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 $r=0$

Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

In this paper, we establish the existence and stability of cylindrical transonic shock solutions under three dimensional perturbations of the incoming flows and the exit pressure without any further restrictions on the background transonic shock solutions. The strength and position of the perturbed transonic shock are completely determined by the incoming flows and the exit pressure. The optimal regularity is obtained for all physical quantities, and the velocity, the Bernoulls's function, the entropy and the pressure share the same regularity. The problem is reduced to solve a nonlinear free boundary value problem for a hyperbolic-elliptic mixed system. There are two main ingredients in our analysis. One is to use the deformation-curl decomposition to the steady Euler system introduced by the authors in \cite{wx19,w19} to effectively decouple the hyperbolic and elliptic modes. Another one is the reformulation of the Rankine-Hugoniot conditions, which determines the shock front by an algebraic equation and also gives an unusual second order differential boundary conditions on the shock front for the deformation-curl system. After homogenizing the curl system and introducing a potential function, the solvability of the boundary value problem of the deformation-curl system for the velocity field is reduced to a second order elliptic equation for the potential function with a nonlocal term involving only the trace of the potential function on the shock front. This simplification follows essentially from an oblique boundary condition for the potential function on the shock front which is obtained by solving the Poisson equation on the shock front with the homogeneous Neumann boundary conditions on the intersection of the shock front and the cylinder walls.Comment: 50 pages. Any comments are welcom

Smooth transonic flows with nonzero vorticity to a quasi two dimensional steady Euler flow model

This paper concerns studies on smooth transonic flows with nonzero vorticity in De Laval nozzles for a quasi two dimensional steady Euler flow model which is a generalization of the classical quasi one dimensional model. First, the existence and uniqueness of smooth transonic flows to the quasi one-dimensional model, which start from a subsonic state at the entrance and accelerate to reach a sonic state at the throat and then become supersonic are proved by a reduction of degeneracy of the velocity near the sonic point and the implicit function theorem. These flows can have positive or zero acceleration at their sonic points and the degeneracy types near the sonic point are classified precisely. We then establish the structural stability of the smooth one dimensional transonic flow with positive acceleration at the sonic point for the quasi two dimensional steady Euler flow model under small perturbations of suitable boundary conditions, which yields the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity and positive acceleration to the quasi two dimensional model. The positive acceleration of the one dimensional transonic solutions plays an important role in searching for an appropriate multiplier for the linearized second order mixed type equations. A deformation-curl decomposition for the quasi two dimensional model is utilized to deal with the transonic flows with nonzero vorticity.Comment: 54 page