Asymptotics and lower bound for the lifespan of solutions to the Primitive Equations

Frédéric Charve (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

This article generalizes a previous work in which the author obtained a large lower bound for the lifespan of the solutions to the Primitive Equations, and proved convergence to the 3D quasi-geostrophic system for general and ill-prepared (possi-bly blowing-up) initial data that are regularization of vortex patches related to the potential velocity. These results were obtained for a very particular case when the kinematic viscosity ν is equal to the heat diffusivity ν ′ , turning the diffusion operator into the classical Laplacian. Obtaining the same results without this assumption is much more difficult as it involves a non-local diffusion operator. The key to the main result is a family of a priori estimates for the 3D-QG system that we obtained in a companion paper.

Mots clés: Primitive Equations; 3D Quasi-geostrophic model; non-local diffusion; Strichartz estimates; Besov spaces; Sobolev spaces; singular integrals

Domaines: Equations aux dérivées partielles [math.AP]

Frédéric Charve. Asymptotics and lower bound for the lifespan of solutions to the Primitive Equations. Acta Applicandae Mathematicae, inPress. ⟨hal-01086979⟩