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

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

2008-06-25

arxiv.org/abs/0806.4083v1

Mathematics

Analysis of PDEs

Bulletin de la Societe Mathematique de France, in press

Scientific paper

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

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