Global Well-Posedness of High Dimensional Maxwell-Dirac for Small Critical Data [1 ed.] 9781470458089, 9781470441111

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Number 1279

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data Cristian Gavrus Sung-Jin Oh

March 2020 • Volume 264 • Number 1279 (second of 6 numbers)

Number 1279

Global Well-Posedness of High Dimensional Maxwell–Dirac for Small Critical Data Cristian Gavrus Sung-Jin Oh

March 2020 • Volume 264 • Number 1279 (second of 6 numbers)

Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/memo/1279

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25 24 23 22 21 20

Contents Chapter 1. Introduction

1

Chapter 2. Preliminaries

7

Chapter 3. Function spaces

11

Chapter 4. Decomposition of the nonlinearity

19

Chapter 5. Statement of the main estimates

27

Chapter 6. Proof of the main theorem

31

Chapter 7. Interlude: Bilinear null form estimates

43

Chapter 8. Proof of the bilinear estimates

57

Chapter 9. Proof of the trilinear estimates

69

Chapter 10. Solvability of paradifferential covariant half-wave equations

83

Bibliography

93

iii

Abstract In this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on R1+d (d ≥ 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 KriegerSterbenz-Tataru (2015)), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

Received by the editor December 12, 2016, and, in revised form, April 12, 2017. Article electronically published on March 25, 2020. DOI: https://doi.org/10.1090/memo/1279 2010 Mathematics Subject Classification. Primary 35L45, 35Q41, 35Q61. C. Gavrus is affiliated with the Department of Mathematics, Johns Hopkins University. Email: [email protected]. S.-J. Oh is affiliated with the Department of Mathematics, University of California, Berkeley. Email: [email protected]. The authors thank Daniel Tataru for many fruitful conversations. C. Gavrus was supported in part by the NSF grant DMS-1266182. S.-J. Oh was supported by the Miller Research Fellowship. Part of this work was carried out during the trimester program ‘Harmonic Analysis and PDEs’ at the Hausdorff Institute of Mathematics in Bonn. c 2020 American Mathematical Society

v

CHAPTER 1

Introduction Let R1+d be the (d + 1)-dimensional Minkowski space with the metric η = diag(−1, +1, . . . , +1) in the rectilinear coordinates (x0 , x1 , . . . , xd ). Associated to the Minkowski metric η are the gamma matrices, which are N × N complex-valued matrices γ μ (μ = 0, 1, . . . , d) satisfying the anti-commutation relations 1 μ ν (γ γ + γ ν γ μ ) = −η μν I, 2 where I is the N × N identity matrix, and also the conjugation relations (1.1)

(1.2)

(γ μ )† = γ 0 γ μ γ 0 .

On R1+d , the rank of the gamma matrices γ μ in the standard representation is d+1 N = 2 2  [10, Appendix E]. A spinor field ψ is a function on R1+d (more generally, open subsets of R1+d ) that takes values in CN , on which γ μ acts as multiplication. Given a real-valued 1-form Aμ (connection 1-form), we introduce the gauge covariant derivative on spinors Dμ ψ := ∂μ ψ + iAμ ψ, (which acts componentwisely on ψ) and the associated curvature 2-form Fμν := (dA)μν = ∂μ Aν − ∂ν Aμ . The Maxwell–Dirac system is a relativistic Lagrangian field theory describing the interaction between a connection 1-form Aμ , representing an electromagnetic potential, and a spinor field ψ, modeling a charged fermionic field (e.g., an electron). Its action (i.e., the space-time integral of the Lagrangian) takes the form  1 − Fμν F μν + iγ μ Dμ ψ, γ 0 ψ − mψ, ψ dtdx. S[Aμ , ψ] = 4 1+d R Here ψ 1 , ψ 2  := (ψ 2 )† ψ 1 is the usual inner product on CN , where ψ † denotes the hermitian transpose. Furthermore, we use the standard convention of raising and lowering indices using the Minkowski metric η, and the Einstein summation convention of summing repeated upper and lower indices. The Euler–Lagrange equations for S[Aμ , ψ] take the form  ∂ ν Fμν = − ψ, αμ ψ, (MD) iαμ Dμ ψ =mβψ. where αμ = γ 0 γ μ and β = γ 0 . Henceforth, we will refer to (MD) as the Maxwell– Dirac equations. 1

2

1. INTRODUCTION

A basic feature of (MD) is invariance under gauge transformations. That is, given any solution (A, ψ) of (MD) and a real-valued function χ (gauge transforma˜ = (A − dχ, eiχ ψ) of (A, ψ) is also a ˜ ψ) tion) on I × Rd , the gauge transform (A, solution to (MD). In order to make (MD) a (formally) well-posed system, we need to remove the ambiguity arising from this invariance. To this end, we impose in our paper the (global) Coulomb gauge condition, which reads divx A =

(1.3)

d 

∂j Aj = 0.

j=1

In this paper, we show global well-posedness and scattering for massless (MD) (i.e., m = 0) on the Minkowski space R1+d with d ≥ 4 under the Coulomb gauge condition, for initial data which are small in the scale-critical Sobolev space. When restricted to the massless case, (MD) is invariant under the scaling (λ > 0)   3 (Aμ , ψ) → λ−1 Aμ (λ−1 t, λ−1 x), λ− 2 ψ(λ−1 t, λ−1 x) . For the sake of concreteness, we focus on the case d = 4, which is the most difficult. Theorem 1.1 (Critical small data global well-posedness and scattering on R1+4 ). Consider (MD) on R1+4 with m = 0. There exists a universal constant

∗ > 0 such that the following statements hold. (1) Let (ψ(0), Aj (0), ∂t Aj (0)) be a smooth initial data set satisfying the Coulomb condition (1.3) and the smallness condition ψ(0)H˙ 1/2 (R4 ) + sup (Aj , ∂t Aj )(0)H˙ 1 ×L2 (R4 ) < ∗ .

