Chaplygin气体Euler方程组的轴对称解

Axisymmetric Solutions of the Euler System for the Chaplygin Gas

构造了两维Chaplygin气体Euler方程组的三参数、自相似的弱解.在自相似和轴对称的假设下,两维Chaplygin气体Euler方程组可以化为无穷远边值的常微分方程组,由此得到了解的存在性和解的结构.与多方气体不同的是Chaplygin气体的Euler方程组是完全线性退化的.即使在轴向速度大于零的时候解也会出现间断现象.这些解展示了宇宙演化过程中的一些现象,例如黑洞的形成与演化以及宇宙的暴涨和膨胀.

In this paper, a three parameter family of self similar weak solutions are con structed in two dimensional space for all positive time to the axisymmetric Euler equations for the Chaplygin gas. Under the self similar and axisymmetry assumptions, the equations are reduced to a system of three ordinary differential equations with boundary value at in finity, from which the detailed structures of solutions as well as their existence are obtained. Different from polytropic gas, the Euler equations for the Chaplygin gas is fully linear degen erate. Discontinuities exist even though the velocity of radially direction is positive. These solutions exhibit some phenomena, such as black hole, expansion and explosive expansion, in the evolution of universe.