Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

Nicolas Ginoux, Simone Murro

Abstract

In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary.

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

Abstract

In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided. Keywords: symmetric hyperbolic systems, symmetric positive systems, admissible boundary conditions, Dirac operator, normally hyperbolic operator, Klein-Gordon operator, heat operator, reaction-diffusion operator, globally hyperbolic manifolds with timelike boundary.

Paper Structure

This paper contains 27 sections, 19 theorems, 130 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. Friedrichs systems of constant characteristic
  4. Admissible boundary conditions
  5. The forward and the backward Cauchy problem
  6. Energy Inequality
  7. $\mathbf L^{\mathbf 2}$-well-posedness in a time strip
  8. Weak solutions
  9. Strong solutions
  10. Differentiability of the solutions for symmetric hyperbolic systems
  11. Global well-posedness
  12. Examples of symmetric hyperbolic systems
  13. The Euler momentum equation
  14. The classical Dirac operator
  15. Lorentzian chirality boundary conditions.
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\mathsf{M}$ be a globally hyperbolic manifold with timelike boundary and let $t\colon\mathsf{M}\to{\mathbb{R}}$ be a Cauchy temporal function. For any $0<T\in{\mathbb{R}}$ denote with $\mathsf{M}_\mathsf{T}:=t^{-1}((0,T))$ a time strip. Let $\mathsf{S}$ be a Friedrichs system with constant char Furthermore, if the bilinear form $\prec \sigma_\mathsf{S} (dt) \cdot\,|\, \cdot \succ_p$ is posi

Figures (2)

  • Figure 1: Finite propagation of speed -- $\mathcal{V}\cap \mathsf{T}$.
  • Figure 2: Fermi coordinates on each Cauchy surface.

Theorems & Definitions (56)

  • Theorem 1.1: Strong solutions
  • Theorem 1.2: Smooth solutions
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Ake-Flores-Sanchez-18, Theorem 1.1
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Lemma 2.8
  • ...and 46 more