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

Hartmut Pecher

doi:10.3934/cpaa.2014.13.673

Abstract

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.

Highlights

Introduction and main resultsConsider the Cauchy problem for the nonlinear Dirac equation in two space dimensions i(∂t + α · ∇)ψ + M βψ = − βψ, ψ βψ [1]with initial data ψ(0) = ψ0 . [2]Here ψ is a two-spinor field, i.e. ψ : R1+2 → C2, M ∈ R and ∇ = (∂x1, ∂x2), α · ∇ = α1∂x1 + α2∂x2 . α1, α2, β are hermitian (2 × 2)-matrices satisfying β2 = (α1)2 = (α2)2 = I, αj β + βαj = 0, αj αk + αkαj = 2δjkI . ·, · denotes the C2 - scalar product
In quantum field theory the nonlinear Dirac equation is a model of selfinteracting Dirac fermions
It was originally formulated in one space dimension known as the Thirring model [T] and in three space dimensions [So]

Summary

Introduction and main results

Consider the Cauchy problem for the nonlinear Dirac equation in two space dimensions i(∂t + α · ∇)ψ + M βψ = − βψ, ψ βψ [1]. In the case of three space dimensions Escobedo and Vega [EV] showed local well-posedness in Hs for s > 1, which is almost critical with respect to scaling The Cauchy problem for the Dirac equation [1], [2] has a unique local solution ψ for data ψ0 ∈ Hs(R2), if s > 1/2. Cauchy problem (−i∂t ± |D|)ψ± = F , ψ±(0) = f for data F ∈ X±s,b−1+δ[0, T ] and f ∈ Hs has a unique solution ψ± ∈ X±s,b[0, T ]. For a ∈ R and ǫ > 0 we denote by a+, a + +, a−, a − − numbers with a − ǫ < a − − < a− < a < a+ < a + + < a + ǫ

Proof of the Theorems

Delgado