A priori estimates for the 3D quasi-geostrophic system

The present article is devoted to the 3D dissipative quasi-geostrophic system (QG  ). This system can be obtained as a limit model of the Primitive Equations in the asymptotics of strong rotation and stratification, and involves a non-radial, non-local, homogeneous pseudo-differential operator of order 2 denoted by Γ (and whose semigroup kernel reaches negative values). After a refined study of the non-local part of Γ, we prove a priori estimates (in the general LpLp setting) for the 3D QG-model. The main difficulty of this article is to study this highly non-classical operator Γ and the commutator of Γ with a Lagrangian change of variable. An important application of these a priori estimates, providing bound from below to the lifespan of the solutions of the Primitive Equations for ill-prepared blowing-up initial data, can be found in a companion paper.