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Geometric scattering for nonlinear wave equations on the Schwarzschild metric

Pham Truong Xuan

Abstract

In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface $Σ_0 = \{ t = 0 \}$ and that through the null conformal boundaries $\mathfrak{H}^+ \cup \scri^+$ (respectively, $\mathfrak{H}^- \cup \scri^-$). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data.

Geometric scattering for nonlinear wave equations on the Schwarzschild metric

Abstract

In this paper, we establish a conformal scattering theory for defocusing semilinear wave equations on Schwarzschild spacetime. We combine the energy and pointwise decay results for solutions obtained in \cite{Yang} with a Sobolev embedding on spacelike hypersurfaces to derive two-sided energy estimates between the energy flux of solutions through the Cauchy initial hypersurface and that through the null conformal boundaries (respectively, ). By combining these estimates with the well-posedness of the Cauchy and Goursat problems for nonlinear wave equations, we construct a bounded linear and locally Lipschitz scattering operator that maps past scattering data to future scattering data.

Paper Structure

This paper contains 10 sections, 8 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Schwarzschild spacetime and nonlinear wave equations
  3. Schwarzschild metric, coordinates and foliation
  4. The nonlinear wave equation and conservation law
  5. Energy fluxes, energy decay and pointwise decay
  6. Energy fluxes through spacelike and null hypersurfaces
  7. Pointwise and energy decays and two sides of energy estimates
  8. Cauchy and Goursat problems and scattering operator
  9. Cauchy problem and an injective trace operator
  10. Goursat problem and conformal scattering operator

Key Result

Theorem 1

Consider the Cauchy problem for the rescaled nonlinear wave equation rescaledequation with smooth and compactly supported initial data on $\Sigma_0 = \{ t = 0 \}$. We can define the energy fluxes of the rescaled solution $\hat{\psi}$ through the null conformal boundary $\mathfrak{H}^+ \cup {\mathscr Furthermore, we have the following equality:

Figures (3)

  • Figure 1: Conformal compactification diagram and foliation $\left\{ \mathcal{S}_\tau \right\}_\tau$ of $\mathcal{I}^+(\mathcal{S}_0)$
  • Figure 2: Forward mapping from data on $\Sigma_0$
  • Figure 3: Backward mapping from data on $\mathfrak{H}^+\cup \mathscr{I}^+$

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Definition 1
  • ...and 11 more