Formation of singularities of solutions to the three-dimensional Euler–Boltzmann equations in radiation hydrodynamics

Abstract

The Cauchy problem for the three-dimensional Euler-Boltzmann equations of a polytropic, ideal and isentropic fluid in radiation hydrodynamics is considered. The formation of singularities in smooth solutions is studied. It is proved that some C 1 solutions, regardless of the size of the initial disturbance, will develop singularities in a finite time provided that the initial disturbance satisfies certain conditions.

On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum

“…Compared with the results obtained in [5] [6] on the formation of singularities away from vacuum for Euler-Boltzmann equations, via making full use of the mathematical structure of our system in vacuum domain, an important observation about the propagation of radiation effect has been shown in this paper as follows. If the initial velocity also vanishes in our vacuum region V (see Definition 3.1) where the initial mass density vanishes, then there exists a critical time T c = 2R 0 c such that beyond T c , the radiation effect on the fluid propagates completely from the interior of B(t) (see Definition 3.2) to the exterior of B(t).…”

“…For Euler-Boltzmann equations, when the initial density is away from vacuum, Jiang-Zhong [6] obtained the local existence of C 1 solutions for the Cauchy problem. Jiang-Wang [5] showed that some C 1 solutions will blow up in finite time regardless of the size of the initial perturbation.…”

Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics

“…Due to its complexity, there are few mathematical results on the radiation hydrodynamical system (1.1)-(1.4). For the local existence of C 1 solutions and finite-time formation of singularities in solutions to the system of radiation hydrodynamics (1.1)-(1.4), see [10,11]. For the studies of some simplified systems of radiation hydrodynamic models, see [1,12,13].…”

Existence Results and Blow-Up Criterion of Compressible Radiation Hydrodynamic Equations

“…For Euler-Boltzmann equations, recently, Jiang-Zhong [19] obtained the local existence of C 1 solutions for the Cauchy problem away from vacuum. Jiang-Wang [18] showed that some C 1 solutions will blow up in finite time, regardless of the size of the initial disturbance. Li-Zhu [21] established the local existence of Makino-Ukai-Kawashima type's regular solution (see [24]) with vacuum, and also proved that the regular solutions with compact density will blow up in finite time.…”