Global well-posedness of the Chern-Simons-Higgs equations with finite energy

We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global well-posedness for more regular data. Moreover, we prove local well-posedness even below the energy regularity, using the the null structure of the system in Lorenz gauge and bilinear space-time estimates for wave-Sobolev norms.


Introduction
The (2+1)-dimensional abelian Chern-Simons-Higgs model was proposed by Hong, Kim and Pac [5] and Jackiw and Weinberg [8] in the study of vortex solutions in the abelian Chern-Simons theory. The Lagrangian for the model is on the Minkowski space-time R 1+2 = R t × R 2 x with metric g µν = diag(1, −1, −1). Here D µ = ∂ µ −iA µ is the covariant derivative associated to the gauge field A µ ∈ R, F µν = ∂ µ A ν − ∂ ν A µ is the curvature, φ ∈ C is the Higgs field, V (|φ| 2 ) ∈ R is a Higgs potential, κ > 0 is a Chern-Simons coupling constant, and ǫ µνρ is the skewsymmetric tensor with ǫ 012 = 1. Greek indices range from 0 to 2, Latin indices from 1 to 2, and repeated upper/lower indices are implicitly summed.
There is a conserved energy, and the equations are invariant under the gauge transformations hence we may impose an additional gauge condition. In this paper we rely on the Lorenz condition ∂ µ A µ = 0.
We are interested in the Cauchy problem for the non-topological case, which received considerable attention recently. Local well-posedness for low-regularity data was studied in [6,1,11,7], but the energy regularity was not quite reached; Huh [7] came arbitrarily close to energy using the Coulomb gauge. In this paper we close the remaining gap, using the Lorenz gauge. In fact, we prove that the problem is locally well posed not only at the energy regularity but even a little below it. From the local finite-energy well-posedness we get the corresponding global result by exploiting the conservation of energy and the residual gauge freedom within Lorenz gauge. In particular, we improve the earlier result of Chae and Choe [2], who proved global well-posedness for more regular data, namely with one derivative extra in L 2 compared with energy.
In order to pose the Cauchy problem one should know the observables F µν , J ρ and E at time t = 0, so it suffices to specify φ(0) and D µ φ(0). Since we are interested in the non-topological case we assume V (0) = 0. Moreover we assume that V ′ (r) has polynomial growth, hence E(0) is absolutely convergent if since by the Hardy-Littlewood-Sobolev inequality on R 2 , HereḢ s =Ḣ s (R 2 ), |s| < 1, is the completion of S(R 2 ) with respect to the norm f Ḣs = |ξ| Here s > −1 ensures that S ⊂ L 2 (|ξ| 2s dξ) (densely), and s < 1 ensures that functions in L 2 (|ξ| 2s dξ) are tempered, so the inverse Fourier transform can be applied. We also need the inhomogeneous space H s = H s (R 2 ), which is the completion of where |∇| = (−∆) 1/2 and ∇ = (1 − ∆) 1/2 . In particular, H 1 ⊂ L p for all 2 ≤ p < ∞, and The notation a b stands for a ≤ Cb.

Main results
Since the value of the positive constant κ is irrelevant for our analysis, we shall set κ = 1. Augmented with the Lorenz gauge condition ∂ µ A µ = 0, (1.1) reads where J ρ = 2 Im φD ρ φ and ǫ jk is the skew-symmetric tensor with ǫ 12 = 1.
We pose the Cauchy problem in terms of data for A µ and (φ, ∂ t φ). The question then arises: What are the natural data spaces, given that (1.3)-(1.5) should hold? To answer this, first note that the Lorenz condition leaves some gauge freedom, since it is preserved by (1.2) So formally, at least, we may impose the initial constraints for if these are not already satisfied, they will be after a gauge transformation (1.2) with gauge function χ satisfying But from (2.2) and (2.1b) we get so A j (0) should be inḢ 1/2 , recalling (1.5). Then from (1.3) and (1.4) we infer that (φ, ∂ t φ)(0) ∈ H 1 × L 2 , since ∂ t φ(0) = D 0 φ(0) and ∂ j φ(0) = D j φ(0) + iA j (0)φ(0), and the last term is in L 2 by (1.9). So now we know what the correct data spaces for A µ and (φ, ∂ t φ) are. Note, however, that (2.1b) imposes an initial constraint. The following lemma shows that given any data for (φ, ∂ t φ) in H 1 × L 2 , there exists an initial potential A µ (0) satisfying this constraint as well as the finite energy requirements (1.3) and (1.4).
More generally, we shall prove local well-posedness for any data satisfying (2.1b) initially: with norm bounded in terms of the norm of (2.5), in view of (1.9).
and a solution (A, φ) of (2.1) on (−T, T ) × R 2 with the regularity The solution is unique in a certain subset of this regularity class. Moreover, the solution depends continuously on the data, and higher regularity persists. In particular, if the data are smooth, then so is the solution.
The proof is given in Section 4. Our plan is now to show that (i) the time T in fact only depends on I(0), where and (ii) I(t) is a priori controlled for all time in terms of E(0) and φ(0) L 2 . Then it will of course follow that the solutions extend globally in time.
Proof. The solution of (2.3) is First, if the Fourier transform of A µ (0) is supported in {ξ ∈ R 2 : |ξ| ≥ 1}, then g ∈ H 1/2 , and (2.8) has a unique solution f ∈ H 3/2 , so χ ∈ C(R; H 3/2 ) ⊂ C(R 1+2 ), and ∂ µ χ ∈ C(R; H 1/2 ). Now assume that A µ (0) has Fourier support in {ξ ∈ R 2 : |ξ| < 1}. Then A µ (0) is smooth, but it is not obvious that (2.8) has a solution (what is clear is that the solution, if it exists, will be smooth). Formally, f should be given by, with but it is not clear that this is meaningful. However, if we take the gradient we get something well-defined: is a smooth vector field on R 2 with zero curl: We also need the covariant Sobolev inequality, proved in [4],

