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Book DetailsMemoirs of the American Mathematical SocietyVolume: 264; 2020; 94 ppMSC: Primary 35
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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Table of Contents
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Chapters
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1. Introduction
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2. Preliminaries
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3. Function spaces
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4. Decomposition of the nonlinearity
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5. Statement of the main estimates
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6. Proof of the main theorem
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7. Interlude: Bilinear null form estimates
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8. Proof of the bilinear estimates
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9. Proof of the trilinear estimates
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10. Solvability of paradifferential covariant half-wave equations
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Additional Material
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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.
-
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
