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Global existence for semi-linear hyperbolic equations in a neighbourhood of future null infinity

J. Arturo Olvera-Santamaria

TL;DR

This work proves global existence for a semi-linear class of hyperbolic equations in $3+1$ dimensions under a bounded weak null condition by constructing a conformal compactification of the future null cone and formulating a conformal symmetric hyperbolic Fuchsian system on a bounded manifold. The method relies on transforming the original wave system into a Global Initial Value Problem via a conformal map, differentiating and rescaling to obtain a symmetric Fuchsian system, and removing the leading singular quadratic term through an asymptotic flow that defines a new variable $Y$. A thorough extension of the system and a careful decomposition into differentiated, asymptotic, and complementary components yields a complete Fuchsian framework, enabling application of the BOOS:2020 theory to obtain global solutions for small initial data near future null infinity. The results extend prior work on weak null conditions by accommodating non-compact initial data and providing a robust approach to analyze singular hyperbolic systems in non-compact, asymptotic regimes, with potential applications to nonlinear wave phenomena in general relativity and geometric analysis.

Abstract

In this paper, we establish the global existence of a semi-linear class of hyperbolic equations in 3+1 dimensions, that satisfy the bounded weak null condition. We propose a conformal compactification of the future directed null-cone in Minkowski spacetime, enabling us to establish the solution to the wave equation in a neighbourhood of future null infinity. Using this framework, we formulate a conformal symmetric hyperbolic Fuchsian system of equations. The existence of solutions to this Fuchsian system follows from an application of the existence theory developed in [1], and [2].

Global existence for semi-linear hyperbolic equations in a neighbourhood of future null infinity

TL;DR

This work proves global existence for a semi-linear class of hyperbolic equations in dimensions under a bounded weak null condition by constructing a conformal compactification of the future null cone and formulating a conformal symmetric hyperbolic Fuchsian system on a bounded manifold. The method relies on transforming the original wave system into a Global Initial Value Problem via a conformal map, differentiating and rescaling to obtain a symmetric Fuchsian system, and removing the leading singular quadratic term through an asymptotic flow that defines a new variable . A thorough extension of the system and a careful decomposition into differentiated, asymptotic, and complementary components yields a complete Fuchsian framework, enabling application of the BOOS:2020 theory to obtain global solutions for small initial data near future null infinity. The results extend prior work on weak null conditions by accommodating non-compact initial data and providing a robust approach to analyze singular hyperbolic systems in non-compact, asymptotic regimes, with potential applications to nonlinear wave phenomena in general relativity and geometric analysis.

Abstract

In this paper, we establish the global existence of a semi-linear class of hyperbolic equations in 3+1 dimensions, that satisfy the bounded weak null condition. We propose a conformal compactification of the future directed null-cone in Minkowski spacetime, enabling us to establish the solution to the wave equation in a neighbourhood of future null infinity. Using this framework, we formulate a conformal symmetric hyperbolic Fuchsian system of equations. The existence of solutions to this Fuchsian system follows from an application of the existence theory developed in [1], and [2].

Paper Structure

This paper contains 18 sections, 3 theorems, 182 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Fuchsian Method
  3. The bounded weak null condition
  4. Conformal mapping of Minkowski spacetime near future null infinity
  5. Expansion formulas for the tensor components.
  6. The conformal wave equation
  7. The wave equation
  8. The extended system
  9. Initial data transformations
  10. Construction of the complete Fuchsian system
  11. The differentiated system
  12. The asymptotic equation
  13. Asymptotic flow assumptions
  14. The complementary variable $X^K$
  15. The complete Fuchsian system
  16. ...and 3 more sections

Key Result

Proposition 4.1

Suppose the bounded weak null condition holds (see Definition bwnc). Then there exists a $R_0\in (0,\mathcal{R}{}_0)$ such that the flow $\mathscr{F}{}(t,t_0,y,\xi)$ of the asymptotic equation aeqi2 satisfies the flow assumptions flowassump.1-flowassump.2 for this choice of $R_0$ and any choice of $

Figures (2)

  • Figure 1: In this diagram we plot time-like geodesics of the form $\bar{t}{}=m\bar{r}{},$ from Minkowski spacetime and represented in the $(t,r,\theta,\phi)$ coordinates, here $m \geq 1$. The red curve represents the time-like hyper-surface $\bar{r}{}=0$. In the limit $m \nearrow \infty$ , the time-like curves accumulate near the parabola $t=\Bigl(\frac{r}{1-r}\Bigr)^2$. Note that all the time like curves end at the point $t=0,\ r=0.$
  • Figure 3: In this diagram we plot the family of space-like geodesics $\bar{t}{}=k$ from Minkowski spacetime represented in the $(t,r,\theta,\phi)$ coordinates, where $k$ is a positive constant and each curve corresponds to a different value of $k$ . In the limit when $k \nearrow \infty$ the space-like geodesics accumulate near $\mathscr{I}{}^+$.

Theorems & Definitions (5)

  • Definition 1.1
  • Proposition 4.1
  • Lemma 4.2
  • Theorem 4.3
  • proof