Smooth axisymmetric transonic irrotational flows to steady Euler system with an external force in cylinders

For a class of external forces, we first prove the existence and uniquenessof one dimensional smooth transonic flows to the steady Euler system, which issubsonic at the inlet and flows out at supersonic speed after smoothlyaccelerating through the sonic point. We then investigate the structuralstability of the one-dimensional smooth transonic flows with positiveacceleration under axisymmetric perturbations of suitable boundary conditions,and establish the first existence and uniqueness result for smooth axisymmetrictransonic irrotational flows. The key point lies on the analysis of a linearsecond order elliptic-hyperbolic mixed differential equation of Keldysh typewith a singular term. Some weighted Sobolev spaces $H_r^m(D) (m=2,3,4)$ areintroduced to deal with the singularities near the axis. Compared with thestability analysis in the two dimensional case \cite{WX23}, there are severalinteresting new observations about the structure of the linear mixed typeequation which play crucial roles in establishing the $H^4_r(D)$ estimate.