On a classification of steady solutions to two-dimensional Euler equations

TL;DR

This paper classifies steady solutions to 2D Euler equations based on flow angles, revealing that bounded flows in the plane are mostly shear flows, with additional classes in strips and half-planes, and derives related Liouville-type theorems.

Contribution

It provides a comprehensive classification of steady Euler flows in various domains and introduces new insights into flow angle sets and curvature properties.

Findings

Flow angles in bounded plane flows cover the entire circle unless shear.

Additional flow classes exist in strips and half-planes with semicircular flow angle sets.

Liouville-type theorems are established for related elliptic equations.

Abstract

In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. As consequences, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines.