Global Existence Results for the Anisotropic Boussinesq System in Dimension Two

TL;DR

This paper establishes global existence and well-posedness of solutions for certain two-dimensional anisotropic Boussinesq systems with horizontal viscosity, highlighting the role of directional diffusion in mathematical fluid dynamics.

Contribution

It provides the first global weak solutions for these anisotropic Boussinesq systems with large data, emphasizing the impact of horizontal diffusion on solution regularity.

Findings

Global weak solutions constructed for large data

Horizontal diffusion enables global well-posedness for regular data

Diffusion perpendicular to buoyancy is crucial for analysis

Abstract

We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.