LOW REGULARITY SOLUTIONS OF THE CHERN-SIMONS-HIGGS EQUATIONS IN THE LORENTZ GAUGE

Electronic Journal of Differential Equations, Vol. 2009(2009), No. 4, pp. 0. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LOW REGULARITY SOLUTIONS OF THE CHERN-SIMONS-HIGGS EQUATIONS IN THE LORENTZ GAUGE NIKOLAOS BOURNAVEAS Abstract. We prove local well-posedness for the 2 + -dimensional Chern- Simons-Higgs equations in the Lorentz gauge with initial data of low regularity. Our result improves earlier results by Huh [0, ].. Introduction The Chern-Simon-Higgs model was proposed by Jackiw and Weinberg [2] and Hong, Pac and Kim [9] in the context of their studies of vortex solutions in the abelian Chern-Simons theory. Local well-posedness of low regularity solutions was recently studied in Huh [0, ] using a null-form estimate for solutions of the linear wave equation due to Foschi and Klainerman [8] as well as Strichartz estimates. Our aim in this paper is to improve the results of [0, ] in the Lorentz gauge. For this purpose we use estimates in the restriction spaces X s,b introduced by Bourgain, Klainerman and Machedon. A key ingredient in our proof is a modified version of a null-form estimate of Zhou [9] and product rules in X s,b spaces due to D Ancona, Foschi and Selberg [6, 7] and Klainerman and Selberg [3]. The Higgs field has fractional dimension (see below for details), a common feature of systems involving the Dirac equation, see for example Bournaveas [, 2], D Ancona, Foschi and Selberg [6, 7], Machihara [4, 5], Machihara, Nakamura, Nakanishi and Ozawa [6], Selberg and Tesfahun [7], Tesfahun [8]. The Chern-Simon-Higgs equations are the Euler-Lagrange equations corresponding to the Lagrangian density L κ 4 ɛµνρ A µ F νρ + D µ φ D µ φ V ( φ 2). Here A µ is the gauge field, F µν µ A ν ν A µ is the curvature, D µ µ ia µ is the covariant derivative, φ is the Higgs field, V is a given positive function and κ is a positive coupling constant. Greek indices run through {0,, 2}, Latin indices run through {, 2} and repeated indices are summed. The Minkowski metric is defined 2000 Mathematics Subject Classification. 35L5, 35L70, 35Q40. Key words and phrases. Chern-Simons-Higgs equations; Lorentz gauge; null-form estimates; low regularity solutions. c 2009 Texas State University - San Marcos. Submitted February 22, 2009. Published September 2, 2009.

2 N. BOURNAVEAS EJDE-2009/4 by (g µν ) diag(,, ). We define ɛ µνρ 0 if two of the indices coincide and ɛ µνρ ± according to whether (µ, ν, ρ) is an even or odd permutation of (0,, 2). We define Klainerman s null forms by Q µν (u, v) µ u ν v ν u µ v, Q 0 (u, v) g µν µ u ν v. (.a) (.b) Let I µ 2Im ( φd µ φ ). Then the Euler-Lagrange equations are (we set κ 2 for simplicity) F µν 2 ɛ µναi α, (.2a) D µ D µ φ φv ( φ 2). The system has the positive conserved energy given by 2 E D µ φ 2 + V ( φ 2 ) dx. R 2 µ0 (.2b) We are interested in the so-called non-topological case in which φ 0 as x +. For the sake of simplicity we follow [0, ] and set V 0. It will be clear from our proof that for various classes of V s the term φv ( φ 2 ) can easily be handled. Under the Lorentz gauge condition µ A µ 0 the Euler-Lagrange equations (.2) become 0 A j j A 0 + 2 ɛ iji i, A 2 2 A + 2 I 0, 0 A 0 A + 2 A 2, D µ D µ φ 