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Softcover ISBN:  9781470441111 
Product Code:  MEMO/264/1279 
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Softcover ISBN:  9781470441111 
eBook ISBN:  9781470458089 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 264; 2020; 94 ppMSC: Primary 35
In this paper, the authors prove global wellposedness of the massless Maxwell–Dirac equation in the Coulomb gauge on \(\mathbb{R}^{1+d} (d\geq 4)\) for data with small scalecritical 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 MaxwellKleinGordon (for which an analogous result was proved earlier by KriegerSterbenzTataru, which says that the most difficult part of Maxwell–Dirac takes essentially the same form as MaxwellKleinGordon.

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 halfwave equations


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In this paper, the authors prove global wellposedness of the massless Maxwell–Dirac equation in the Coulomb gauge on \(\mathbb{R}^{1+d} (d\geq 4)\) for data with small scalecritical 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 MaxwellKleinGordon (for which an analogous result was proved earlier by KriegerSterbenzTataru, which says that the most difficult part of Maxwell–Dirac takes essentially the same form as MaxwellKleinGordon.

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 halfwave equations