Some topics on compressible flows in nozzles

First, we study global subsonic and subsonic-sonic potential flows through a general infinitely long two dimensional or three dimensional axially symmetric nozzle. It is proved that there exists a critical value for the incoming mass flux so that a global uniformly subsonic flow exists in the nozzle as long as the incoming mass flux is less than the critical value. Furthermore, we establish some uniform estimates for the deflection angles and the minimum speed of the subsonic flows by combining the hodograph transformation and the comparison principle for elliptic equations. With the help of these properties and a compensated compactness framework, we prove the existence of a global subsonic-sonic flow solution in the case of the critical incoming mass flux.;Second, global existence of steady subsonic Euler flows through infinitely long nozzles is established when the variation of Bernoulli's constant in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value. One of the main difficulties lies in that the full steady Euler system is a hyperbolic-elliptic coupled system in a subsonic region. A key point is to use stream function formulation for compressible Euler equations. By this formulation, Euler equations are equivalent to a quasilinear second order equation for stream function. We obtain existence of solution to the boundary value problem for stream function with the help of estimate for elliptic equation of two variables. Asymptotic behavior for the stream function is obtained via a blow up argument and energy estimate. This asymptotic behavior, together with some refined estimates on the stream function, yields the consistency of the stream function formulation and the original Euler equations.;Finally, we discuss some open problems closely related to the results obtained in this thesis and give some perspectives.