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

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-B$\acute{\mathrm{e}}$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).