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
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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.