(1.4)

j=1,...,4

Then there exists a unique global smooth solution (ψ, A) to the system (MD) under the Coulomb gauge condition (1.3) on R1+4 with these data. (2) For any T > 0, the data-to-solution map (ψ, Aj , ∂t Aj )(0) → (ψ, Aj , ∂t Aj ) extends continuously to H˙ 1/2 × H˙ 1 × L2 (R4 ) ∩ {(1.4) holds} → C([0, T ]; H˙ 1/2 × H˙ 1 × L2 (R4 )). The same statement holds on the interval [−T, 0]. (3) For each sign ±, the solution (ψ, A) exhibits modified scattering as t → ±∞, in the sense that there exist a solution (ψ ±∞ , A±∞ ) to the linear system j  A±∞ =0, j ±∞ =0, αμ DB μψ

such that )[t]H˙ 1 ×L2 (R4 ) → 0 (ψ − ψj±∞ )(t)H˙ 1/2 (R4 ) + (Aj − A±∞ j

as t → ±∞.

Here, B0 = 0 and Bj can be taken to be either the solution Af ree to Af ree = 0 . with data Afj ree [0] = Aj [0], or Bj = A±∞ j In the general case d ≥ 4, the same theorem holds with the spaces H˙ 1/2 (R4 ) and d−3 d−2 d−4 H˙ × L2 (R4 ) are replaced by and H˙ 2 (Rd ) and H˙ 2 × H˙ 2 (Rd ), respectively. We refer to Remarks 5.7, 6.6, 8.8, 9.4 and 10.8 for the necessary modifications in the argument when d ≥ 5. 1

1.2. MAIN IDEAS

3

Remark 1.2. Although the theorem is stated only for Coulomb initial data sets, it may be applied to an arbitrary smooth initial data set satisfying the smallness condition (1.4) by performing a gauge transform. Indeed, given an arbitrary connection 1-form Aj (0) on Rd , there exists a gauge transformation d ˙ 1,d ∩ BM O(Rd ) such that the gauge transform A(0) ˜ χ ∈ H˙ 2 ∩ W = A(0) − dχ obeys the Coulomb gauge condition (4.7). Moreover, the small data condition (1.4) is preserved up to multiplication by a universal constant for ∗ small enough. Such a gauge transformation can be found by solving the Poisson equation Δχ = divx Aj (0). We remark that our method do not apply to the case of nonzero mass m = 0, although the observations made in this paper suggest that it would likely follow from a corresponding result for the massive Maxwell–Klein–Gordon equations; see ‘Parallelism with Maxwell–Klein–Gordon’ in Section 1.2. The physically interesting case of d = 3, with or without mass, remains open.

1.1. Previous work A brief survey of previous results on (MD) and related equations is in order. After early work on local well-posedness of (MD) on R1+3 by Gross [13] and Bournaveas [4], D’Ancona–Foschi–Selberg [8] established local well-posedness of (MD) on R1+3 in the Lorenz gauge ∂ μ Aμ = 0 for data ψ(0) ∈ H  , Aμ [0] ∈ H 1/2+ ×H −1/2+ , which is almost optimal. In the course of their proof, a deep system null structure of (MD) in the Lorenz gauge was uncovered. Although we work in a different gauge, our work develop upon many ideas from [8]. D’Ancona–Selberg [9] extended this approach to (MD) on R1+2 and proved global well-posedness in the charge class. Regarding (MD) on R1+3 , we also mention [6, 11, 12, 25] on global wellposedness for small, smooth and localized data, [1, 20] on the non-relativistic limit and [21] on unconditional uniqueness at regularity ψ ∈ Ct H 1/2 , Ax [·] ∈ Ct (H 1 ×L2 ) in the Coulomb gauge. A scalar counterpart of (MD) is the Maxwell–Klein–Gordon equations (MKG). An analogue of Theorem 1.1 for (MKG) was proved by Krieger–Sterbenz–Tataru [18]. As we will explain in Section 1.2, [18] may be regarded as one of the direct predecessors of the present work. In the energy critical case d = 4, global wellposedness of (MKG) for arbitrary finite energy data was recently established by the second author and Tataru [22–24], and independently by Krieger–L¨ uhrmann [17]. In contrast, although (MD) is also energy critical on R1+4 , the energy for (MD) is not coercive; whether our Theorem 1.1 may be extended to the large data case is therefore unclear. Finally, we note that optimal small data global well-posedness was proved recently for the cubic Dirac equation in R1+2 and R1+3 by Bejenaru–Herr [2, 3] (massive) and Bournaveas–Candy [5] (massless). This equation features a spinorial null structure similar to what is considered in this work.

1.2. Main ideas We now provide an outline of the main ideas of this paper.

4

1. INTRODUCTION

Null structure of (MD) in Coulomb gauge. Null structure arises in equations from mathematical physics which exhibit covariance properties. It manifests through the vanishing of resonant components of the nonlinearities of such equations, and its presence is fundamental in obtaining well-posedness at critical regularity. An important component of our proof is uncovering the null structure of (MD) in the Coulomb gauge (MD-CG), which involves both classical (i.e., scalar) and spinorial null forms. A classical null form for scalar inputs refers to a linear combination of Qαβ (φ, ψ) = ∂α φ∂β ψ − ∂β φ∂α ψ,

0≤α