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We are concerned with the so-called Boussinesq equations with partialviscosity. These equations consist of the ordinary incompressible Navier-Stokesequations with a forcing term which is transported {\it with no dissipation} bythe velocity field. Such equations are simplified models for geophysics (inwhich case the forcing term is proportional either to the temperature, or tothe salinity or to the density). In the present paper, we show that thestandard theorems for incompressible Navier-Stokes equations may be extended toBoussinesq system despite the fact that there is no dissipation or decay atlarge time for the forcing term. More precisely, we state the global existenceof finite energy weak solutions in any dimension, and global well-posedness indimension $N\geq3$ for small data. In the two-dimensional case, the finiteenergy global solutions are shown to be unique for any data in $L^2(\R^2).$

The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity