Regularization of planar vortices for the incompressible flow

Abstract

In this paper, we continue to construct stationary classical solutions for the incompressible planar flows approximating singular stationary solutions of this problem. This procedure is carried out by constructing solutions for the following elliptic equations {-Δu=λ∑j=1k1Bδ(xO,j)(u-kj)+p, in Ω,u=0,on ∂Ωwhere 0 <p < 1, ⊂ ℝ2 is a bounded simply-connected smooth domain, κ (i, 1, …, k) is prescribed positive constant. The result we prove is that for any given non-degenerate critical point x0 = (x0,1, … x0, k) of the Kirchhoff-Routh function defined on Ωk corresponding to (κ1, …, κk), there exists a stationary classical solution approximating stationary k points vortex solution. Moreover, as λ → + ∞, the vorticity set {y: uλ > κj} ∩ Bδ(x0, j) shrinks to {x0, j}, and the local vorticity strength near each x0, j approaches κj, j = 1, …, k. This result makes the study of the above problem with p ≥ 0 complete since the cases p 1, p = 1, p = 0 have already been studied in [11, 12] and [13] respectively.