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This paper concerns studies on smooth transonic flows with nonzero vorticityin De Laval nozzles for a quasi two dimensional steady Euler flow model whichis a generalization of the classical quasi one dimensional model. First, theexistence and uniqueness of smooth transonic flows to the quasi one-dimensionalmodel, which start from a subsonic state at the entrance and accelerate toreach a sonic state at the throat and then become supersonic are proved by areduction of degeneracy of the velocity near the sonic point and the implicitfunction theorem. These flows can have positive or zero acceleration at theirsonic points and the degeneracy types near the sonic point are classifiedprecisely. We then establish the structural stability of the smooth onedimensional transonic flow with positive acceleration at the sonic point forthe quasi two dimensional steady Euler flow model under small perturbations ofsuitable boundary conditions, which yields the existence and uniqueness of aclass of smooth transonic flows with nonzero vorticity and positiveacceleration to the quasi two dimensional model. The positive acceleration ofthe one dimensional transonic solutions plays an important role in searchingfor an appropriate multiplier for the linearized second order mixed typeequations. A deformation-curl decomposition for the quasi two dimensional modelis utilized to deal with the transonic flows with nonzero vorticity.

Smooth transonic flows with nonzero vorticity to a quasi two dimensional steady Euler flow model