Theorem 2.2. The solution (A, φ) from Theorem 2.1 exists up to a time T > 0 which is a continuous and decreasing function of
Proof. Given data (2.5) satisfying (2.6), apply the gauge transformation (1.2) with χ as in Lemma 2.2. Then (1.2) preserves the regularity (2.7), as does its inverse, obtained by replacing χ by −χ. In the new gauge, , and by the latter combined with (2.6) (which is gauge invariant), Since we know that A ′ j (0) belongs toḢ 1/2 , and since in general ∆f = 0 implies where we applied (2.9) in the last step. Moreover, where we used (2.9) and (2.10). Thus, applying Theorem 2.1 we get the solution (A ′ , φ ′ ) up to a time T > 0 which is a continuous and decreasing function of I(0). Finally, reverse the gauge transformation to get the solution (A, φ).
Finally, we show that the solutions extend globally in time. In view of Theorem 2.2 it suffices to show that is a priori bounded on every finite time interval. For this, we rely of course on the conservation of energy (which is satisfied since our local solutions are limits of smooth solutions with compact spatial support). First we note, using E(t) = E(0) and the assumption V (r) ≥ −α 2 r, that By (2.11) and (2.12) we control I(t), and Theorem 2.3 is proved. It remains to prove Theorem 2.1. Note that in Lorenz gauge, and this is the system we actually solve.
Then we have to check that, conversely, (2.13) implies (2.1a) and (2.1b), assuming that the latter two are satisfied at t = 0. But then , and ∂ µ J µ = 0 (which follows from the last equation in (2.13)), one finds and these vanish at t = 0 since v j and w do. Taking another time derivative gives Since the data vanish, we conclude that v j = w = 0, so (2.1a) and (2.1b) hold. Before proving Theorem 2.1, we consider in the following section the problem of local well-posedness with minimal regularity, and it turns out that we can get below the energy regularity. Here we take data for A µ in inhomogeneous Sobolev spaces.

