Global Well-posedness of the 3D Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion

TL;DR

This paper proves the global regularity and well-posedness of strong solutions to the 3D primitive equations with partial vertical turbulence mixing heat diffusion, under specific boundary conditions, ensuring solutions depend continuously on initial data.

Contribution

It establishes the global existence and regularity of strong solutions for the primitive equations with partial vertical diffusion, a significant step in understanding their long-term behavior.

Findings

Global strong solutions exist for all time

Solutions depend continuously on initial conditions

The model remains regular under specified boundary conditions

Abstract

The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.