Steady solutions for the Euler system in an infinitely long nozzle

Congming Li, Yingshu Lv, Chunjing Xie

Published 2022-03-16Version 1

A recent prominent result asserts that steady incompressible Euler flows strictly away from stagnation in a two-dimensional infinitely long strip must be shear flows. On the other hand, flows with stagnation points, very challenging in analysis, are interesting and important phenomenon in fluids. In this paper, we study flows with stagnation points, such as Poiseuille flows, and establish the uniqueness and existence of steady solutions for the Euler system in an infinitely long nozzle. First, we prove a Liouville type theorem for steady Euler system with Poiseuille flows as upstream far field state in an infinitely long strip. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non-Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the uniqueness of flows with positive horizontal velocity inside the general nozzles was established. Finally, a class of steady incompressible Euler flows, tending to Poiseuille flows in the upstream, are proved to exist in an infinitely long nozzle via variational approach. One of very interesting phenomena we observe is that the free boundary may appear for flows with stagnation points.