Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles
Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles
复制标题
DOI：
10.1137/09076667x
复制
发表时间：
2009-07
期刊：
影响因子：
--
通讯作者：
Chunjing Xie;Z. Xin
中科院分区：
文献类型：
--
作者：
Chunjing Xie;Z. Xin
在本文中，我们在不假设无旋性的情况下研究了通过无限长喷管的定常亚声速欧拉流的整体存在性。结果表明，当上游的伯努利函数的变化足够小且质量通量处于具有上临界值的合适范围时，对于一般的变截面喷管，在合适的类别中存在唯一的整体亚声速解。对于一般的定常欧拉流，主要困难之一是控制方程即使对于均匀亚声速流也是一个混合的椭圆 - 双曲型方程组。我们理论的一个关键点是对可压缩欧拉方程使用流函数形式。通过这种形式，欧拉方程等价于一个关于流函数的拟线性二阶方程，从而已经涉及到粒子路径的双曲性。借助于二元椭圆方程的估计，得到了流函数边值问题解的存在性。通过爆破论证和能量估计得到了流函数的渐近行为。这种渐近行为，连同对流函数的一些精细估计，得出了流函数形式与原始欧拉方程的一致性。
In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value, then there exists a unique global subsonic solution in a suitable class for a general variable nozzle. One of the main difficulties for the general steady Euler flows, the governing equations are a mixed elliptic-hyperbolic system even for uniformly subsonic flows. A key point in our theory is to use a stream function formulation for compressible Euler equations. By this formulation, Euler equations are equivalent to a quasilinear second order equation for a stream function so that the hyperbolicity of the particle path is already involved. The existence of solution to the boundary value problem for stream function is obtained with the help of the estimate for elliptic equation of two variables. The asymptotic behavior for the stream function is obtained via a blow up argument and energy estimates. This asymptotic behavior, together with some refined estimates on the stream function, yields the consistency of the stream function formulation and thus the original Euler equations.