REGULARIZATION OF POINT VORTICES FOR THE EULER EQUATION IN DIMENSION TWO

Abstract. In this paper, we construct stationary classical solutions of the incompress-ible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem{ −ε2∆u = (u − q − κ2pi ln 1ε)p+, x ∈ Ω, u = 0, x ∈ ∂Ω, where p> 1, Ω ⊂ R2 is a bounded domain, q is a harmonic function. We showed that if Ω is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh functionW(x1, · · · , xm) with the same strength κ> 0, there is a stationary classical solution approximating stationary m points vortex solution of incompressible Euler equations with vorticity mκ. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen in [20]