Global existence and uniqueness for a non linear Boussinesq system in dimension two

TL;DR

This paper proves the global existence and uniqueness of solutions for a two-dimensional Boussinesq system with fractional diffusion in temperature, for initial data with critical regularity when the fractional order exceeds one.

Contribution

It establishes the global well-posedness of a coupled Boussinesq system with fractional diffusion for the first time in the critical regularity setting.

Findings

Global solutions exist and are unique for  > 1.

Solutions are valid for initial data with critical regularities.

The result extends understanding of Boussinesq systems with fractional diffusion.

Abstract

We study the global well-posedness of a two-dimensional Boussinesq system which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion of type $|\DD|^{\alpha}$ for the temperature. We prove that for $\alpha>1,$ there exists a unique global solution for initial data with critical regularities.