定常Euler-Poisson方程组的适定性研究

The steady Euler-Poisson equations are typical examples for mixed type equations which appear in physics, biology and engineering applications. The equations can be hyperbolic, elliptic, or even parabolic. This proposal devotes to study subsonic, supersonic and transonic shock solutions for steady Euler-Poisson equations. We will investigate wellposedness of subsonic irrotational solutions of multidimensional (with dimension bigger than or equal to 2) Euler-Poisson equations, the wellposedness of subsonic solutions for Euler-Poisson equations in general two dimensional nozzles, wellposedness of supersonic solutions for two dimensional Euler-Poisson equations, and structural stability of transonic shocks for Euler-Poisson equations. The delicate structure of Euler-Poisson equations will be used to establish a priori estimates and a special transformation will be used to deal with the free boundary. We believe that these studies will not only improve our understanding of theories of partial differential equations but also provide some theoretical support for numerical computations and engineering applications.

定常Euler-Poisson方程组是非常典型的混合型方程组，同时又具有非常强的物理、生物及工程背景。这个方程组既有双曲特征，也有椭圆特征，甚至抛物特征。本项目将对定常Euler-Poisson方程组的亚音速，超音速和跨音速激波解进行系统的研究。我们将研究高维（二维，三维）Euler-Poisson方程组的无旋亚音速解的适定性，一般二维管道中的Euler-Poisson方程组的亚音速解的适定性，二维Euler-Poisson方程组的超音速解的适定性，以及跨音速激波的结构稳定性。我们将运用Euler-Poisson方程组精致的结构来得到先验估计和一类特殊的变换来处理自由边界。通过这些研究，我们相信既能充实偏微分方程理论，也能为数值计算和工程应用提供理论支持。

本项目着重研究Euler-Poisson方程组的定常解的适定性问题，特别是其亚音速解、超音速解以及跨音速解的稳定性问题。在项目执行期间，我们得到了一些Euler和Euler-Poisson方程组高维解的适定性结果：证明了三维有限长管道中Euler方程组的亚音速解的适定性，得到了一类具有大旋度的Euler方程组的亚音速解的适定性，证明了具有角点的管道中的亚音速解的适定性与结构稳定性，证明了Euler-Poisson的亚音速解在高维扰动下的结构稳定性，得到了超音速解在高维扰动下的稳定性。我们在项目执行期间正式发表SCI文章5篇，有两篇接受发表，另有两篇文章在投稿中。项目执行期间有一名硕士研究生毕业，目前有两名博士研究生在读。我们得到的结果为全面理解高维Euler-Poisson方程组的定常解奠定了基础。同时也将帮助我们理解高维守恒律方程组的适定性问题。