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

TL;DR

This paper extends previous results on the lifespan and convergence of solutions to the Primitive Equations, removing restrictive assumptions and dealing with complex non-local diffusion operators.

Contribution

It generalizes earlier work by establishing lower bounds for solution lifespan without assuming equal viscosity and diffusivity, using new a priori estimates for the 3D-QG system.

Findings

Established lower bounds for solution lifespan.

Proved convergence to the 3D quasi-geostrophic system.

Extended results to non-local diffusion operators.

Abstract

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 (possibly 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 $\nu$ is equal to the heat diffusivity $\nu '$, 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.