A ``horizontal" hyper--diffusion $3-D$ thermocline planetary geostrophic model: well-posedness and long time behavior

Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation

The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is ``no'' in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: ``If the level set topology in the temperature field for the 2D quasi-geostrophic act…

广义准地转方程的双曲爆破存在性

Existence of hyperbolic blow-up to the generalized quasi-geostrophic equation

本文针对参数范围β∈(1,2)的二维广义表面准地转方程，基于双曲鞍点设定证明了光滑解存在双曲爆破并给出爆破时间下界，是该类方程奇点形成的首个严格证明结果。

Regularity "in Large" for the 3D Salmon's Planetary Geostrophic Model of Ocean Dynamics

It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the literal boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no heat flux physical boundary condition. Consequently, the second order parabolic heat equation is over de…