Local well-posedness for the nonlinear Dirac equation in two space dimensions

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in H^s for s>1/2. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.

In the case of one space dimension global existence for data in H 1 was proven by Delgado [D]. For less regular data Selberg and Tesfahun [ST] showed local wellposedness in H s for s > 0, unconditional uniqueness in C 0 ([0, T ], H s ) for s > 1/4 and global well-posedness for s > 1/2. Recently T.Candy [C] was able to show global well-posedness in L 2 , which is the critical case with respect to scaling.
In the case of three space dimensions Escobedo and Vega [EV] showed local well-posedness in H s for s > 1, which is almost critical with respect to scaling. Moreover they considered more general nonlinearities, too. Global solutions for small data in H s for s > 1 were shown to exist by Machihara, Nakanishi and Ozawa [MNO]. Machihara, Nakamura, Nakanishi and Ozawa [MNNO] proved global existence for small data in H 1 under some additional regularity assumptions for the angular variables.
In the present paper we now consider the case of two space dimensions where the critical space is H 1/2 . We show local well-posedness in H s for s > 1/2, which is optimal up to the endpoint, and unconditional uniqueness for s > 3/4. We construct the solutions in spaces of Bougain-Klainerman-Machedon type, using that the nonlinearity satisfies a null condition. Our proof uses the approach to the corresponding problem for the Dirac-Klein-Gordon equations by d 'Ancona, Foschi and Selberg [AFS], [AFS1]. The crucial estimates for the cubic nonlinearity can then be reduced to bilinear Strichartz type estimates for the wave equation which were given by S. Selberg [S] and D. Foschi and S. Klainerman [FK].
. Fundamental for our results are the following bilinear Strichartz type estimates, which we state for the two-dimensional case.
Corollary 1.2. Under the assumptions of the proposition the following estimate holds: The main result reads as follows: Theorem 1.1. The Cauchy problem for the Dirac equation (1), (2) has a unique local solution ψ for data ψ 0 ∈ H s (R 2 ), if s > 1/2. More precisely there exists a T > 0 and a unique solution

This solution has the property
We also get the following uniqueness result.
Theorem 1.2. The solution of Theorem 1.1 is (unconditionally) We use the following well-known linear estimates (cf. e.g. [AFS], Lemma 5). T ] with an implicit constant independent of T .

Proof of the Theorems
Proof of Theorem 1.1. Using Prop. 1.4 a standard application of the contraction mapping principle reduces the proof to the estimates for the nonlinearity in the following Proposition 2.1.
The null structure of the Dirac equation has the following consequences (we here follow closely [AFS] and [AFS1]). Denoting we remark that by orthogonality this quantity vanishes if ±η and ± 3 ζ line up in the same direction whereas in general (cf. [AFS1], Lemma 1): where ∠(η, ζ) denotes the angle between the vectors η and ζ.
We also need the following elementary estimates which can be found in [AFS], section 5.1 or [GP], Lemma 3.2 and Lemma 3.3.
Using (6) it is thus sufficient to prove where we assume w.l.o.g. that the Fourier transforms are nonnegative. Defining we thus have to show In order to prove (7) let us first of all consider the low frequency case |η − ξ| ≤ 1. We simply use |Θ ±,±3 |, |Θ ±1,±2 | 1 and estimate crudely . From now on we assume |η − ξ| ≥ 1. The estimates for I depend on the different signs which have to be considered. Part I: We start with the case where all the signs ±, ± 3 , ± 1 , ± 2 are + -signs. Analogously one can treat all the cases where ± and ± 3 as well as ± 1 and ± 2 have the same sign. Besides the trivial bounds Θ +,+ , Θ +,+ 1 we make in the following repeated use of the following estimates which immediately follow from Lemma 2.2: In the case |C + | ≥ |A|, |B + | we also use We consider several cases depending on the relative size of the terms in the right hand sides of (8) and (9). We may assume by symmetry in (7) that for the rest of the proof we have |D ±1 | ≥ |E ±2 |, which reduces the number of cases.
Using (12) we obtain the same estimate as in case 1.2.
We arrive at x .
which implies the desired bound. which is further estimated by use of Cor. 1.1 with β 0 = 0 , β − = 1 2 , α 1 = 0 , α 2 = 1. Case 6.2: |η| ≥ |ξ|. We obtain in this case the same estimate as in Part I, Case 5.2 with E + replaced by E − , so that the proof is now complete.