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

Yachun Li; Shengguo Zhu

doi:10.3934/dcds.2015.35.3059

Article Contents

2015, Volume 35, Issue 7: 3059-3086. Doi: 10.3934/dcds.2015.35.3059

This issue Previous Article Time-dependent singularities in the Navier-Stokes system Next Article Multiple solutions to elliptic inclusions via critical point theory on closed convex sets

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

Yachun Li^1, and
Shengguo Zhu^2,

1.
Department of Mathematics and Key Lab of Scientific and Engineering Computing (MOE), Shanghai Jiao Tong University, Shanghai 200240
2.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received: August 31, 2013

Revised: October 31, 2014

Published: May 2017

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Abstract

In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of a unique local regular solution with vacuum by the theory of quasi-linear symmetric hyperbolic systems and some techniques dealing with the complexity caused by the coupling between fluid and radiation field under some physical assumptions for the radiation quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent $1<\gamma\leq 3$ via some observations on the propagation of the radiation field. Compared with [11][15][20], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.

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Mathematics Subject Classification: Primary: 35Q35, 35D35; Secondary: 35A01, 35B44.

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