| Abstract: |
| In this talk, we want to present the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. A new approach is introduced for the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by the compensated compactness argument. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic-sonic limit of the solutions is also shown. Finally, through the incompressible limit, the existence and uniqueness of incompressible Euler flows is established in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solution with vortex sheets. This is the joint work with Gui-Qiang G. Chen, Fei-Min Huang, and Wei Xiang. |
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