Steady Euler Flows with Large Vorticity and Characteristic Discontinuities in Arbitrary Infinitely Long Nozzles

TL;DR

This paper proves the existence and uniqueness of steady Euler flows with large vorticity and discontinuities in infinitely long nozzles, covering both compressible and incompressible fluids, and introduces new methods for such complex flow problems.

Contribution

It presents the first global existence results for multidimensional steady compressible Euler equations with free boundaries and large vorticity, using novel approaches without sign assumptions.

Findings

Existence of smooth solutions with large vorticity in arbitrary nozzles.

Construction of weak solutions with vortex sheets and entropy waves.

Validation of subsonic-sonic and incompressible limits.

Abstract

We establish 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. We first develop a new approach to establish 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 nonlinear arguments. 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, we establish the existence and uniqueness of incompressible Euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets. The methods and techniques developed here will be useful for solving other problems involving similar difficulties.