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

Abstract

In this paper we prove global well-posedness and modified scattering for the massive Maxwell-Klein-Gordon equation in the Coulomb gauge on R 1+d (d ≥ 4) for data with small critical Sobolev norm. This extends to the general case m 2 > 0 the results of Krieger-Sterbenz-Tataru (d = 4, 5) and Rodnianski-Tao (d ≥ 6), who considered the case m = 0.We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein-Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon-Sterbenz. To treat it one needs sharp L 2 null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru-Herr. To overcome logarithmic divergences we rely on an embedding property of −1 in conjunction with endpoint Strichartz estimates in Lorentz spaces.

Ill-posedness of the Maxwell–Dirac system below charge in space dimension three and lower

“…Indeed, premultiplying the Dirac equation in (10) by iψ = iψ * γ 0 , taking real parts, and using the fact that M and the A μ are real, and that γ 0 and γ 0 γ j are hermitian, one obtains the conservation law ∂ t ρ + ∂ 1 j = 0, where ρ = ψ * ψ = |ψ| 2 and j = ψ * γ 0 γ 1 ψ. Integration then gives (11). We now state the global well-posedness result in the charge class.…”

“…with its self-induced electromagnetic field. Our interest here is in the Cauchy problem with prescribed initial data at time t = 0, ψ(0, x) = ψ 0 (x), A μ (0, x) = a μ (x), ∂ t A μ (0, x) = b μ (x), (2) and the question of local or global solvability, which has received some attention in recent years; see [1,4,7,14,15,18] for the case of one space dimension and [3,5,6,8,9,[11][12][13]16] for higher dimensions, and the references therein.…”

“…We remark that our proof of ill-posedness works also in dimensions d ≥ 4, but then the critical regularity is above the charge, so this is not really of much interest. In dimensions d ≥ 4, global existence and modified scattering for data with small scaling-critical norm has been proved in [11].…”

“…To extend the local result globally in time, it suffices to obtain an a priori bound on the solution (ψ, A, ∂ t A)(t) in L 2 (R) × AC(R) × L 1 (R). For ψ, this bound is directly provided by the conservation of charge, (11). The latter also provides the necessary bound for (A, ∂ t A), via the linear estimate (48) and the bilinear estimate (49).…”

Untitled

“…We note, however, that the large data problem for the energy critical, massive (i.e., m = 0) Maxwell-Klein-Gordon equation is still open; see the recent work[5] for the small energy case.3 Another excellent example of such an analogy would be the Einstein equation/the (Riemannian) Einstein metric condition/Ricci flow. However, due to their severe nonlinearity as well as their richer geometric properties, the general discussion in this subsection does not seem to apply directly to these equations.…”