0. Alternatively, they can be written as a system of two nonlinear wave equations: A α 2 ɛαβγ Im(D γ φd β φ D β φd γ φ) + 2 ɛαβγ ( β A γ γ A β ) φ 2, φ 2iA α α φ + A α A α φ. (.3a) (.3b) (.3c) (.3d) (.4a) (.4b) We prescribe initial data in the classical Sobolev spaces A µ (0, x) a µ 0 (x) Ha, t A µ (0, x) a µ (x) Ha, φ(0, x) φ 0 (x) H b, t φ(0, x) φ (x) H b. Dimensional analysis shows that the critical values of a and b are a cr 0 and b cr 2. It is well known that in low space dimensions the Cauchy problem may not be locally well posed for a and b close to the critical values due to lack of decay at infinity. Observe also that φ has fractional dimension. From the point of view of scaling it is natural to take b a + 2. With this choice it was shown in Huh [0] that the Cauchy problem is locally well posed for a 3 4 + ɛ and b 5 4 + ɛ. This was improved in Huh [] to a 3 4 + ɛ, b 9 8 + ɛ (.5) (slightly violating b a + 2 ). The proof relies on the null structure of the right hand side of (.4a). Indeed, D γ φd β φ D β φd γ φ Q γβ (φ, φ) + i ( A γ β ( φ 2 ) A β γ ( φ 2 ) ). On the other hand, since in (.3) the A µ satisfy first order equations and φ satisfies a second order equation it is natural to investigate the case b a +. It turns out

EJDE-2009/4 CHERN-SIMONS-HIGGS EQUATIONS 3 that this choice allows us to improve on a at the expense of b. It is shown in Huh [] that we have local well posedness for a 2, b 3 2. (.6) To prove this result Huh uncovered the null structure in the right hand side of equation (.4b). Indeed, if we introduce B µ by µ B µ 0 and µ B ν ν B µ ɛ µνλ A λ, then the equations take the form: B γ Im ( φd γ φ ) Im ( φ γ φ ) + iɛ µνγ µ B ν φ 2, φ iɛ αµν Q µα (B ν, φ) + Q 0 (B µ, B µ )φ + Q µν (B µ, B ν )φ. (.7a) (.7b) In this article we shall prove the Theorem stated below which corresponds to exponents a 4 + ɛ and b 5 4 + ɛ. This improves (.6) by 4 ɛ derivatives in both a and b. Compared to (.5), it improves a by 2 derivatives at the expense of 8 derivatives in b. Theorem.. Let n 2 and 4 < s < 2. Consider the Cauchy problem for the system (.7) with initial data in the following Sobolev spaces: B γ (0) b γ 0 Hs+ (R 2 ), t B γ (0) b γ Hs (R 2 ), (.8a) φ(0) φ 0 H s+ (R 2 ), t φ(0) φ H s (R 2 ). (.8b) Then there exists a T > 0 and a solution (B, φ) of (.7)-(.8) in [0, T ] R 2 with B, φ C 0 ([0, T ]; H s+ (R 2 )) C ([0, T ]; H s (R 2 )). The solution is unique in a subspace of C 0 ([0, T ]; H s+ (R 2 )) C ([0, T ]; H s (R 2 )), namely in H s+,θ, where 3 4 < θ < s + 2 (the definition of Hs+,θ is given in the next section). Finally, we remark that the problem of global existence is much more difficult. We refer the reader to Chae and Chae [4], Chae and Choe [5] and Huh [0, ]. 2. Bilinear Estimates In this Section we collect the bilinear estimates we need for the proof of Theorem.. We shall work with the spaces H s,θ and H s,θ defined by where Λ and Λ are defined by H s,θ {u S : Λ s Λ θ u L 2 (R 2+ )}, H s,θ {u H s,θ : t u H s,θ } Λ s u(τ, ξ) ( + ξ 2 ) s/2 ũ(τ, ξ), Λ θ u(τ, ξ) ( + (τ 2 ξ 2 ) 2 ) θ/2ũ(τ, ξ). + τ 2 + ξ 2 Notice that the weight ( + (τ 2 ξ 2 ) 2 +τ 2 + ξ 2 ) θ/2 is equivalent to the weight w (τ, ξ) θ, where we define w ± (τ, ξ) + τ ± ξ. We define the norms u H s,θ ξ s w (τ, ξ) θ ũ(τ, ξ) L2 (R 2+ ), u H s,θ u H s,θ + t u H s,θ.

