Generalized 2D Euler–Boussinesq equations with a singular velocity

This paper studies the global (in time) regularity problem concerning a system of equations generalizing the two-dimensional incompressible Boussinesq equations. The velocity here is determined by the vorticity through a more singular relation than the standard Biot–Savart law and involves a Fourier multiplier operator. The temperature equation has a dissipative term given by the fractional Laplacian operator −Δ. We establish the global existence and uniqueness of solutions to the initial-value problem of this generalized Boussinesq equations when the velocity is “double logarithmically” more singular than the one given by the Biot–Savart law. This global regularity result goes beyond the critical case. In addition, we recover a result of Chae, Constantin and Wu [8] when the initial temperature is set to zero.