On the Well-Posedness of Reduced 3D Primitive Geostrophic Adjustment Model with Weak Dissipation

Chongsheng Cao,Edriss S Titi,Quyuan Lin

doi:10.1007/s00021-020-00495-6

Abstract

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization.

Highlights

It is commonly believed that the dynamics of ocean and atmosphere adjusts itself toward a geostrophic balance
Where the velocity field (u, v, w), the temperature T and the pressure p are the unknown functions of horizontal variable x, vertical variable z, and time t, and f0 is the Coriolis parameter
System (1)–(5) is reduced from the 3D inviscid primitive equations model by assuming that the flow is independent of the third spatial variable

Summary

Introduction

It is commonly believed that the dynamics of ocean and atmosphere adjusts itself toward a geostrophic balance. The first systematically mathematical studies of the viscous primitive equations (PEs) were carried out in the 1990s by Lions–Temam–Wang [45, 46, 47] They considered the PEs with both full viscosities and full diffusivities and established the global existence of weak solutions. 150], by introducing the linear (Rayleigh-like friction) damping in both horizontal momentum equation (6) and vertical hydrostatic approximation (7), we consider the following system: ut + u ux + wuz + 1u − f0 v + px − νuzz = 0,. Unlike the strong horizontal dissipation, i.e., horizontal viscosity, we regard the linear damping term 2w as a weak dissipation With this weak dissipation, we are able to prove the local well-posedness and global well-posedness with small initial data.

Preliminaries

L2 ds u0

Convergence

Blow-up Criterion