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In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value, then there exists a unique global subsonic solution in a suitable class for a general variable nozzle. One of the main difficulties for the general steady Euler flows is that the governing equations are a mixed elliptic-hyperbolic system even for uniformly subsonic flows. A key point in our theory is to use a stream function formulation for compressible Euler equations. By this formulation, Euler equations are equivalent to a quasilinear second order equation for a stream function so that the hyperbolicity of the particle path is already involved. The existence of a solution to the boundary value problem for stream function is obtained with the help of the estimate for an elliptic equation of two variables. The asymptotic behavior for the stream function is obtained via a blowup argument and energy estimates. This asymptotic behavior, together with some refined estimates on the stream function, yields the consistency of the stream function formulation and thus the original Euler equations.

In this paper, we prove the existence and stability of subsonic flows for a steady full Euler-Poisson system in a two-dimensional nozzle of finite length when imposing the electric potential difference on a noninsulated boundary from a fixed point at the entrance, and prescribing pressure at the exit of the nozzle. The Euler-Poisson system for subsonic flow is a hyperbolicelliptic coupled nonlinear system. One of the crucial ingredients of this work is the combination of Helmholtz decomposition for the velocity field and stream function formulation. In terms of the Helmholtz decomposition, the Euler-Poisson system is rewritten as a second order nonlinear elliptic system of three equations and transport equations for entropy and pseudo-Bernoulli's invariant. The associated elliptic system in a Lipschitz domain with nonlinear boundary conditions is solved with the help of the estimates developed in [M. Bae, B. Duan, and C. J. Xie, Existence and Stability of Multidimensional Steady Potential Flows for Euler-Poisson Equations, preprint, arXiv:1211.5234, 2012 based on its nice structure. The transport equations are resolved via the flow map induced by the stream function formulation. Furthermore, the delicate estimates for the flow map give the uniqueness of the solutions.

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method. More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π /2; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.

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