A note on the Chern-Simons-Dirac equations in the Coulomb Gauge

We prove that the Chern-Simons-Dirac equations in the Coulomb gauge are locally well-posed from initial data in H^s with s>1/4 . To study nonlinear Wave or Dirac equations at this regularity generally requires the presence of null structure. The novel point here is that we make no use of the null structure of the system. Instead we exploit the additional elliptic structure in the Coulomb gauge together with the bilinear Strichartz estimates of Klainerman-Tataru.


Introduction
Chern-Simons gauge theories form an important component of the relativistic theory of planar physics.
In particular they are used to model physical phenomena such as the fractional quantum hall effect, and have been well studied by physicists, see for instance [15,5,4] and the references therein. Mathematically, Chern-Simons terms were first introduced in [3] in connection with certain geometric invariants.
Repeated indices are summed over µ = 0, ..., 2 and raised and lowered using the Minkowski metric g = diag (1, −1, −1), ǫ µνρ is the completely anti-symmetric tensor with ǫ 012 = 1, D µ = ∂ µ − iA µ is the covariant derivative, and F νρ = ∂ ν A ρ − ∂ ρ A ν denotes the curvature of the connection A µ . The equations are coupled using the Dirac current J ν = ψγ ν ψ where ψ = ψ † γ 0 is the Dirac adjoint, and ψ † denotes the conjugate transpose. The Gamma matrices γ µ are 2 × 2 complex matrices which satisfy the relations We take the representation The CSD equations are derived from the Lagrangian is also a solution for any sufficiently regular map θ : R 1+2 → R. Thus to obtain a well-posed Cauchy problem, we need to couple the CSD system (1) with a choice of gauge. Common choices are the Coulomb gauge Solutions to the CSD equation also satisfy conservation of charge and if m = 0, are invariant under the rescaling (ψ, A µ )(t, x) → λ ψ, A µ (λt, λx). This rescaling leaves the L 2 norm unchanged, and so the CSD equation is charge critical. Thus ideally we would like to prove local well-posedness from initial data in L 2 . This would be particulary interesting in view of the conservation of charge (2).
Recently the local and global well-posedness of Chern-Simons systems has received considerable attention, see for instance [1,2,9,8,7,10,14,16]. In particular, it was shown by Huh-Oh [10] that if we couple the Chern-Simons-Dirac equations with the Lorenz gauge condition ∂ µ A µ = 0, then we have local well-posedness for initial data in H s with s > 1 4 . This improved earlier work of Huh [6] where local well-posedness was obtained for s > 1 2 in the Coulomb gauge, s > 5 8 Lorenz gauge, and s > 3 4 in the Temporal gauge.
A crucial component in the proof of local well-posedness of Huh-Oh in [10] was the presence of null structure. Here null structure refers to the fact that, from the point of view of bilinear estimates, the nonlinear terms in (1) behave better than generic bilinear terms such as |ψ| 2 . More precisely, if we consider a nonlinear wave equation of the general form then in general, we have ill-posedness if s < 3 4 due to the counterexamples of Lindblad [13]. On the other hand, if we replace the nonlinearity u∇u with a null form such as Q ij (|∇| −1 u, u) where then we have well-posedness for s > 1 4 , see for instance [11]. Note that the nonlinearities u∇u and Q ij (|∇| −1 u, u) are roughly of the same "strength" in terms of derivatives. Now if we write the CSD equations as a system of nonlinear wave equations, then schematically the CSD equations are of the form (3). Thus, at least at first glance, it appears that null structure is essential to get LWP below 3 4 .
In the current article we show that, if we couple the system (1) with the Coulomb Gauge condition then LWP holds for s > 1 4 . This extends the recent results of Huh-Oh from the Lorenz gauge to the Coulomb gauge. The advantage of the Coulomb gauge is that no null structure is needed. This is somewhat surprising in light of the schematic form of the CSD equation, and the counterexamples of Lindblad mentioned above.
Our main result is the following.
and the solution is unique in this class.
Remark 1. In the result of Huh [6], local well-posedness in the Coulomb gauge was obtained under the x where the initial data should satisfy the constraints This is in contrast to Theorem 1 where we only provide initial data for the spinor ψ. This apparent ambiguity is reconciled by the fact that the initial data for ψ, completely determines A µ (0) via the constraint equations. Thus there is no need to specify data for the gauge A µ (0). See Section 2 below.
The key observation in the proof of Theorem 1 is that, the equations for the gauge, coupled with the Coulomb gauge condition, mean that A µ satisfies elliptic equations of the form

Elliptic Structure
We start by examining the equations for the gauge A µ , for this we need a little preliminary notation.
Define the "curl" ∇ ⊥ = (−∂ 2 , ∂ 1 ) and recall the identity where B : R 2 → C 2 . Define the projections P cf , P df by It is easy to see that P cf and P df are orthogonal projections on L 2 (R 2 ) and ∇ · P df = ∇ ⊥ · P cf = 0. Let A = (A 1 , A 2 ) denote the spatial component of the gauge A µ . Then the gauge equations in (1) can be written as Note that, unlike in the Maxwell or Yang-Mills gauge theories, we have an elliptic component independent of the choice of gauge. If now enforce the Coulomb gauge condition we see that we must have A cf = 0 and therefore, the equations for the gauge (A 0 , A) are Taking ∇ ⊥ of both sides of the equation for A 0 , and adding the equations for the spinor ψ, we see that the CSD equations in the Coulomb gauge are

