Global well-posedness for the 2D Euler-Boussinesq-B$\rm\acute{e}$nard equations with critical dissipation

TL;DR

This paper proves the global well-posedness of the 2D Euler-Boussinesq-Bénard equations with critical dissipation, establishing unique solutions for Yudovich-type initial data and resolving a previously open regularity problem.

Contribution

It demonstrates the global existence and uniqueness of solutions for the 2D Euler-Boussinesq-Bénard equations with critical dissipation, addressing an open question in the field.

Findings

Global unique solutions exist for Yudovich's data.

The global regularity problem is resolved.

The model's well-posedness is established for critical dissipation.

Abstract

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-B$\rm\acute{e}$nard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (J. Differential Equations 253:100--125, 2012).