On the Regularity Conditions for the Dissipative Quasi-geostrophic Equations

We obtain regularity conditions for solutions of the dissipative quasi-geostrophic equation. The first one imposes on the integrability of the magnitude of the temperature gradient, and corresponds to the Serrin type of condition in the theory of Navier--Stokes equations. The other one incorporates the direction of normals to the level curves and the magnitude of the temperature gradient simultaneously. For the proof of the second result, in particular, we use geometric properties of the nonlinear term as well as the estimates using the Triebel--Lizorkin type of norms.