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Energy estimates for the good-bad-ugly model

Miguel Duarte

TL;DR

This work studies energy estimates for the good-bad-ugly toy model that mimics nonlinearities in the Einstein equations written in generalized harmonic gauge. By linking the ugly equation to well-understood good/bad equations via the rescaled incoming-null derivative $R\nabla_{\ul{\psi}}$, the authors establish $L^2$ and $L^\infty$ bounds on Cauchy slices using the Klainerman-Sobolev inequality, and clarify how logarithmic terms $\log R$ arise and can be controlled. They then perform a first-order reduction and radial compactification in flat space to derive an energy estimate on hyperboloidal slices, providing a rigorous step toward energy estimates for the hyperboloidal initial-value problem of compactified Einstein equations in generalized harmonic gauge. These results lay groundwork for extending energy methods to more general asymptotically flat metrics and multiple fields, informing approaches to include null infinity in numerical evolutions of GR.

Abstract

We establish a relationship between the equations that constitute the so-called good-bad-ugly model, whose nonlinearities are known to mimic those present in the Einstein field equations in generalized harmonic gauge. This relationship between ugly fields and good and bad ones stems from the fact that one can write the equation for the rescaled derivative of an ugly along an incoming null direction as a good or a bad equation depending on whether there are source terms or not. This provides a new interpretation of the logarithms of the radial coordinate that show up in expansions of solutions to ugly equations near null infinity. This furthermore allows us to use the Klainerman-Sobolev inequality for the standard wave equation on Cauchy slices to show uniform boundedness for the ugly equation. In the second part of this paper we perform a first order reduction of the ugly equation with given sources in flat space and we radially compactify the coordinates in order to show an energy estimate for that equation on hyperboloidal slices. This result is an important first step towards establishing energy estimates for the hyperboloidal initial value problem of the first order compactified Einstein field equations in generalized harmonic gauge.

Energy estimates for the good-bad-ugly model

TL;DR

This work studies energy estimates for the good-bad-ugly toy model that mimics nonlinearities in the Einstein equations written in generalized harmonic gauge. By linking the ugly equation to well-understood good/bad equations via the rescaled incoming-null derivative , the authors establish and bounds on Cauchy slices using the Klainerman-Sobolev inequality, and clarify how logarithmic terms arise and can be controlled. They then perform a first-order reduction and radial compactification in flat space to derive an energy estimate on hyperboloidal slices, providing a rigorous step toward energy estimates for the hyperboloidal initial-value problem of compactified Einstein equations in generalized harmonic gauge. These results lay groundwork for extending energy methods to more general asymptotically flat metrics and multiple fields, informing approaches to include null infinity in numerical evolutions of GR.

Abstract

We establish a relationship between the equations that constitute the so-called good-bad-ugly model, whose nonlinearities are known to mimic those present in the Einstein field equations in generalized harmonic gauge. This relationship between ugly fields and good and bad ones stems from the fact that one can write the equation for the rescaled derivative of an ugly along an incoming null direction as a good or a bad equation depending on whether there are source terms or not. This provides a new interpretation of the logarithms of the radial coordinate that show up in expansions of solutions to ugly equations near null infinity. This furthermore allows us to use the Klainerman-Sobolev inequality for the standard wave equation on Cauchy slices to show uniform boundedness for the ugly equation. In the second part of this paper we perform a first order reduction of the ugly equation with given sources in flat space and we radially compactify the coordinates in order to show an energy estimate for that equation on hyperboloidal slices. This result is an important first step towards establishing energy estimates for the hyperboloidal initial value problem of the first order compactified Einstein field equations in generalized harmonic gauge.

Paper Structure

This paper contains 16 sections, 1 theorem, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. Representation of the metric
  4. Our model and assumptions
  5. The good and the ugly
  6. Energy estimates on Cauchy slices
  7. The standard ugly case
  8. Pushing logs down in the expansion
  9. Finding estimates in $L^2$
  10. Finding estimates in $L^\infty$
  11. Radially compactified ugly
  12. First order reduction
  13. Radial compactification
  14. Symmetrizer
  15. Estimates for uglies on hyperboloids
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\phi_1\in C^{\infty}([0,\infty)\times \mathbb{R}^3)$ such that $\phi_1(T)\in\mathbb{R}^3$ is a function of compact support for any $T\geq 0$. Then the following estimate holds for each $T\geq 0$ and $X^{\ul{i}}\in \mathbb{R}^3$,

Figures (2)

  • Figure 1: A schematic of our geometric setup.
  • Figure 2: A schematic of our geometric setup. We truncate the hyperboloidal slices at $r=r_1$ and then take the limit $r_1\rightarrow r_{\mathscr{I}}$.

Theorems & Definitions (1)

  • Theorem 1