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Characteristic initial value problem for nonlinear wave equation with singular initial data

Wei Dai, Shiwu Yang

TL;DR

This work addresses local well-posedness for nonlinear wave equations with singular characteristic data on a conic light cone in $\mathbb{R}^{1+3}$, where the energy can be infinite and blow-up may occur at the cone point. The authors develop a vector-field method with a novel multiplier and weighted energy framework, decomposing the solution into a linear part and a nonlinear correction to prove a contraction and local existence in a neighborhood of the cone. They further apply these results to the Maxwell-Klein-Gordon system by employing conformal compactification to translate scattering data from future null infinity into characteristic data, yielding improved regularity results for inverse scattering and connecting near-null infinity dynamics to timelike infinity. The approach provides a robust scheme for handling singular data and extends the reach of local well-posedness and scattering theory in nonlinear hyperbolic equations and gauge theories.

Abstract

In this paper, we study the characteristic initial value problem for a class of nonlinear wave equations with data on a conic light cone in the Minkowski space $\mathbb{R}^{1+3}$. We show the existence of local solution for a class of singular initial data in the sense that the standard energy could be infinite and the solution may blow up at the conic point. As an application, we improve our previous result on the inverse scattering problem for the Maxwell-Klein-Gordon equations with scattering data on the future null infinity.

Characteristic initial value problem for nonlinear wave equation with singular initial data

TL;DR

This work addresses local well-posedness for nonlinear wave equations with singular characteristic data on a conic light cone in , where the energy can be infinite and blow-up may occur at the cone point. The authors develop a vector-field method with a novel multiplier and weighted energy framework, decomposing the solution into a linear part and a nonlinear correction to prove a contraction and local existence in a neighborhood of the cone. They further apply these results to the Maxwell-Klein-Gordon system by employing conformal compactification to translate scattering data from future null infinity into characteristic data, yielding improved regularity results for inverse scattering and connecting near-null infinity dynamics to timelike infinity. The approach provides a robust scheme for handling singular data and extends the reach of local well-posedness and scattering theory in nonlinear hyperbolic equations and gauge theories.

Abstract

In this paper, we study the characteristic initial value problem for a class of nonlinear wave equations with data on a conic light cone in the Minkowski space . We show the existence of local solution for a class of singular initial data in the sense that the standard energy could be infinite and the solution may blow up at the conic point. As an application, we improve our previous result on the inverse scattering problem for the Maxwell-Klein-Gordon equations with scattering data on the future null infinity.
Paper Structure (13 sections, 6 theorems, 115 equations, 1 figure)

This paper contains 13 sections, 6 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Additional notations
  4. Technical tools
  5. The vector field method
  6. Proof for the main Theorem \ref{['Tm:M1']}
  7. Energy estimate for the linear solution
  8. Energy estimate for the iteration
  9. End of the proof for Theorem Theorem \ref{['Tm:M1']}
  10. Proof of Theorem \ref{['Tm:M2']}
  11. Conformal compactification
  12. Reducing to the characteristic initial value problem
  13. Proof for Theorem \ref{['Tm:M2']}

Key Result

Theorem 1.1

Consider the characteristic initial value problem to the semilinear wave equation Eq:Me with Eq:toF. Assume that the initial data $\varphi_0$ on the initial cone $\mathcal{H}_0$ is bounded in the following sense for some positive constant $M_0$ and a small positive constant $\delta$. Then there exist positive constants $M_0'$, $\varepsilon$, depending only on $M_0$ and $\delta$, such that the cha

Figures (1)

  • Figure :

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof