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The Maxwell-Klein-Gordon equation with scattering data

Wei Dai, He Mei, Dongyi Wei, Shiwu Yang

TL;DR

This work develops a nonlinear scattering theory for the Maxwell-Klein-Gordon system on Minkowski space by solving a characteristic initial-value problem with data on future null infinity. Using a conformal transformation and a gauge-invariant vector-field approach, the authors construct global solutions that scatter to prescribed large radiation data, including nonzero total charge, and establish quantitative energy bounds. A key novelty is treating large-data scattering by a two-region exterior analysis (upper and lower exterior) with weighted conformal multipliers that absorb the charge-induced indeterminacy, extending He’s small-data results to a broad class of initial data. The results provide a robust nonlinear map from scattering data on ${ m I}^+$ to the corresponding data on past null infinity, with uniqueness up to gauge and explicit control of charged and chargeless parts, highlighting significant progress in nonlinear gauge theory scattering theory and radiation data problems.

Abstract

It has been shown in [Yang-Yu 2019] that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space $\mathbb{R}^{1+3}$ decay like linear solutions. One hence can define the associated radiation field on the future null infinity as the limit of $(r\underlineα, rφ)$ along the out going null geodesics. In this paper, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem for the MKG system with general large data by using gauge invariant vector field method. We in particular extend the small data result of He in \cite{MR4299134} to a class of general large data.

The Maxwell-Klein-Gordon equation with scattering data

TL;DR

This work develops a nonlinear scattering theory for the Maxwell-Klein-Gordon system on Minkowski space by solving a characteristic initial-value problem with data on future null infinity. Using a conformal transformation and a gauge-invariant vector-field approach, the authors construct global solutions that scatter to prescribed large radiation data, including nonzero total charge, and establish quantitative energy bounds. A key novelty is treating large-data scattering by a two-region exterior analysis (upper and lower exterior) with weighted conformal multipliers that absorb the charge-induced indeterminacy, extending He’s small-data results to a broad class of initial data. The results provide a robust nonlinear map from scattering data on to the corresponding data on past null infinity, with uniqueness up to gauge and explicit control of charged and chargeless parts, highlighting significant progress in nonlinear gauge theory scattering theory and radiation data problems.

Abstract

It has been shown in [Yang-Yu 2019] that general large solutions to the Cauchy problem for the Maxwell-Klein-Gordon system (MKG) in the Minkowski space decay like linear solutions. One hence can define the associated radiation field on the future null infinity as the limit of along the out going null geodesics. In this paper, we show the existence of a global solution to the MKG system which scatters to any given sufficiently localized radiation field with arbitrarily large size and total charge. The result follows by studying the characteristic initial value problem for the MKG system with general large data by using gauge invariant vector field method. We in particular extend the small data result of He in \cite{MR4299134} to a class of general large data.
Paper Structure (64 sections, 15 theorems, 249 equations, 5 figures)

This paper contains 64 sections, 15 theorems, 249 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Related results
  4. Strategy for the proof
  5. Organization of the paper
  6. Preparation
  7. Notation
  8. Basic calculation results
  9. The multiplier vector fields and energy equality
  10. Some induced formulas of the MKG equation
  11. The analysis in interior region
  12. The conformal transform
  13. The equation on ${\tilde{{\mathcal{D}}}}_{0,1}^{0,1}$ and its data
  14. The preparation for local existence
  15. The uniformly boundedness of $\Psi_j$ in ${\mathcal{D}}_{0,\delta}^{0,1}$
  16. ...and 49 more sections

Key Result

Theorem 1.1

For any given scattering data $({\underline{\mathcal{A}}},\Phi)$ such that $\|({\underline{\mathcal{A}}},\Phi)\|_{S\!N_{U_*}}$ is finite for any $U_*\in \mathbb{R}$, there exists a solution $(F,\phi)$ to the MKG system Eq:MKGe-Fphi which scatters to $({\underline{\mathcal{A}}},\Phi)$ in the sense of in which the weighted energy $\|(r{\underline\alpha},r\phi)\|_{S\!N_{U_*}'}$ is similarly defined o

Figures (5)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 3.1
  • Claim 3.2
  • Remark 3.1
  • Remark 3.2
  • Claim 4.1
  • ...and 24 more