Three-Dimensional Global Supersonic Euler Flows in the Infinitely Long Divergent Nozzles

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.