On Blow-up of Regular Solutions to the Isentropic Euler and Euler-Boltzmann Equations with Vacuum

In this paper,the authors study the Cauchy problem of n-dimensional isentropic Euler equations and Euler-Boltzmann equations with vacuum in the whole space.They show that if the initial velocity satisfies some condition on the integral J in the"isolated mass group"(see(1.13)),then there will be finite time blow-up of regular solutions to the Euler system with J≤0(n≥1)and to the Euler-Boltzmann system with J<0(n≥1)and J=0(n≥2),no matter how small and smooth the initial data are.It is worth mentioning that these blow-up results imply the following:The radiation is not strong enough to prevent the formation of singularities caused by the appearance of vacuum,with the only possible exception in the case J=0 and n=1 since the radiation behaves differently on this occasion.