Existence and regularity of rotating global solutions for the generalized surface quasi-geostrophic equations

Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global rotating solutions for the generalized surface quasi-geostrophic equation for $\alpha \in [1,2)$. We also show $C^{\mathrm{\infty }}$ regularity of their boundary for all $\alpha \in (0,2)$.