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

Citation:

Yue CAO,Yachun LI.On Blow-up of Regular Solutions to the Isentropic Euler and Euler-Boltzmann Equations with Vacuum*[J].Chinese Annals of Mathematics B,2021,42(4):495~510
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Authors:

Yue CAO; Yachun LI

Foundation:

National Natural Science Foundation of China (Nos. 11831011,11571232) and China Scholarship Council (No. 201806230126).
Abstract: 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.

Keywords:

Euler and Euler-Boltzmann equations, Finite time blow-up, Multidimensional, Regular solutions, Vacuum

Classification:

35A21, 35B44, 35A09, 78A40, 35Q35
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