A priori estimates for the 3D quasi-geostrophic system

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• 作者：Fré ; dé ; ric Charve ^{frederic.charve@u-pec.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：3D Quasi-geostrophic model ; A priori estimates ; Besov spaces ; Non-local diffusion operator ; Lagrangian flow ; Singular integrals
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：444
• 期：2
• 页码：911-946
• 全文大小：611 K

文摘

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 L^p$L^p$ 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.