A priori estimates for the 3D quasi-geostrophic system

2014/11/26 by Charve, Frederic · 1 citation
#35B25 #35B45 #35Q35 #35S30 #76U05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1411.7132

Abstract

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 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 apriori estimates (in the general L^p setting) for the 3D QG-model. The main difficulty of this article is to study 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.

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• Mixed 3D-2D asymptotics for the weak and strong solutions of the rotating magnetohydrodynamic system

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