On 2D steady Euler flows with small vorticity near the boundary

Abstract

In this paper, we investigate steady incompressible Euler flows with nonvanishing vorticity in a planar bounded domain. Let q be a harmonic function that corresponds to an irrotational flow. This paper proves that if q has k isolated local extremum points on the boundary, then there exist two kinds of steady Euler flows with small vorticity supported near these k points. For the first kind, near each maximum point the vorticity is positive and near each minimum point the vorticity is negative. For the second kind, near each minimum point the vorticity is positive and near each maximum point the vorticity is negative. Moreover, near these k points, the flow is characterized by a semilinear elliptic equation with a given profile function in terms of the stream function. The results are achieved by solving a certain variational problem for the vorticity and studying the limiting behavior of the extremizers.