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

Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Preliminaries -- Chapter 3. Function spaces -- Chapter 4. Decomposition of the nonlinearity -- Chapter 5. Statement of the main estimates -- Chapter 6. Proof of the main theorem -- Chapter 7. Interlude: Bilinear null form estimates -- Chapter 8. Proof of the bilinear estimates -- Chapter 9. Proof of the trilinear estimates -- Chapter 10. Solvability of paradifferential covariant half-wave equations -- Bibliography -- Back Cover.

In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \mathbb{R}^{1+d} (d\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' 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, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.