining we obtain the equivalent system We obtain the following result: (12),(13),(14), (15) with (9) and Cauchy conditions

It has the properties
This result is proven in section 2. In section 3 we show the following theorem and its corollary as a consequence of it.
(21) The solution of Theorem 1.1 is the unique solution of the Cauchy problem for the system where which fulfills the Lorenz condition ∂ µ A µ = 0 .

,(21). Then the solution of (22),(23),(24) with initial conditions (26) under the Lorenz condition
. Combined with the existence result of Yuan [Y] we obtain local well-posedness and in energy space and above (s ≥ 1) global well-posedness.
Fundamental for the proof of our theorem are the following bilinear estimates in wave-Sobolev spaces which were proven by d'Ancona, Foschi and Selberg in the two-dimensional case n = 2 in [AFS] in a more general form which include many limit cases which we do not need.

Proof of Theorem 1.1
An application of the contraction mapping is by well-known arguments reduced to suitable multilinear estimates of the right hand sides of (1), (2) and (3). The linear terms are easily treated and therefore omitted here.
The right hand side of (13) can be handled in the same way. It remains to consider the right hand side of (14). We start with the most interesting quadratic term, where the null conditions come into play, namely 2ieA µ ∂ µ φ . Defining the modified Riesz transforms R j := ∇ −1 ∂ j and splitting A j into divergence-free and curl-free parts and a smooth remainder we obtain The first two terms on the right hand side are of null form type, where for the first term we have to use the Lorenz condition (these arguments go back to Selberg-Tesfahun). We calculate which can be estimated by where Θ(ξ, η) denotes the angle between two vectors ξ, η ∈ R 2 . Next the Lorenz condition gives Our aim is to prove the following estimate in the case 1 2 < s ≤ 5 8 : and in the case 5 8 < s < 1: We first estimate the last term on the right hand side of (27) in the whole range 1 2 < s < 1 . We have the sufficient estimate where we used (14) in the last line. This implies ∂ 1 Im(φD 1 φ) + ∂ 2 Im(φD 2 φ) = Im(φD 2 0 φ) + eIm(iφW φ) − Im(φU φ (|φ| 2 , N )) = Im(φD 2 0 φ) + eW |φ| 2 using Im(φU (|φ| 2 , N )) = 0 . Furthermore Collecting all these calculations we finally arrive at ∆W = ∂ t V − 2e 2 |φ| 2 W , so that (42) is proven and We remind that W (0) = 0 , (∂ t W )(0) = V (0) = 0. For sufficiently regular φ this Cauchy problem for a linear wave equation is uniquely solvable by W (t) ≡ 0 . Thus by (42) also V (t) ≡ 0, especially the Lorenz condition is satisfied. But the solutions obtained in Theorem 1.1 are continously depending on the data and also persistence of higher regularity holds. Thus by regularization of the data we may assume here φ, ∂ t φ ∈ C ∞ 0 ([0, t] × R 2 ) and A µ , ∂A µ ∈ C ∞ ([0, T ] × R 2 ) , which justifies our calculations.
Summarizing we have shown that the solution of Theorem 1.1 is a solution of the Cauchy problem for (1),(2),(3) satisfying the Lorenz condition. As remarked earlier already the reverse is also true, so that uniqueness holds.