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The present article is devoted to the 3D dissipative quasi-geostrophic system(QG). This system can be obtained as limit model of the Primitive Equations inthe asymptotics of strong rotation and stratification, and involves anon-radial, non-local, homogeneous pseudo-differential operator of order 2denoted by $\Gamma$ (and whose semigroup kernel reaches negative values). Aftera refined study of the non-local part of $\Gamma$, we prove apriori estimates(in the general L^p setting) for the 3D QG-model. The main difficulty of thisarticle is to study the commutator of $\Gamma$ with a Lagrangian change ofvariable. An important application of these a priori estimates, providing boundfrom below to the lifespan of the solutions of the Primitive Equations forill-prepared blowing-up initial data, can be found in a companion paper.

A priori estimates for the 3D quasi-geostrophic system