Planar vortex patch problem in incompressible steady flow

In this paper, we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain Ω   of R2R2. Let k   be a positive integer and let κjκj be a positive constant, j=1,…,kj=1,…,k. For any given non-degenerate critical point x0=(x0,1,…,x0,k)x0=(x0,1,…,x0,k) of the Kirchhoff–Routh function defined on ΩkΩk corresponding to (κ1,…,κk)(κ1,…,κk), we prove the existence of a planar flow, such that the vorticity ω of this flow equals a large given positive constant λ   in each small neighborhood of x0,jx0,j, j=1,…,kj=1,…,k, and ω=0ω=0 elsewhere. Moreover, as λ→+∞λ→+∞, the vorticity set {y:ω(y)=λ}{y:ω(y)=λ} shrinks to ⋃j=1k{x0,j}, and the local vorticity strength near each x0,jx0,j approaches κjκj, j=1,…,kj=1,…,k.