The 2D Boussinesq equations with logarithmically supercritical velocities

Abstract: This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq–Navier–Stokes equations. The velocity in this system is related to the vorticity through the relations and , which reduces to the standard velocity–vorticity relation when . When either or , the velocity is more singular. The “quasi-velocity” determined by satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when and . In addition, the vorticity is shown to be globally bounded in several functional settings such as for in a suitable range. [Copyright &y& Elsevier]