一维具有阻尼和摩擦项的可压缩流体欧拉方程组当压力消失时黎曼解的极限

摘要/Abstract

摘要：

该文研究带有复合源项的一维可压缩流体欧拉方程组的黎曼问题, 其中源项可以是摩擦项, 也可以是阻尼项, 也可以是阻尼和摩擦两者都具有. 与齐次型不同, 非齐次守恒律方程组的黎曼解是非自相似的. 当绝热指数$ \gamma\rightarrow1$即压力消失时, 讨论带有复合源项的一维可压缩流体欧拉方程组的黎曼解中集中现象和真空状态的形成, 证明包含两条激波的黎曼解收敛于零压下的delta激波解, 包含两条稀疏波的黎曼解收敛于零压下的两条接触间断解, 其中连接两条接触间断解的中间状态是真空状态.

关键词: 消失压力极限, 可压缩流体欧拉方程组, 复合源项, Delta激波, 真空状态, 黎曼问题

Abstract:

In this paper, we study the Riemann problem for the Euler equations of compressible fluid flow with a composite source term. The source can cover a Coulomb-like friction or a damping or both. Different from the homogeneous system, Riemann solutions of the inhomogeneous system are non self-similar. Concentration and cavitation in the pressureless limit of solutions to the Riemann problem for the Euler equations of compressible fluid flow with a composite source term are investigated in detail as the adiabatic exponent tends to one. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a composite source term, and the intermediate density between the two shocks tends to a weighted $ \delta$-mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a composite source term, whose intermediate state between the two contact discontinuities is a vacuum state.

Key words: Pressureless limit, Euler equations of compressible fluid flow, Composite source term, Delta shock wave, Vacuum state, Riemann problem

中图分类号:

O175.29

邵志强. 一维具有阻尼和摩擦项的可压缩流体欧拉方程组当压力消失时黎曼解的极限[J]. 数学物理学报, 2022, 42(4): 1150-1172.

Zhiqiang Shao. Concentration and Cavitation in the Pressureless Limit of Euler Equations of Compressible Fluid Flow with Damping and Friction[J]. Acta mathematica scientia,Series A, 2022, 42(4): 1150-1172.

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