Location of concentrated vortices in planar steady Euler flows

TL;DR

This paper analyzes steady 2D incompressible Euler flows with concentrated vorticity regions, showing they must be near critical points of a domain-dependent function, and establishes nonexistence in convex domains.

Contribution

It proves that concentrated vortices in steady Euler flows are located near critical points of the Kirchhoff-Routh function, linking vortex positions to domain geometry, and shows nonexistence in convex domains.

Findings

Vortices are near critical points of the Kirchhoff-Routh function.

Nonexistence of multiple vortex flows in convex domains.

Vorticity regularity is in $L^{4/3}$, optimal for weak solutions.

Abstract

In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics. The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff-Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3},$ which is the optimal regularity for weak solutions to make sense. As a by-product, we prove a nonexistence result for concentrated multiple vortex flows in convex domains.