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

For the generalized surface quasi-geostrophic equation { a m p ; ∂ t θ + u ⋅ ∇ θ = 0 , in R 2 × ( 0 , T ) , a m p ; u = ∇ ⊥ ψ , ψ = ( − Δ ) − s θ in R 2 × ( 0 , T ) , {equation*} \{ {aligned} & ∂ _t θ +u· ∇ θ =0, in R^2 × (0,T), \\ & u=∇ ^⊥ ψ , ψ = (-Δ )⁻ˢθ in R^2 × (0,T) , {aligned} . {equation*} 0 > s > 1 0>s>1 , we consider for k ≥ 1 k≥ 1 the problem of finding a family of k k </