Global well-posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

TL;DR

This paper proves the global existence and uniqueness of solutions for 2D nonlinear Boussinesq equations with mixed partial viscosities and thermal diffusivity, covering all cases without restrictive assumptions.

Contribution

It establishes the first comprehensive global well-posedness results for 2D anisotropic Boussinesq equations with various viscosity and diffusivity combinations, including previously unresolved cases.

Findings

Global well-posedness for three viscous combinations

Unique solutions under minimal regularity for other cases

Results are new even for simplified models

Abstract

In this paper, we discuss with the global well-posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well-posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model.