Shear flows of an ideal fluid and elliptic equations in unbounded domains

TL;DR

This paper proves that steady, boundary-tangent flows of an ideal incompressible fluid in certain unbounded domains are shear flows, using geometric and elliptic equation symmetry methods, with broader implications for n-dimensional slabs.

Contribution

It establishes that steady ideal fluid flows with no stationary points in specific unbounded domains are necessarily shear flows, extending symmetry results to higher dimensions.

Findings

Steady flows in a 2D strip are shear flows.

Similar results hold for bounded flows in a half-plane.

Includes rigidity results for n-dimensional slabs.

Abstract

We prove that, in a two-dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half-plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n-dimensional slabs in any dimension n.