The global existence and blowup of the classical solution to the relativistic dust in a FLRW geometry
Xianshu Ju, Xiangkai Ke, Changhua Wei
TL;DR
This work analyzes the global existence versus blowup for the relativistic Euler equations with $p=0$ (dust) on a fixed FLRW background, clarifying how the expansion rate $a(t)$ dictates the life span of solutions. By reducing to a relativistic Burgers system and applying the method of characteristics, the authors derive explicit velocity and flow maps and establish a determinant-based criterion, $\det(\partial x/\partial \alpha)>0$, that governs regularity, with a full classification according to four scale-factor regimes $H_1$–$H_4$. The paper proves global existence for accelerated or sufficiently fast expansion ($H_1$, $H_2$) and demonstrates finite-time blowup of velocity (and sometimes density) for slower expansion ($H_3$, $H_4$) under appropriate initial data; it also extends the analysis to the density field, showing positivity and decay in favorable regimes and examining spherically symmetric cases where density concentration can accompany gradient blowup. Overall, the results quantify the precise influence of cosmological expansion on the life span of relativistic dust solutions and provide sharp blowup criteria via flow-map determinants and characteristic methods.
Abstract
This paper is concerned with the global existence and blowup of the classical solution to the Cauchy problem of the relativistic Euler equation with $ p=0 $ in a fixed Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. The aim of this work is to study clearly the effect of the expansion rate of the spacetime on the life span of the classical solution to the pressureless fluid. Since the density and the velocity of the relativistic dust admits the same principal part, we can obtain much more accurate results by the characteristic method rather than energy estimates.