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Whether or not global solutions of the 2D Navier–Stokes–Cahn–Hilliard (NS–CH) system without full viscosity and mobility can develop finite time singularities is a difficult issue. A major result of this paper deals with global regularity of strong solutions for the NS–CH system with mixed partial viscosity and mobility. In addition, the 2D NS–CH system without viscosity but with full mobility is investigated. In this case, we also prove the global existence and uniqueness of classical solutions.

In this paper we study a three dimensional thermocline planetary geostrophic "horizontal" hyper-diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite dimen- sional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper-diffusion model does not exhibit any of the nonphysical instabilities near the literal boundary which are observed numerically in other models.