SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 133-180, January 2021.
In this paper, we are concerned with the global existence and stability of a smooth supersonic Euler flow with vacuum state at infinity in a three-dimensional (3-D) infinitely long divergent nozzle. The flow is described by 3-D compressible steady Euler equations, which are quasilinear multidimensional hyperbolic with respect to the supersonic direction. By the mass conservation of gases and the geometric property of the divergent nozzle, the moving gases in the nozzle will gradually become rarefactive and tend to the vacuum state at infinity, which means that the compressible Euler equations are degenerate at infinity. For such an expansive supersonic Euler flow and for small initial perturbations, we show that the 3-D Euler flow is globally stable and there are no vacuum domains in the nozzle.

中文翻译：

---

无限长发散喷嘴中的三维全局超音速欧拉流

SIAM数学分析杂志，第53卷，第1期，第133-180页，2021年1月。
在本文中，我们关注在三维（3-D）无限长发散喷嘴中具有无限真空状态的光滑超声速Euler流的整体存在和稳定性。用3-D可压缩的稳态Euler方程描述流，该方程是关于超音速方向的准线性多维双曲线。通过气体的质量守恒和发散喷嘴的几何特性，喷嘴中的运动气体将逐渐变得稀疏并趋于无穷大的真空状态，这意味着可压缩的欧拉方程在无穷大时退化。对于这种膨胀的超音速欧拉流和较小的初始扰动，我们表明3-D欧拉流是全局稳定的，并且喷嘴中没有真空域。