Null structure and almost optimal local well-posedness of the Maxwell-Dirac system

Abstract

We uncover the full null structure of the Maxwell-Dirac system in Lorenz gauge. This structure, which cannot be seen in the individual component equations, but only when considering the system as a whole, is expressed in terms of tri-and quadrilinear integral forms with cancellations measured by the angles between spatial frequencies. In the 3D case, we prove frequencylocalized L 2 space-time estimates for these integral forms at the scale invariant regularity up to a logarithmic loss, hence we obtain almost optimal local wellposedness of the system by iteration.

Global well-posedness of the Maxwell–Dirac system in two space dimensions

“…In particular, the two largest of N 0 , N 1 and N 2 must be comparable, and N 012 min N 012 max ∼ N 0 N 12 min . As shown in [12],…”

“…We concentrate first on the quadrilinear estimate (5.5), proved in the next four sections by adapting the proof of the analogous estimate in 3d from [12]. We make a dyadic decomposition, use the null structure of the quadrilinear form in the integral, reduce to various L 2 bilinear estimates, and finally sum the dyadic pieces to obtain (5.5).…”

“…which by the definition of the data (a μ ,ȧ μ ) satisfies the Lorenz gauge condition ∂ μ A μ = 0 (see [12]). Now define…”

“…Decisive improvements in the local theory have been made possible through the discovery, by the authors in joint work with Damiano Foschi, of the complete null structure of first DKG, in [11], and then MD, in [12], permitting significant progress compared to earlier local results such as [18,4,6,24,14,2], where at most partial null structure was used.…”

Local well-posedness for the space–time Monopole equation in Lorenz gauge

“…Traditionally, for nonlinear hyperbolic systems with a gauge freedom such as Maxwell-Klein-Gordon or Maxwell-Dirac, the gauge was chosen to satisfy the Coulomb condition ∂ j A j = 0, but more recently null structure has been discovered in the Lorenz gauge as well [6,13]. In the Coulomb gauge, the system (1.1) can be written as a nonlinear system of wave equations for (A 1 , A 2 , φ) coupled with a nonlinear elliptic equation for A 0 .…”