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

Existence and uniqueness of axially symmetric compressible subsonic jet impinging on an infinite wall