CRITICAL POINTS OF THE N-VORTEX HAMILTONIAN IN BOUNDED PLANAR DOMAINS AND STEADY STATE SOLUTIONS OF THE INCOMPRESSIBLE EULER EQUATIONS

We prove the existence of critical points of the N-vortex Hamiltonian H-KR(x(1),..., x(N)) = Sigma(N)(i=1)Gamma(2)(i) h(x(i)) +Sigma(i,j=1) (N)(j not equal k) Gamma(i)Gamma(j)G(x(i), x(j))+ 2 Sigma(N)(i=1) Gamma(i)psi(0)(x(i)) in a bounded domain Omega subset of R-2 which may be simply or multiply connected. Here G denotes the Green function for the Dirichlet Laplace operator in Omega, more generally a hydrodynamic Green function, and h the Robin function. Moreover psi(0) is an element of C-1((Omega) over bar) is a harmonic function on Omega. We obtain new critical points x = (x(1),..., x(N)) for N = 3 or N = 4 under conditions on the vorticities Gamma(i) is an element of R \ {0}. These critical points correspond to point vortex equilibria of the Euler equation in vorticity form. The case Gamma(i) = (-1)(i) of counterrotating vortices with identical vortex strength is included. The point vortex equilibria can be desingularized to obtain smooth steady state solutions of the Euler equations for an ideal fluid. The velocity of these steady states will be irrotational except for N vorticity blobs near x(1),..., x(N).