Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions

Abstract

We show that in dimensions $n\ge 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm ${\dot{H}}^{n/2-1}\times {\dot{H}}^{n/2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.

49 pages, no pictures, to appear, CMP. A minor problem with a Fourier angular projection causing a certain phase function to no longer be real has now been fixed

References in corpus (3)

• Global regularity of wave maps I. Small critical Sobolev norm in high dimension
• On the Global Regularity of Wave Maps in the Critical Sobolev Norm
• Rough solution for the Einstein Vacuum equations

Cited by in corpus (21)

• Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation
• Global well-posedness for the Maxwell-Klein Gordon equation in 4+1 dimensions. Small energy
• Hodge-Teichmueller planes and finiteness results for Teichmueller curves
• Strong Cosmic Censorship in the presence of matter: the decisive effect of horizon oscillations on the black hole interior geometry
• The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions
• Smoothing - Strichartz Estimates for the Schrodinger Equation with small Magnetic Potential
• Spinning Q-balls in Abelian Gauge Theories with positive potentials: existence and non existence
• Global solution for massive Maxwell-Klein-Gordon equations
• Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge
• Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation
• Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space
• Energy dispersed solutions for the (4+1)-dimensional Maxwell-Klein-Gordon equation
• Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equations in Lorenz gauge
• On global behavior of solutions of the Maxwell-Klein-Gordon equations
• Local well-posedness for low regularity data for the higher-dimensional Maxwell-Klein-Gordon system in Lorenz gauge
• Well-posedness of a gauge-covariant wave equation with space-time white noise forcing
• Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity
• Global Low Regularity Solutions of Quasi-linear Wave Equations
• Local well-posedness of Yang-Mills equations in Lorenz gauge below the energy norm
• On a Calderon-Zygmund commutator-type estimate
• Global well-posedness of high dimensional Maxwell-Dirac for small critical data