The quasi-geostrophic equation in the Triebel-Lizorkin spaces

We prove the local-in-time well-posedness in the Triebel-Lizorkin spaces for the two-dimensional quasi-geostrophic equation. We also obtain a sharp finite time blow-up criterion of solutions both in the super-critical and the critical cases, which improve the previous one by Constantin et al (1994 Nonlinearrily 7 1495-533). In the proof of the results, we use Littlewood-Paley decomposition and the paradifferential calculus applied directly to the equation.