3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler-Poisson system in a cylinder supplemented with non small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl(=angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler-Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis $r=0$

Subsonic Euler flows in a three-dimensional finitely long cylinder with arbitrary cross section

This paper concerns the well-posedness of subsonic flows in a three-dimensional finitely long cylinder with arbitrary cross section. We establish the existence and uniqueness of subsonic flows in the Sobolev space by prescribing the normal component of the momentum, the vorticity, the entropy, the Bernoulli's quantity at the entrance and the normal component of the momentum at the exit. One of the key points in the analysis is to utilize the deformation-curl decomposition for the steady Euler system introduced in \cite{WX19} to deal with the hyperbolic and elliptic modes. Another one is to employ the separation of variables to improve the regularity of solutions to a deformation-curl system near the intersection between the entrance and exit with the cylinder wall