Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations

In this paper,we study the existence of corotating and counter-rotating pairs of simply connected patches for Euler equations and the(SQG)_αequations with α∈(0,1).From the numerical experiments implemented for Euler equations in Deem and Zabusky(Phys Rev Lett40(13):859–862,1978),Pierrehumbert(J Fluid Mech 99:129–144,1980),Saffman and Szeto(Phys Fluids 23(12):2339–2342,1980)it is conjectured the existence of a curve of steady vortex pairs passing through the point vortex pairs.There are some analytical proofs based on variational principle(Keady in J Aust Math Soc Ser B 26:487–502,1985;Turkington in Nonlinear Anal Theory Methods Appl9(4):351–369,1985);however,they do not give enough information about the pairs, such as the uniqueness or the topological structure of each single vortex. We intend in this paper to give direct proofs confirming the numerical experiments and extend these results for the(SQG)_α equation when α∈(0,1).The proofs rely on the contour dynamics equations combined with a desingularization of the point vortex pairs and the application of the implicit function theorem.