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This present paper is dedicated to the study of the Cauchy problem of thetwo-dimensional Euler-Boussinesq-B$\rm\acute{e}$nard equations which couple theincompressible Euler equations for the velocity and a transport equation withcritical dissipation for the temperature. We show that there is a global uniquesolution to this model with Yudovich's type data. This settles the globalregularity problem which was remarked by Wu and Xue (J. Differential Equations253:100--125, 2012).

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