γ>2时欧拉-泊松方程组平衡解的存在唯一性

Existence and Uniqueness of Stationary Solutions to Euler-poisson Equations with γ>2

本文主要研究速度和熵函数满足质量守恒和能量守恒时方程组的平衡解(即与时间t无关的解).作者在绝热常数γ>2和熵函数满足一定的光滑性条件下,采用变量变换将方程组转化成一个半线性椭圆型方程,运用上下解方法得到了方程组平衡解的存在性,并证明了平衡解的唯一性.

In this article, we consider stationary solutions of the Euler-poisson equations, i.e., the solutions independent of time t, for some velocity fields and smooth entropy functions that solve the conservation of mass and energy. When γ 〉 2 and the entropy function satisfies some smooth property, we introduce a nonlinear transformation to turn the Euler-poisson system into a semilinear elliptic equation, and then obtain the existence and uniqueness of the stationary solutions by upper-lower-solution method.