THREE-DIMENSIONAL FULL EULER FLOWS WITH NONTRIVIAL SWIRL IN AXISYMMETRIC NOZZLES

We are concerned with the unique existence of three-dimensional steady compressible full Euler flows through arbitrary infinitely long axisymmetric and piecewise smooth nozzles with nontrivial swirl. We develop a new approach to prove the nondegeneracy of the axial velocity based on the observation of the potential flow. A modified argument is also employed to handle the stagnation at the corner points. It is the first result on the three-dimensional compressible Euler flow with more than one nonzero and large vorticity. In order to show it, one new stream-conserved quantity is constructed. Finally, the minimum flux limits and the incompressible limits are considered. Via the incompressible limit, we also establish the unique existence of incompressible Euler flows with nontrivial swirl. The methods and techniques developed in this paper are also helpful to other related problems.