Integral geometry of Euler equations

TL;DR

This paper introduces an integral geometric framework for stationary Euler equations, deriving a linear differential equation for a function on the Grassmannian, and shows that local solutions with compact support must be trivial.

Contribution

It develops a novel integral geometric approach to analyze stationary Euler equations and establishes conditions under which solutions with compact support are necessarily zero.

Findings

Derived a linear differential equation for the geometric function w

Proved that local compactly supported solutions of steady Euler equations are trivial

Established a connection between integral geometry and fluid dynamics

Abstract

We develop an integral geometry of stationary Euler equations defining some function $w$ on the Grassmannian of affine lines in the space. This function depends on a putative compactly supported solution $v$ of the system, and we deduce a linear differential equation for $w$. We prove also that the purported annulation of $w$ implies that locally supported solutions of the steady Euler equation in $\mathbb R^3$ are zero.