  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 $\pi/2$; 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.