Low regularity local well-posedness
The system (2.13) expands to Here we added A µ and φ to each side of (3.1a) and (3.1c), respectively, to get the operator + 1; this is done to avoid a singularity in (3.7) below. We specify data The data for ∂ t A µ are given by the constraints where J k = 2 Im φD k φ = 2 Im φ∂ k φ + 2A k |φ| 2 , hence J k (0) ∈ H s−1 with norm bounded in terms of the norm of (3.2), as follows from: Lemma 3.1. If s > 0, the following estimates hold: Proof. This follows from the H s product law in two dimensions (see, e.g., [3]), which states that, for s 0 , s 1 , s 2 ∈ R, the estimate holds if and only if (i) s 0 + s 1 + s 2 ≥ 1, (ii) s 0 + s 1 + s 2 ≥ max(s 0 , s 1 , s 2 ) and (iii) at most one of (i) and (ii) is an equality. In particular, for s > 0 this implies (3.5), as well as f g H s−1/2 f H s g H s−1/2 and the latter combined with (3.5) gives (3.6).
The solution is unique in a certain subset of this regularity class. Moreover, the solution depends continuously on the data, and higher regularity persists. In particular, if the data are smooth, then so is the solution.
To prove this we iterate in X s,b -spaces, so by standard methods we reduce to proving estimates for the right hand sides in (3.1). The most difficult terms are the two bilinear ones, for which null structure is needed. The first term on the right hand side of (3.1a) is already a null form, whereas the first term on the right hand side of (3.1c) appears also in the Maxwell-Klein-Gordon system in Lorenz gauge, and we showed in [10] that it has a null structure. To reveal this structure we transform the variables: Then (3.1) transforms to We split the spatial part A = (A 1 , A 2 ) of the potential into divergence-free and curl-free parts and a smoother part: where B 2 = A df · ∇φ was shown in [9] to be a null form: In [10] we found that B 1 also has a null structure: By the Lorenz condition (3.1b) we have Taking into account (3.13) and (3.15), we rewrite (3.8) as with B 1 and B 2 given by (3.15) and (3.13), and The initial data are The systems (3.16) and (3.1) are equivalent via the transformation (3.7), so it suffices to solve (3.16). The Lorenz condition (3.16b) reduces to an initial constraint, since if (A + , A − , φ + , φ − ) is a solution of (3.16), then setting A = A + + A − and φ = φ + +φ − we have ( +1)A µ = M µ , so (3.1a) is satisfied, i.e., A µ = −ǫ µνρ ∂ ν J ρ . Thus, u = ∂ µ A µ satisfies u = 0, and u(0) = ∂ t u(0) = 0 by (3.3) and (3.4).
We prove local well-posedness of (3.16) by iterating in the X s,b -spaces adapted to the operators i∂ t ± ∇ , so by standard arguments (see, e.g., [10] for more details) the proof of Theorem 3.1 reduces to proving, for some b, b ′ ∈ (1/2, 1), m ≥ 2, and ε > 0, the estimates Here u(τ, ξ) is the space-time Fourier transform of u(t, x). Note that −τ ± |ξ| is comparable to −τ ± ξ . We also need the wave-Sobolev norms Frequent use will be made of the fact that u X a,α ± ≤ u H a,α if α ≤ 0, and that the reverse inequality holds if α ≥ 0. In particular, it suffices to prove (3.18) and (3.19) with the X-norms on the left hand sides replaced by the corresponding H-norms.
3.3. Proof of (3.18) for M µ,3 = A µ . Trivially, for j = 1, 2, 3. For B 2 given by (3.13) we reduce to , and proceeding as we did for (3.22), we further reduce to all of which hold by Theorem 4.1 provided that and ε > 0 is sufficiently small. Thus we have (3.28) for B 2 . For B 1 given by (3.15), the estimate (3.28) reduces to The left hand side is bounded by I(τ, ξ) L 2 τ,ξ , where I is given by (3.24) with so we reduce to (3.29) and two additional estimates: Finally, the estimate for B 3 = ∇ −2 A · ∇φ reduces to (3.32).
3.5. Proof of (3.19) for N 2 = A µ A µ φ. For this we need But two applications of Theorem 4.1 give assuming (3.30) holds.

Local well-posedness for finite-energy data
Here we prove Theorem 2.1, or rather the following equivalent statement: Theorem 4.1. If s = 1/2, the analogue of Theorem 3.1 holds with the data space H 1/2 ×H −1/2 for (A µ , ∂ t A µ ) replaced by its homogeneous counterpartḢ 1/2 ×Ḣ −1/2 , and we allow any potential V ∈ C ∞ (R + ; R) such that V (0) = 0 and all derivatives of V have polynomial growth.
We remark that for existence one only needs that V ′ (r) has polynomial growth, but to get persistence of higher regularity one must take additional derivatives of the equations, hence the same assumption is required on all higher derivatives.
The proof follows closely that of Theorem 3.1. We do not add A µ to each side of the wave equation for A µ , but use whereas the data for ∂ t A µ are given by the constraints (3.3) and (3.4), hence they belong toḢ −1/2 , recalling from Section 1 that J k (0) ∈Ḣ −1/2 , with norm bounded in terms of the norm of (4.1). We modify (3.7) by setting 2A µ,± = A µ ± i −1 |∇| −1 ∂ t A µ . The splitting of A = (A 1 , A 2 ) into divergence-free and curl-free parts now reads A = A df + A cf , where A df and A cf are still given by (3.10) and (3.11), but now R j is the Riesz bounded on L p , 1 < p < ∞. Then (3.12) remains valid, but without the term B 3 , and with B 1 and B 2 given by (3.15) and (3.13). Thus we obtain the system with B 1 and B 2 given by (3.15) and (3.13). Here it is understood that The initial data are Local well-posedness reduces to proving Let P |ξ|<1 and P |ξ|≥1 be the multipliers with symbols χ |ξ|<1 and χ |ξ|≥1 , which we use to split f (x) into low-and high-frequency parts: f = P |ξ|<1 f + P |ξ|≥1 f .
But (4.7) is easily proved by applying (4.4): where we used the Sobolev embeddingḢ 1/2 ⊂ L 4 on R 2 . Now consider the high-frequency case where we replace |∇| −1/2 on the left hand side by |∇| −1/2 P |ξ|≥1 . This case obviously reduces to and if u is replaced by P |ξ|≥1 u, these in turn reduce to (3.25) and (3.26). On the other hand, if u is replaced by P |ξ|<1 u, (4.8) and (4.9) both follow from where we used (4.5).
For the low-frequency part we reduce to where we used (4.5).