Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data

Abstract

In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on R 1+d (d ≥ 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru [18]), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.The authors thank Daniel Tataru for many fruitful conversations. C. Gavrus was supported in part by the NSF grant DMS-1266182. S.-J. Oh is a Miller Research Fellow, and acknowledges support from the Miller Institute. Part of this work was carried out during the trimester program 'Harmonic Analysis and PDEs' at the Hausdorff Institute of Mathematics in Bonn.such that the gauge transformÃ(0) = A(0)−dχ obeys the Coulomb gauge condition (4.7). Moreover, the small data condition (1.4) is preserved up to multiplication by a universal constant for ǫ * small enough. Such a gauge transformation can be found by solving the Poisson equation ∆χ = div x A j (0).We remark that our method do not apply to the case of nonzero mass m = 0, although the observations made in this paper suggest that it would likely follow from a corresponding result for the massive Maxwell-Klein-Gordon equations; see 'Parallelism with Maxwell-Klein-Gordon' in Section 1.2. The physically interesting case of d = 3, with or without mass, remains open. 1.1. Previous work. A brief survey of previous results on (MD) and related equations is in order. After early work on local well-posedness of (MD) on R 1+3 by Gross [13] and Bournaveas [4], D'Ancona-Foschi-Selberg [8] established local well-posedness of (MD) on R 1+3 in the Lorenz gauge ∂ µ A µ = 0 for data SMALL CRITICAL DATA GWP FOR MAXWELL-DIRAC 5Another type of null structure that arise in our work is spinorial null forms. These are bilinear forms with the symbol Π ± (ξ)Π ∓ (η), which were first uncovered by D'Ancona, Foschi, Selberg for the Dirac-Klein-Gordon system in [7]. These authors further investigated the spinorial null forms in the study of the Maxwell-Dirac equation on R 1+3 in the Lorenz gauge (in [8]; see also [9]). In the work of Bejenaru-Herr [3,2] and Bournaveas-Candy [5], these null forms were used in the proof of global well-posedness of the cubic Dirac equation for small critical data.A more detailed exposition of the null structure of MD-CG is given in Section 7.4 below. At this point we simply note that the null structure alone is insufficient to close the proof of Theorem 1.1 due to the presence of nonperturbative nonlinearity, which is the next topic of discussion.Presence of nonperturbative nonlinearity. As in many previous works on low regularity well-posedness, we take a paradifferentia...

Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data

“…The idea of these estimates is taken from [18]. For a proof of these bounds as stated here, see [7,Section 7.5]. We can invoke that proof since we have Corollary 8.8 and Lemma 8.9.…”

“…The above idea was also implemented for covariant Dirac operators in [7] in the context of the Maxwell-Dirac equation.…”

“…Recent works on Yang-Mills include: [17], [19], [23], [24]. We also mention the related R 4+1 Maxwell-Dirac result [7] at critical regularity.…”

The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

“…Other related works. In related developments, one should also note the works [4,5] on the closely related cubic Dirac equation, as well as the massive Dirac-Klein-Gordon system, as well as [13] on the Maxwell-Dirac equation and [12] on the massive Maxwell-Klein-Gordon system.…”

“…By Proposition A.7 and the smallness properties of A, e −ē (as well an extra iteration procedure to include the term [Ω α , A α ]), we may find a unique solution Ω to (5.5) such that Ω ∈ L 5 and ∇Ω ∈ L 2Ḣ 1 2 . Integrating the system of ODEs O −1 ∂ α O = Ω, for which the curl condition serves as the compatibility condition needed for integrability 13 , we find a gauge transformation O in B with regularity…”