4 N. BOURNAVEAS EJDE-2009/4 The last norm is equivalent to ξ s w + (τ, ξ)w (τ, ξ) θ ũ(τ, ξ) L 2 (R 2+ ). We can now state the null form estimate we are going to use in the proof of Theorem.. Proposition 2.. Let n 2, 4 < s < 2, 3 4 < θ < s + 2. Let Q denote any of the null forms defined by (.). Then for all sufficiently small positive δ we have Q(φ, ψ) H s,θ +δ φ H s+,θ ψ H s+,θ. (2.) If Q Q 0 there is a better estimate. Proposition 2.2. Let n 2, s > 0 and let θ and δ satisfy 2 < θ min{, s + 2 }, Then 0 δ min{ θ, s + 2 θ}. Q 0 (φ, ψ) H s,θ +δ φ H s+,θ ψ H s+,θ (2.2) For a proof of the above proposition, see [3, estimate (7.5)]. For Q Q ij, Q 0j estimate (2.) should be compared (if we set θ s + 2 and δ 0) to the following estimate of Zhou [9]: N s,s (Q αβ(φ, ψ)) N 2 s+,s+ (φ)n 2 s+,s+ (ψ), (2.3) 2 where 4 < s < 2 and Ns,θ(u) w+(τ, ξ) s w (τ, ξ) θ ũ(τ, ξ) L2 τ,ξ. (2.4) The spaces in estimate (2.) are different, with φ and ψ slightly less regular in the sense that u H s,θ N s,θ (u). Moreover we have to account for the extra hyperbolic derivative of order δ on the left hand side. Proof of Proposition 2.. We only sketch the proof for Q Q 0j. Q Q ij is similar. Let F (τ, ξ) ξ s w + (τ, ξ)w θ (τ, ξ) φ(τ, ξ), G(τ, ξ) ξ s w + (τ, ξ)w θ (τ, ξ) ψ(τ, ξ). The proof for Let H(τ, ξ) be a test function. We may assume F, G, H 0. We need to show: ξ + η s w θ +δ (τ + λ, ξ + η) τη j λξ j ξ s w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η) (2.5) F (τ, ξ)g(λ, η)h(τ + λ, ξ + η)dτ dλ dξ dη Using F L 2 G L 2 H L 2. ξ + η s ξ s + η s we see that we need to estimate the following integral (and a symmetric one): w θ +δ (τ + λ, ξ + η) τη j λξ j F (τ, ξ)g(λ, η)h(τ + λ, ξ + η) w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ dτ dλ dξ dη (2.6) η)

EJDE-2009/4 CHERN-SIMONS-HIGGS EQUATIONS 5 We restrict our attention to the region where τ 0, λ 0. The proof for all other regions is similar. We use τη λξ ( ξ η η ξ) + (τ ξ )η (λ η )ξ ( ξ η η ξ) + ( τ ξ )η ( λ η )ξ to see that, we need to estimate the following three integrals: R + ξ η η ξ F (τ, ξ)g(λ, η)h(τ + λ, ξ + η)dτdλ dξ dη w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η), T + τ ξ η F (τ, ξ)g(λ, η)h(τ + λ, ξ + η)dτdλ dξ dη w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η), L + λ η ξ F (τ, ξ)g(λ, η)h(τ + λ, ξ + η)dτdλ dξ dη w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η). We start with R +. We have η ξ ξ η ξ /2 η /2 ( ξ + η ) /2 ( τ + λ ξ + η + τ ξ + λ η ) /2. (2.7) Indeed, η ξ ξ η 2 2 η ξ ( ξ η ξ η) We have ξ + η + ξ + η 2 ( ξ + η ) and η ξ ( ξ + η + ξ + η ) ( ξ + η ξ + η ). ξ + η ξ + η (τ + λ ξ + η ) (λ η ) (τ ξ ) τ + λ ξ + η + λ η + τ ξ, therefore (2.7) follows. Following Zhou [9] we use (2.7) to obtain η ξ ξ η η ξ ξ η 2s η ξ ξ η 2s Therefore, where R + η ξ ξ η 2s ξ /2 s η /2 s ( ξ + η ) /2 s τ + λ ξ + η /2 s + η ξ ξ η 2s ξ /2 s η /2 s ( ξ + η ) /2 s τ ξ /2 s + η ξ ξ η 2s ξ /2 s η /2 s ( ξ + η ) /2 s λ η /2 s. R + R + + R+ 2 + R+ 3, η ξ ξ η 2s ξ /2 s η /2 s ( ξ + η ) /2 s τ + λ ξ + η 2 s w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η) η ξ ξ η 2s ( ξ + η ) /2 s w θ (τ, ξ)w θ (λ, η) ξ s+/2 η 2s+/2

