We are concerned with global steady subsonic flows through general infinitely long nozzles for the full Euler equations. The problem is formulated as a boundary value problem in the unbounded domain for a nonlinear elliptic equation of second order in terms of the stream function. It is established that, when the oscillation of the entropy and Bernoulli functions at the upstream is sufficiently small in $C^{1,1}$ and the mass flux is in a suitable regime, there exists a unique global subsonic solution in a suitable class of general nozzles. The assumptions are required to prevent the occurrence of supersonic bubbles inside the nozzles. The asymptotic behavior of subsonic flows at the downstream and upstream, as well as the critical mass flux, has been clarified.

We are concerned with the unique existence of three-dimensional steady compressible full Euler flows through arbitrary infinitely long axisymmetric and piecewise smooth nozzles with nontrivial swirl. We develop a new approach to prove the nondegeneracy of the axial velocity based on the observation of the potential flow. A modified argument is also employed to handle the stagnation at the corner points. It is the first result on the three-dimensional compressible Euler flow with more than one nonzero and large vorticity. In order to show it, one new stream-conserved quantity is constructed. Finally, the minimum flux limits and the incompressible limits are considered. Via the incompressible limit, we also establish the unique existence of incompressible Euler flows with nontrivial swirl. The methods and techniques developed in this paper are also helpful to other related problems.

This paper is devoted to the study of the wellposedness of the radiative Euler equations. By employing the anti-derivative method, we show the unique global-in-time existence and the asymptotic stability of the solutions of the radiative Euler equations for the composite wave of two viscous shock waves with small strength. This method developed here is also helpful to other related problems with similar analytical difficulties.