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Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data
About this Title
Cristian Gavrus, Department of Mathematics, UC Berkeley, Berkeley, California 94720 and Sung-Jin Oh, Korea Institute for Advanced Study, Seoul, Republic of Korea 02455
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 264, Number 1279
ISBNs: 978-1-4704-4111-1 (print); 978-1-4704-5808-9 (online)
DOI: https://doi.org/10.1090/memo/1279
Published electronically: March 25, 2020
MSC: Primary 35L45, 35Q41, 35Q61
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Function spaces
- 4. Decomposition of the nonlinearity
- 5. Statement of the main estimates
- 6. Proof of the main theorem
- 7. Interlude: Bilinear null form estimates
- 8. Proof of the bilinear estimates
- 9. Proof of the trilinear estimates
- 10. Solvability of paradifferential covariant half-wave equations
Abstract
In this paper, we 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 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 (2015)), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.- Philippe Bechouche, Norbert J. Mauser, and Sigmund Selberg, On the asymptotic analysis of the Dirac-Maxwell system in the nonrelativistic limit, J. Hyperbolic Differ. Equ. 2 (2005), no. 1, 129–182. MR 2134957, DOI 10.1142/S0219891605000415
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