  This paper is concerned with the well-posedness theory of the impact of a
subsonic axially symmetric jet emerging from a semi-infinitely long nozzle,
onto a rigid wall. The fluid motion is described by the steady isentropic Euler
system. We showed that there exists a critical value $M_{cr}>0$, if the given
mass flux is less than $M_{cr}$, there exists a unique smooth subsonic axially
symmetric jet issuing from the given semi-infinitely long nozzle and hitting a
given uneven wall. The surface of the axially symmetric impinging jet is a free
boundary, which detaches from the edge of the nozzle smoothly. It is showed
that a unique suitable choice of the pressure difference between the chamber
and the atmosphere guarantees the continuous fit condition of the free
boundary. Moreover, the asymptotic behaviors and the decay properties of the
impinging jet and the free surface in downstream were also obtained. The main
results in this paper solved the open problem on the well-posedness of the
compressible axially symmetric impinging jet, which has proposed by A. Friedman
in Chapter 16 in [FA2]. The key ingredient of our proof is based on the
variational method to the quasilinear elliptic equation with the Bernoulli's
type free boundaries.
