Steady vortex flows of perturbation type in a planar bounded domain

TL;DR

This paper proves the existence of steady Euler flows with localized vorticity near boundary maximum points in a bounded domain, using an adapted vorticity method and semilinear elliptic equations.

Contribution

It introduces a novel approach to constructing steady Euler flows with vorticity concentrated near boundary maxima, extending previous methods to include multiple localized vortices.

Findings

Existence of steady flows with vorticity near boundary maxima.

Construction of flows with multiple localized vortices.

Vorticity supported in small neighborhoods of maximum points.

Abstract

In this paper, we investigate steady Euler flows in a two-dimensional bounded domain. By an adaption of the vorticity method, we prove that for any nonconstant harmonic function $q$, which corresponds to a nontrivial irrotational flow, there exists a family of steady Euler flows with small circulation in which the vorticity is continuous and supported in a small neighborhood of the set of maximum points of $q$ near the boundary, and the corresponding stream function satisfies a semilinear elliptic equation with a given profile function. Moreover, if $q$ has $k$ isolated maximum points $\{\bar{x}_1,\cdot\cdot\cdot,\bar{x}_k\}$ on the boundary, we show that there exists a family of steady Euler flows whose vorticity is continuous and supported in $k$ disjoint regions of small diameter, and each of them is contained in a small neighborhood of $\bar{x}_i$, and in each of these small regions the stream function satisfies a semilinear elliptic equation with a given profile function.