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 the local uniqueness of flow whose stream function psi(epsilon) satisfies { -epsilon(2) triangle psi epsilon = Sigma(i=1) (k) 1(B delta) (z(0,i))(psi(epsilon) -mu(epsilon),i)gamma+, in Omega,psi(epsilon) = 0, on theta Omega,with epsilon -> 0(+) the scale parameter of vortices, gamma is an element of (0, infinity), Omega subset of R-2 a bounded simply connected Lipschitz domain, z(0),(i) is an element of Omega the limiting location of i(th) vortex, and mu(epsilon,i) the flux constants unprescribed. Our proof is achieved by a detailed description of asymptotic behavior for psi(epsilon) and Pohozaev identity technique. For k = 1, we prove the nonlinear stability of corresponding vorticity in L-p norm, provided that z(0,1) is a non-degenerate minimum point of Robin function. This stability result can be generalized to the case k >= 2, and (z(0,1), ..., z(0,k)) is an element of theta Omega(k) being a non-degenerate minimum point of the Kirchhoff-Routh function.