Local Solution Of Three-Dimensional Axisymmetric Supersonic Flow In A Nozzle

In this paper,we have studied the existence of local solution to compressible steady isentropic irrotational supersonic Euler flow in a three-dimensional axisymmetric nozzle.Under the assumptions of three-dimensional axisymmetric nozzle,the non-homogeneous terms of Euler’s equations have singularity in the axis of symmetry,and for the solid wall condition,the minimal characterizing number condition of the local solution existence theorem corresponding to the quasilinear hyperbolic system boundary value problems cannot be satisfied.In this paper,a special case is studied,assuming a steady supersonic flow into the nozzle throat,considering the local solution of the supersonic flow at the nozzle throat.On the one hand,because the incoming flow is steady,the equations is naturally satisfied near the symmetry axis of the nozzle throat,which avoids the difficulty of the singularity of the symmetry axis.On the other hand,by performing a linear transformation of the boundary value condition of the problem,we introduce a transformed form of the minimal characterizing number,which ensures that the assumptions can meet the sufficient conditions of the local solution existence theorem,regardless of the incoming supersonic flow’s velocity.This technique can generalize the solvability condition of the boundary value problems of quasilinear hyperbolic systems to more general cases(see theorem 2.4,2.5).