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Global well-posedness of the viscous Boussinesq equations. (English) Zbl 1274.76185
Summary: We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in $$H^m$$ with $$m\geq 3$$. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the velocity equation, but there is no viscosity in the density equation. Using sharp and delicate energy estimates, we prove global existence and strong regularity of this viscous Boussinesq system for general initial data in $$H^m$$ with $$m\geq 3$$.

MSC:
 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35Q35 PDEs in connection with fluid mechanics
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