Some three dimensional smooth transonic flows for the steady Euler equations with an external force

We establish the existence and uniqueness of some smooth accelerating transonic flows governed by the three dimensional steady compressible Euler equations with an external force in cylinders with arbitrary cross sections, which include both irrotational flows and Beltrami flows with nonuniform proportionality factors. One of the key ingredients in the analysis of smooth transonic irrotational flows is the well-posedness theory of classical solutions in $H^4$ to a linear elliptic-hyperbolic mixed second order differential equation of Keldysh type in cylinders with mixed boundary conditions. This is achieved by extending the problem to an auxiliary linear elliptic-hyberbolic-elliptic mixed problem in a longer cylinder where the governing equation becomes elliptic at the exit of the new cylinder, so that one can use the multiplier method and the cut-off techniques to derive the $H^2$ and higher order estimates in transonic regions. It is further shown that the energy estimate can be closed in the $H^4$ framework. For smooth transonic Beltrami flows, we solve a transport equation for the proportionality factor and a type-changing enlarged deformation-curl system with mixed boundary conditions. The compatibility conditions for the $H^4$ estimate to the enlarged deformation-curl system near the intersection between the entrance and the cylinder wall play a crucial role in the analysis.