Subsonic flows passing a duct for three-dimensional steady compressible Euler system with friction

For the three-dimensional steady non-isentropic compressible Euler system with friction, we show existence of a class of symmetric subsonic, supersonic and transonic-shock solutions in a straight duct with constant square-section. Such flows are called Fanno flow in engineering. We formulate a boundary value problem for subsonic flows, and study their stability under multidimensional small perturbations of boundary conditions. Since the subsonic Euler system is of elliptic-hyperbolic composite-mixed type, this is achieved by using the framework established in [L. Liu; G. Xu; H. Yuan: Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations. Adv. Math. 291 (2016), 696--757], and establishing an iteration scheme, which involves solving a second order nonlocal elliptic equation.