6 N. BOURNAVEAS EJDE-2009/4 (we have used the fact that w s+ 2 θ δ (τ + λ, ξ + η). Indeed, s + 2 θ δ > 0 for small δ, because θ < s + 2.) η ξ ξ η R 2 + 2s ξ /2 s η /2 s ( ξ + η ) /2 s τ ξ /2 s w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η) η ξ ξ η 2s ( ξ + η ) /2 s w θ+s 2 (τ, ξ)w (λ, θ η) ξ s+/2 η 2s+/2 (we have used the fact that w θ δ (τ + λ, ξ + η). Indeed, θ δ 0 for small δ because θ < s + 2 <.) R + 3 η ξ ξ η 2s ξ /2 s η /2 s ( ξ + η ) /2 s λ η /2 s w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η) η ξ ξ η 2s ( ξ + η ) /2 s w (τ, θ ξ)w θ+s 2 (λ, η) ξ s+/2 η 2s+/2 We present the proof for R 2 +. The proofs for R+ and R+ 3 are similar. We change variables τ u : τ ξ τ ξ and λ v : λ η λ η and we use the notation to get f u (ξ) F (u + ξ, ξ), g v (η) G(v + η, η), H u,v (τ, ξ ) H(u + v + τ, ξ ) R + 2 [ η ξ ξ η 2s ( ξ + η ) ( + u ) θ+s 2 ( + v ) θ ξ s+/2 η 2s+/2 ] f u (ξ)g v (η)h u,v ( ξ + η, ξ + η) dξ dη du dv. /2 s We have η ξ ξ η 2 2 ξ η ( ξ η ξ η) therefore ( ξ η ξ η) s ( ξ + η ) /2 s [ ] f ξ /2 η s+/2 u (ξ)g v (η)h u,v ( ξ + η, ξ + η)dξdη ( f u (ξ) 2 g v (η) 2 dξdη ) /2K /2 where f u L2 (R 2 ) g v L2 (R 2 )K /2, ( ξ η ξ η) 2s ( ξ + η ) 2s K ξ η 2s+ H u,v ( ξ + η, ξ + η) 2 dξ dη ( ξ η η (ξ η) η) 2s ( ξ η + η ) 2s ξ η η 2s+ H u,v ( ξ η + η, ξ ) 2 dξ dη.

EJDE-2009/4 CHERN-SIMONS-HIGGS EQUATIONS 7 We use polar coordinates η ρω to get ( ξ ρω + ρ ξ ω) 2s ( ξ ρω + ρ) 2s K ξ ρω H u,v ( ξ ρω + ρ, ξ ) 2 dξ dρ dω. For fixed ξ and ω, we change variables ρ τ : ξ ρω + ρ to get [ K τ 2s ] S (τ ξ ω) 2s dω H(τ, ξ ) 2 dξ dτ. From [9, estimate (3.22)] we know that τ 2s S (τ ξ dω ; ω) 2s therefore K H 2. Putting everything together we get: Ã ( R 2 + f u )( L 2 (R 2 ) g v ) L du 2 (R 2 ) ( + u ) θ+s 2 ( + v ) θ dv H. Since 2θ + 2s > 2 3 4 + 2 3 4 and 2θ > 2 4 > we can use the Cauchy-Schwarz inequality to conclude: R + 2 f u L2 (R 2 ) L 2 u g v L2 (R 2 ) L 2 v H F G Ã H Ã. This completes the estimates for R 2 +. Next we estimate T +. We use τ ξ w + (τ, ξ) θ w (τ, ξ) θ to get T + τ ξ η F (τ, ξ)g(λ, η)h(τ + λ, ξ + η)dτdλ dξ dη w θ δ (τ + λ, ξ + η)w + (τ, ξ)w (τ, θ ξ) η s w + (λ, η)w (λ, θ η) F (τ, ξ)g(λ, η)h(τ + λ, ξ + η) ξ θ η s w (λ, θ dτdλ dξ dη. η) Changing variables τ u : τ ξ τ ξ and λ v : λ η λ η we have T + ξ θ η s [ F (u + ξ, ξ)g(v + η, η)h(u + v + ξ + η, ξ + η) ] ( + v ) θ du dv dξ dη. For fixed ξ and η we apply [9, Lemma A] in the (u, v)-variables to get T + ξ θ η s F (u + ξ, ξ) L 2 G(v + η, η) u L 2 v H(w + ξ + η, ξ + η) L 2 w dξdη ξ θ η s F (, ξ) L 2 (R) G(, η) L2 (R) H(, ξ + η) L2 (R) dξ dη. Now we do the same in the (ξ, η)-variables to get T + F (, ξ) L 2 (R) L 2 ξ G(, η) L 2 (R) L 2 η H(, ξ ) L 2 (R) L 2 ξ F Ã G Ã H Ã. The proof for L + is similar. We are also going to need the following product rules in H s,θ spaces.

