Local uniqueness of vortices for 2D steady Euler flow in a bounded domain

We study the 2D Euler equation in a bounded simply-connected domain, and establish local uniqueness of flow whose stream function ψε satisfies{−ε2Δψε=∑i=1k1Bδ(z0,i)(ψε−με,i)+γ,inΩ,ψε=0,on∂Ω, with ε→0+ scale parameter vortices, γ∈(0,∞), Ω⊂R2 simply connected Lipschitz z0,i∈Ω limiting location ith vortex, με,i flux constants unprescribed. Our proof is achieved by detailed description asymptotic behavior for Pohozaev identity technique. For k=1, we prove nonlinear stability corresponding vorticity Lp norm, provided that z0,1 non-degenerate minimum point Robin function. This result can be generalized to case k≥2, (z0,1,⋯,z0,k)∈Ωk being Kirchhoff-Routh