Well-posedness of anisotropic and homogeneous solutions to the Einstein-Boltzmann system with a conformal-gauge singularity

Abstract

We consider the Einstein-Boltzmann system for massless particles in the Bianchi I space-time with scattering cross-sections in a certain range of soft potentials. We assume that the space-time has an initial conformal gauge singularity and show that the initial value problem is well posed with data given at the singularity. This is understood by considering conformally rescaled equations. The Einstein equations become a system of singular ordinary differential equations, for which we establish an existence theorem which requires several differentiability and eigenvalue conditions on the coefficient functions together with the Fuchsian conditions. The Boltzmann equation is regularized by a suitable choice of time coordinate, but still has singularities in momentum variables. This is resolved by considering singular weights, and the existence is obtained by exploiting singular moment estimates.

arXiv comment75 pages, errata corrected and references added

References in corpus (17)

• The Einstein-Vlasov system/Kinetic theory
• Isotropic cosmological singularities 2: The Einstein-Vlasov system
• Isotropic cosmological singularities 3: The Cauchy problem for the inhomogeneous conformal Einstein-Vlasov equations
• Isotropic cosmological singularities 1: Polytropic perfect fluid spacetimes
• The geometry of the tangent bundle and the relativistic kinetic theory of gases
• Isotropic cosmological singularities: other matter models
• An introduction to the relativistic kinetic theory on curved spacetimes
• Stable Big Bang formation for Einstein's equations: The complete sub-critical regime
• Some Remarks on the $C^0$-(in)extendibility of Spacetimes
• Global existence of solutions for the relativistic Boltzmann equation with arbitrarily large initial data on a Bianchi type I space-time
• Future global existence and asymptotic behaviour of solutions to the Einstein-Boltzmann system with Bianchi I symmetry
• Isotropic cosmological singularities in spatially-homogeneous models with a cosmological constant
• An extension theorem for conformal gauge singularities
• The massless Einstein-Boltzmann system with a conformal-gauge singularity in an FLRW background
• Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes
• Small solutions of the Einstein-Boltzmann-scalar field system with Bianchi symmetry
• The spatially homogeneous Boltzmann equation for massless particles in an FLRW background

Cited by in corpus (1)

• Homogeneous solutions to the Einstein-matter equations with a magnetic field and a conformal gauge singularity