8 N. BOURNAVEAS EJDE-2009/4 Proposition 2.3. Let n 2. Then provided that uv H c, γ u H a,α v H b,β, (2.8) a + b + c > (2.9) a + b 0, b + c 0, a + c 0 (2.0) α + β + γ > /2 (2.) α, β, γ 0. (2.2) Proof. If a, b, c 0, the result is contained in [3, Proposition A]. If not, observe that, due to (2.0), at most one of the a, b, c is negative. We deal with the case c < 0, a, b 0. All other cases are similar. Observe that ξ c τ ξ γ ũv(τ, ξ) τ ξ γ ξ η c ũ(τ λ, ξ η) ṽ(λ, η) dλdη + τ ξ γ ũ(τ λ, ξ η) η c ṽ(λ, η) dλdη, therefore uv H c, γ Uv H 0, γ + u V H 0, γ, where Ũ(τ, ξ) ξ c ũ(τ, ξ), ũ (τ, ξ) ũ(τ, ξ), Ṽ (τ, ξ) ξ c ṽ(τ, ξ), ṽ (τ, ξ) ṽ(τ, ξ). Since a + c 0, we have Uv H 0, γ U H a+c,α v H b,β u H a,α v H b,β. Since b + c 0, we have u V H 0, γ u H a,α V H b+c,β u H a,α v H b,β. The result follows. Proposition 2.4. Let n 2. If s > and 2 < θ s 2, then Hs,θ is an algebra. For the proof of the above proposition, see [3, Theorem 7.3]. Proposition 2.5. Let n 2, s >, 2 < θ s 2. Assume that θ α 0 s a < s + α. (2.3) Then H a,α H s,θ H a,α. (2.4) The proof can be found in [3, Theorem 7.2].

EJDE-2009/4 CHERN-SIMONS-HIGGS EQUATIONS 9 Proof of Theorem.. Theorem. follows by well known methods from the following a-priori estimates (together with the corresponding estimates for differences): For any space-time functions B, B, φ, φ H s+,θ and any γ, µ {0,, 2} we have: φ γ φ H s,θ +δ φ H s+,θ φ H s+,θ, (2.5) ( µ B) φ φ H s,θ +δ B H s+,θ φ H s+,θ φ H s+,θ, (2.6) Q µα (B, φ) H s,θ +δ B H s+,θ φ H s+,θ, (2.7) Q 0 (B, B )φ H s,θ +δ B H s+,θ B H s+,θ φ H s+,θ, (2.8) Q µν (B, B )φ H s,θ +δ B H s+,θ B H s+,θ φ H s+,θ. (2.9) Here 4 < s < 2, 3 4 < θ < s + 2 and δ is a sufficiently small positive number. To prove (2.5) we use Proposition 2.3 to get: φ γ φ H s,θ +δ φ H s+,θ γ φ H s,θ φ H s+,θ φ H s+,θ. Similarly, for (2.6) we have ( µ B) φ φ H s,θ +δ µ B H s,θ φ φ H s+,θ B H s+,θ φ φ H s+,θ. By Proposition 2.4 and our assumptions on s and θ it follows that the space H s+,θ is an algebra. Therefore, φ φ H s+,θ φ H s+,θ φ H s+,θ, and estimate (2.6) follows. Estimate (2.7) follows from Proposition (2.). Finally, we consider estimates (2.8) and (2.9). We use the letter Q to denote any of the null forms Q 0, Q µν. We have Q(B, B ) φ H s,θ +δ Q(B, B ) H s,θ +δ φ H s+,θ. (2.20) This follows from Proposition 2.5 with s replaced by s+ and α replaced by θ +δ. Next, by (2.), therefore (2.8) and (2.9) follow. Q(B, B ) H s,θ +δ B H s+,θ B H s+,θ (2.2) References [] Bournaveas, N.; Low regularity solutions of the Dirac Klein-Gordon equations in two space dimensions. Comm. Partial Differential Equations 26 (200), no. 7-8, 345 366. [2] Bournaveas, N.; Local existence of energy class solutions for the Dirac-Klein-Gordon