# A Liouville theorem for the Euler equations in the plane

## TL;DR

This paper proves that bounded steady incompressible flows in the plane without stagnation points must be shear flows, with all streamlines parallel, based on geometric analysis of stream functions and growth estimates.

## Contribution

It establishes a Liouville theorem characterizing the structure of such flows, showing they are necessarily shear flows, which is a new geometric result.

## Key findings

- All bounded steady flows without stagnation points are shear flows.
- Streamlines are all parallel lines.
- Flow argument grows at most logarithmically.

## Abstract

This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at most logarithmic growth of the argument of the flow. These estimates lead to the conclusion that the streamlines of the flow are all parallel lines.