Proof of Well-posedness
Here we prove Theorem 1. By taking the equations for the gauge A µ , and substituting them into the Dirac component, we see that to prove Theorem 1, it is enough to prove well-posedness for the cubic Dirac equation where N is the bilinear operator Once we have the solution ψ to (7), we then reconstruct the gauge A µ by solving the elliptic equations The proof of local well-posedness for (7) will rely on the following bilinear refinement of the classical Strichartz estimates for the wave equation due to Klainerman-Tataru [12]. This has the following useful consequence.
Corollary 3. Let 1 4 < s 1 2 and I ⊂ R with |I| < ∞. Let B be as in (8) and assume that ψ = e ±it|∇| f , φ = e ±it|∇| g. Then Proof. To obtain the L 4 t L 2 x bound we just note that an application of Proposition 2 with q = 4, r = 2 gives 1 On the other hand, for the L 2 t L ∞ x bound, we start by writing where P <1 is the projection onto frequencies |ξ| < 1. To deal with the low frequency component we use the assumption |I| < ∞ to obtain where the sum is over dyadic λ ∈ 2 Z , λ 1, and the P λ are the standard Littlewood-Paley projections onto frequencies |ξ| ≈ λ.
On the other hand, for the high frequency piece we use Sobolev embedding followed by an application of Holder in time to deduce that where a = 1 − 3 r < 1 − 2 r (so we can apply Sobolev embedding) and q > 2 such that 1 q + 1 2r = 1 2 where r < ∞ is to be chosen later. An application of Proposition 2 then gives Result now follows by taking r sufficiently large.
We also require the following version of the product rule for H s . Then Proof. See the Appendix. 1 Whenever we multiply two spinors together, i.e. ψφ, we really mean i,j ψ i φ j where ψ i , ψ j are the components of the spinor.
The intuition here is that when g is higher frequency than f , we should have |∇| s (f g) ≈ f |∇| s g, which is essentially the first term. On the other hand, when f is higher frequency than g, we should be able to shift derivatives from g onto f , or f |∇| α g (|∇| α f )g, since it is much worse to have a derivative fall on a high frequency piece rather than a low frequency term. To make this more precise requires a straightforward application of Littlewood-Paley theory.
Fix T > 0. The proof of Theorem 1 will proceed by the standard iteration argument using the Duhamel norm where I = [0, T ]. It is easy to see that we have the energy inequality . Moreover we have the following version of the transference principle.
Lemma 5. Let s ∈ R and 1 q, r ∞. Suppose that we have for any f ∈ H s where M is a Fourier multiplier acting only on the spatial variable x ∈ R 2 . Then for any Proof. Let U (t)f denote the solution operator for the Dirac equation iγ µ ∂ µ ψ = 0 with initial data ψ(0) = f . An easy computation shows that U (t − s) = U (t)U (s) and where L ± are bounded, time-independent, Fourier multipliers on H s for all s ∈ R. Now given any where F (s) = iγ µ ∂ µ ψ. Hence using (11) we obtain Π m j=1 f j H s x immediately implies that We now come to the proof of local well-posedness for the cubic Dirac equation (7). A standard iteration argument using the energy inequality, followed by Holder in time, and Lemma 5, shows that to prove local well-posedness for (7) it is enough to prove the estimate where we assume ψ j = e ±jit|∇| f j is a homogeneous wave with data f j ∈ H s x . To prove (12) we start by considering the low frequency case |ξ| < 1. Then by Corollary 3 we obtain We can now replace the H s x norm on the left hand side of (12) with the homogeneous versionḢ s and hence, via an application of the Sobolev product rule in Proposition 4 (with α = 1 2 ), we deduce that where we used the bilinear estimates in Corollary 3, together with the linear L 4 t L ∞ x Strichartz estimate.
To complete the proof of Theorem 1, it only remains to reconstruct the gauge A µ by using (9). To compute the correct regularity for the gauge, note that from (9) we have and since we are assuming the spinor ψ ∈ L ∞ t H s x we need the product estimate The required conditions for product estimates inḢ s to hold, are given by the following.
Proposition 6. Assume s 1 + s 2 + s 3 = n 2 with s j + s k > 0 for j = k. Then We omit the proof of Proposition 6 since it is well known. However for the special case that we use below, namely s 1 = 1 − 2s, s 2 = s 3 = s, we note that, provided 0 < s < 1 2 , the estimate follows by a simple application of Sobolev embedding It we now return to estimating the gauge A µ , we observe that if we want to put A µ ∈ L ∞ tḢ r , in light of (13) and Proposition 6, we need r − 1 + 1 = 2s, =⇒ r = 2s and consequently the correct regularity for the gauge is A µ ∈ L ∞ tḢ 2s . Note that this required the assumption s < 1, if s 1, then the same argument puts the gauge A µ ∈Ḣ r for 0 < r < s + 1.

Appendix -Proof of Proposition 4
Proof. The first step is to write For the high-low piece, we use the fact that the Fourier support of f λ g ≪λ is contained inside the annulus |ξ| ≈ λ and so The low-high piece follows an identical argument, essentially just repeat the previous reasoning but replace f with g, g with f and take α = 0.
Rearranging we obtain α > n(1 − 1 q ′ ) = n q which holds provided q is sufficiently large.
where to obtain the last equality we just relabeled our sequence to start at 1 4 instead of µ (∈ 2 Z ). Now using a similar argument to before where the last line follows by again relabeling the sequence.