equations. Comm. Partial Differential Equations 24 (999), no. 7-8, 67 93. (Reviewer: Shu- Xing Chen) 35Q53 (35L70) [3] Bournaveas, N.; Local existence for the Maxwell-Dirac equations in three space dimensions. Comm. Partial Differential Equations 2 (996), no. 5-6, 693 720. [4] Chae, D., ; Chae, M.; The global existence in the Cauchy problem of the Maxwell-Chern- Simons-Higgs system. J. Math. Phys. 43 (2002), no., 5470 5482. [5] Chae, D., ; Choe, K.; Global existence in the Cauchy problem of the relativistic Chern- Simons-Higgs theory. Nonlinearity 5 (2002), no. 3, 747 758. [6] D Ancona, P., Foschi, D., Selberg, S.; Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions. J. Hyperbolic Differ. Equ. 4 (2007), no. 2, 295 330. [7] D Ancona, P., Foschi, D., Selberg, S.; Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system. J. Eur. Math. Soc. 9 (2007), no. 4, 877 899. [8] Foschi, D., Klainerman, S.; Bilinear space-time estimates for homogeneous wave equations. Ann. Sci. Ecole Norm. Sup. (4) 33 (2000), no. 2, 2 274.

0 N. BOURNAVEAS EJDE-2009/4 [9] Hong, J., Kim, Y., Pac, P. Y. ; Multivortex solutions of the abelian Chern-Simons-Higgs theory. Phys. Rev. Lett. 64 (990), no. 9, 2230 2233. [0] Huh, H.; Low regularity solutions of the Chern-Simons-Higgs equations. Nonlinearity 8 (2005), no. 6, 258 2589. [] Huh, H.; Local and global solutions of the Chern-Simons-Higgs system. J. Funct. Anal. 242 (2007), no. 2, 526 549. [2] Jackiw, R. Weinberg, E. J.; Self-dual Chern-Simons vortices. Phys. Rev. Lett. 64 (990), no. 9, 2234 2237. [3] Klainerman, S., Selberg, S.; Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4 (2002), no. 2, 223 295. [4] Machihara, S.; The Cauchy problem for the -D Dirac-Klein-Gordon equation. NoDEA Nonlinear Differential Equations Appl. 4 (2007), no. 5-6, 625 64. [5] Machihara, S.; Small data global solutions for Dirac-Klein-Gordon equation. Differential Integral Equations 5 (2002), no. 2, 5 57. [6] Machihara, S., Nakamura, M., Nakanishi, K., Ozawa, T.; Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 29 (2005), no., 20 [7] Selberg, S., Tesfahun, A.; Low regularity well-posedness of the Dirac-Klein-Gordon equations in one space dimension. Commun. Contemp. Math. 0 (2008), no. 2, 8 94. [8] Tesfahun, A.; Low regularity and local well-posedness for the + 3 dimensional Dirac-Klein- Gordon system. Electron. J. Differential Equations 2007, No. 62, 26 pp. [9] Zhou, Yi; Local existence with minimal regularity for nonlinear wave equations. Amer. J. Math. 9 (997), no. 3, 67 703. Nikolaos Bournaveas University of Edinburgh, School of Mathematics, James Clerk Maxwell Building, King s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK E-mail address: n.bournaveas@ed.ac.uk