含摩擦效应的三维直管中定常可压缩亚音速Euler流

This paper studies steady motion of gas in a rectilinear duct with square cross-sections, governed by the three-dimensional (3-d) non-isentropic compressible Euler equations with a friction term. Such flows are called Fanno flows in engineering. We construct respectively special subsonic flows, supersonic flows and transonic shocks in the duct. Since the 3-d steady compressible Euler equations are of quasi-linear hyperbolic- elliptic composite type for subsonic flows, and there is no general theory up to now, we formulate a boundary value problem arising from studies of transonic shocks, and prove the well-posedness of this problem by showing that the special subsonic flows constructed above are stable under small multi-dimensional perturbations. The proof depends on separation of the elliptic and hyperbolic parts in the Euler equations, and designation of a suitable nonlinear iteration scheme. Particularly, there are strong interactions between the elliptic part and the hyperbolic part due to the appearance of friction, and we deduce a linear mixed boundary value problem of a second-order elliptic equation with an integral-type nonlocal term. Its well-posedness is established by applying methods of Fourier analysis and regularity theory of second-order elliptic equations.