Supersonic flows of the Euler–Poisson system with nonzero vorticities in three-dimensional cylinders

Abstract

We prove the unique existence of three-dimensional supersonic solutions to the steady Euler–Poisson system in cylindrical nozzles. First, we establish the unique existence of irrotational solutions in a cylindrical nozzle with an arbitrary cross-section with using weighted Sobolev norms. Then, we establish the unique existence of axisymmetric solutions with nonzero vorticity in a circular cylinder. Several technical issues, including the issue of nonlinear hyperbolic–elliptic mixed type partial differential equation (PDE) system and corner singularities in a Lipschitz domain, are carefully addressed.