Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation

For the generalized surface quasi-geostrophic equation {equation presented} 0 < s < 1, we consider for k ≥ 1 the problem of finding a family of k-vortex solutions Θ _ϵ(x, t) such that as ϵ → 0 {equation presented} for suitable trajectories for the vortices x = ζ _j(t). We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem {equation presented} whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42].