Desingularization of vortices for 2D steady Euler flows via the vorticity method

Abstract

In this paper, we consider steady Euler flows in a planar bounded domain in which the vorticity is sharply concentrated in a finite number of disjoint regions of small diameter. Such flows are closely related to the point vortex model and can be regarded as desingularization of point vortices. By an adaption of the vorticity method, we construct a family of steady Euler flows in which the vorticity is concentrated near a global minimum point of the Robin function of the domain, and the corresponding stream function satisfies a semilinear elliptic equation with a given profile function. Furthermore, for any given isolated minimum point $(\bar{x}_1,\cdot\cdot\cdot,\bar{x}_k)$ of the Kirchhoff-Routh function of the domain, we prove that there exists a family of steady Euler flows whose vorticity is supported in $k$ small regions near $\bar{x}_i$, and near each $\bar{x}_i$ the corresponding stream function satisfies a semilinear elliptic equation with a given profile function.