Asymptotics and Lower Bound for the Lifespan of Solutions to the Primitive Equations

Abstract

In a previous work we 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 blowing-up data, when the kinematic viscosity $\nu $ is equal to the heat diffusivity $\nu '$ , turning the diffusion operator $\varGamma $ into the classical Laplacian. Obtaining the same results in the general case is much more difficult as it involves a homogeneous non-local non-radial diffusion operator $\varGamma $ whose semi-group and singular integral form kernels present sign changes. Every classical result related to non-local operators, or to Navier-Stokes system then becomes more involved here and the key ingredient will be new transport-diffusion estimates obtained in a companion paper and a precise use of the quasi-geostrophic decomposition.