TRAVELLING AND ROTATING SOLUTIONS TO THE GENERALIZED INVISCID SURFACE QUASI-GEOSTROPHIC EQUATION

For the generalized surface quasi-geostrophic equation {partial derivative(t)theta + u center dot del theta = 0, in R-2 x (0, T), u = del(perpendicular to)psi, psi = (-Delta)(-s)theta in R-2 x (0,T), 0 < s < 1, we consider for k >= 1 the problem of finding a family of k-vortex solutions theta(epsilon)(x,t) such that as epsilon -> 0 theta(epsilon)(x,t) -> Sigma(k)(j=1)m(j)delta(x - xi(j)(t)) for suitable trajectories for the vortices x = xi(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 (-Delta)W-s = (W - 1)(+)(gamma), in R-2, 1 < gamma < 1 + s/1 - s whose existence and uniqueness have recently been proven in Chan, del Mar Gonzalez, Huang, Mainini, and Volzone

Automated summary

The article investigates k-vortex solutions in the generalized surface quasi-geostrophic equation. In special cases, vortices can move along an axis with constant speed or rotate around the origin with the same speed, reducing the problem to handling a fractional elliptic equation. A key element in the construction is proving the non-degeneracy of the radial ground state for the fractional plasma problem.